Deterministic Shift Extension of Affine Models for Variance Derivatives

DETERMINISTIC SHIFT EXTENSION OF
AFFINE MODELS FOR VARIANCE DERIVATIVES
DETERMINISTICSHIFTEXTENSIONOF
AFFINEMODELSFORVARIANCEDERIVATIVES
GABRIELE POMPA
GABRIELEPOMPA
PhD in INSTITUTIONS, MARKETS AND TECHNOLOGIES -
TRACK in COMPUTER, DECISION AND SYSTEMS SCIENCE
CURRICULUM in MANAGEMENT SCIENCE
2016
< la costola può variare in larghezza

IMT School for Advanced Studies, Lucca
Lucca, Italy
Deterministic Shift Extension of Afﬁne Models
for Variance Derivatives
PhD Program in Computer Decision and System Science,
curriculum: Management Science
XXVIII Cycle
By
Gabriele Pompa
2016

The dissertation of Gabriele Pompa is approved.
Program Coordinator: Prof. Rocco De Nicola, IMT School for Advanced
Studies Lucca
Supervisor: Prof. Fabio Pammolli, IMT School for Advanced Studies
Lucca
Supervisor: Prof. Roberto Ren`o, University of Verona
The dissertation of Gabriele Pompa has been reviewed by:
Prof. Martino Grasselli, University of Padova
Prof. Fig`a Talamanca, University of Perugia
IMT School for Advanced Studies, Lucca
2016

Ho sempre pensato che guardarsi indietro non facesse bene, che ri-
leggere il proprio tempo non potesse che ricordare le battaglie perse, le
occasioni non colte, le persone un tempo importanti e presenti, oggi las-
ciate al bivio appena passato, destinate su una strada che difficilmente
s’incrocerà un’altra volta alla mia.
E tuttavia mi sbagliavo. Qualcosa rimane. Ci sono state le soddis-
fazioni. Ciò che non ho colto è ancora l`ı, o è stato un abbaglio ed oggi è
opaco e ingrigito, e ne colgo i contorni, e non m’acceca più. E chi c’era
ancora c’è e mi ha scelto, e l’ho scelto. E il bivio è il prossimo, e m’attende
l`ı, e l’attendo.
A chi mi ha porto la penna quando ancora non la tenevo in mano, a
chi non ha mi ha corretto la mancina, a chi m’ha insegnato a scrivere il
mio tempo. A chi mi ha insegnato il mio indirizzo.
A chi me l’ha tolta e mi ha alzato la testa, e m’ha mostrato che potevo
viverlo. E mi si è seduta accanto.
A chi mi ha mostrato cosa non è Scienza e quanto sèrva, meschina e
vile sia quand’è un lavoro. A chi mi ha ricordato la sua bellezza, e mi ha
insegnato che ci si può vivere e viverla assieme, senza mai confondere il
mezzo col fine.
A tutto ciò che ho vissuto e vivo. Che se non è solo per capire chi
sono, è già qualcosa poterlo essere.
Dedico questa tesi alle persone importanti della mia vita, che si sono
già lette tra le righe, e a chiunque voglia leggersi.
Gabriele, 10 Marzo 2016.
Io... Ma si che me la cavo.

Contents
Abstract ix
1 Affine Models: preliminaries 4
1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 VIX and VIX derivatives 20
2.1 Markets: definitions and empirical facts . . . . . . . . . . . 22
2.1.1 VIX Index . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 VIX Futures . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.3 VIX Options . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Models: standalone and consistent approach . . . . . . . . . 30
2.2.1 Standalone models of VIX . . . . . . . . . . . . . . . 31
2.2.2 Consistent models of S&P500 and VIX . . . . . . . . 37
3 The Heston++ model 61
3.1 Pricing VIX derivatives with the Heston++ model . . . . . 64
3.1.1 Model specification . . . . . . . . . . . . . . . . . . . 64
3.1.2 Nested models . . . . . . . . . . . . . . . . . . . . . 66
3.1.3 SPX and VIX derivatives pricing . . . . . . . . . . . 67
3.2 A general displaced affine framework for volatility . . . . 71
3.2.1 Affine modeling of VIX index . . . . . . . . . . . . . 81
3.2.2 Affine modeling of VIX derivatives . . . . . . . . . 85
vii

4 The Heston++ model: empirical analysis 89
4.1 Empirical analysis . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 Calibration results . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.1 Impact of the short-term . . . . . . . . . . . . . . . . 111
4.2.2 Analysis with Feller condition imposed . . . . . . . 124
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A Mathematical proofs and addenda 133
A.1 Conditional characteristic functions of H models . . . . . . 133
A.2 Proof of Proposition 4: CH++
SP X (K, t, T) . . . . . . . . . . . . 136
A.3 Proof of Proposition 5: V IXH++
t . . . . . . . . . . . . . . . 136
A.4 Proof of Proposition 6: FH++
V IX (t, T) and CH++
V IX (K, t, T) . . . 136
A.5 Proof of proposition 9: EQ
hR T
t
Xsds Ft
i
. . . . . . . . . . 139
A.6 Proof of proposition 11: FV IX(t, T) and CV IX(K, t, T) un-
der the displaced afﬁne framework . . . . . . . . . . . . . . 142
A.7 Afﬁnity conservation under displacement transformation
of instantaneous volatility . . . . . . . . . . . . . . . . . . . 144
References 153
viii

Abstract
The growing demand for volatility trading and hedging has
lead today to a liquid market for derivative securities writ-
ten on it, which made these instruments a widely accepted
asset class for trading, diversifying and hedging. This grow-
ing market has consistently driven the interest of both prac-
titioner and academic researchers, which can ﬁnd in VIX and
derivatives written on it a valuable source of informations
on S&P500 dynamics, over and above vanilla options. Their
popularity stems from the negative correlation between VIX
and SPX index, which make these instruments ideal to take a
pure position on the volatility of the S&P500 without neces-
sarily taking a position on its direction. In this respect futures
on VIX enable the trader to express a vision of the markets
future volatility and call options on VIX offer protection from
market downturns in a clear-cut way. From the theoretical
point of view, this has lead to the need of a framework for
consistently pricing volatility derivatives and derivatives on
the underlying, that is the need to design models able to ﬁt
the observed cross-section of option prices of both markets
and properly price and hedge exotic products. The consistent
pricing of vanilla options on S&P500 and futures and options
on VIX is a requirement of primary importance for a model
to provide an accurate description of the volatility dynamics.
Since equity and volatility markets are deeply related, but at
the same time show striking differences, the academic debate
around the relevant features should a model incorporate in
order to be coherent with both markets is still ongoing. In
this thesis we leverage on the growing literature concerning
the developing of models for consistently pricing volatility
ix

derivatives and derivatives on the underlying and propose
the Heston++ model, which is an affine model belonging to
the class of models analyzed by Duffie et al. (2000) with a
multi-factor volatility dynamics and a rich jumps structure
both for price and volatility. The multi-factor Heston (1993)
structure enables the model to better capture VIX futures term
structures along with maturity-dependent smiles of options.
Moreover, both correlated and idiosyncratic jumps in price
and volatility factors help in reproducing the positive sloping
skew of options on VIX, thanks to an increased level of the
skewness of VIX distribution subsumed by the model. The
key feature of our approach is to impose an additive displace-
ment, in the spirit of Brigo and Mercurio (2001), on the instan-
taneous volatility dynamics which, acting as lower bound for
its dynamics, noticeably helps in capturing the term structure
of volatility. Both increasing the fit to the at-the-money term
structure of vanilla options, as already pointed out in Pacati
et al. (2014), and remarkably capturing the different shapes
experienced by the term structure of futures on VIX. More-
over, we propose a general affine framework which extends
the affine volatility frameworks of Leippold et al. (2007), Egloff
et al. (2010) and Branger et al. (2014) in which the risk-neutral
dynamics of the S&P500 index features several diffusive and
jump risk sources and two general forms of displacement char-
acterize the dynamics of the instantaneous variance process,
which is affine in the state vector of volatility factors. The
instantaneous volatility is modified according to a general
affine transformation in which both an additive and a mul-
tiplicative displacement are imposed, the first supporting its
dynamics, the second modulating its amplitude. We calibrate
the Heston++ model jointly and consistently on the three mar-
kets over a sample period of two years, with overall abso-
lute (relative) estimation error below 2.2% (4%). We analyze
the different contributions of jumps in volatility. We add two
x

sources of exponential upward jumps in one of the two volatil-
ity factors. We first add them separately as an idiosyncratic
source of discontinuity (as in the SVVJ model of Sepp (2008b))
and then correlated and synchronized with jumps in price (as
in the SVCJ model of Duffie et al. (2000)). Finally, we let the
two discontinuity sources act together in the full-specified
model. For any model considered, we analyze the impact
of acting a displacement transformation on the volatility dy-
namics. In addition, we perform the analysis restricting fac-
tor parameters freedom to satisfy the Feller condition. Our
empirical results show a decisive improvement in the pric-
ing performance over non-displaced models, and also pro-
vide strong empirical support for the presence of both price-
volatility co-jumps and idiosyncratic jumps in the volatility
dynamics.
xi

Introduction
The recent ﬁnancial crisis has raised the demand for derivatives directly
linked to the volatility of the market. This growing demand has lead
today to a liquid market for VIX derivatives, futures and options written
on the CBOE VIX volatility index Carr and Lee (2009).
Their popularity stems from the negative correlation between VIX
and SPX index, which make these instruments ideal to take a position
on the volatility of the S&P500 without necessarily taking a position on
its direction. In this respect futures on VIX enable the trader to express
a vision of the market’s future volatility and call options on VIX offer
protection from market downturns in a clear-cut way.
This growing market has consequently driven the interest of both
practitioner and academic researchers, ﬁnding in volatility and deriva-
tives written on it a valuable source of informations on the returns dy-
namics over and above vanilla options (Andersen et al., 2002; Bardgett
et al., 2013; Chung et al., 2011; Kaeck and Alexander, 2012; Menc´ıa and
Sentana, 2013; Song and Xiu, 2014).
Indeed equity and volatility markets are deeply related, but at the
same time show striking differences. The academic debate around the
relevant features should a model incorporate in order to be consistent
with both markets is still ongoing (Bardgett et al., 2013; Branger et al.,
2014; Menc´ıa and Sentana, 2013).
In this thesis we leverage on the growing literature concerning the
building of models for consistent pricing volatility derivatives and deriva-
tives on the underlying and propose the Heston++ model, which is a an
1

affine model with a multi-factor volatility dynamics and a rich jumps
structure both in price and volatility.
The multi-factor Heston (1993) structure enables the model to better
capture futures term structures along with maturity-dependent smiles.
Moreover, both correlated and idiosyncratic jumps in price and volatil-
ity factors help in reproducing the positive sloping skew of options on
VIX, thanks to an increased level of the skewness of VIX distribution
subsumed by the model.
Nevertheless, the key feature of our approach is to impose an ad-
ditive displacement, in the spirit of Brigo and Mercurio (2001), on the
instantaneous volatility dynamics which, acting as lower bound for its
dynamics, noticeably helps in capturing the term structure of volatility
expressed both through the ATM term structure of vanilla options, as al-
ready pointed out in Pacati et al. (2014), and through the term structures
of futures on VIX.
Moreover, we propose a general affine framework which allows for a
general affine transformation of the instantaneous volatility, both impos-
ing an additive displacement which support its dynamics, and a multi-
plicative displacement which modules its amplitude.
Overall, we conduct an extensive analysis with the Heston++ model
and its several nested specifications and we find an outstanding ability
in fitting the two SPX and VIX options surfaces together, along with very
different shapes of the term structure of VIX futures, with an overall ab-
solute (relative) pricing error of 2.2% (4%), showing a decisive improve-
ment in the pricing performance over non-displaced models.
Moreover the remarkable ability of the Heston++ model in capturing
features of the VIX options skew, without compromising the ability in fit
the smile of the vanilla surface, provide a strong empirical support for
the presence of two sources of upward jumps in volatility, one synchro-
nized and correlated with the price dynamics, the second one indepen-
dent and idiosyncratic.
The thesis is structured as follows:
• Chapter 1 introduces the mathematical infrastructure of affine mod-
els in the footsteps of Duffie et al. (2000).
2

• Chapter 2 presents the market of VIX and volatility derivatives and
the growing contributions of the literature.
• Chapter 3 introduces the Heston++ model and gives closed-form
pricing formulas for SPX and VIX derivatives. Moreover, the gen-
eral displaced afﬁne framework is introduced.
• Chapter 4 describes the empirical analysis conducted with the He-
ston++ model and its nested speciﬁcations and presents the results.
3

Chapter 1
Affine Models:
preliminaries
(With a bit of philosophy). The problem of valuing financial securities,
describing the dynamics of the term structure of interest rates, pricing
options and estimating credit-risk instruments would in general depend
on (and require) the knowledge with certainty of just an infinite amount
of state variables X describing the system under exam. A reductionist
approach is a must. If one accepts the idea of giving at most a proba-
bilistic description (and in addition of only a few) of the state variables
really driving the quantities to be evaluated, then interestingly a partic-
ular property of the dynamics of X is able to dramatically reduce the
complexity of the problem. This is the affinity property.
(Keeping discussion informal). An affine process is a stochastic process
X in which the drift vector, which governs the deterministic component
of the dynamics of X, the instantaneous covariance matrix, which de-
scribes how diffusive randomness enters in each component Xi of X and
spreads through the others Xj, and the jump arrival intensities, responsi-
ble for discontinuities in the dynamics of X, are all very simple functions
of the value of X at that time, that is affine functions.
(Taking it seriously). Prominent examples among affine processes in
term-structure literature are the Gaussian model of Vasicek (1977) and
4

square-root CIR model of Cox et al. (1985). Duffie and Kan (1996) in-
troduce a general multivariate class of affine jump diffusion models of
interest-rates term-structures. Concerning the option pricing literature
most of subsequent models built on the particular affine stochastic-volatility
model for currency and equity prices proposed by Heston (1993). These
were, among many, the models proposed by Bates (1996), Bakshi et al.
(1997), Duffie et al. (2000), Eraker (2004) and Christoffersen et al. (2009),
that brought successively jumps in returns and in volatility factor(s), ei-
ther idiosyncratic or simultaneous and correlated, while maintaining the
simple property that the (logarithm of the) characteristic function, which
- entirely and univocally - describes the statistical and dynamical prop-
erties of the state vector X, is an affine function of X itself. A property
that is crucial and guarantees an otherwise usually hopeless mathemati-
cal tractability of asset-pricing and risk-measures problems.
In this respect, a truly breakthrough has been made by Duffie et al. (2000),
who study in full generalities the properties of affine jump diffusion
models, from their characterization, to the problem of pricing, deriv-
ing in particular closed-form expressions for a wide variety of valuation
problems, trough a transform analysis.
This opens the way to richer - but still tractable - models both for equity
and other derivatives, such as those written on the volatility of an under-
lying process, that will be introduced in the next Chapter.
This Chapter is structured as follows: in Section 1.1 we will introduce
affine processes, substantially following Duffie et al. (2000), then in Sec-
tion 1.2 we will derive their pricing formula for call options on equity,
which is based on Fourier transform analysis, and connect it to the widespread
formula of Geman et al. (1994), which is based on a change of numeraire
technique.
1.1 Definition
We will refer to the notation in Duffie et al. (2000). The n-dimensional
jump-diffusion process (Duffie et al., 2000, sec. 2.2) Xt = (x1, x2, ..., xn)T
,
5

solving the SDE
dXt = µ(t, Xt)dt + (t, Xt)dWt + dZt (1.1)
where Wt is an n-dim standard Wiener and Zt is a n-dim pure jump
process, is said an affine jump-diffusion (AJD) process if the following
dependencies holds:
drift vector : µ(t, X) = K0 + K1X for (K0, K1) 2 Rn
⇥ Mn⇥n
covariance matrix : ( (t, X) T
(t, X))ij = (H0)ij + (H1)ij · X
= (H0)ij +
Pn
k=1(H
(k)
1 )ijXk
for (H0, H1) 2 Mn⇥n ⇥ Tn⇥n⇥n with H(k)
2 Mn⇥n
jump intensities : (t, X) = 0 + 1 · X for ( 0, 1) 2 R ⇥ Rn
short rate : R(t, X) = R0 + R1 · X for (R0, R1) 2 R ⇥ Rn
(1.2)
A more formal definition of affine process Xt can be found in Duffie
et al. (2003), where a (regular) affine process is characterized in three
equivalent ways: stating the form of its infinitesimal generator A (Theo-
rem 2.7), giving the expression for its characteristic triplet (Theorem 2.12)
and requiring the infinite decomposability property of its associated dis-
tribution (Theorem 2.15). In particular, the previous requirements corre-
sponds to their characterization of admissible parameters, as given in their
Definition 2.6. In this thesis we will deal in particular with 3-dimensional
state vectors Xt consisting of log-price xt and two volatility factors 2
i,t,
as:
Xt =
0
@
xt
2
1,t
2
2,t
1
A (1.3)
or eventually permutations of these components. According to the speci-
fication analysis developed in Dai and Singleton (2000), under some non-
degeneracy conditions and a possible reordering of indices (associated to
a permutation of the components of the state vector), it is sufficient for
affinity of the diffusion (1.1) that the volatility matrix (t, X) is of the
6

following canonical form:
(t, X)n⇥n = ⌃n⇥n
p
V n⇥n = ⌃n⇥n
0
B
B
B
B
@
p
V1(X) 0 · · · 0
0
p
V2(X) · · · 0
...
...
...
...
0 0 · · ·
p
Vn(X)
1
C
C
C
C
A
(1.4)
where ⌃, V 2 Mn⇥n with Vii = Vi(X) = ai + bi · X, with ai 2 R and bi 2
Rn
. This sufficient conditions have been extended by Collin-Dufresne
et al. (2008) and Cheridito et al. (2010) to allow for the possibility of a
number m of independent Wiener processes possibly greater than the
number of state variables n  m:
(t, X)n⇥m = ⌃n⇥m
p
V m⇥m = ⌃n⇥m
0
B
B
B
B
@
p
V1(X) 0 · · · 0
0
p
V2(X) · · · 0
...
...
...
...
0 0 · · ·
p
Vm(X)
1
C
C
C
C
A
(1.5)
where ⌃ 2 Mn⇥m (n  m) and V 2 Mm⇥m, with diagonal elements
defined as before. The extended canonical form (1.5) is not the most gen-
eral condition, but in the present contest it will be sufficient. Indeed, we
will consider only affine models in which the state vector’s components
follow only CIR (Cox et al., 1985) diffusions (+ jumps) and no Gaussian
components will be present (Cheridito et al., 2010; Collin-Dufresne et al.,
2008).
At any time t 2 [0, T] the distribution of Xt, as well as the effects of
any discounting, is described by the characteristic (K, H, , jumps, R)
w.r.t. which expectations are taken. A generalized transform is introduced
(u = (u1, ..., un)T
)
(u, Xt, t, T) = E
"
exp
Z T
t
R(s, Xs)ds
!
eu·XT
Ft
#
(1.6)
where u 2 Cn
which, for affine processes, may be expressed in the famil-
iar exponential-affine form (Duffie et al., 2000, Prop. 1):
(u, Xt, t, T) = e↵(t)+ (t)·Xt
(1.7)
7

where ↵(t) and each component k(t) (k = 1, ..., n) of (t) solve the set
of equations:
˙↵(t) = ⇢0 K>
0
1
2
T
H0 0 (✓( ) 1) (1.8)
˙k(t) = ⇢1 KT
1
1
2
nX
i,j=1
i(H
(k)
1 )ij j 1 (✓( ) 1) (1.9)
with terminal conditions:
↵(T) = 0 (1.10)
(T) = u (1.11)
This can be proved by applying Ito’s lemma to ﬁnd d (Xt), with dXt
given as in (1.1). Unless jump intensities are constant ( (t, Xt) ⌘ 0),
equations (1.9) are coupled, with different components of mixed. There-
fore ↵ and will have the following dependencies in general:
↵ = ↵(t, T, u = (u1, ..., un)T
) (1.12)
k = k(t, T, u = (u1, ..., un)T
) (1.13)
Function ✓(c), which is in fact the moment generating function of jump
amplitudes Z, is called jump transform:
✓(c) =
Z
⌦
ec·Z
d⌫(Z) (1.14)
with c = (c1, c2, ..., cn)T
2 Cn
, ⌦ ✓ Rn
and ⌫(Z = (z1, z2..., zn)T
) denot-
ing the multivariate jump-size distribution under the measure associated
to . The ﬁrst component z1 will usually denotes the jump-size of the
log-price and c1 its conjugated variable, whereas zi and ci, with i > 1,
are associated with volatility factors.1
The payoff function Ga,b(·), a, b 2 Rn
, y 2 R is introduced as follows
Ga,b(y, Xt, t, T, ) = E
"
exp
Z T
t
R(s, Xs)ds
!
ea·XT
Ib·XT y
#
(1.15)
which has a clear pricing interpretation if the chosen measure is the risk-
neutral one ( = Q): Ga,b(y, Xt, t, T, Q) is the price at time t of a claim
which pays at time T the amount ea·XT
if the claim is in-the-money at
time T (that is if b · XT  y).
1Unless a permutation of indexes has been performed.
8

1.2 Pricing
From (1.15), it is clear that the risk-neutral evaluation of the price at time
t of an European call option (of maturity T and strike: K) may be written
in the log-price xt = log St as (✏(1)i = 1 if i = 1 and 0 otherwise)
C(t, T, K) = EQ

exp
✓ Z T
t
R(s, Xs)ds
◆
(exT
K)+
(1.16)
= G✏(1), ✏(1)( log K, Xt, T, Q) KG0, ✏(1)( log K, Xt, T, Q)
where (x)+
= max(x, 0).2
Interestingly, they found a closed-form expres-
sion for Ga,b(y) in terms of the transform, via inversion of its Fourier
transform Ga,b(z):
Ga,b(y, Xt, t, T, ) =
(a, Xt, t, T)
2
1
⇡
Z 1
0
Im
⇥
e izy
(a + izb, Xt, t, T)
⇤
z
dz
(1.17)
Proof. Given in (Duffie et al., 2000, App. A).
This last expression allows to give a closed-form expression for the
price of a large class of securities in which the state process is an AJD.
In this Section we will elaborate on the connection between the Duffie
et al. (2000) generalized transform and payoff function on a side, and on
the S-martingale and T-forward measures and characteristic functions of
the general option pricing formula of Geman et al. (1994) on the other
side. We start with a simple Lemma concerning change of numeraire
transformations. We will state it as a Lemma to be self-contained in the
present exposition, but it is in fact a manipulation of (Geman et al., 1994,
Corollay 2 of Theorem 1) and the notation is borrowed from Björk (Bjork,
1998, Prop. 26.4).
Lemma 1. Assume that there exist two equivalent (on FT ) martingale mea-
sures Q0
and Q1
, whose associated numeraire processes are S0 and S1, respec-
tively. Then, the likelihood process L1
0(t) of the change of numeraire transfor-
mation Q0
! Q1
verifies:
S0(t)
S0(T)
=
S1(t)
S1(T)
·
L1
0(T)
EQ0
h
L1
0(T)|Ft
i 0  t  T (1.18)
2This expression must be changed in case permutations of the components of Xt apply:
✏(1) have to be replaced by ✏(i) if the i-th component of Xt is the log-price xt.
9

Proof. According to (Geman et al., 1994, Cor. 2), the likelihood process
L1
0(t) defined in (Bjork, 1998, Eq. 26.18) and recalled here (0  t  T)
takes the form
L1
0(t) =
S0(0)
S1(0)
S1(t)
S0(t)
(1.19)
Therefore
S0(t)
S0(T)
=
S1(t)
S1(T)
L1
0(T)
L1
0(t)
(1.20)
and thus the thesis holds since L1
0(t), as defined in (1.19), is a Q0
-martingale.
We will make use of this Lemma in order to connect, via the Abstract
Bayes’ Formula (Bjork, 1998, Prop. B.41), expectations under a given Q0
measure with those under an ad hoc chosen Q1
measure. The general
context of application is illustrated in the following Q0
-compound ex-
pectation of the variable X:
EQ0
h S0(t)
S0(T)
· X|Ft
i
=
EQ0
h
S1(t)
S1(T ) · XL1
0(T)|Ft
i
EQ0
h
L1
0(T)|Ft
i = EQ1
h S1(t)
S1(T)
· X|Ft
i
(1.21)
We will specialize Lemma 1 to transformations of the risk-neutral mea-
sure Q0
= Q, which is the martingale measure associated to the riskless
money account
B(t) = B(0)exp
⇣ Z t
0
R(s, Xs)ds
⌘
(1.22)
where we have defined the (possibly stochastic) short rate consistently
with the AJD notation above. In particular, we will consider, as ad hoc Q1
measures, the following two equivalent martingale measures:
• S-martingale measure, QS
: whose associated numeraire is the price
process S(t) of the asset and, according to definition (1.19), the like-
lihood process LS
(t), 0  t  T of the change of numeraire trans-
formation Q ! QS
is
LS
(t) =
B(0)
S(0)
S(t)
B(t)
(1.23)
10

• T-forward measure, QT
: whose associated numeraire is the price pro-
cess of a zero-coupon bond maturing at time T
p(t, T) = EQ
h B(t)
B(T)
|Ft
i
(1.24)
which is worth p(T, T) = 1 at maturity. Correspondingly, the like-
lihood process LT
(t), 0  t  T of the change of numeraire trans-
formation Q ! QT
takes the form
LT
(t) =
B(0)
p(0, T)
p(t, T)
B(t)
(1.25)
Corollary 1. Consider the risk-neutral measure (Q) and the equivalent (on
FT ) martingale measures S-martingale (QS
) and T-forward (QT
). Then the
discounting factor may be expressed as follows (0  t  T):
exp
⇣ Z T
t
R(s, Xs)ds
⌘
=
8
>>><
>>>:
S(t)
S(T)
·
LS
(T)
EQ[LS(T)|Ft]
if Q ! QS
p(t, T) ·
LT
(T)
EQ[LT (T)|Ft]
if Q ! QT
(1.26)
Proof. Straightforward from the definition of riskless money account (1.22)
and specializing Lemma 1 to the likelihood processes LS
(t) and LT
(t) in
(1.23) and (1.25).
Corollary 1 will be needed in order to relate the risk-neutral specifi-
cation Q
of the generalized transform3
(defined in (1.6))
Q
(u, Xt, t, T) = EQ
"
exp
Z T
t
R(s, Xs)ds
!
eu·XT
Ft
#
(1.27)
of the state vector process Xt with its characteristic functions under the S-
martingale and T-forward measures, as presented in the following Propo-
sition 1. Let us first introduce the conditional characteristic functions of
the log-price process xt at time T under QS
and QT
, respectively:
f1(z; Xt) = EQS
h
eizxT
|Ft
i
(1.28)
f2(z; Xt) = EQT
h
eizxT
|Ft
i
(1.29)
3Under Q, is Q and note that E Q [·] has exactly the same meaning of EQ[·], so we
have preferred the latter notation, which is more familiar to every body.
11

where the dependencies on the entire state vector process Xt = (xt, 2
1,t, 2
2,t, ...)T
is in general legitimate. These can be extended to the entire process n-
dimensional Xt process (at time T) as follows:
F1(z; Xt) = EQS
h
eiz·XT
|Ft
i
(1.30)
F2(z; Xt) = EQT
h
eiz·XT
|Ft
i
(1.31)
We have not change notation as it should be clear by the context, but to
be crystal-clear: in (1.28) and (1.29) the Fourier variable is z 2 R, whereas
in the general versions (1.30) and (1.31) it is z 2 Rn
.
Proposition 1. Consider the risk-neutral measure (Q) and the equivalent (on
) and T-forward (QT
). Then, the
risk-neutral specification Q
(1.27) of the generalized transform (1.6) may
be expressed as follows (u 2 Cn
):
Q
(u, Xt, t, T) =
8
>><
>>:
S(t)EQS

eu·XT
S(T)
|Ft
p(t, T)EQT

eu·XT
|Ft
(1.32)
at any time 0  t  T. Moreover, expressing u = Re(u) + i Im(u), with
Re(u), Im(u) 2 Rn
, we have in particular:
• if the log-price xt is the i-th component of Xt and if Re(u) = ✏(i), then
Q
verifies:
Q
⇣
✏(i) + i Im(u), Xt, t, T
⌘
= S(t)F1
⇣
Im(u); Xt
⌘
(1.33)
where the XT conditional characteristic function (w.r.t. QS
) F1(z; Xt) is
defined as in (1.30).
• if evaluated on the pure-imaginary sub-space u = i Im(u), Q
verifies:
Q
⇣
i Im(u), Xt, t, T
⌘
= p(t, T)F2
⇣
Im(u); Xt
⌘
(1.34)
where the XT conditional characteristic function (w.r.t. QT
) F2(z; Xt) is
defined as in (1.31).
12

These results are invariant under permutations of the components of the state
vector Xt.
Proof. By definition (1.27) of Q
, applying the first of (1.26), we get:
Q
(u, Xt, t, T) =
EQ
h
S(t)
S(T ) eu·XT
LS
(T)|Ft
i
EQ
h
LS(T)|Ft
i
= EQS

S(t)
S(T)
eu·XT
|Ft
= S(t)EQS

eu·XT
S(T)
|Ft (1.35)
which is the first of (1.32). Last equality holds since the asset price S(t)
at time t is Ft-measurable. Concerning with the second of (1.32), consid-
ering again (1.27) and the second of (1.26), we get:
Q
(u, Xt, t, T) =
EQ
h
p(t, T)eu·XT
LT
(T)|Ft
i
EQ
h
LT (T)|Ft
i
= EQT

p(t, T)eu·XT
|Ft
= p(t, T)EQT

eu·XT
|Ft (1.36)
where last equality holds as the zero-coupon bond price p(t, T) at time t
is Ft-measurable.
Equations (1.33) and (1.34) are particular cases of (1.32) and are obtained
expressing u 2 Cn
as u = Re(u) + i Im(u), exploiting conditions on
real/imaginary parts and substituting definitions (1.30) and (1.31) into
(1.42) and (1.43), respectively. The invariance under permutations is ach-
ieved since equations (1.32), as well as the condition resulting into the
(1.34), concern only scalar products4
; moreover the condition Re(u) =
✏(i), resulting into the (1.33), accounts explicitly for any possible reshuf-
fling of the components of Xt.
4The scalar product is unaffected by the same reshuffling of the components of the vec-
tors involved in the product. If the reshuffled vectors have the form x0 = ⇡x, then since the
permutation matrix ⇡ must be unitary (in the present Real context it is simply orthogonal):
x0
· y0
= (⇡x) · (⇡y) = (⇡T
⇡x) · y = (⇡ 1
⇡x) · y = x · y. (1.37)
13

Dealing with (risk-neutral) pricing evaluation of an European call op-
tion (1.16), we will have to evaluate the payoff function Ga,b(y) with 1-
dimensional specifications of vectors a and b. Thus the vector u 2 Cn
(on which the Q
transform have to be evaluated), will have only one
nonzero component, the first one or - if permutations of Xt apply - the
one corresponding to the log-price component xt.
The following Corollary of Proposition 1 provides a match of the
relevant-for-pricing evaluation of the generalized transform of Duffie
et al. (2000), with the conditional characteristic functions associated to
the S-martingale and T-forward distributions of the log-price appearing
in the general option pricing formula (Geman et al., 1994, Th. 2).
Corollary 2. Consider the setting of Proposition (1) and in particular if the
log-price xt is the i-th component of Xt. Then:
• if Re(u) = ✏(i) and Im(u) = z✏(i), then Q
verifies:
Q
⇣
✏(i) + iz✏(i), Xt, t, T
⌘
= S(t)f1(z; Xt) (1.38)
where the conditional characteristic function f1(z; Xt) (w.r.t. QS
) of the
log-price xT is defined as in (1.28).
• if Im(u) = z✏(i), then Q
verifies:
Q
⇣
iz✏(i), Xt, t, T
⌘
= p(t, T)f2(z; Xt) (1.39)
where the conditional characteristic function f2(z; Xt) (w.r.t. QT
) of the
log-price xT is defined as in (1.29).
vector Xt.
Proof. From definitions (1.28) and (1.29) of the log-price characteristic
functions, equations (1.38) and (1.39) are straightforward specializations
of (1.32) (first and the second, respectively). The invariance w.r.t. permu-
tations of the components of the state vector Xt is explicitly accounted in
the ✏(i) notation.
14

The following Proposition for the risk-neutral specification of the pay-
off function
Ga,b(y, Xt, t, T, Q) = EQ
"
exp
Z T
t
R(s, Xs)ds
!
ea·XT
Ib·XT y
#
(1.40)
parallels equations (1.32) of Proposition (1) for the generalized transform
Q
.
Proposition 2. Consider the risk-neutral measure (Q) and the equivalent (on
) and T-forward (QT
). Then, the
risk-neutral specification (1.40) of the payoff function (1.15) may be expressed
as follows (a, b 2 Rn
, y 2 R):
Ga,b(y, Xt, t, T, Q) =
8
>><
>>:
S(t)EQS

ea·XT
S(T)
Ib·XT y|Ft
p(t, T)EQT

ea·XT
Ib·XT y|Ft
(1.41)
vector Xt.
Proof. The proof is an application of Lemma 1. From the risk-neutral
specification (1.40) of Ga,b and applying the first of (1.26), we get:
EQ
h
S(t)
S(T ) ea·XT
Ib·XT yLS
(T)|Ft
i
EQ
h
LS(T)|Ft
i
= EQS

S(t)
S(T)
ea·XT
Ib·XT y|Ft
= S(t)EQS

ea·XT
S(T)
Ib·XT y|Ft (1.42)
which is the first of (1.41). Last equality holds since the asset price S(t)
at time t is Ft-measurable. Concerning with the second of (1.41), consid-
15

ering again (1.40) and the second of (1.26), we get:
EQ
h
p(t, T)ea·XT
Ib·XT yLT
(T)|Ft
i
EQ
h
LT (T)|Ft
i
= EQT

p(t, T)ea·XT
Ib·XT y|Ft
= p(t, T)EQT

ea·XT
Ib·XT y|Ft (1.43)
where last equality holds as the zero-coupon bond price p(t, T) at time t
is Ft-measurable. The invariance under permutations holds since equa-
tions (1.41) involve only scalar products.
From the risk-neutral evaluation (1.16) of the price of an European
call, if the log-price xt is the i-th component of Xt, then it becomes:
C(t, T, K) = EQ
"
exp
Z T
t
R(s, Xs)ds
!
(S(T) K)
+
#
(1.44)
= G✏(i), ✏(i)( log K, Xt, T, Q) KG0, ✏(i)( log K, Xt, T, Q)
As it represents a price, whose numerical value must be independent
from the speciﬁc evaluation setting, equation (1.44) must coincide with
the general option pricing formula given in Theorem 2 of Geman et al.
(1994)
C(t, T, K) = S(t)QS
⇣
S(T) K
⌘
Kp(t, T)QT
⇣
S(T) K
⌘
(1.45)
which is written in its general formulation, allowing for the possibility
of a stochastic short rate R. The next Corollary to Proposition 8 closes
the circle, as it states the correspondence between the risk-neutral pricing
formula in the DPS setting and the general one of GKR. It parallels Corol-
lary 2, which links the generalized transform under Q with the charac-
teristic functions under QS
and QT
.
Proposition 3. Consider the setting of Proposition (8). In particular if the
log-price xt is the i-th component of Xt. Then:
G✏(i), ✏(i)( log K, Xt, T, Q) = S(t)QS
⇣
S(T) K
⌘
(1.46)
G0, ✏(i)( log K, Xt, T, Q) = p(t, T)QT
⇣
S(T) K
⌘
(1.47)
16

vector Xt.
Proof. Equations (1.46) and (1.47) are straightforward specializations of
(1.41) (first and the second, respectively). The invariance w.r.t. permuta-
tions of the components of the state vector Xt is explicitly accounted in
the ✏(i) notation.
In the context of GKR, once the characteristic functions (1.28) and
(1.29) have been found, the pricing formula (1.45) can be explicitly (nu-
merically) evaluated as follows:
QS
⇣
S(T) K
⌘
=
1
2
+
1
⇡
Z 1
0
Re

e iz log(K)
f1(z; Xt)
iz
dz (1.48)
QT
⇣
S(T) K
⌘
=
1
2
+
1
⇡
Z 1
0
Re

e iz log(K)
f2(z; Xt)
iz
dz (1.49)
whereas, recalling the expression in (1.17) for Ga,b, in the Duffie, Pan
and Singleton setting the pricing formula (1.44) lead us to evaluate the
following integrals (if the log-price is the i-th component of Xt):
G✏(i), ✏(i)( log K, Xt, T, Q) =
Q
(✏(i))
2
1
⇡
Z 1
0
Im
⇥
eiz log(K) Q
(✏(i) iz✏(i))
⇤
z
dz
(1.50)
G0, ✏(i)( log K, Xt, T, Q) =
Q
(0)
2
1
⇡
Z 1
0
Im
⇥
eiz log(K) Q
( iz✏(i))
⇤
z
dz
(1.51)
where 0 is a n-vector of zeros. In order to prove (1.46) and (1.47) we
can demonstrate that the integral expressions (1.50) and (1.51) in fact co-
incides with (1.48) and (1.49) respectively. And this is indeed the case,
thanks to Corollary 2.
Observe that for any complex valued function5
g : C ! C we have
Im(g(z)) = Re(g(z)/i) and Im(z) = Im(z⇤
) 8z 2 C. Moreover, re-
calling the definition (1.6), under complex conjugation (denoted with the
5and thus a fortiori this holds for a real-valued one
17

⇤):
⇤
(u, Xt, t, T) = ⇤
(Re(u) + i Im(u), Xt, t, T) (1.52)
= (Re(u) i Im(u), Xt, t, T) (1.53)
= (u⇤
, Xt, t, T) (1.54)
This lead us to (only relevant dependencies written explicitly):
Ga,b(y) =
(a)
2
1
⇡
Z 1
0
Im
⇥
e izy
(a + izb)
⇤
z
dz
=
(a)
2
+
1
⇡
Z 1
0
Im
⇥
eizy ⇤
(a + izb)
⇤
z
dz
=
(a)
2
+
1
⇡
Z 1
0
Re

eizy ⇤
(a + izb)
iz
dz
=
(a)
2
+
1
⇡
Z 1
0
Re

eizy
(a izb)
iz
dz (1.55)
In addition we observe that:
Q
(✏(i), Xt, t, T)
1.33
= S(t) (1.56)
Q
(0, Xt, t, T)
1.34
= p(t, T) (1.57)
Therefore, beginning with (1.50), we have:
G✏(i), ✏(i)( log K, Xt, T, Q) =
Q (✏(i))
2
1
⇡
Z 1
0
Im
h
eiz log(K) Q (✏(i) iz✏(i))
i
z
dz
1.55
=
Q (✏(i))
2
+
1
⇡
Z 1
0
Re

e iz log(K) Q (✏(i) + iz✏(i))
iz
dz
1.38,1.56
= S(t)
(
1
2
+
1
⇡
Z 1
0
Re

e iz log(K)
f1(x; Xt)
iz
dz
)
1.48
= S(t)QS
⇣
S(T) K
⌘
(1.58)
18

and for (1.51) we have
G0, ✏(i)( log K, Xt, T, Q) =
Q
(0)
2
1
⇡
Z 1
0
Im
⇥
eiz log(K) Q
( iz✏(i))
⇤
z
dz
1.55
=
Q
(0)
2
+
1
⇡
Z 1
0
Re

e iz log(K) Q
(iz✏(i))
iz
dz
1.39,1.57
= p(t, T)
(
1
2
+
1
⇡
Z 1
0
Re

e iz log(K)
f2(x; Xt)
iz
dz
)
1.49
= p(t, T)QT
⇣
S(T) K
⌘
(1.59)
19

Chapter 2
VIX and VIX derivatives
The growing demand for trading volatility and managing volatility risk
has lead today to a liquid market for derivative securities whose payoff
is explicitly determined by the volatility of some underlying. Deriva-
tives of this kind are generically referred to as volatility derivatives and
include, among many, variance swaps, futures and options written on a
volatility index known as VIX (Carr and Lee, 2009).
Variance swaps were the first volatility derivatives traded in the over-
the-counter (OTC) market, dating back to the first half of the 80s. These
are swap contracts with zero upfront premium and a single payment at
expiration in which the long side pays a positive dollar amount, the vari-
ance swap rate, agreed upon at inception. In front of this fixed payment,
the short side agrees to pay the annualized average of squared daily log-
arithmic returns of an underlying index. The amount paid by the floating
leg is usually called realized variance.
By the end of 1998, both practitioner and academic works had already
suggested that variance swaps can be accurately replicated by a strip of
out-of-the-money (OTM) vanilla options (Britten-Jones and Neuberger,
2000; Demeterfi et al., 1999). The high implied volatilities experienced in
that years contributed to the definitive take off of these instruments, with
hedge funds taking short positions in variance and banks, on the other
side, buying it and contextually selling and delta-hedging the variance
replicating strip.
With the 2000s, the OTC market for volatility kept increasing, with
several innovative contracts introduced, such as options on realized vari-
20

ance, conditional and corridor variance swaps1
in 2005, and timer op-
tions2
in 2007.
On the exchanges side of the market, the Chicago Board Options Ex-
change (CBOE) introduced in 1993 the VIX volatility index. In an early
formulation, the volatility index - today known as VXO - was an average
of the Black and Scholes (1973) implied volatility of eight near term OEX
American options (calls and puts on the S&P100 index).
In 2003, CBOE completely revised the definition of VIX index under
several aspects: the underlying index was switched to be the S&P500
(SPX) and the flat implied volatility methodology of Black and Scholes
(1973) was left in favor of a robust replication of the variance swap rate
Exchange (2009), in the footprints of results in the literature (Britten-
Jones and Neuberger, 2000; Carr and Madan, 2001; Demeterfi et al., 1999).
The market definition of VIX will be presented in the next Section.
Derivatives written on VIX index were introduced in the second half
of 2000s: VIX futures in 2004 and VIX options in 2006. Their popularity
stems from the well-known negative correlation between VIX and SPX
index, which made these instruments a widely accepted asset class for
trading, diversifying and hedging. In this respect, SPX and VIX indexes,
together with options on both markets, provide a valuable source of in-
formation to better specify and understand the dynamics of volatility.
This has lead to the need of a framework for consistent pricing volatil-
ity derivatives and derivatives on the underlying, that is the need to de-
sign models able to fit the observed cross-section of option prices of both
markets and properly price and hedge exotic products.
The present chapter is organized as follows: next Section introduces
the CBOE VIX index, whereas derivatives written on it are presented
in sections 2.1.2 and 2.1.3. Market definitions and the unique empirical
properties of VIX futures and options, which make volatility a peculiar
asset class, are therein discussed. Section 2.2 is an account of the aca-
demic and practitioner contributions to VIX and VIX derivatives litera-
ture. In particular, standalone and consistent approaches are distinguished
and respectively reviewed in sections 2.2.1 an 2.2.2. The first approach
models the VIX index as a separated independent process, whilst the lat-
1These are swaps in which the floating leg pays the variance realized only during days
in which a condition is satisfied by the return process. The exact specification of the payout
of these swaps differs from firm to firm (Allen et al., 2006; Carr and Lewis, 2004).
2Exotic options whose maturity is a random stopping time, corresponding to a known
amount of cumulated realized volatility being surpassed; product of this kind had been
popularized by Société Générale Corporate and Investment Banking (SC BIC).
21

ter derives it from a model for the S&P500 returns.
2.1 Markets: definitions and empirical facts
2.1.1 VIX Index
The VIX volatility index measures the 30-day expected volatility of the
S&P500 index (Exchange, 2009). It is computed by CBOE as a model-free
replication of the realized variance over the following T = 30 days using
a portfolio of short-maturity out-of-the-money options on the S&P500
index over a discrete grid of strike prices. At time t, the quantity
2
t,T =
2
T t
X
i
Ki
K2
i
er(T t)
Q(Ki, t, T)
2
T t
✓
F(t, T)
K0
1
◆2
(2.1)
is computed and the corresponding VIX index value is
V IXt = 100 ⇥ t,T (2.2)
The sum runs over a set of strikes of OTM options of price Q(Ki, t, T)
with common expiry at time T, the risk-free rate r is the bond-equivalent
yield of the U.S. T-bill maturing closest to the expiration date of the SPX
options and Ki is the strike of the i-th option. Ki = (Ki+1 Ki)/2 is the
interval between two consecutive strikes3
and F(t, T) denotes the time-t
forward SPX index level deduced by put-call parity as
F(t, T) = K⇤
+ er(T t)
[C(K⇤
, t, T) P(K⇤
, t, T)] . (2.3)
The strike K⇤
is the strike at which the price difference between an OTM
call C(K⇤
, t, T) and put P(K⇤
, t, T) is minimum
K⇤
= Ki⇤
i⇤
= min
i
|C(Ki, t, T) P(Ki, t, T)|
(2.4)
and K0 is the first strike below the level of F(t, T).
Since V IXt is expressed in annualized terms, investors typically di-
vide it by
p
256 in order to gauge the expected size of the daily move-
ments in the stock markets implied by this index (Rhoads, 2011). Be-
ing an industry standard, several technical details apply to VIX calcula-
tion, for which we refer to the CBOE VIX white paper (Exchange, 2009).
3For the lowest (highest) strikes, Ki is defined as the difference between that strike
and next higher (lower) one.
22

Among these, the time to expiration T t is measured in calendar days
and in order to replicate the precision that is commonly used by profes-
sional option and volatility traders, each day is divided into minutes and
the annualization is consistently referred to the minutes in the year.
Moreover, the components of the VIX calculation are near- and next-
term put and call options with, respectively, more than T1 = 23 days
and less than T2 = 37 days to expiration. For these two maturity buck-
ets, formula (2.1) is applied with appropriate risk-free rates R1, R2, and
forward SPX index levels F(t, T1), F(t, T2), computed as in (2.3). The
volatility levels t,T1
, t,T2
are then consistently obtained. The effective
variance level 2
t,30 to be considered in VIX calculation is the weighted
average of 2
t,T1
and 2
t,T2
2
t,30 =

T1
2
t,T1
✓
NT2 N30
NT2
NT1
◆
+ T2
2
t,T2
✓
N30 NT1
NT2
NT1
◆
⇥
N365
N30
(2.5)
where NT denotes the number of minutes to settlement of option in the
near-/next-term maturity bucket and N30 (N365) is the number of min-
utes in 30 (365) days. Finally, the VIX index value effectively computed
is
V IXt = 100 ⇥ t,30 (2.6)
CBOE began disseminating the price level information about VIX using
the methodology exposed here from September 22, 2003, but price data
are available, back-calculated, since 1990.
Figure 1 shows thirteen years of historical closing prices of S&P500
and VIX, in which is evident the inverse relation between the two in-
dexes, with VIX spiking when the S&P500 index falls and then slowly
mean-reverting toward lower levels. Figure 2 presents the empirical VIX
closing price distribution obtained with data from 1990 to 2013. The dis-
tribution is positively skewed and leptokurtic, which is evidently in con-
trast with the negatively skewed distribution of returns, which is a styl-
ized fact commonly found in market data.
The ﬁnancial press has usually referred to VIX as the fear gauge and it is
currently considered as a reliable barometer of investor sentiment and
market volatility. The interest expressed by several investors in trading
instruments related to the market’s expectation of future volatility has
lead CBOE to introduce futures and options written on VIX index, re-
spectively in 2004 and 2006.
23

Figure 1: S&P500 and VIX index daily closing values from January 1990, to Decem-
ber 2003. Source: Bloomber and CBOE.
2.1.2 VIX Futures
The idea of a futures contract on VIX is to provide a pure play on the
volatility level, independently of the direction of S&P500. These con-
tracts are currently traded at the Chicago Futures Exchange (CFE), intro-
duced in 2003 by the CBOE expressly to provide exchange-traded volatil-
ity derivatives.
VIX futures contracts settle on the Wednesday that is thirty days prior
to the third Friday of the calendar month immediately following the
month in which the applicable VIX futures contract expires. From ﬁgure
5, for example, the May 2004 (labelled as K4) contract settled on Wednes-
day, May 19, 2004.
The underlying is the VIX index and each contract is written on $1,000
times the VIX. The date-t settlement value FV IX(t, T) of a futures of tenor
T is calculated with a so called Special Opening Quotation (SOQ) of VIX,
which is obtained from a sequence of opening prices of the SPX options
considered for the VIX calculation at date T. An extensive discussion of
the settlement procedures and market conventions of VIX futures can be
found in the paper of Zhang et al. (2010).
24

Figure 2: VIX closing price distribution. Sample is from January 1990 to March 2013.
Source: Six Figure Investing blog.
From a pricing perspective, since the VIX index is not the price of any
traded asset, but just a risk-neutral volatility forecast, there is no cost-of-
carry relationship, arbitrage free, between VIX futures price FV IX(t, T)
and the underlying V IXt (Gr¨unbichler and Longstaff, 1996; Zhang et al.,
2010)
FV IX(t, T) 6= V IXter(T t)
(2.7)
and, differently from commodity futures, there is no convenience yield
either. In absence of any other market information, the model price of
futures (and options) on VIX have to be computed according the risk
neutral evaluation formula
FV IX(t, T) = EQ
[V IXT |Ft] (2.8)
where Q denotes the martingale pricing measure and the V IXt dynam-
ics is described by some model, either directly (standalone approach) or
implied by the S&P500 dynamics (consistent approach), as will be dis-
cussed in the next section. The term structure of VIX Futures is the graph
obtained as a map
T =) FV IX(t, T) (2.9)
25

Figure 3: VIX futures term structure, as observed on Monday, 29 June 2009. VIX
futures settle prices are in US$ and tenor T is expressed in years.
and its shape provides interesting insights on market expectations. Fig-
ure 3 provides an example of humped term structure, in which a contango
market for lower tenors, in which investors expect future VIX (and, there-
fore, volatility) to rise, is followed by a backwardation phase in which mar-
ket expects volatility to calm down somehow in the future. In figure 4,
the term structure of VIX futures is plotted against date between Febru-
ary 2006 and December 2010, spanning a period before, during and after
the financial crisis. The level of prices remains low and the shape of the
term structure upward sloping until mid-2007, suggesting a too low per-
ceived value of the VIX index. The period of the crisis then raised the
overall level of the prices, but the backwarding shapes suggests that mar-
ket expected high volatility in the short-period, but not in the medium-
to long- term. The sample period in figure 4 ends just before the begin-
ning of the Greek debt crisis. By definition of futures contract, as date t
approaches the settlement date T, the price of the futures converges to
the spot VIX value and at settlement
FV IX(T, T) = V IXT (2.10)
Figure 5 provides an example of this convergence with the price time
series of four different contracts expiring between May and November
26

Figure 4: VIX futures term structure, as observed between February 2006 and De-
cember 2010. VIX futures settle prices are in US$. Source: Menc´ıa and Sentana (2013).
2004, starting from values relatively far from the corresponding VIX level
and gradually converging to its level at expiration.
In light of the present analysis of displaced affine models, a consider-
ation is useful for future reference: a hump in the term structure is hard to
get reproduced by Heston-like affine models if calibrated consistently on
both VIX futures, SPX and VIX options, unless the instantaneous volatil-
ity process t is extended with the introduction of a so-called displace-
ment t, a positive deterministic function which acts as a lower bound
for the volatility process, that we found able to dramatically increase the
fit to the term structure of futures on VIX.
2.1.3 VIX Options
Call options on VIX with maturity T and strike K are European-style op-
tions paying the amount (V IXT K)+
at maturity.Since they expire the
same day of a futures on VIX and subsume the same volatility reference
period of 30 days starting from the maturity date, from equation (2.10)
they can be regarded as options on a VIX futures contract FV IX(t, T)
sharing expiry date with the option. This implies that VIX call (put)
prices CV IX(K, t, T) (PV IX) can be priced according to the risk-neutral
27

Figure 5: Pattern of VIX index value and four VIX futures settle prices: May 04, Jun
04, Aug 04 and Nov 04, settling respectively on 19 May, 16 June, 18 August and 17
November 2004. Source: The New Market for Volatility Trading (Zhang et al., 2010).
evaluation4
CV IX(K, t, T) = e r⌧
EQ
h
(FV IX(T, T) K)
+
Ft
i
PV IX(K, t, T) = e r⌧
EQ
h
(K FV IX(T, T))
+
Ft
i (2.11)
where ⌧ = T t and satisfy the following put-call parity relation (Lian
and Zhu, 2013, eq. 25)
CV IX(K, t, T) PV IX(K, t, T) = e r(T t)
(FV IX(t, T) K) (2.12)
Moreover, no arbitrage conditions can be expressed with respect to VIX
futures price (Lin and Chang, 2009)
⇣
e r(T t)
(FV IX(t, T) K)
⌘+
 CV IX(K, t, T)  e r(T t)
FV IX(t, T)
⇣
e r(T t)
(K FV IX(t, T))
⌘+
 PV IX(K, t, T)  e r(T t)
K
(2.13)
4As it is usually assumed in the VIX derivative literature, the short rate r is held ﬁxed
and deterministic (Menc´ıa and Sentana, 2013).
28

Given the price of a call option on VIX, C⇤
V IX(K, t, T), the implied volatil-
ity Blk
V IX(K, T) at time t is inverted through the Black (1976) formula
solving the equation (Papanicolaou and Sircar, 2014, Sec. 2.2)
CBlk
V IX(K, t, T; FV IX(t, T), r, Blk
V IX(K, T)) = C⇤
V IX(K, t, T) (2.14)
where
CBlk
V IX(K, t, T; F, r, ) = e r(T t)
(FN(d1) KN(d2))
d1 =
log F
K + 1
2
2
(T t)
p
T t
d2 = d1
p
T t
(2.15)
and N(·) denotes the CDF of the standard normal distribution function.
The empirical observation of S&P500 vanilla and VIX option implied
volatility surfaces conveys relevant informations on the different nature
of the two markets. As an example of the most evident differences be-
tween the two markets, in figure 6 we plot the Black and Scholes (1973)
implied volatility surface observed on Monday, 29 June 2009 and in fig-
ure 7 the VIX implied surface of call options observed on the same date.
Both options datasets have been filtered using standard procedures
(A¨ıt-Sahalia and Lo, 1998; Bakshi et al., 1997), as will be detailed for our
empirical analysis in Chapter 4. Since VIX call options are fairly more
liquid than put options, only the former have been reported in figure 7,
and the price of an illiquid in-the-money (ITM) call option has been in-
ferred from the corresponding put price via put-call parity (2.12). The
SPX implied volatility surface observed in figure 6 presents typical fea-
tures: a negative skew more pronounced at lower maturities with OTM
calls much more cheaper than corresponding puts. The VIX surface
of figure 7 instead, shows rather peculiar characteristics: the implied
volatility smile is upward sloping and the volatility level is overall higher
compared to vanilla options.
OTM call options on VIX are much more liquid (and are traded at higher
premiums) than OTM puts, showing an opposite scenario with respect to
options on S&P500, in which OTM puts are more expensive and heavily
traded. A possible explanation for this dichotomy is the following: both
puts on S&P500 and calls on VIX provide insurance from equity market
downturns. On the buy-side, investors use OTM S&P500 put options
to protect their portfolios against sharp decreases in stock prices and in-
creases in volatility (Branger et al., 2014). On the sell-side, market makers
29

Figure 6: Black and Scholes (1973) implied volatility surface of european calls and
puts on S&P500, as observed on Monday, 29 June 2009. Asterisk (triangle) markers are
for mid (bid/ask) price implied vols. Maturities are expressed in days and volatilities
are in % points.
that have net short positions on OTM S&P500 index puts require net long
positions on OTM VIX calls to hedge their volatility risk (Chung et al.,
2011). Moreover, by holding VIX derivatives investors can expose their
portfolio to S&P500 volatility without need to delta hedge their option
open positions with positions on the stock index. Due to this possibil-
ity, VIX options are the only asset in which open interests are highest for
OTM call strikes (Rhoads, 2011).
2.2 Models: standalone and consistent approach
Theoretical approaches for VIX modeling can be broadly divided in two
categories: a consistent and a standalone approach. The contributions con-
sidered most relevant for this thesis will be reviewed in this section.
30

Figure 7: Black implied volatility surface Black (1976) of call options on VIX, as
observed on Monday, 29 June 2009. Asterisk (triangle) markers are for mid (bid/ask)
price implied vols. Maturities are expressed in days and volatilities are in % points.
2.2.1 Standalone models of VIX
In the earlier standalone approach, the volatility is directly modeled, sep-
arated from the underlying stock index process. This approach only fo-
cuses on pricing derivatives written on VIX index without considering
SPX options. A risk-neutral dynamics for V IXt is usually assumed and
pricing formulas as well as calibration to VIX futures and options can be
easily obtained. Within this stream of literature, theoretical contributions
in modeling VIX index and pricing VIX derivatives appeared well before
the opening of the corresponding markets.5
The GBM model of Whaley (1993)
In 1993, when VIX deﬁnition was still Black-Scholes based (i.e. VIX was
what is today known as VXO), Whaley (1993) modeled V IXt as a Geo-
5In this Section we mostly follow the review of Menc´ıa and Sentana (2013), though re-
deﬁning the notation in order to normalize it to the rest of the thesis.
31

metric Brownian Motion (GBM) under the martingale measure Q
dV IXt
V IXt
= rdt + dWt (2.16)
The pricing formula for a VIX call option CGBM
V IX(K, t, T) under the model
(2.16) is the Black-76 formula Black (1976), as presented in equation (2.15)
and that of a futures is
FGBM
V IX(t, T) = EQ
[V IXT |Ft] = V IXter(T t)
(2.17)
The GBM dynamics is both too simple to capture the dynamics of VIX,
since it does not allow for mean-reversion, and to reproduce the positive
implied skew of VIX options, since it yields a flat implied volatility.
The observed mean-reversion property of VIX was introduced in the
subsequent models of Grünbichler and Longstaff (1996) and Detemple
and Osakwe (2000).
The SQR model of Grünbichler and Longstaff (1996)
Grünbichler and Longstaff (1996) modeled the standard deviation of stock
index returns as a square-root mean reverting model (Cox et al., 1985)
dV IXt = ↵ ( V IXt) dt + ⇤
p
V IXtdWt (2.18)
where is the long-term mean-reverting level, ↵ the rate of mean-reversion
and ⇤ the constant vol-of-vol parameter. Under the SQR model, the VIX
index is proportional to a non-central 2
variable with 2q + 2 degrees of
freedom and parameter of non-centrality 2u, that is at any point in time
the outcome of the volatility index process is distributed according to
2cV IXT
|Ft
⇠ 2
(2q + 2, 2u) (2.19)
with
c =
2↵
⇤2 (1 e ↵⌧ )
u = cV IXte ↵⌧
v = cV IXT
q =
2↵
⇤2
1
(2.20)
32

The transition pdf of V IXt is therefore known in closed form
pQ
V IX(V IXT | V IXt) = ce u v
⇣v
u
⌘q/2
Iq(2
p
uv) ⇥ I {V IXT 0}
(2.21)
where Iq(·) is a modified Bessel function of the first kind of order q, ⌧ =
T t and the indicator function is defined as I {x 0} = 1 if x 0 and
0 otherwise. As a result, the price of a VIX futures is simply (Menc´ıa and
Sentana, 2013, eq. 4)
FSQR
V IX(t, T) = EQ
[V IXT |Ft] = + (V IXt ) e ↵⌧
(2.22)
and options on VIX can be obtained in terms of the CDF FNC 2 (·; k, )
of a non-central 2
random variable with k degrees of freedom and non-
centrality parameter (Menc´ıa and Sentana, 2013, eq. 5)
CSQR
V IX(K, t, T) = V IXte (↵+r)⌧
⇥
1 FNC 2 (2cK; 2q + 6, 2u)
⇤
+ 1 e ↵⌧
⇥
1 FNC 2 (2cK; 2q + 4, 2u)
⇤
e r⌧
Ke r⌧
⇥
1 FNC 2 (2cK; 2q + 2, 2u)
⇤
(2.23)
The LOU model of Detemple and Osakwe (2000)
Detemple and Osakwe (2000) modeled the log V IXt as an Ornstein-Uhlenbeck
process (LOU)
d log V IXt = ↵ ( log V IXt) dt + ⇤dWt (2.24)
which subsumes a log-normal conditional distribution for V IXt,
V IXT
|Ft
⇠ LogN µ(t, T), 2
(⌧) (2.25)
where
µ(t, T) = + (log V IXt ) e ↵⌧
2
(⌧) =
⇤2
2↵
1 e 2↵⌧
(2.26)
and therefore, as in the SQR model, and ↵ are the long-run mean
and mean-reversion parameters, respectively. Futures on VIX are easily
priced as conditional mean of a LogN variable
FLOU
V IX(t, T) = EQ
[V IXT |Ft] = eµ(t,T )+ 1
2
2
(⌧)
(2.27)
33

and the price of a call option on VIX can be expressed as a Black (1976)
formula (Menc´ıa and Sentana, 2013, eq. 7), given in (2.15)
CLOU
V IX(K, t, T) = CBlk
V IX(K, t, T; FLOU
V IX(t, T), r, (⌧)) (2.28)
which presents a flat implied volatility across strikes, but depending on
the maturity of the options, due to the time-dependent volatility param-
eter (⌧).
Both SQR and LOU have been extensively studied in literature: Zhang
and Zhu (2006) analyzed the SQR pricing errors on VIX futures and Dot-
sis et al. (2007) studied the gains of adding jumps. The hedging effec-
tiveness of SQR and LOU specifications have been tested by Psychoyios
and Skiadopoulos (2006), and Wang and Daigler (2011) added options
on VIX to the testing sample. Overall, as confirmed by the extensive
analysis conducted by Menc´ıa and Sentana (2013), who considered his-
torical VIX and VIX derivatives data6
from February 2006 (opening of
VIX options market) to December 2010, the LOU dynamics yields lower
pricing errors compared to the SQR. Their performance tends to deterio-
rate during the 2008-09 financial crisis and the underlying assumption of
an exponentially fast rate of mean reversion towards the long-run mean,
poses both SQR and LOU models at odds with the empirical evidence,
especially during bearish stock markets when VIX takes long periods to
revert from high levels. Moreover, both models are unable to reproduce
the positive skew observed in VIX options, the LOU yielding a flat im-
plied volatility w.r.t. strike (for each maturity), and the SQR a negative
skew.
The SQR and LOU extensions of Menc´ıa and Sentana (2013)
The restriction of an exponential rate of mean reversion in the SQR mo-
del, is relaxed introducing the concatenated CSQR model (Bates, 2012)
dV IXt = ↵ ( t V IXt) dt + ⇤
p
V IXtdWV IX
t
d t = ¯↵ ¯ t dt + ¯⇤
p
tdWt
(2.29)
where corr(dWV IX
t , dWt ) = 0. This extension features a stochastic mean
reverting level t, which in turn reverts toward a long-rung level ¯. The
6They use also historical data on the VIX index itself in order to estimate SQR and LOU
models under both under real and risk-neutral measures. Since in this thesis our focus is
on derivative pricing, we do not consider explicitly real measure specifications.
34

stochastic central tendency t directly affects the conditional mean of
EQ
[V IXT | Ft], that is the futures price (Menc´ıa and Sentana, 2013, eq.
10 and 11)
FCSQR
V IX (t, T) = ˆ + (⌧)( t
¯) + (V IXt t) e ↵⌧
(⌧) =
↵
↵ ¯↵
e ¯↵⌧ ¯↵
↵ ¯↵
e ↵⌧
(2.30)
but seems to be unable to reproduce the positive skew of VIX options,
priced according to Amengual and Xiu Amengual and Xiu (2012)
CCSQR
V IX (K, t, T) =
e r⌧
⇡
Z 1
0
Re

fCSQR
V IX (z; ⌧)
e Kz
z2
d Im(z)
Re(z) < ⇣c(⌧) :=
2↵
⇤2
1
1 e ↵⌧
(2.31)
where ⌧ = T t and
fCSQR
V IX (z; ⌧) = EQ
⇥
eizV IXT
Ft
⇤
(2.32)
with z = Re(z) + i Im(z) 2 C, is the conditional characteristic function of
VIX (Menc´ıa and Sentana, 2013, App. B).
Extensions of the LOU model are ﬁrst considered separately.
• A CTOU model extends the log V IXt dynamics with a time-varying
central tendency
d log V IXt = ↵ ( t log V IXt) dt + ⇤dWV IX
t
d t = ¯↵ ¯ t dt + ¯⇤dWt
(2.33)
where corr(dWV IX
t , dWt ) = 0.
• In the LOUJ model, compensated intense exponential jumps in-
troduce non-normality in the conditional distribution of log V IXt
d log V IXt = ↵ ( log V IXt) dt + ⇤dWV IX
t + dMt
dMt = cdNt
↵
dt
(2.34)
where Nt is an independent Poisson process and c ⇠ Exp( ).
35

• The constant spot volatility assumption is relaxed with the LOUSV
d log V IXt = ↵ ( log V IXt) dt + !2
t dWt
d!2
t = !2
t dt + cdNt
(2.35)
where Nt is an independent Poisson process, with intensity and
c ⇠ Exp( ). The advantage of the chosen speciﬁcation for the
stochastic volatility !2
t , as compared for example with a square root
dynamics, is that it allows to price futures and options on V IXt by
means of Fourier inversion of its conditional CF.
Then in combination.
• Combining time-varying central tendency and jumps, the CTOUJ
model is obtained
d log V IXt = ↵ ( t log V IXt) dt + ⇤dWV IX
t + dMt
d t = ¯↵ ¯ t dt + ¯⇤dWt
dMt = cdNt
↵
dt
(2.36)
where corr(dWV IX
t , dWt ) = 0 and jumps are as in the LOUJ mo-
del.
• If time-varying central tendency is combined with stochastic volatil-
ity, the CTOUSV model is obtained
d log V IXt = ↵ ( t log V IXt) dt + !tdWV IX
t
d t = ¯↵ ¯ t dt + ¯⇤dWt
d!2
t = !2
t dt + cdNt
(2.37)
where corr(dWV IX
t , dWt ) = 0 and stochastic volatility !2
t is as in
the LOUSV model.
All the ·OU· extensions of the basic LOU model belong to the class of the
AJD processes analyzed in Dufﬁe et al. (2000), as shown in App. A of
Menc´ıa and Sentana (2013). As a consequence, VIX derivative prices can
be obtained computing the conditional CF of the log V IXt process
f·OU·
log V IX(z; t, T) = EQ
⇥
eiz log V IXT
Ft
⇤
(2.38)
36

detailed in App. C of Menc´ıa and Sentana (2013) for all ·OU· specifica-
tions. Therefore, VIX futures are easily obtained as
F·OU·
V IX = f·OU·
log V IX( i; t, T) ⌘ EQ
[V IXT | Ft] (2.39)
and VIX options can be priced applying the results of Carr and Madan
(1999)
C·OU·
V IX =
e ↵ log K
⇡
Z 1
0
e iu log K
↵(u)du (2.40)
where
↵(u) =
e r⌧
f·OU·
log V IX(u (1 + ↵)i; t, T)
↵2 + ↵ u2 + i(1 + 2↵)u
(2.41)
Their findings show that the time-varying central tendency has a deep
impact in pricing futures, whereas the time-varying stochastic volatility
of VIX reduces pricing errors on VIX options and the CTOUSV model
yields the overall best fit in both markets. They find that jumps almost
do not change futures prices and provide a minor improvement for VIX
options. In conclusion, they give empirical support to a model of spot
(log) VIX featuring time-varying central tendency and stochastic volatil-
ity, needed to capture the level and shape of VIX futures term structure,
as well as the positive slope of options on VIX.
2.2.2 Consistent models of S&P500 and VIX
Although closed-form expressions for VIX derivatives prices are readily
obtainable with the standalone approach, the tractability comes at the ex-
pense of consistency with vanilla options. Since the same volatility pro-
cess underlies both equity and volatility derivatives, a reasonable model
should be able to consistently price both vanilla on S&P500 and deriva-
tives on VIX. A feature that is difficult to test if the volatility dynamics
is directly modeled. Moreover, VIX index itself is computed by CBOE
with a portfolio of liquid out of the money SPX vanilla, but modeling it
directly does not necessarily presumes the requested replicability.
Consistent approaches retain the inherent relationship between S&P500
and VIX index. Given a risk-neutral dynamics for the S&P500 index St,
the expression for the VIX index in continuous time has been derived in
a model-free way in terms of the risk neutral expectation of a log contract
(Lin, 2007, App. A)
✓
V IXt
100
◆2
=
2
¯⌧
EQ

log
✓
St+¯⌧
F(t, t + ¯⌧)
◆
Ft (2.42)
37

where ¯⌧ = 30/365 and F(t, t+¯⌧) = Ste(r q)¯⌧
denotes the forward price of
the underlying SPX (Duan and Yeh, 2010; Zhang et al., 2010). This expres-
sion links the SPX dynamics with that of the VIX volatility index and will
be at the base of VIX derivatives pricing. Assuming a stochastic volatility
affine specification ·SV·,7
as it is predominant within this stream of liter-
ature, the expression (2.42) takes a simple form: it is an affine function of
the stochastic volatility factors 2
i,t driving the dynamics of St
✓
V IX·SV·
t
100
◆2
=
1
¯⌧
nX
i=1
ai
2
i,t + bi
!
(2.43)
where (ai, bi) depend on the risk neutral drift of the volatility factors in
the [t, t + ¯⌧] time interval and, eventually, on the presence of jumps (both
in St and/or in 2
i,t), but not on the specification of the martingale com-
ponent of the factors (Egloff et al., 2010; Leippold et al., 2007, Corollary
1).8
Consistent models of VIX futures
Early contributions focused on the replication of the term structure of
VIX futures. Zhang and Zhu (2006), assumed a risk-neutral Heston (1993)
stochastic volatility SV model for the SPX dynamics St
dSt = rStdt + St tdWS
t
d 2
t = ↵( 2
t )dt + ⇤ tdWt
(2.44)
where corr dWS
t , dWt = ⇢dt. Zhu and Zhang (2007), extended the
(2.44) dynamics allowing for a time-dependent mean reverting level t
which can be calibrated to the term structure of the forward variance
EQ
[VT | Ft] = Vte ↵(T t)
+ ↵
Z T
t
e ↵(T s)
sds (2.45)
The time-varying mean reverting level t is made stochastic in the SMRSV
model of Zhang et al. (2010), where
d t = ¯⇤dWt (2.46)
7SV is for Stochastic Volatility, the dots are to synthetically include the generalization of
the basic SV model of Heston (1993) that will be considered in the following.
8The expression in (3.15) can be derived for any ·SV· model, given the dynamics of St. It
will be given for any model reviewed here, will be explicitly deduced for our 2-SVCVJ++
model in Chapter 3 and will be generalized to a broad class of affine models for volatility
derivatives in Proposition 10.
38

with corr(dWt , dWt ) = 0 and can be calibrated to the observed VIX
futures term structure observed in a given day. The effect of jumps in the
S&P500 and volatility dynamics has been analyzed by Lin (2007), who
considered the SVCJ model9
for xt = log St, introduced in Dufﬁe et al.
(2000)
dxt =
✓
r q ¯µ
1
2
2
t
◆
dt + tdWS
t + cxdNt
d 2
t = ↵( 2
t )dt + ⇤ tdWt + c dNt
(2.47)
where corr dWS
t , dWt = ⇢dt. The SVCJ model features correlated co-
jumps, driven by the compound Poisson process Nt, with state-dependent
intensity = 0 + 1
2
t , exponentially distributed volatility jumps c ⇠
Exp(µco, ), jumps in price conditionally normally distributed cx ⇠ N(µco,x+
⇢J c , 2
co,x) | c . The characteristic function of the jump size is given by
✓co
(zx, z ) = EQ
⇥
eicxzx+ic z
⇤
=
eiµco,xzx
1
2
2
co,xz2
x
1 iµco, (z + ⇢J zx)
(2.48)
and the compensator process is ¯µt, with ¯µ = EQ
[ecx
1] = ✓co
( i, 0).
In these models the VIX squared is as in (3.15), where
• under the SV model in Zhang and Zhu (2006):
a(¯⌧) =
1 e ¯⌧↵
↵
b(¯⌧) =
⇣
¯⌧ a(¯⌧)
⌘ (2.49)
• under the time-dependent mean-reverting model MRSV in Zhu
and Zhang (2007)
a(¯⌧) =
1 e ¯⌧↵
↵
b(t, t + ¯⌧) =
Z t+¯⌧
t
⇣
1 e (t+¯⌧ s)
⌘
sds
(2.50)
• under the stochastic mean-reverting model SMRSV in Zhang et al.
(2010), the VIX index depends on the instantaneous mean-reverting
9SVCJ is for Stochastic Volatility with Correlated Jumps in price and volatility.
39

level t
a(¯⌧) =
1 e ¯⌧↵
↵
b(t, ¯⌧) = t
⇣
¯⌧ a(¯⌧)
⌘ (2.51)
• under the SVCJ model in Lin (2007), the VIX index will depend
also on the jump sizes and correlation10
a(¯⌧) =
1 e ¯⌧↵
↵
b(¯⌧) =
↵ + µco,
↵
⇣
¯⌧ a(¯⌧)
⌘
+ 2
h
¯µ (µco,x + ⇢J µco, )
i
(2.52)
As already noted for the SQR standalone model, outcomes of a CIR pro-
cess (Cox et al., 1985) are proportional to a non-central 2
random vari-
able. Therefore, knowing the transition function pQ
( 2
T | 2
t ) (which has
the same functional form as the pQ
V IX(V IXT |V IXt) in (2.21)), VIX fu-
tures prices under the SV model of Zhang and Zhu (2006) can be com-
puted taking the expected value of the VIX at expiration
FSV
V IX(t, T) = EQ
[V IXT | Ft] = 100 ⇥
Z 1
0
p
a(¯⌧) + b(¯⌧)y pQ
(y|x)dy
(2.53)
where (a, b) are given in (2.49). In the same way can be priced futures
under the MRSV in Zhu and Zhang (2007), but pQ
( 2
T | 2
t ) has to be eval-
uated Fourier-inverting its conditional CF f (z; 2
t , t, T)
pQ
(y|x) =
1
⇡
Z 1
0
e izy
f (z; x, t, T)dz (2.54)
Zhang et al. (2010) and Lin (2007) adopted approximated expressions.11
In the SMRSV model in (2.46), they approximated T with t under the
expectation FV IX(t, T) = EQ
[V IXT | Ft], omitting O(¯⇤2
) terms, and
then made a third order expansion around EQ
⇥ 2
T Ft
⇤
, leveraging on
the availability of closed-form expressions for the moments of the CIR
model. For the SVCJ in (2.47), Lin proposed a convexity adjustment to
10For reasons of brevity, and as this will the speciﬁcation with which we will mostly work
with, we report only the expressions for = 0 and 1 = 0. The complete expressions
with 1 > 0 can be found in (Lin, 2007, eq 7).
11We refer to the papers for detailed derivations.
40

overcome the non linear relation between squared VIX, which is a known
affine function of the stochastic volatility 2
t according to (2.52), and VIX
futures price (Lin, 2007, eq. 8 and 9)
FSVCJ
V IX (t, T) = EQ
[V IXT | Ft]
⇡
q
EQ [V IX2
T | Ft]
var[V IX2
T |Ft]
8(EQ [V IX2
T | Ft])3/2
(2.55)
With calibration performed on VIX futures data from May 2004 to Novem-
ber 2008, Zhang et al. (2010) find reasonable good in sample results and,
with the mean-reverting level t calibrated on the term structure ob-
served in date t, the SMRSV model predicts one day lag t + 1 out-of-
sample changes in the term structure rather reliably (out-of-sample pe-
riod ending on February 2009). The SVCJ model of Lin (2007) evidenced
that contribution of jumps in St is determinant (with respect to a SV
specification) in pricing the medium- to long-term structure of futures
(sample from May 2004 to April 2006), while the inclusion of jumps in
volatility 2
t (possibly with a state-dependent intensity) reduce the out-
of-sample pricing error on short-term dated futures on VIX.
Nevertheless, the two approximations proposed in Lin (2007) and
Zhang et al. (2010) have been criticized by Zhu and Lian (2012), who
showed that those approximations could be often inaccurate. Moreover,
they found an exact analytical pricing formula for futures on VIX which
is applicable to any model as long as the conditional CF f (z; 2
t , t, T) is
computable. Taking as example the SVCJ model (with constant = 0),
the Zhu and Lian (2012) VIX futures pricing formula is12
FSVCJ
V IX (t, T) = EQ
[V IXT | Ft]
= 100 ⇥
1
2
p
⇡
Z 1
0
1 e sb(¯⌧)/¯⌧
f (isa(¯⌧)/¯⌧; 2
t , t, T)
s3/2
ds
(2.56)
where (a, b) are given in (2.52) and (⌧ = T t)
fSVCJ
(z; 2
t , t, T) = EQ
h
eiz 2
T Ft
i
= eA (z;⌧)+B (z;⌧) 2
t +Cco(z;⌧)
(2.57)
is the conditional CF of 2
t (Zhu and Lian, 2012, eq. A3), with coefficients
A , B and Cco satisfying the following set of ODEs (Zhu and Lian, 2012,
12 Recalling the identity
p
x = 1
2
p
⇡
R 1
0
1 e sx
s3/2 ds (Zhu and Lian, 2012, eq. A9) and
Fubini theorem.
41

eq. A4)
@A (z; ⌧)
@⌧
= ↵ B (z; ⌧)
@B (z; ⌧)
@⌧
=
1
2
⇤2
(B (z; ⌧))
2
↵B (z; ⌧)
@Cco(z; ⌧)
@⌧
=
⇣
✓co
(0, iB (z, ⌧)) 1
⌘
(2.58)
with initial conditions A (z; 0) = Cco(z; 0) = 0, B (z; 0) = iz, and closed
form solutions (Zhu and Lian, 2012, eq. A6)
A (z; ⌧) =
2↵
⇤2
log
✓
1 iz
⇤2
2↵
1 e ↵⌧
◆
B (z; ⌧) =
ize ↵⌧
1 iz ⇤2
2↵
(1 e ↵⌧ )
Cco(z; ⌧) = ⇥(z; ⌧, µco, )
⇥(z; ⌧, µ) =
2µ
⇤2 2↵µ
log
✓
1
iz
1 izµ
⇤2
2↵µ
2↵
1 e ↵⌧
◆
(2.59)
Consistent introduction of VIX options
On the wave of the increasing demand for volatility trading in the years
of the financial crisis, the academic interest has moved consistently to-
ward the rather new market of options written on the volatility process of
stock indexes, mostly focusing on the widespread CBOE options on VIX.
However, the transition was not easy at all and even today there is ongo-
ing debate about which specification is better at capturing the structural
novelties presented by the volatility surface implied by VIX options: an
upward sloping smile, more pronounced for shorter maturities and flat-
tening at the longer, with considerable time-variation on daily scales.
A clear-cut observation, which evidenced the deep distinction between
SPX vanilla options and those written on its VIX volatility index, was
made by Gatheral (2008). He pointed out that VIX options truly consti-
tute a discriminant for stochastic volatility models: even though Heston
(1993) model performs fairly well to price S&P500 option, it totally fails
to price VIX options, usually producing a negatively skewed surface. (an
example is presented in figure 8).
From the technical perspective, the transition from the linear payoffs
of VIX futures, toward the piecewise linear one of options on VIX, to-
gether with the widespread lack of known transition pdf of volatility
42

Figure 8: Market and SV model Heston (1993) implied volatilities for VIX options
(four maturities) on October 20th, 2010 (date t) plotted with respect to log-moneyness
log(K/FV IX (t, T)). Maturities T are in year fractions. The market (resp. model) im-
plied volatilities are represented by the blue crosses (resp. the solid green line). These
ﬁts are obtained by minimizing relative errors between market implied volatilities
and the Heston model implied volatility. Source: Inferring volatility dynamics and risk
premia from the S&P 500 and VIX markets (Bardgett et al., 2013, version of July 21st,
2013).
pQ
featured by the newly introduced models,13
has strongly pushed the
mathematical development and numerical implementation of sophisti-
cated techniques, commonly based on the Fourier inversion (Carr and
Madan, 1999; Lewis, 2000, 2001) or series development (Bardgett et al.,
2013; Fang and Oosterlee, 2008) of the conditional characteristic function
of volatility f , which, for the wide class of afﬁne models has a closed
13The dynamics of which is far richer than that of a SV model Heston (1993).
43

Figure 9: A Comparison of the VIX steady-state density distributions obtained with
SV, SVJ, SVVJ and SVCJ models and empirical VIX frequency. The SV model is
the Heston (1993) model considered by Zhang and Zhu (2006) and defined in (2.44).
The SVCJ model (here denoted with the alternative label SVJJ) is the one-factor cor-
related co-jump model introduced in Duffie et al. (2000), considered in (Lin, 2007,
setting 1 ⌘ 0), Zhu and Lian (2012) and Lian and Zhu (2013) and defined in (2.47).
The SVJ model features jumps in price only, is considered in the equity pricing liter-
ature in (Bakshi et al., 1997; Bates, 1996, among many), is defined in equation (2.67)
and is nested in the SVCJ model taking c ⌘ 0. The SVVJ model features jumps in
variance only, is introduced in Duffie et al. (2000) as nested in the SVCJ model taking
cx ⌘ 0 and is nested in the model of Sepp (2008b), defined in equation (2.71), setting
to the constant 1 the local volatility term t. The model implied steady-state distri-
bution is taken from the transition density pQ
V IX0 (y|x)/100 in (2.63) (or equivalently
(Zhu and Lian, 2012, eq. 8)) in the limit ⌧ = T t ! 1. Data sample: VIX close levels
between March 2004 and July 2008. Sampling frequency: daily. Model parameters:
taken from (Zhu and Lian, 2012, Table 2). Source: An analytical formula for VIX futures
and its applications Zhu and Lian (2012).
form expression (Chen and Joslin, 2012; Duffie et al., 2000).
Elaborating on the observation of Gatheral (2008) and on the empiri-
cal properties of the VIX options’ surface it can be concluded that the
pQ
volatility distribution implied by VIX options has more mass at high
volatility and less mass at lower volatility levels than the non-central 2
pdf of a SV Heston (1993) model (an example is given in figure 9).
Right skewness can be primarily induced by jumps in the volatility factor
44

2
t , as essentially proposed with the SVVJ model14
by Sepp (2008a,b) and
as featured (together with correlated co-jumps in returns) by the SVCJ
model in Duffie et al. (2000), considered in the VIX option pricing context
by (Lian and Zhu, 2013, among many).
Alternatively, one can model the S&P500 index dynamics with stochastic
volatility 2
t and a stochastic volatility of volatility !2
t positively corre-
lated to the SPX volatility dynamics. This model is likely to produce
a positive sloping skew in VIX options as it implies that low values of
the S&P500 index (market downturns) are followed by high values of
its volatility and, in turn, of its volatility of volatility. This possibility
has been considered by Branger et al. (2014), with the 2-SVSVJ model,15
which features stochastic volatility of variance, together with Gamma
distributed jumps in variance.
Multi-factor specifications, as the 2-SV model proposed by Christoffersen
et al. (2009), were already found relevant in the context of equity pric-
ing: e.g. providing stochastic leverage correlation between the return
and variance processes, better capturing the volatility term structure and
enhancing the model ability to fit maturity-dependent smiles (Ander-
sen et al., 2002; Kaeck and Alexander, 2012; Menc´ıa and Sentana, 2013,
among many). Additional factors have been added in various ways:
• as an additional independent volatility factor 2
2,t in the 2-SVCJ
model of Chen and Poon (2013) and Lo et al. (2013);
• as a stochastic volatility of variance factor !2
t in the 2-SVSVJ of
Branger et al. (2014);
• as a stochastically mean-reverting level t in the 2-SMRSVCJ16
of
Bardgett et al. (2013).
These affine specifications, together with some non-affine models pro-
posed, will be reviewed in what follows. We will first focus on mod-
els focused (and calibrated) in reproducing the empirical properties of
VIX options, then we will consider the few truly consistent models that
tackle the problem of jointly calibrating the S&P500 and VIX options sur-
faces. We anticipate that, with the exclusion of Kokholm et al. (2015),17
14SVVJ is for Stochastic Volatility with Jumps in its stochastic Volatility.
152-SVSVJ is for Stochastic Variance of Stochastic Volatility with Jumps in volatility.
162-SMRSVCJ is for 2-factors Stochastic Mean-Reversion of Stochastic Volatility with
Correlated Jumps in price and volatility.
17Whose most specified model is a one factor SVCJ model which yields not satisfactory
results, thus claiming for more flexibility.
45

we are the ﬁrst to include also VIX futures in the joint calibration of the
2-SVCVJ++ model, to be introduced in the next chapter.
Consistent models of VIX options
Lian and Zhu (2013): a general (simple) pricing formulas for SVCJ (SV) model
Lian and Zhu (2013), considered a SVCJ model given in equation (2.47),
as in Zhu and Lian (2012) and Lin (2007)18
dxt =
✓
r q ¯µ
1
2
2
t
◆
dt + tdWS
t + cxdNt
d 2
t = ↵( 2
(2.60)
and derive a closed-form expression (derived in Appendix of Lian and
Zhu (2013)) for futures FSVCJ
V IX (t, T) and call options CSVCJ
V IX (K, t, T) on VIX
(V IXt = x, K0
= K/100, ¯⌧ = 30/365, ⌧ = T t)
FSVCJ
V IX
100
=
¯⌧
2a
p
⇡
Z 1
0
Re

eizb/a
f
✓
iz;
¯⌧x2
b
a
, t, T
◆
1
(iz¯⌧/a)3/2
d Re(z)
CSVCJ
V IX
100
=
¯⌧e r⌧
2a
p
⇡
Z 1
0
Re
"
eizb/a
f
✓
iz;
¯⌧x2
b
a
, t, T
◆
1 erf(K0
p
iz¯⌧/a)
(iz¯⌧/a)3/2
#
d Re(z)
(2.61)
where z = Re(z) + i Im(z) 2 C. The integrals are performed along a
straight line parallel to the Re(z) axis, selecting 0 < Im(z) < ⇣c(⌧), where
the critical value is
⇣c(⌧) = min
1
µco,
,
1
⇤2
2↵ (1 e ↵⌧ ) + µco, e ↵⌧
!
(2.62)
as given in (Lian and Zhu, 2013, eq. A7). Moreover, (a, b) are as in (2.52),
f is the conditional characteristic function on 2
t , deﬁned in (2.57), and
erf(z) = 2p
⇡
R z
0
e s2
ds is the complex error function. The formula lever-
ages on the fact that - for one factor models - the transition pdf pQ
V IX0 of
the scaled index V IX0
= V IX/100 is in one-to-one correspondence with
pQ
and thus with f , by Fourier-inversion. From VIX expression (3.15), if
18If 1 ⌘ 0.
46

n = 1,
pQ
V IX0 (y | x) =
2¯⌧y
a
pQ
✓
¯⌧y2
b
a
¯⌧x2
b
a
◆
=
¯⌧y
a⇡
Z
R
e
iz
⇣
¯⌧y2 b
a
⌘
f
✓
z;
¯⌧x2
b
a
, t, T
◆
dz
=
2¯⌧y
a⇡
Z 1
0
Re

e
iz
⇣
¯⌧y2 b
a
⌘
f
✓
z;
¯⌧x2
b
a
, t, T
◆
d Re(z)
(2.63)
as detailed in eq. 7 and Appendix of Lian and Zhu (2013)). This one-to-
one relation is lost in multi-factor models (consider again equation (3.15)
if n 2), and thus the formula proposed by Lian and Zhu cannot be
extended directly to multi-factor afﬁne models Lian and Zhu (2013).
If the SVCJ model is restricted to the Heston SV dynamics in (2.44),
leveraging on (2.63) and on the fact that, under CIR diffusion Cox et al.
(1985),
2c 2
T
|Ft
⇠ 2
(2q + 2, 2u) (2.64)
with c, q and u given in (2.20) (with V IX· replaced by 2
· ) and transition
density pQ
( 2
T | 2
t ) given by
pQ
(y | x) = ce u v
⇣v
u
⌘q/2
Iq(2
p
uv) ⇥ I {y 0} (2.65)
as in (2.21), they show that the price of futures and options on VIX can
be computed by direct integration of their payoff19
FSV
(t, T) = 100 ⇥
Z 1
p
b(¯⌧)/¯⌧
y pQ
V IX0 (y | x)dy
CSV
(K, t, T) = 100 ⇥ e r⌧
Z 1
max(K0,
p
b(¯⌧)/¯⌧)
(y K0
)
+
pQ
V IX0 (y | x)dy
(2.66)
where ·0
= ·/100 and (x)+
= max(x, 0) and the integration domain has
been restricted, considering the effective support of the integrands.
19The formula for VIX futures is analogous to (2.53), as given in Zhang and Zhu (2006).
47

Kokholm et al. (2015): a simple pricing formula for the SVJ model
In a recent publication, Kokholm et al. (2015) extend the last kind of pric-
ing formulas to the SVJ model20
(Bakshi et al., 1997; Bates, 1996)
dxt =
✓
r q ¯µ
1
2
2
t
◆
dt + tdWS
t + cxdNt
d 2
t = ↵( 2
t )dt + ⇤ tdWt
(2.67)
where corr dWS
t , dWt = ⇢dt. The SVJ model features idiosyncratic
jumps in price only, driven by the compound Poisson process Nt, with
constant intensity . Jump sizes are normally distributed cx ⇠ N(µx, 2
x).
The characteristic function of the jump size is given by
✓x
(zx) = EQ
⇥
eicxzx
⇤
= eiµxzx
1
2
2
xz2
x (2.68)
and the compensator process is ¯µt, with ¯µSVJ = EQ
[ecx
1] = ✓x
( i)
1. The SVJ model is nested in the SVCJ model in (2.47), imposing c ⌘ 0.
The proposed pricing formula leverage on the observation of Baldeaux
and Badran (2014) that the introduction of jumps in returns imply a sim-
ple translation of the distribution of V IXt, in particular:
pQ,SVJ
V IX0 (y | x; bSVJ
(¯⌧)) = pQ,SV
⇣
y | x; bSV
(¯⌧) + 2 (✓x
( i) 1 µx)
⌘
in (2.65)
(2.69)
where we have explicitly written model dependencies and bSV
(¯⌧) has
been deﬁned in (2.49). To conclude, also if the St dynamics features
jumps, we have for the SVJ model:
FSVJ
(t, T) = FSV
(t, T)
CSVJ
(K, t, T) = CSV
(K, t, T)
b(¯⌧) =) b(¯⌧) + 2 (✓x
( i) 1 µx)
(2.70)
with FSV
, CSV
given in (2.66). These expressions are evidently simpler to
implement and faster to execute than the corresponding general formu-
las in (2.61) implemented with c ⌘ 0, that is reducing the SVCJ model
to the SVJ.
They perform a calibration on few days quotes of SPX and VIX op-
tions and VIX futures term structures. Their results are unsatisfactory,
especially considering the ability of the SVCJ model of capturing the
20SVJ is for Stochastic Volatility with Jumps in price.
48

different shapes of the term structure (Kokholm et al., 2015, figure 8). We
conclude that they are probably facing the need of an additional volatil-
ity factor.
Sepp (2008b): the SVVJ model with jumps in variance only and deterministic
ATM volatility term structure
Sepp (2008b) proposes a one factor SVVJ model in which the stochastic
volatility features positive upward jumps with the aim of better capture
the right skewness of the VIX distribution.21
dxt =

r q ¯µ
1
2
t
2
t dt +
q
t
2
t dWS
t
d 2
t = ↵( 2
(2.71)
The risk-neutral correlation is constant corr(dWS
t , dWt ) = ⇢dt and jumps
in variance are driven by the compound Poisson process with constant
intensity and the sizes distributed according to an exponential distri-
bution c ⇠ Exp(µ ), whose characteristic function is given by
✓ (z ) =
1
1 iµ z
(2.72)
Since VIX derivatives are not driven directly by returns dynamics, but
mostly by their volatility dynamics,22
the dynamics of the asset price pro-
cess is left purely continuous. In this model, the istantaneous variance is
a time-dependent affine function of the volatility level
Vc( 2
t ) = t
2
t (2.73)
Given the continuous dynamics of xt = log St, the VIX volatility index
coincides with the annualized expected diffusive quadratic variation (al-
ternatively named expected realized variance) (Sepp, 2008b, eq. 3,4 and
21To compare with the original notation in (Sepp, 2008b, eq. 2) t corresponds to 2(t)
and the volatility dynamics dV (t) corresponds to d 2
t if = 1 and 2
0 = 1. Other variables
are simply renamed.
22We will come back to this point in Section 3.2.2 (in particular in Proposition 11) when
we will present our pricing formulas (which hold for a general affine displaced volatility
framework) in which it will be clear that the price of VIX derivatives is essentially driven by
the volatility distribution, through the characteristic function of the volatility state vector.
49

8,9)
✓
V IXt
100
◆2
=
1
¯⌧
EQ
Z t+¯⌧
t
Vc( 2
s )ds Ft
=
1
¯⌧
Z t+¯⌧
t
s

1 +
µ
↵
⇣
1 e ↵(s t)
⌘
ds ⇥ 2
t
= a(t, t + ¯⌧; [t,t+¯⌧]) + b(t, t + ¯⌧; [t,t+¯⌧]) 2
t
(2.74)
where (a(t), b(t)) are (m1(t), m2(t)) given in (Sepp, 2008b, eq. 8). Futures
on VIX can be therefore expressed as a t-expectation of the (square root
of) forward realized variance23
(Sepp, 2008b, eq. 10)
FSVVJ
V IX (t, T)
100
= EQ
2
4
s✓
V IXT
100
◆2
Ft
3
5
= EQ
⇥
a(T, T + ¯⌧; [T,T +¯⌧]) + b(T, T + ¯⌧; [T,T +¯⌧]) 2
T Ft
⇤
(2.75)
Function t is a piece-wise constant deterministic function that Sepp in-
terprets coherently as an at-the-money volatility. Indeed, from the last
equation, t can be calibrated to any observed VIX futures term struc-
ture F⇤
V IX(t, T). We anticipate here that the SVVJ model is a model that
evidently belongs to our general framework of Section (3.2) for affine
models featuring a continuous spot variance Vc(Xt) which is an affine
function
Vc(Xt) = >
t Xt + t (2.76)
of the possibly multi-factor volatility state vector Xt 2 Rn
. Here Xt = 2
t
and Vc is as in (2.73). In this perspective, the expression for the VIX /
realized variance of equation (2.74) is a particular case of Proposition (8),
equation (3.63) and Proposition (10). Moreover, Sepp (2008b) presents
a pricing formula for a general class of derivatives written on volatility
(Sepp, 2008b, eq. 5,6). We skip it from the present discussion as it is
mathematically strictly related24
to the general pricing formula for VIX
derivatives that will be presented in Proposition 6 for our 2-SVCJ++ mo-
del and, more in general, in Proposition 11 for our displaced affine frame-
work for volatility.
23Which is denoted with ¯I(t, T) = EQ
h R T
t
2(t0)V (t0)dt0 Ft
i
in the original notation
of the paper (Sepp, 2008b, eq. 3).
24Thought slightly more general, since formulas (Sepp, 2008b, eq. 5,6) are not restricted
to derivatives written directly on volatility, but e.g. on the realized variance too.
50

Lo et al. (2013) and Chen and Poon (2013): Is it better to add jumps to 2
t or a
second 2
2,t factor?
The contribution of the working paper Lo et al. (2013) is twofold: it pro-
poses efﬁcient numerical approximations to compute the price of VIX
derivatives under the 2-SVCJ model and then, relying separately on VIX
futures and options data, examines the relative contribution of jumps
in volatility and of an additional volatility factor. The 2-SVCJ model
combines the SVCJ model of Dufﬁe et al. (2000) with the 2-SV model of
Christoffersen et al. (2009)
dxt =

r q ¯µ
1
2
2
1,t + 2
2,t dt + 1,tdWS
1,t + 2,tdWS
2,t + cxdNt
d 2
1,t = ↵1( 1
2
1,t)dt + ⇤1 1,tdW1,t + c dNt
d 2
2,t = ↵2( 2
2
2,t)dt + ⇤2 2,tdW2,t
(2.77)
where the jump structure is the same of the SVCJ model in (2.47) and the
two volatility factors are mutually independent and correlated with the
returns process as follows
corr(dWS
i,t, dWi,t) = ⇢idt for i = 1, 2
corr(dWS
i,t, dWj,t) = 0 if i 6= j
(2.78)
In this model, the squared VIX of (3.15) is given by
✓
V IX2-SVCJ
t
100
◆2
=
1
¯⌧
2X
i=1
ai
2
i,t + bi
!
(2.79)
with
ai(¯⌧) =
1 e ¯⌧↵i
↵i
for i = 1, 2
b1(¯⌧) =
↵1 1 + µco,
↵1
⇣
¯⌧ a1(¯⌧)
⌘
+ 2
h
¯µ (µco,x + ⇢J µco, )
i
b2(¯⌧) = 2
⇣
¯⌧ a1(¯⌧)
⌘
(2.80)
51

Their approximation is based on the following identity (·0
= ·/100)
C2-SVCJ
V IX (K, t, T)
100
= e r⌧
EQ
⇥
(V IX0
T K)+
Ft
⇤
= e r⌧
EQ
⇣p
B + VT K0
⌘+
Ft
= e r⌧
h
˜G(K0
) K0
G0, 1 B K02
i
(2.81)
where
B = (b1(¯⌧) + b2(¯⌧)) /¯⌧
Vt = a1(¯⌧) 2
1,t + b2(¯⌧) 2
2,t /¯⌧
(2.82)
and
˜G(K) = EQ
hp
B + VT I VT K2
B Ft
i
Ga,b(y) = EQ
⇥
eaVT
I {bVT  y} Ft
⇤ (2.83)
The expression in (2.81) is exact and similar to the standard representa-
tion of European options payoff, due to Duffie, Pan and Singleton (Duffie
et al., 2000, eq. 1.6), except for the non linear function ˜G(·) of the Vt pro-
cess (which is a linear combination of the variance factors 2
i,t). Their idea
is to reconnect ˜G(K) to a Ga,b(y) function approximating the non-linear
payoff with an exponential curve fitted in the (k, N) interval [V
(k)
0 , V
(k)
N ]
covering k standard deviation around the mean EQ
[VT | Ft]. The approx-
imation is based on a series expansion in terms of the form Gcn, 1( V
(k)
n )
with n = 0, ..., N and is given in eq. 15 of Lo et al. (2013). The VIX call op-
tion is given in their Proposition 1 and it is exact in the limit (k, N) ! 1.
Futures are priced setting K = r = 0 in the corresponding call option
formula. We refer to the paper for details and the lengthy expressions.
Moreover, together with nested specifications, they consider the SVCJ
model as a representative of discontinuous volatility dynamics, and the
2-SV model as representative of multi-factor specification. Their dataset
is made of daily VIX futures settle prices and VIX options end-of-the-day
quotes from January 2007 to December 2010. Under their approximated
pricing setting, they perform separately the calibration on the data of
the two markets. Their results pointed out that one-factor specifications
significantly under-perform (compared to two-factors model), in repro-
ducing humped VIX futures term structures (an is example given in fig-
ure 3). The intuition they provide for this is that (approximately) the
term structure FV IX(t, t + ⌧) produced by one-factor models can only
52

be monotonically increasing (decreasing) in the horizon ⌧ = T t when
2
t is smaller (greater) than the long-run effective mean (Lo et al., 2013,
Lemma in App. B)
eff = +
µco,
↵
(2.84)
Moreover, one-factor models like the SVCJ model can only generate
monotonic daily changes between term structures FV IX(t, ·) and FV IX(t+
1, ·) (Lo et al., 2013, Prop. 2 in App. B). Coming to the pricing of VIX
options, using the nested SVJ model in (2.67) as benchmark, they ﬁnd
that adding jumps in 2
t provide only a minor improvement in terms of
pricing error, whereas the introduction of an additional factor produces
remarkably lower RMSEs. From the experience of this thesis, we think
that the conclusions in this paper showing an almost negligible impact of
jumps in variance are likely to be biased from the separate use of the two
VIX derivatives datasets and since the authors do not consider at all op-
tions on S&P500, thought the strong need for a multi-factor structure in
volatility is perfectly in line with literature (Andersen et al., 2002; Kaeck
and Alexander, 2012; Menc´ıa and Sentana, 2013, among many).
The 2-SVCJ model (along with nested speciﬁcations) have been consid-
ered also by Chen and Poon (2013). They concentrate on the term struc-
ture of the correlation between VIX futures F(t, T) of different maturities,
which is stantaneously
⇢1,2
t =
EQ
[dF(t, T1)dF(t, T2)| Ft]
p
EQ [dF(t, T1)dF(t, T1)| Ft] EQ [dF(t, T2)dF(t, T2)| Ft]
(2.85)
where, under the 2-SVCJ, F(t, T) solve the SDE in (Chen and Poon, 2013,
Sec. 4.2.2). This has a direct implication on the effectiveness of hedging
strategies, as it is possible to hedge a futures contract on VIX with other
futures contracts of different maturities, and futures contracts on VIX are
the somehow natural hedging tool for options on VIX, being their under-
lying. Their study analytically shows that one-factor models always im-
ply a perfect correlation ⇢1,2
t ⇡ 1 between VIX futures of different maturi-
ties (at odds with the market), whereas the addition of another volatility
factor is able to enrich considerably the possible shapes of correlation
term structure produced by the model.
53

Branger et al. (2014): the stochastic volatility of variance 2-SVSVJ model
In their analysis, Branger et al. (2014) propose an affine framework to
price volatility derivatives and specialize it considering a model with
stochastic volatility of variance and gamma distributed jumps in vari-
ance. We will come back on their general framework in Section 3.2, when
we will introduce our displaced affine framework. The volatility dynam-
ics of the 2-SVSVJ model is
d 2
t = ↵( 2
t )dt + ⇤!tdW1,t + c dNt
d!2
t = ↵!( ! !2
t )dt + ⇤!!tdW!
t
(2.86)
where the risk neutral correlation between variance 2
t and its stochastic
volatility !2
t is described by
corr(dWt , dW!
t ) = ⇢ !dt (2.87)
and jumps in variance are driven by the compound Poisson process with
intensity affine = 0, + 1,
2
t and the sizes are distributed according
to Gamma distribution of shape ⌫ and mean µ
c ⇠
⇣
⌫,
µ
⌫
⌘
(2.88)
whose characteristic function is given by
✓ (z ) =
1
1 iµ
⌫ z
⌫ (2.89)
In their empirical analysis, the authors state that the returns log St dy-
namics lacks of jumps, but leave it otherwise deliberately unspecified.25
They test the 2-SVSVJ model on the average VIX option implied volatil-
ity surface of the period from February 2006 to December 2011.
Their results show that both variance jumps and a stochastic volatility
of variance are important to reconcile empirical regularities with the the-
oretical models. Positive shocks to the instantaneous variance increase
both its mean and volatility, contributing both to increase the overall
level of the surface, and to make the skew upward sloping. In terms
of VIX distribution, jumps in variance introduce right skewness. This is
25The correlation structure between the Wieners WS
t and Wt is not required in an analy-
sis based on VIX derivatives only. We will come back on the role of the correlation structure
of the model in Appendix A.7.
54

particularly pronounced26
since the Gamma distribution with shape pa-
rameter ⌫ < 1 has higher variance, skewness and kurtosis compared to
the nested exponential distribution (which is a Gamma with ⌫ ⌘ 1).
Moreover, the presence of stochastic volatility of variance factor !2
t
increases the persistency of the effect of the shocks due to jumps, which
has an impact on the long-term options and overally contributing to in-
creasing the kurtosis of the VIX distribution (more weight on both tails of
the pdf, compared to a model with !2
t = 2
t ). Finally, the strong positive
correlation ⇢ ! between variance 2
t and its stochastic volatility !2
t (they
ﬁnd ⇢ ! = 0.88) makes OTM options on VIX rather expensive,27
in turn
contributing to the upward sloping smile.
Bardgett et al. (2013): the stochastic mean-reverting level of volatility 2-SMRSVCJ
model
Bardgett et al. (2013) leverage on the widespread literature results that
have shown the inadequate limitations to the volatility dynamics induced
by one-factor models and that adding an additional factor to the Heston
(1993) model, thought increasing the complexity of the model, is a need
to provide an accurate description of the volatility dynamics (Andersen
et al., 2002; Bates, 2012; Egloff et al., 2010; Kaeck and Alexander, 2012;
Menc´ıa and Sentana, 2013, among many). They overcome the limitations
of one-factor models in their two-factor model 2-SMRSVCJ (Bardgett
et al., 2013, Sec. 2.1)
dxt =
✓
r q ¯µ
1
2
2
t
◆
dt + tdWS
t + cxdNt
d 2
t = ↵( t
2
d t = ¯↵(¯ t)dt + ¯⇤
p
tdWt + c dN0
t
(2.90)
The only nonzero correlation is corr(dWS
t , dWt ) = ⇢dt and jumps are
driven by the two independent compound Poisson processes Nt and N0
t
26As compared for example to a SVCJ model, which features exponentially distributed
jumps in 2
t .
27In particular more expensive compared to their benchmark Black (1976) prices, as com-
puted with a Whaley (1993) model (2.16) with comparable volatility of V IXt. This is some-
how the specular of the phenomenon observed in equity options in which a negative corre-
lation ⇢ between returns log St and stochastic volatility 2
t in a Heston (1993) model makes
the prices of ITM equity options higher than the benchmark Black and Scholes (1973) prices
with comparable returns volatility.
55

with affine intensities = 0 + 1
2
t + 2 t and 0
= 0
0 + 0
1 t, re-
spectvely. The sizes are independent and distributed according to cx ⇠
N(µco,x, 2
co,x), c ⇠ Exp(µco, ) and c ⇠ Exp(µ ). The characteristic
function of gaussian price jumps, ✓x
(zco,x), was already given for the
SVJ model in (2.68) (the compensator term is ¯µ = ✓x
( i) 1), whereas
exponential jumps in volatility are described by the same jump charac-
teristic function of the SVVJ model of Sepp (2008b), given in (2.72).
They leverage on the Fourier Cosine Expansion, introduced by Fang
and Oosterlee (2008), to develop in Fourier series the VIX call option pay-
off, in such a way that under their model (Bardgett et al., 2013, eq. 15)
C2-SMRSVCJ
V IX (K, t, T) = 100 ⇥ e r⌧ 1
2
A0U0 +
NX
n=1
AnUn
!
(2.91)
Coefficients An depend marginally on the jump structure of returns, through
the affine expression of the squared VIX index (Bardgett et al., 2013, eq.
9,10), and strongly on the conditional CF of the volatility state vector
( 2
t , t)>
f , (z , z ; 2
t , t, t, T) = EQ
h
eiz 2
T +iz T
Ft
i
(2.92)
where z , z 2 C. f , is computable in closed-form and takes the usual
exponential-affine form (Bardgett et al., 2013, Prop. 2.2 and App. A and
C). Coefficients Un are Cosine transforms of the rescaled payoff (x =
V IX0
= V IX/100, k = K0
= K/100)
wC(x2
)
100
=
⇣p
x2 k
⌘+
(2.93)
which have the functional form
Un =
Z b
a
(
p
x k)+
cos(!n(x a))dx
=
2
b a
Re
(
e i!na
"p
be i!nb
i!n
+
p
⇡
erf(
p
i!nb) erf(k
p
i!n)
2( i!n)3/2
#)
(2.94)
if n 1 and similarly for U0. The parameters !n = n⇡/b a are angular
frequencies and the expansion interval [a, b] is a support interval for the
distribution pQ
V IX0 of V IX0
T that have to be selected. The pricing formula
56

is exact in the limits of N, a, b ! 1. For details and derivation refer to
their Proposition 2.3 and Appendix B.
For their empirical analysis they consider a continuous t factor (c ⌘
0). The dataset for daily calibrations consists of closing prices of Eu-
ropean SPX and VIX options from March 2006 to October 2010. They
jointly calibrate the 2-SMRSVCJ, together with several nested specifica-
tions, to the cross Section of prices in some chosen dates.28
From their
analysis it can be concluded that jumps in the return log St and variance
2
t processes are needed to better reproduce the right tail of the variance
distribution and short-maturity options. Moreover, the introduction of a
stochastic level of reversion t for the variance helps to better represent
the tails of the returns distribution and the term structure of S&P 500 and
VIX option prices.
Consistent non-affine models
We will now give an account of the main non-affine models aiming at
reproducing the peculiar properties of the VIX options surface and/or
at jointly calibrating the two SPX and VIX markets. We usually refer
to the original papers for the details concerning the pricing formulas as
are usually involved and often require rather sophisticated Monte-Carlo
techniques.to get implemented.
Gatheral (2008) and Bayer et al. (2013): the double mean-erverting CEV model
DMR
Gatheral (2008) proposes a double mean-reverting model, in which each
volatility factor follows a CEV dynamics
dSt
St
= tdWS
t
d 2
t = ↵( t
2
t )dt + ⇤( 2
t ) 1
dWt
d t = ¯↵(¯ t)dt + ¯⇤( t) 2
dWt
(2.95)
where the Wieners are allowed to be correlated. The DMR model features
a short term variance level 2
t that reverts to a moving level t at rate ↵.
28Their analysis goes far beyond a simple calibration exercise. They make jointly use also
of times series data of the S&P500 and VIX indexes and estimate real P and risk-neutral
Q parameters, along with equity and variance risk-premia, adopting a particle-filtering
methodology (Pitt and Shephard, 1999). Ours is a deliberately partial review of their con-
tribution.
57

t reverts to the long term level ¯ at the slower rate ¯↵ < ↵. This model
reduces to the 2-SV model of Christoffersen et al. (2009) if 1 = 2 = 0.5
and to a double log-normal model if 1 = 2 = 1. Testing calibrations
performed on daily SPX and VIX surfaces suggest that 1 ⇡ 1, which
is consistent with the stylized fact that volatility should be roughly log-
normally distributed and that the implied VIX distribution of the 2-SV
model presents too few right skew and a too fat left tail around 0.
Closed-form pricing expressions are not available and the calibration
of the DMR model is rather involved: parameters (↵, ¯↵, ¯) are calibrated
interpolating/extrapolating/integrating the t-time series of option strips
that replicate the fair value SWt,T of variance swaps (check the variance
swap rate replication in (3.58)), which, under the diffusive dynamics St,
can be expressed as the realized variance
SWt,T =
1
T t
[log S]c
t,T =
1
T t
EQ
"Z T
t
2
s ds Ft
#
(2.96)
which can be easily computed as under the DMR model (Bayer et al.,
2013, eq. 2.3). This allows also to estimate the volatility state variables
2
t , t with a linear regression of SWt,T . While the elasticity parameters
( 1, 2) can be estimated through a SABR calibration, the other parame-
ters (vol-of-vol , ¯ and correlations between Wieners) are Monte-Carlo
estimated.
Cont and Kokholm (2013): a consistent framework for index options and
volatility derivatives
We give an extremely partial and untechnical review of their work Cont
and Kokholm (2013). The fundamental object of their framework is the
Forward Variance Swap rate V i
t , seen at time t for the forward interval
[Ti, Ti+1]. In continuous time it is the time-t expected value of the for-
ward total quadratic variation [log S]Ti,Ti+1
of the returns process St in
the [Ti, Ti+1] interval29
V i
t =
1
Ti+1 Ti
EQ
⇥
[log S]Ti,Ti+1 Ft
⇤
(2.97)
Imposing a Lévy specification for the dynamics of V i
t = V i
0 eXi
t , this in
turn imposes restrictions on the compatible dynamics of the return pro-
29See also equation (3.34), in which we defined the spot variance swap rate SWt,T (which
is obtained from V i
t if Ti = t, Ti+1 = T), and discussion in Section 3.2.1.
58

cess log St. Having directly modeled a quantity related to volatility, this
enables closed form solutions for futures and options on VIX, as long as
the conditional characteristic function of the exponent Xi
t is available (it
is given for various jump speciﬁcations in their Appendix). Options on
the underlying St index instead, need Monte-Carlo simulations of the
path of V i
t to be priced (Cont and Kokholm, 2013, eq. 3.17).
Papanicolaou and Sircar (2014): sharp regime-shifts make Heston smiling
Papanicolaou and Sircar (2014) extend the familiar Heston (1993) model
adding sharp-regime shifts to the realized volatility which has also im-
pact on jumps in price, featuring a regime-dependent jump structure.
dxt =
✓
r
1
2
f2
(✓t) 2
t ⌫(✓t )
◆
dt + f(✓t) tdWS
t (✓t)cxdNt
d 2
t = ↵( 2
t )dt + ⇤ tdWt
(2.98)
The discrete variable ✓t 2 {1, 2, 3} represents the state (low, medium and
high) of volatility and is driven by a Markov-Chain Qmn with -slow
time scale
d
dt
pQ
✓ (✓t = n) =
3X
m=1
QmnpQ
✓ (✓t = m) n = 1, 2, 3 (2.99)
These changes of state enters in the returns’ dynamics via the purely dis-
continuous process dNt = I {✓t 6= ✓t } and modulate the realized vari-
ance through function f(✓t). Jump sizes are driven, both in amplitude
and in direction, by function (✓t), which modulates positive exponen-
tial jumps cx ⇠ 1. Function ⌫(✓t ) compensate jumps.
The tractability of their model arise from the fact that options prices
P can be approximated around the original Heston price P0 by a power
series of the small time scale (Papanicolaou and Sircar, 2014, eq. 9)
P ⇡ P0 + P1 + 2
P2 (2.100)
Stock options are easily in power series of the price Fourier transform ˆP
(Papanicolaou and Sircar, 2014, eq. 12 and 13)
ˆP ⇡ ˆP0 + ˆP1 + 2 ˆP2 (2.101)
To price options on VIX, it is possible to write down explicitly the tran-
sition density of the effective volatility process which is, as a density, the
59

product of two independent densities: the pdf pQ
of the diffusion 2
t and
the Markov Chain transition density pQ
✓ of the state process ✓t (Papanico-
laou and Sircar, 2014, Sec. 4.2)
pQ 2
T = y 2
T = y ⇥ pQ
✓ (✓T = m| ✓T = n) (2.102)
This pdf, integrated against the payoff of the option w.r.t. y and summed
over the final possible states m = 1, 2, 3, gives the conditional expected
value of the payoff, that is the price.
A joint SPX and VIX option calibration performed on few selected
dates, shows that regime shifts helps capturing the positive sloping
skew of options on VIX, consistently with the SPX negative one.
To conclude, other model which is ought to mention are:
• the affine Lévy model of Kallsen et al. (2011) which allows to joint
price derivatives on the underlying and it volatility;
• the 3/2 consistent stochastic volatility model of Baldeaux and Bad-
ran (2014) which is able to capture the upward sloping smile of VIX
options and, augmented with jumps in price, is able to consistently
fit short-term vanilla options too;
• the standalone analysis of Goard and Mazur (2013) which test the
3/2 diffusion as a direct specification for the VIX index dynamics,
in which the changes in vol-of-vol are more sensible to the actual
level of the index.
60

Chapter 3
The Heston++ model
The empirical evidences and the results of the literature, discussed and
reviewed in the previous Chapter, enable us to design and motivate our
model, whose ﬁrst objective is the consistently pricing of both vanilla
S&P500 options and VIX derivatives. We make the following require-
ments to our candidate model:
• Reliability: it should be able to express an outstanding ability in
matching market prices and to guarantee it in several different mar-
ket scenarios.
• Consistency: being reliable, it should be able to accommodate con-
sistently and in a ﬁnancially convincing way the rather different
features of the equity and volatility markets.
• Tractability: being consistent with both markets, it should still pre-
serve the tractability usually featured by models designed for pric-
ing equity only and extend it to the class of volatility derivatives.
The consistency requirement induces us to exclude the standalone models
presented in Section 2.2.1 as we primarily require an adequate descrip-
tion and control of the S&P500 dynamics. In this, models that directly
specify the dynamics of V IXt are not necessarily incompatible with the
SPX vanilla surface (Menc´ıa and Sentana, 2013). Nevertheless, the re-
quested replicability of the VIX index in terms of vanilla options is not
guaranteed modeling directly its dynamics (Branger et al., 2014). We
therefore decided to opt for a consistent model for the underlying S&P500
61

index dynamics St. This in turn induces a dynamics for the V IXt index
which is by default consistent with the market definition of VIX.1
As discussed in Section 2.2.2, the academic (and practitioner) interest
around consistent models is primarily concerned with accommodating
the rather new features presented by derivatives written on VIX with
models designed for the equity market. The several different term struc-
tures experienced by futures, the high implied volatility of options on
VIX and the upward sloping smile of their implied surface, severely chal-
lenged the consistency and reliability of often standard and benchmark
models such as the Heston model (Gatheral, 2008). One-factor models
pose too strict limitation to the volatility dynamics and an additional
volatility factor is likely to provide a more accurate description (Ander-
sen et al., 2002; Bates, 2012; Egloff et al., 2010; Kaeck and Alexander, 2012;
Menc´ıa and Sentana, 2013, among many).
Multi-factor models have been found able to better capture the dif-
ferent shapes and correlation of the VIX term structure (Chen and Poon,
2013; Lo et al., 2013) and a second factor, added to a Heston dynamics,
has been introduced as a stochastic volatility of variance factor (Branger
et al., 2014) or as a stochastic mean reverting level (Bardgett et al., 2013).
The enhanced specification of the volatility of volatility provided by the
additional factor is likely to produce the upward sloping smile of the VIX
implied surface and/or to better capture its term structure.
Moreover, the distribution of VIX has been found empirically more
skewed than a 2
like distribution induced by the CIR dynamics of the
stochastic volatility factor of a Heston model and a direct channel to in-
crease the right skewness of the model distribution of VIX can be repre-
sented by the addition of jumps in the volatility dynamics (Sepp, 2008b).
Our model is in the line of the consistent approach. We specify a single
dynamics for the price process, and use this dynamics to price vanilla op-
tions together with VIX futures and options and employ an affine multi-
factor specification with jumps.
We augment the time homogeneous dynamics of the model with a
deterministic shift extension t (also called a displacement) to the stochas-
tic volatility 2
t , as already introduced by (Pacati et al., 2014), so that the
effective instantaneous volatility Vt driving the model is given by
Vt = 2
t + t (3.1)
The class of models obtained with the extension is labelled Heston++,
1Or at least with its continuous time limit, as discussed in Section 3.2.1.
62

since it parallels the structure of the CIR++ model of Brigo and Mer-
curio (2001), in which a deterministic function t is added to a time-
homogeneous spot-rate model xt, such that the instantaneous short rate
described by the model is
rt = xt + t (3.2)
and the extension is meant to fit the term structure of interest rates. Pacati
et al. (2014) show that the deterministic shift can dramatically improve
the calibration of the term structure of at-the-money vanilla options, thus
improving sensibly the fit of the whole surface of vanilla.
In this paper, we extend their model (by adding jumps in volatility)
and show that the deterministic shift t provides the necessary flexibil-
ity to describe the term structure of VIX futures and the surface of VIX
options, without compromising the fit on vanilla options, which makes
our model eventually reliable. Moreover, this flexibility comes at no ad-
ditional expense in terms of both analytical and numerical complexity,
compared to a non-displaced specification, which makes it also tractable.
Further, the success of our proposed specification to jointly fit the
vanilla and VIX surfaces (two ”smiles” at once) also allows to exploit the
additional information content provided by variance derivatives to learn
about the features of the price dynamics. Overall, we provide strong sup-
port for the contemporaneous presence of two kinds of jumps in volatil-
ity, the first being correlated with jumps in the index (typically, account-
ing for market downturns accompanied by a spike in volatility, as also
empirically supported by Todorov and Tauchen (2011) and Bandi and
Renò (2015)), and the second being independent from price movements
and accounting for spikes in volatility not accompanied by changes in
the index. Our empirical findings suggest then that traders in option
markets hedge against both sources of risk. In particular, idiosyncratic
jumps in volatility appear to be particularly relevant for the pricing of
VIX options.
The Chapter is structured as follows. In Section 3.1 we specify the
model adopted in our empirical investigations together with the closed-
form pricing expressions for SPX vanilla options and VIX index and
derivatives. In Section 3.2 we introduce a general affine framework
which allows for a general affine transformation of the instantaneous
volatility.
63

3.1 Pricing VIX derivatives with the Heston++
model
In this Section we introduce the Heston++ model for the dynamics of the
underlying price. It is an affine model with a deterministic shift exten-
sion in the spirit of Brigo and Mercurio (2001). We then provide pricing
formulas for equity and VIX futures and options.
3.1.1 Model specification
We consider a filtered probability space (⌦, F, {Ft}t 0 , Q), satisfying
usual assumptions. Under the risk-neutral measure Q, we specify the
evolution of the logarithmic price of the underlying xt = log St as fol-
lows
dxt =

r q ¯µ
1
2
2
1,t + t + 2
2,t dt +
q
2
1,t + t dWS
1,t + 2,tdWS
2,t + cxdNt
d 2
1,t = ↵1( 1
2
1,t)dt + ⇤1 1,tdW1,t + c dNt + c0
dN0
t
d 2
2,t = ↵2( 2
2
2,t)dt + ⇤2 2,tdW2,t
(3.3)
where r is the short rate, q is the continuously compounded dividend
yield rate, and in which the risk-neutral dynamics of the index is driven
by continuous and discontinuous shocks, modeled by the Wiener pro-
cesses WS
1,2, W1,2 and the independent Poisson processes N, N0
respec-
tively. The short rate and the dividend rate are kept constant for simplic-
ity, but could be easily be made time-varying, for example as in Bakshi
et al. (1997). The first volatility factor is displaced, as in Pacati et al. (2014),
by a sufficiently regular deterministic function t which verifies:
t 0 and 0 = 0, (3.4)
and ↵i, i, ⇤i are non-negative constants. In this (and following) Chapter
we generically label this model as Heston++. Alternatively, we refer to it
also as 2-SVCVJ++ model, stressing its dynamical properties2
and even-
tually to distinguish from the several nested specifications that will be
discussed in Section 3.1.2 and will be as well part of the empirical anal-
ysis presented in the next Chapter. The corresponding dynamics of the
22-factor Stochastic Volatility model with Co-jumps between price and volatility and
idiosyncratic Volatility jumps.
64

index St is, by It¯o’s lemma:
dSt
St
= (r q ¯µ) dt+
q
2
1,t + tdWS
1,t+ 2,tdWS
2,t+(ecx
1) dNt (3.5)
All correlations among Wiener processes are zero, with the exception of
the following ones, which are defined as
corr(dWS
1,t, dW1,t) = ⇢1
s
2
1,t
2
1,t + t
dt (3.6)
corr(dWS
2,t, dW2,t) = ⇢2dt (3.7)
where ⇢1, ⇢2 2 [ 1, 1] are constants. This choice guarantees that the mo-
del is affine according to the specification analysis of Dai and Singleton
(2002), extended by (Cheridito et al., 2010; Collin-Dufresne et al., 2008).
Indeed, with this correlation structure imposed, it is possible to write the
diffusion matrix (t, 2
1,t, 2
2,t) in the extended canonical form:
(t,
2
1,t,
2
2,t) = ⌃
q
V (t, 2
1,t, 2
2,t)
=
0
B
@
⇤1 0 0 0
0 0 ⇤2 0
⇢1
q
1 ⇢2
1 ⇢2
q
1 ⇢2
2
1
C
A
0
B
B
B
B
B
B
B
@
q
2
1,t 0 0 0
0
r
2
1,t + t
1 ⇢2
1
0 0
0 0
q
2
2,t 0
0 0 0
q
2
2,t
1
C
C
C
C
C
C
C
A
(3.8)
Appendix A.7, under the extended displaced affine framework intro-
duced in Section 3.2, elaborates on the meaning of the restrictions in
the correlation structure, such as (3.6), to be imposed in displaced affine
models in order to preserve and extend the affinity of the un-displaced
model toward the displaced specification.
The Poisson processes Nt and N0
t are independent (between them)
and also independent from all the Wiener processes. Their intensities
are given by the constant parameters and 0
respectively. They drive
jumps in price and jumps in volatility. The first Poisson process Nt is
responsible for correlated jumps, occurring simultaneously in price and
volatility, with sizes cx and c respectively. The second Poisson process
N0
t is instead responsible for idiosyncratic jumps in volatility, with size
c0
, independent from all other shocks. Jumps in volatility are exponen-
tially distributed, with parameters µco, and µid, expressing the mean of
correlated and idiosyncratic jumps respectively. Jumps in price are con-
ditionally (to jumps in volatility) normally distributed with conditional
65

mean µx +⇢J c and variance 2
x. The characteristic functions of the jump
sizes are thus given by:
✓co
(zx, z ) = EQ
⇥
eicxzx+ic z
⇤
=
eiµxzx
1
2
2
xz2
x
1 iµco, (z + ⇢J zx)
✓id
(z0
) = EQ
⇥
eic0
z0 ⇤
=
1
1 iµid, z0
(3.9)
where zx, z , z0
2 C. Jumps characteristic functions in equation (3.9) can
be extended to the complex plane as long as, respectively
Im(z + ⇢J zx) > 1/µco,
Im(z0
) > 1/µid,
(3.10)
This lead to the parameter restriction ⇢J < 1/µco, which is assumed
throughout the present analysis and which is often a fortiori satisfied by
market calibrated correlation parameter, as it is usually found ⇢J  0.
We define ¯µ = EQ
[ecx
1] = ✓co
( i, 0) 1, so that the price jump com-
pensator is ¯µt.
3.1.2 Nested models
The Heston++ model (3.3) belongs to the affine class of (Duffie et al.,
2000). In case of no displacement ( t ⌘ 0), the model nests several mod-
els already analyzed in the literature and introduced in Section 2.2.1.
Imposing 2,t ⌘ 0, several one-factor specifications can be obtained: the
standard Heston SV model of (2.44) if Nt ⌘ N0
t ⌘ 0 is additionally im-
posed; if N0
t ⌘ z ⌘ 0 (i.e. allowing for log-normal jumps in price only)
the SVJ model in (2.67) is recovered, which is considered for example by
(Bakshi et al., 1997; Bates, 1996) and introduced by Duffie et al. (2000) as
a nested specification of the SVCJ model in (2.47), which features corre-
lated co-jumps in price and volatility.
The SVCJ model, extensively studied in the equity pricing literature
(Broadie et al., 2007; Eraker, 2004; Eraker et al., 2003, among many), is
obtained by switching off the N0
t Poisson process and imposing 2,t ⌘ 0.
This model is considered for the pricing of futures and options on VIX
by Kokholm et al. (2015); Lian and Zhu (2013); Lin (2007); Zhu and Lian
(2012).
If Nt ⌘ 2,t ⌘ 0, we obtain the SVVJ model of equation (2.71) which fea-
tures idiosyncratic jumps in volatility and is introduced in Duffie et al.
66

(2000) as nested in the SVCJ model switching off jumps in price. The
SVVJ model is adopted by Sepp (2008b) for VIX option pricing extended
with a local volatility term.
Two-factor specifications can be obtained letting 2,t > 0: the double He-
ston 2-SV model of Christoffersen et al. (2009) is obtained imposing no
jumps Nt ⌘ N0
t ⌘ 0. If N0
t ⌘ z ⌘ 0, the 2-SVJ of Bates (2000) with con-
stant jump intensity is obtained. Finally if N0
t ⌘ 0 we obtain the 2-SVCJ
model of equation (2.95) considered by Chen and Poon (2013); Lo et al.
(2013) for VIX derivatives pricing. The corresponding displaced models
are obtained letting t 0 and are labelled as their t ⌘ 0 counterparts,
with the suffix ++. Without restrictions, we label the model by 2-SVCVJ.
The unrestricted model has in total 17 parameters, that can be schemati-
cally described as follows
St :
N jumps
z }| {
(µx, 2
x) ( , ⇢J )
| {z }
2
1,t : ↵1, 1, ⇤1, 2
1,0
| {z }
SVfactor
co
z }| {
µco,
idiosync
z }| {
( 0
, µid, )
| {z }
Exp jumps
2
2,t :
z }| {
↵2, 2, ⇤2, 2
2,0
plus the function t.
3.1.3 SPX and VIX derivatives pricing
In this Section we generically label with H the model 2-SVCVJ and its
nested specifications described in the previous Section and with H++
the 2-SVCVJ++ model in (3.3) and its nested specifications (H models
with t 0). The analytical tractability of the displaced models H +
+ directly stems from the properties of non-displaced specifications H.
The following Lemma summarizes the relation among the log-price and
volatility characteristic functions of the H and H + + models. All proofs
and mathematical detailes are contained in Appendix A.1.
Lemma 2. Under the H++ models, the conditional characteristic function of the
price returns fH++
x (z) = EQ
⇥
eizxT
Ft
⇤
and of the two stochastic volatility
67

factors fH++
(z1, z2) = EQ
h
eiz1
2
1,T +iz2
2
2,T Ft
i
are given by:
fH++
x (z; xt, 2
1,t, 2
2,t, t, T, ) = fH
x (z; xt, 2
1,t, 2
2,t, ⌧)e
1
2 z(i+z)I (t,T )
fH++
(z1, z2; 2
1,t, 2
2,t, ⌧) = fH
(z1, z2; 2
1,t, 2
2,t, ⌧)
(3.11)
where ⌧ = T t, z, z1, z2 2 C and I (t, T) =
R T
t sds.
We can thus provide closed-form pricing formulas for vanilla options
and VIX derivatives for any of the H++ model based on the conditional
characteristic functions of the log-index and volatility factors under the
corresponding H model. For both classes of derivatives, we use the re-
sults of Lewis (2000, 2001) which turn out to be convenient for numerical
implementation.
Proposition 4. Under the H++ models, the arbitrage-free price at time t of
a European call option on the underlying St, with strike price K and time to
maturity ⌧ = T t, is given by
CH++
SP X (K, t, T) = Ste q⌧ 1
⇡
p
StKe
1
2
(r+q)⌧
Z 1
0
Re

eiuk
fH
x
✓
u
i
2
◆
e (u2
+ 1
4 )I (t,T )
u2 + 1
4
du
(3.12)
where k = log St
K + (r q)⌧.
The price dynamics under the H++ models also determines the dy-
namics of the volatility index. In practice, as will be discussed in Sec-
tion 3.2.1, the VIX quotation at time t is computed by CBOE as a model-
free replication of the integrated variance over the following 30 days. In
the present analysis we will adopt a standard deﬁnition for the volatil-
ity index, expressed as the risk-neutral expectation of a log-contract, as
given in equation (2.42), which we rewrite for convenience (Duan and
Yeh, 2010; Lin, 2007; Zhang et al., 2010):
✓
V IXt
100
◆2
=
2
¯⌧
EQ

log
✓
St+¯⌧
F(t, t + ¯⌧)
◆
Ft (3.13)
where Ft,t+¯⌧ = e(r q)¯⌧
St denotes the forward index quotation. For the
CBOE VIX, ¯⌧ = 30 days. The following Proposition gives the expression
of V IXt under the H models and the effect of the displacement t on the
index dynamics.
68

Proposition 5. Under the H++ models,
V IXH++
t
100
!2
=
✓
V IXH
t
100
◆2
+
1
¯⌧
I (t, t + ¯⌧) (3.14)
where (V IXH
t /100)2
is the corresponding quotation under H models, which is
an afﬁne function of the volatility factors 2
1,t and 2
2,t
✓
V IXH
t
100
◆2
=
1
¯⌧
0
@
X
k=1,2
ak
2
k,t + bk
1
A (3.15)
where I (t, t + ¯⌧) =
R t+¯⌧
t sds and the exact forms of ak(¯⌧) and bk(¯⌧) are
provided in Appendix A.3.
Pricing of VIX derivatives is complicated by the non afﬁnity of VIX
with respect to volatility process. As discussed in Section 2.1.2, the
arbitrage-free price FV IX(t, T) at time t of a futures contract with tenor
T written on it cannot be derived as a simple cost-of-carry relationship,
but has to be evaluated as the risk neutral expectation of the VIX at set-
tlement (Bardgett et al., 2013; Zhang et al., 2010)
FV IX(t, T) = EQ
[V IXT | Ft] (3.16)
Call options on VIX with maturity T and strike K are European-style
options paying the amount (V IXT K)+
at maturity. As discussed in
Section 2.1.3, they can be regarded as options on VIX futures price pro-
cess and can be priced according to standard risk-neutral evaluation
CV IX(K, t, T) = e r⌧
EQ
⇥
(V IXT K)+
Ft
⇤
(3.17)
We solve the complications related to the non-linear relation between
VIX and volatility by taking advantage of the analytical tractability of the
conditional characteristic function of the volatility factors fH++
(z1, z2) in
Lemma 2, and on the generalized Fourier transform techniques of Chen
and Joslin (2012); Lewis (2000, 2001). We provide an explicit pricing for-
mula for futures and options on VIX for H + + models in the following
Proposition. Similar results can be found in the literature (Branger et al.,
2014; Lian and Zhu, 2013; Sepp, 2008a,b).
Proposition 6. Under H++ models, the time t value of a futures on V IXt
settled at time T and the arbitrage-free price at time t of a call option on V IXt,
69

with strike price K and time to maturity ⌧ = T t are given respectively by
(not relevant dependencies suppressed and ¯⌧ = 30/365)
F H++
V IX (t, T )
100
=
1
2
p
⇡
Z 1
0
Re
2
4f
H
✓
z
a1
¯⌧
, z
a2
¯⌧
◆
e
iz
⇣P
k=1,2 bk+I (T,T +¯⌧)
⌘
/¯⌧
( iz)3/2
3
5 d Re(z)
(3.18)
and
CH++
V IX (K, t, T )
100
=
e r⌧
2
p
⇡
⇥
Z 1
0
Re
2
4f
H
✓
z
a1
¯⌧
, z
a2
¯⌧
◆
e
iz
⇣P
k=1,2 bk+I (T,T +¯⌧)
⌘
/¯⌧
1 erf(K/100
p
iz)
( iz)3/2
3
5 d Re(z)
(3.19)
where z = Re(z) + i Im(z) 2 C, 0 < Im(z) < ⇣c(⌧), ⇣c(⌧) is given in
Appendix A.4, and erf(z) = 2p
⇡
R z
0
e s2
ds is the error function with complex
argument (Abramowitz and Stegun, 1965).
We have analyzed the effect of the choice of the upper bound ⇣c(⌧)
on the integrand behavior and pricing performance. Figure 10 reports,
for the 2-SVCVJ model, the shape of the two integrands of Proposition 6
on the imaginary z axis Re(z) = 0 and the effect on VIX Options and Fu-
tures model prices when Im(z) is set to different values within the strip
of regularity 0 < Im(z) < ⇣c(⌧) in equation (A.21). Figure 11 reports the
effect of the Im(z) running in the 2-SVCVJ++ model in correspondence
of different ranges of the displacement integral I (T, T + ¯⌧) in Propo-
sition 6. In our empirical analysis we have found convenient to chose
Im(z) = ⇣c(⌧)/2.
We conclude this Section by observing that we are not assuming any ex-
plicit functional form for the displacement function t, but we use it as
an analytically tractable correction for the corresponding pricing formu-
las for the non-displaced models H. The only degrees of freedom of t
determined by SPX and VIX derivatives are its integrals over the life of
the options on price, I (t, T), and those over the ﬁxed forward volatility
horizon of ¯⌧ from the expiry of VIX futures/options onward I (T, T +¯⌧).
These are the quantities that will be effectively calibrated to market data.
Moreover, calibrated integrals are constrained by the non-negativity of
t. For example, if we observe two consecutive SPX vanilla maturities
TSP X
1 , TSP X
2 and an intermediate VIX option expiration TV IX
ordered
as
t < TSP X
1 < TV IX
< TV IX
+ ¯⌧ < TSP X
2 , (3.20)
70

the only ordering in the integrals compatible with t 0 is
I (t, TSP X
2 ) I (t, TSP X
1 ) I (TV IX
, TV IX
+ ¯⌧) 0 . (3.21)
3.2 A general displaced affine framework for
volatility
We will now introduce a general framework, which embeds the H++
models,3
that allows for a more general description of the istantaneous
volatility. The t-displaced (eventually multi-factor) dynamics of the
spot volatility is further extended with a second deterministic displace-
ment function t, which modules the amplitude of the volatility process,
which seems to be a feature already noticed and appreciated in literature
(Papanicolaou and Sircar, 2014; Sepp, 2008b; Zhao, 2013). The general
framework describes in a mathematical compact way these possible gen-
eral deterministic extensions of the volatility process, still preserving the
affinity of the specification.
Our analytical approach builds on the general characterization of affine
models introduced by Duffie et al. (2000), the affine model for variance
swaps of Egloff et al. (2010); Leippold et al. (2007) and on the affine mo-
del of VIX derivatives of Branger et al. (2014). We consider a class of
displaced AJD models in which the risk-neutral dynamics of the S&P500
index features several diffusive and jump risk sources and two general
forms of displacement characterize the dynamics of the instantaneous
variance process, which is affine in the state vector of volatility factors.
Consider a filtered probability space (⌦, F, {Ft}t 0 , Q), where {Ft}t 0
represents the history of the market up to time t and Q denotes the pric-
ing measure. The dynamics of the volatility factor state vector Xt =
( 2
1,t, ..., 2
n,t)>
2 Rn
is described by the affine jump diffusion
dXt = µ(t, Xt)dt + (t, Xt)dWX
t +
mXX
j=1
dZj,t (3.22)
where WX
t is an n-dim standard Wiener process and each Zj,t is a n-
dim compound Poisson process. The affine structure of the process is the
following:
3That is, the 2-SVCVJ++ model introduced in the previous Section together with all its
nested specifications discussed in Section 3.1.2.
71

0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Panel A: VIX Options (45 days)
I m(z)
Integrand
K = 10
K = 40
K = 70
0 1 2 3 4 5 6 7
36.358724
36.358725
36.358726
36.358727
36.358728
Panel B: VIX Options (45 days)
Price
K = 10
0 1 2 3 4 5 6 7
6.35872765
6.35872770
6.35872775
6.35872780
Price
K = 40
0 1 2 3 4 5 6 7
0.08731690
0.08731691
0.08731692
0.08731693
I m(z)
Price
K = 70
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Panel C: VIX Options (6 months)
I m(z)
Integrand
K = 10
K = 40
K = 70
0 0.5 1 1.5 2 2.5 3 3.5 4
37.6485975
37.6485980
37.6485985
Panel D: VIX Options (6 months)
Price
K = 10
0 0.5 1 1.5 2 2.5 3 3.5 4
7.64859980
7.64859985
7.64859990
7.64859995
Price
K = 40
0 0.5 1 1.5 2 2.5 3 3.5 4
0.33476590
0.33476591
0.33476592
0.33476593
0.33476594
I m(z)
Price
K = 70
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Panel E: VIX Futures
I m(z)
Integrand
45 days
3 months
6 months
0 1 2 3 4 5 6 7
46.35864
46.35866
46.35868
46.35870
46.35872
Panel F: VIX Futures
Price
45 days
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
47.08996
47.08998
47.09000
47.09002
Price
3 months
0 0.5 1 1.5 2 2.5 3 3.5 4
47.64848
47.64850
47.64852
47.64854
47.64856
Price
6 months
Figure 10: Left panels A, C and E report the shape of the VIX Options and Fu-
tures integrands in the pricing formulas of Proposition 6 for the 2-SVCVJ model as
a function of Im(z) ranging in the strip of regularity 0 < Im(z) < ⇣c(⌧) and setting
Re(z) = 0. Model parameters are taken from the last column of table 6, interest rate
is set to r = 0 and there is no displacement ( t ⌘ 0). In panel A (C) we consider VIX
options with a maturity of 45 days (6 months), i.e. ⌧ = 45/365 (⌧ = 0.5), and strikes
K = 10, 40 and 70, whereas in panel E VIX Futures with tenors of 45 days, 3 and 6
months, i.e. ⌧ = 45/365, 0.25 and 0.5 (corresponding ⇣c(⌧) bounds are 6.80, 4.69 and
3.89, respectively). Right panels B and D ) present VIX options (Futures) 2-SVCVJ
model prices as a function of 0 < Im(z) ⇣c(⌧) for maturities corresponding to the
right panels A and B (E). Integrals are calculated with the Matlab function quadgk,
with default error tolerance (AbsTol = 10 10 and RelTol = 10 6).
7

0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Panel A: VIX Options (45 days)
I m(z)
Integrand
φt ≡ 0
Iφ = 10−4
Iφ = 10−3
Iφ = 10−2
0 1 2 3 4 5 6 7
6.49137720
6.49137725
6.49137730
Panel B: VIX Options (45 days)
Price
Iφ = 10−4
0 1 2 3 4 5 6 7
7.667986
7.667987
7.667988
7.667989
Price
Iφ = 10−3
0 1 2 3 4 5 6 7
18.103980
18.103981
18.103982
18.103983
I m(z)
Price
Iφ = 10−2
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Panel C: VIX Options (6 months)
I m(z)
Integrand
φt ≡ 0
Iφ = 10−4
Iφ = 10−3
Iφ = 10−2
0 0.5 1 1.5 2 2.5 3 3.5 4
7.7790910
7.7790915
7.7790920
7.7790925
Panel D: VIX Options (6 months)
Price
Iφ = 10−4
0 0.5 1 1.5 2 2.5 3 3.5 4
8.93677530
8.93677540
8.93677550
8.93677560
Price
Iφ = 10−3
0 0.5 1 1.5 2 2.5 3 3.5 4
19.2211292
19.2211294
19.2211296
19.2211298
19.2211300
I m(z)
Price
Iφ = 10−2
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Panel E: VIX Futures (45 days)
I m(z)
Integrand
φt ≡ 0
Iφ = 10−4
Iφ = 10−3
Iφ = 10−2
0 1 2 3 4 5 6 7
46.491380
46.491385
46.491390
46.491395
Panel F: VIX Futures (45 days)
Price
Iφ = 10−4
0 1 2 3 4 5 6 7
47.667986
47.667987
47.667988
47.667989
Price
Iφ = 10−3
0 1 2 3 4 5 6 7
58.10395
58.10400
58.10405
58.10410
I m(z)
Price
Iφ = 10−2
Figure 11: Left panels A, C and E report the shape of the VIX Options and Futures
integrands in the pricing formulas of Proposition 6 for the 2-SVCVJ++ model as a
function of Im(z) ranging in the strip of regularity 0 < Im(z) < ⇣c(⌧) and setting
Re(z) = 0. Model parameters are taken from the last column of table 6, interest rate
is set to r = 0 and displacement parameter I (T, T + ¯⌧) is set to I = 10 4, 10 3
and 10 2. In panel A (C) we consider VIX options with a maturity of 45 days (6
months), i.e. ⌧ = 45/365 (⌧ = 0.5), and strikes K = 40, whereas in panel E VIX
Futures with tenors of 45 days, i.e. ⌧ = 45/365 (corresponding ⇣c(⌧) bound is 6.80).
Right panels B and D (F) present VIX options (Futures) 2-SVCVJ++ model prices as
a function of 0 < Im(z) < ⇣c(⌧) for maturities corresponding to the right panels
A and B (E). Integrals are calculated wit he Matlab function quadgk, with default
error tolerance (AbsTol = 10 10 and RelTol = 10 6).
7

Drift vector: µ(t, X) = K0 + K1Xt where K0 2 Rn
and K1 2 Rn⇥n
;
Variance-covariance matrix: ( (t, X) >
(t, X))ij = (H0)ij + (H1)ij ·
Xt = (H0)ij +
Pn
k=1(H
(k)
1 )ijXk,t where H0 and each H
(k)
1 are sym-
metric n ⇥ n real matrices;
Jump intensities: j(Xt) = 0,j + >
1,jXt where 0,j 2 R and 1,j 2
Rn
for each j = 1, ..., mX;
Jump sizes: random ZX
j 2 Rn4
distributed according to the risk-
neutral jump measure ⌫j of finite variation and independent of
jump timing for each j = 1, ..., mX . The corresponding moment
generating function (MGF) is
✓j(u) =
Z
Rn
eu>
ZX
j d⌫j(ZX
j ) = EQ
[eu>
ZX
j ] (3.23)
where u 2 Cn
.
Under pricing measure Q, the S&P500 index returns process log St fea-
tures n diffusive risk factor contributions and mS jump risk sources, as
follows:
d log St =
0
@r q
1
2
nX
i=1
i,t
2
i,t + i,t
mSX
j=1
¯µj,t
ˆj,t
1
A dt
+
nX
i=1
q
i,t
2
i,t + i,tdWS
i,t +
mSX
j=1
cjdNj,t
(3.24)
where r and q are the constant short-rate and continuously compounded
dividend yield rate, respectively. Each WS
i,t is a scalar standard Wiener
process and each cjNj,t is a scalar compound Poisson process character-
ized by:
Jump intensities: affine in the volatility factor state vector: ˆj,t =
ˆ0,j + ˆ>
1,jXt;
Jump sizes: random cj, with jump measure ˆ⌫j of finite variation and
independent from the jump timing for each j = 1, ..., mS. The MGF
is
ˆ✓j(u) =
Z
R
eucj
dˆ⌫j(cj) = EQ
[eucj
] (3.25)
4That is ZX
i,j 2 R is the random jump size of the i-th volatility factor, induced by the
j-th kind of jump.
74

where u 2 C and compensator process
¯µj,t = EQ
[ecj
1|Ft] = ˆ✓j(1) 1. (3.26)
The multiplicative i,t and additive i,t displacement functions are de-
terministic non-negative functions
i,t 0 and i,t 0 (3.27)
initially set to i,0 = 1 and i,0 = 0 for each i = 1, ..., n. The present
setting is different with respect to the setting in Leippold et al. (2007)
and Egloff et al. (2010), since it allows for uncompensated jumps in
the stochastic volatility factors dynamics. Moreover, it features a time-
varying affine structure of the instantaneous diffusive variance
Vc(Xt) =
nX
i=1
Vc,i( 2
i,t) =
nX
i=1
i,t
2
i,t + i,t = >
t Xt + 1>
t (3.28)
where we have denoted with 1 2 Rn
a vector of ones. A similar setup
is presented also in Branger et al. (2014), but the affinity structure of the
diffusive spot variance is restricted to be constant in time
Vc(Xt) = >
Xt. (3.29)
As observed by Zhao (2013), multi-factor affine models, mostly with con-
stant coefficients, are extensively employed in modeling interest rate dy-
namics (Duffie and Kan, 1996; Duffie et al., 2000), volatility dynamics
(Christoffersen et al., 2009; Egloff et al., 2010) and default rate dynamics
(Duffie and Singleton, 1999). Models with a time-varying affinity struc-
ture of spot variance are less common. In the context of variance deriva-
tives, a n = 2 factor model with
Vc( 2
1,t, 2
2,t) = 1,t
2
1,t + 2,t
2
2,t (3.30)
has been considered by Zhao (2013) in order to fit the term structure of
variance, interpolating between the initial and steady-state mean vari-
ance, with 2,t / 1 1,t playing a role of a damping function. Another
n = 1 factor model with time-dependent multiplicative displacement,
calibrated on futures and options on VIX, with
Vc( 2
t ) = t
2
t (3.31)
75

has been considered by Sepp (2008b) where the function t is calibrated
to the term structure of VIX futures. Our 2-SVCVJ++ model, introduced
in Section 3.1.1, is a particular instance of the present setting, with n = 2
factors and a lower bounded spot variance
Vc( 2
1,t, 2
2,t) = 2
1,t + t + 2
2,t (3.32)
where the function t 0 improves considerably the fit of the VIX fu-
tures term structure, while preserving the other degrees of freedom of
the model for the consistent fit of options on S&P500 and VIX.
The jump structure outlined above allows for the dynamics of Xt and
log St to feature both independent idiosyncratic jumps and simultaneous
correlated co-jumps. For example, in order to model the fact the k-th kind
of jump is a correlated co-jump between the price process St and the i-th
volatility factor Xi,t, one may consider a common Poisson process Nk,t,
driving synchronized jumps in both processes and correlated jump sizes
(ck, ZX
i,k). This is the case of our 2-SVCVJ++ model, in which the first
factor 2
1,t features two kind of jumps, one idiosyncratic and the other
one correlated and syncronized with the underlying S&P500 index pro-
cess. This rich jump specification increases the volatility of variance, that
is the vol-of-vol, and dramatically improves the ability of the model to fit
the positive skew of the VIX options surface.
The jump contribution to the instantaneous variance is given by
Vd(Xt) =
mSX
j=1
EQ
[c2
j ]
⇣
ˆ0,j + ˆ>
1,jXt
⌘
(3.33)
and therefore the overall spot variance is the sum of the two contribu-
tions Vc(Xt) + Vd(Xt). Correspondingly, the total quadratic variation of
the index returns between time t and T, [log S]T [log S]t, which we will
denote as [log S]t,T , is the sum of the diffusive and jump contributions:
[log S]t,T =
Z T
t
Vc(Xs)ds +
Z T
t
Vd(Xs)ds = [log S]c
t,T + [log S]d
t,T (3.34)
where we have defined the diffusive and jumps contributions
[log S]c
t,T =
Z T
t
>
s Xs + 1>
sds (3.35)
[log S]
d
t,T =
mSX
j=1
EQ
[c2
j ] ˆ0,j⌧ + ˆ>
1,j
Z T
t
Xsds
!
(3.36)
76

where ⌧ = T t. As will be discussed in Section 3.2.1, the fair price
SWt,T of a variance swap contract is the (annualized) risk-neutral ex-
pected value of the total quadratic variation at the time t at which the
contract is made:
SWt,T =
1
⌧
EQ
[[log S]t,T | Ft]
=
1
⌧
EQ
⇥
[log S]c
t,T Ft
⇤
+
1
⌧
EQ
⇥
[log S]d
t,T Ft
⇤ (3.37)
We need therefore to compute in the present setting the expected values
of the diffusive and jump contributions (3.35) and (3.36). We will follow
(Egloff et al., 2010; Leippold et al., 2007) and begin introducing the con-
ditional characteristic function (CF) of the diffusive quadratic variation
(3.35).
Proposition 7. Under general integrability conditions,5
the conditional char-
acteristic function of the diffusive quadratic variation (3.35) takes the following
exponential affine form:
EQ
h
eiz[log S]c
t,T Ft
i
= e↵c+ >
c Xt
(3.38)
with z 2 R and coefficients ↵c(z, t, T) 2 R and c(z, t, T) 2 Rn
satisfying the
following ordinary and Riccati differential equations, respectively
˙↵c(z, t, T) = K>
0 c
1
2
>
c H0 c iz1>
(t)
mXX
j=1
0,j (✓j( c) 1)
˙c(z, t, T) = K>
1 c
1
2
>
c H1 c iz (t)
mXX
j=1
1,j (✓j( c) 1)
(3.39)
and the terminal conditions ↵c(z, T, T) = 0 and c(z, T, T) = 0.
Proof. See Proposition 1 of Egloff et al. (2010), though with a slightly dif-
ferent notation. The proof is analogous to the one in (Mortensen, 2005,
App. A.2), thought in the context of intesity-based credit risk models
and is derived in the footsteps of Proposition 1 of Duffie et al. (2000),
which provides the standard transform analysis for CF-like expectations
in affine models. In particular, the closed-form expression of the CF of
an integrated affine process can be found in (Duffie and Singleton, 2012,
5See (Duffie et al., 2000, Prop. 1).
77

App. A.5) and (Duffie and Garleanu, 2001, App. A) in the credit risk
context and in Duffie and Kan (1996), in the general analysis of interest
rates models.
From the characteristic function (3.38), its expectation is easily ob-
tained by differentiation w.r.t. z
EQ
⇥
[log S]c
t,T |Ft
⇤
= i
"
@↵c
@z
+
✓
@ c
@z
◆>
Xt
#
z=0
(3.40)
and partial derivatives may be computed in closed form, as presented in
the next Proposition.
Proposition 8. Under the setting described above, the conditional expectation
of the diffusive quadratic variation in (3.35) is the following affine function of
the volatility factor state vector Xt:
EQ
⇥
[log S]c
t,T |Ft
⇤
= Ac + B>
c Xt + 1>
Z t+¯⌧
t
sds (3.41)
where Ac 2 R and Bc 2 Rn
can be expressed in integral terms6
Ac(t, T; [t,T ]) =
Z T
t
B>
c (s, T; [s,T ])ds
0
@K0 +
mXX
j=1
0,jr✓j(0)
1
A
Bc(t, T; [t,T ]) =
Z T
t
e(K>
1 +
PmX
j=1 1,j r✓>
j (0))(s t)
(s)ds
(3.42)
Proof. Start from the equations for (↵, ) in (3.39) and follow the same
arguments of the proof of Proposition 2 in Leippold et al. (2007).
In the case of constant affinity structure of spot variance, as it is com-
monly assumed in literature, we get the following Corollary.
Corollary 3. If the multiplicative displacement vector is constant t ⌘ , the
6From definition (3.23), r✓j(0) stands for r✓j(u)|u=0 = EQ[ZX
j ].
78

functions Ac and Bc are time homogeneous
Ac(⌧) =
⇥
B>
c (⌧) >
⌧
⇤
0
@K>
1 +
mXX
j=1
1,jr✓>
j (0)
1
A
1 0
@K0 +
mXX
j=1
0,jr✓j(0)
1
A
Bc(⌧) =
h
e(K>
1 +
PmX
j=1 1,j r✓>
j (0))⌧
Idn
i
0
@K>
1 +
mXX
j=1
1,jr✓>
j (0)
1
A
1
(3.43)
where ⌧ = T t and Idn 2 Rn⇥n
is the identity matrix.
If jumps in volatility are compensated, that is if we make the follow-
ing substitution
mXX
j=1
dZj,t =)
mXX
j=1
⇣
dZj,t j(Xt)EQ
[ZX
j ]dt
⌘
(3.44)
in (3.22) and if both multiplicative t and additive t displacements are
constant functions of time, expression (3.41) for the expected integrated
variance EQ
⇥
[log S]c
t,T |Ft
⇤
consistently reduces to the corresponding ex-
pression given by Proposition 2 of Leippold et al. (2007).
We conclude this Section deriving the expected value of the jump-
induced contribution [log S]d
t,T in (3.36) to the total quadratic variation,
which is a linear function of the volatility state vector Xt integrated be-
tween time t and T.
Proposition 9. Under general integrability conditions,7
the conditional ex-
pectation of the jump quadratic variation is a linear function of the integrated
volatility state vector Xt
EQ
⇥
[log S]d
t,T |Ft
⇤
=
mSX
j=1
EQ
[c2
j ] ˆ0,j⌧ + ˆ>
1,jEQ
"Z T
t
Xsds Ft
#!
(3.45)
where
EQ
"Z T
t
Xsds Ft
#
= Ax(⌧) + Bx(⌧)Xt (3.46)
7See (Dufﬁe et al., 2000, Prop. 1).
79

and the time homogeneous functions Ax 2 Rn
and Bx 2 Rn⇥n
are as follows:
Ax(⌧) = [Bx(⌧) Idn⌧]
0
@K1 +
mXX
j=1
r✓j(0) >
1,j
1
A
1 0
@K0 +
mXX
j=1
0,jr✓j(0)
1
A
Bx(⌧) =
h
e(K1+
PmX
j=1 r✓j (0) >
1,j )⌧
Idn
i
0
@K1 +
mXX
j=1
r✓j(0) >
1,j
1
A
1
(3.47)
Proof. This expectation has been carried out in a similar setting in
(Branger et al., 2014, eq. 6), though no proof can be found. A proof can
be easily derived in the following way, from the results in Proposition 7:
1. Consider the conditional CF of the afﬁne process Xt 2 Rn
inte-
grated in [t, T]
Fx(Z; Xt, ⌧) = EQ
h
eiZ>
R T
t
Xsds
Ft
i
(3.48)
where Z 2 Rn
.
2. The CF Fx can be easily derived from the conditional CF of the dif-
fusive quadratic variation [log S]c
t,T , given in equation (3.38), since:
fc(z; Xt, t, t) = EQ
h
eiz[log S]c
t,T Ft
i
= EQ
h
eiz
R T
t
>
s Xs+1>
sds
Ft
i (3.49)
and therefore
Fx(Z; Xt, ⌧) = fc(1; Xt, Z, 0) (3.50)
where 0 2 Rn
is the zero vector.
3. Finally, take the gradient of Fx w.r.t Z and evaluate it at Z = 0:
EQ
"Z T
t
Xsds Ft
#
= irZFx(Z; Xt, ⌧)|Z=0
= Ax(⌧) + Bx(⌧)Xt
(3.51)
In appendix A.5 we apply a different approach, applying the concept of
functional derivative and deriving the expectation of
R T
t
Xsds directly
from the integral
R T
t
>
s Xsds (part of [log S]c
t,T in (3.35)), considering it
as a functional of the function t.
80

3.2.1 Affine modeling of VIX index
In this Section we will consider variance swaps and the VIX volatility in-
dex and study the relation among them. We will first introduce standard
literature results concerning the replication of variance swaps, then we
will step back to the VIX market definition discussed in Section 2.1 and
connect it to a variance swap replication strategy. Finally, we will present
the expression for both instruments under the affine framework outlined
in the previous Section.
Variance swaps are annualized forward contracts written on the annu-
alized realized variance RVt,T of daily (less often, weekly) logarithmic
returns over a time grid t = t0 < t1 < ... < tk = T spanning the fixed
interval of time [t, T] into the future (Bossu et al., 2005; Demeterfi et al.,
1999), which following Cont and Kokholm (2013) can be written as
RVt,T =
1
⌧
kX
i=1
✓
log
Sti
Sti 1
◆2
(3.52)
where ⌧ = T t and St is a price process of the underlying that, for
derivative pricing, we model on a filtered (⌦, F, {Ft}t 0 , Q). At matu-
rity, the payoff of the long side of the swap is equal to the difference
between the realized variance RVt,T and a constant called variance swap
rate SWt,T , determined at inception t (Carr and Wu, 2009, eq. 1)
N (RVt,T SWt,T ) (3.53)
and where N is the notional of the contract. For any semi-martingale,
as the time-grid gets finer (i.e. as supi=1,...,k |ti ti 1| ! 0), the real-
ized variance RVt,T in (3.52) converges to the annualized total quadratic
variation [log S]t,T defined in equation (3.34)
RVt,T =
1
⌧
[log S]t,T (3.54)
Therefore, in the limit of continuous monitoring,8
variance swaps are
contingent claims on the annualized total quadratic variation [log S]t,T
of the log price in that interval (Cont and Kokholm, 2013; Todorov and
Tauchen, 2011). No arbitrage implies zero net value of the contract at the
time of initiation. Therefore, the variance swap rate fair value
SWt,T = EQ
[RVt,T | Ft] (3.55)
8The approximation RVt,T ⇡ 1
⌧
[log S]t,T is still acceptable when the sampling fre-
quency is daily (Broadie and Jain, 2008).
81

for a variance swap signed at time t and maturing at time T is the annu-
alized time-t conditional quadratic variation (3.34) between time t and
T
SWt,T =
1
⌧
EQ
[[log S]t,T | Ft]
=
1
⌧
EQ
⇥
[log S]c
t,T Ft
⇤
+
1
⌧
EQ
⇥
[log S]d
t,T Ft
⇤ (3.56)
Under the affine model introduced in the previous Section, we therefore
have:
SWt,T =
1
⌧
EQ
"Z T
t
>
s Xsds Ft
#
+ 1>
Z T
t
>
s ds
!
+
1
⌧
mSX
j=1
EQ
[c2
j ] ˆ0,j⌧ + ˆ>
1,jEQ
"Z T
t
Xsds Ft
#! (3.57)
Carr and Wu showed that the variance swap rate SWt,T can be repli-
cated by a portfolio of out-of-the-money options Q(K, t, T) maturing at
the same time T of the contract with an infinite continuum of strikes K
plus an error term ✏(t, T) of third order in jump sizes9
SWt,T =
2
⌧
er⌧
Z 1
0
Q(K, t, T)
K2
dK + ✏(t, T) (3.58)
The error term is induced by jumps in the St process and, under our
affine framework, can be written as
✏(t, T) =
2
⌧
mSX
j=1
EQ
"
ecj
1 cj
c2
j
2
#
ˆ0,j⌧ + ˆ>
1,jEQ
"Z T
t
Xsds Ft
#!
(3.59)
Consider the market definition of VIX, as presented in Section 2.1. In
the limit of an infinite continuum of strike prices,10
the square of the VIX
index given in (2.1) and (2.2) approaches the first summand in (3.58)11
9See eq. 5 in Carr and Wu (2009) and (Carr and Madan, 2001; Demeterfi et al., 1999;
Jiang and Tian, 2007).
10And neglecting also the error induced by the interpolation between T1 and T2 option
maturity buckets.
11The second term in (2.1),
⇣
F (t,T )
K0
1
⌘2
, is due to the use of the put-call parity in order
to substitute an ITM call option at K0 with an OTM put at the same strike and it is lost in
the limit of a continuum of strikes, as we assume in the present analysis (Carr and Wu,
2006, App. B).
82

(Carr and Wu, 2006; Cont and Kokholm, 2013)
lim
K!0
Kmax!+1
Kmin=0
✓
V IXt
100
◆2
=
2
¯⌧
er¯⌧
Z 1
0
Q(K, t, t + ¯⌧)
K2
dK (3.60)
where ¯⌧ = 30/365 denotes the annualized 30 days horizon inherent in
VIX definition. We can therefore express the square of the VIX index at
time t as
✓
V IXt
100
◆2
= SWt,t+¯⌧ ✏(t, t + ¯⌧)
=
1
¯⌧
EQ
[[log S]t,t+¯⌧ | Ft] ✏(t, t + ¯⌧)
=
1
¯⌧
EQ
⇥
[log S]c
t,t+¯⌧ Ft
⇤
+
✓
1
¯⌧
EQ
⇥
[log S]d
t,t+¯⌧ Ft
⇤
✏(t, t + ¯⌧)
◆
(3.61)
If we switch off jumps, VIX index and variance swap rate over the same
horizon consistently coincide
✓
V IXt
100
◆2
⌘ SWt,t+¯⌧ St has continuous paths (3.62)
but if we allow the price process St to jump, the effect on the dis-
continuity on the variance swap rate is different from its effect on the
VIX. Equation (3.61) is commonly adopted as a continuous-time defi-
nition of VIX (Zhao, 2013). Comparing to VIX CBOE calculations in
(2.1), (2.2) and (2.5), formula (3.61) is exact up to the discretization er-
ror due to K > 0, the truncation errors due to a finite number of strikes
Kmax < +1, Kmin > 0 and the error introduced ignoring the linear in-
terpolation/extrapolation between the two maturity buckets at T1 and
T2. In our affine framework, comparing the expression in (3.61) with the
one for variance swap rates in (3.57), the VIX index squared at time t can
be explicitly written as
✓
V IXt
100
◆2
=
1
⌧
EQ
"Z T
t
>
s Xsds Ft
#
+ 1>
Z T
t
>
s ds
!
+
2
¯⌧
mSX
j=1
EQ
[ecj
1 cj]
✓
ˆ0,j ¯⌧ + ˆ>
1,jEQ
Z t+¯⌧
t
Xsds Ft
◆
(3.63)
83

and we can see that the difference between VIX and variance swap rate
induced by jumps is consistently of third orders in jumps size (Branger
et al., 2014, App. B.1)
✓
V IXt
100
◆2
SWt,t+¯⌧ = ✏(t, t + ¯⌧) = O EQ
⇥
c3
⇤
(3.64)
which shows that, under our continuous time affine framework, the
error induced approximating the VIX (squared and scaled) with the
variance swap rate is of third order in the jump sizes.
With the results in propositions 8 and 9 we can price instruments
which directly depend on the dynamics of volatility. In particular, the
VIX index can be written as an affine function of the volatility factor
state vector.
Proposition 10. Under the affine framework of Section 3.2, from definition
(3.63), the VIX index squared at time t is
✓
V IXt
100
◆2
=
1
¯⌧
✓
a + b>
Xt + 1>
Z t+¯⌧
t
sds
◆
(3.65)
where functions a 2 R and b 2 Rn
can be expressed as:
a(t, t + ¯⌧; [t,t+¯⌧]) = Ac(t, t + ¯⌧; [t,t+¯⌧]) + 2
mSX
j=1
EQ
[ecj
1 cj]
⇣
ˆ0,j ¯⌧ + ˆ>
1,jAx(¯⌧)
⌘
b(t, t + ¯⌧; [t,t+¯⌧]) = Bc(t, t + ¯⌧; [t,t+¯⌧]) + 2
mSX
j=1
EQ
[ecj
1 cj] B>
x (¯⌧)ˆ1,j
(3.66)
Proof. Straightforward application of definition (3.63), where the affinity
coefficients (Ac, Bc) 2 R ⇥ Rn
of the diffusive quadratic variation have
been defined in (3.42) and the corresponding coefficients (Ax, Bx) 2 Rn
⇥
Rn⇥n
of the integrated volatility factor state vector have been defined in
(3.47).
If the multiplicative displacement vector is constant t ⌘ , the ex-
pected diffusive quadratic variation of proposition 8 is affine in the ex-
pectation of the integrated volatility factor state vector Xt of proposition
9 and the following corollary summarizes how the affinity structure of
VIX squared simplifies
84

Corollary 1. If the multiplicative displacement vector is constant t ⌘ ,
the VIX squared affinity coefficients (a, b) 2 R ⇥ Rn
are time homogeneous
functions
a(¯⌧) = >
Ax(¯⌧) + 2
mSX
j=1
EQ
[ecj
1 cj]
⇣
ˆ0,j ¯⌧ + ˆ>
1,jAx(¯⌧)
⌘
b(¯⌧) = B>
x (¯⌧) + 2
mSX
j=1
EQ
[ecj
1 cj] ˆ1,j
! (3.67)
Under the same setting, from (3.56), the variance SWt,t+¯⌧ at time t,
over the same 30-day horizon of the V IXt, can be obtained replacing
2EQ
[ecj
1 cj] (3.68)
with
EQ
[c2
j ] (3.69)
everywhere in (3.66) and (3.67).
3.2.2 Affine modeling of VIX derivatives
The payoff of a VIX futures contract settled at time T and of a call op-
tion on VIX of strike K and maturing at T are linear functions of the
VIX index value at settle V IXT , respectively wF (V IXT ) = V IXT and
wC(V IXT ) = (V IXT K)+
. As it is clear from Proposition 10, the in-
dex VIX has a non linear (square-root) relation with the volatility vector
Xt. To overcome this issue, together with the other mentioned above,
we rewrite the payoffs as non-linear functions of the scaled squared VIX
index
wF (V IX02
T ) = 100 ⇥
q
V IX02
T
wC(V IX02
T ) = 100 ⇥
✓q
V IX02
T K0
◆+ (3.70)
where V IX0
t = V IXt/100 and K0
= K/100 are, respectively, the index
and strike values expressed in percentage points. The Fourier transforms
85

for these payoffs are available in closed form
ˆwF (z) =
Z 1
0
eizV IX02
T wF (V IX02
T )dV IX02
T
= 100 ⇥
p
⇡
2
1
( iz)3/2
ˆwC(z) =
Z 1
1
eizV IX02
T wC(V IX02
T )dV IX02
T
= 100 ⇥
p
⇡
2
1 erf(K0
p
iz)
( iz)3/2
(3.71)
with z = Re(z) + i Im(z) 2 C and are single-valued regular functions in
the upper half of the complex plane Im(z) > 0. Following the approach
in (Lewis, 2000, 2001), in the next Proposition we will derive a closed-
form expression for the VIX derivative prices in terms of the complex
Fourier transform of futures and options payoffs and the complex CF of
the volatility factor state vector Xt.
Proposition 11. Under the afﬁne framework described above, the time t value
of a futures on VIX settled at time T and the arbitrage-free price at time t of
a call option with strike price K and time to maturity ⌧ = T t are given
respectively by (not relevant dependencies suppressed and ¯⌧ = 30/365)
FV IX (t, T) = 100 ⇥
1
2
p
⇡
Z 1
0
Re
"
f
✓
z
b
¯⌧
◆
e iz(a+1>
I (T,T +¯⌧))/¯⌧
( iz)3/2
#
d Re(z)
(3.72)
and
CV IX(K, t, T) = 100 ⇥
e r⌧
2
p
⇡
⇥
Z 1
0
Re
"
f
✓
z
b
¯⌧
◆
e iz(a+1>
I (T,T +¯⌧))/¯⌧
1 erf(K/100
p
iz)
( iz)3/2
#
d Re(z)
(3.73)
where z = Re(z) + i Im(z) 2 C and the integrals are performed along a line
parallel to the Re(z) axis selecting Im(z) such that 0 < Im(z) < ⇣c(t, T).
The complex valued erf(z) = 2p
⇡
R z
0
e s2
ds is the error function with complex
argument, the integrated additive displacement vector is
I (T, T + ¯⌧) =
Z T +¯⌧
T
sds (3.74)
86

and the VIX index affinity coefficients a 2 R and b 2 Rn
are given in Proposi-
tion 10 and have to be evaluated at time T.12
Finally, the function
f (Z; Xt, t, T) = EQ
h
eiZ>
XT
Ft
i
(3.75)
with Z = Re(Z) + i Im(Z) 2 Cn
, is the risk-neutral conditional characteristic
function of Xt 2 Rn
.
Proof. See Appendix A.6.
The upper bound ⇣c(t, T) depends on the CF f of the specific model
considered and is derived explicitly for our 2-SVCVJ++ model in Ap-
pendix A.4. Similar results can be found in the literature (Branger et al.,
2014; Lian and Zhu, 2013; Sepp, 2008a,b). The previous Proposition is
completely specified once the conditional CF of the stochastic volatility
process Xt is known. As shown in Duffie et al. (2000) and as will be pre-
sented in the next Proposition, an affine process Xt always has a CF and
its functional form is exponential affine in Xt.
Proposition 12. Under the affine framework described above and under techni-
cal regularity conditions13
, the risk-neutral conditional characteristic function
of Xt 2 Rn
is the following exponential affine function of Xt
f (Z; Xt, t, T) = eA +B>
Xt
(3.76)
where the functions A (Z; t, T) 2 C and B (Z; t, T) 2 Cn
satisfy the follow-
ing ordinary and Riccati differential equations, respectively14
˙A (Z, t, T) = K>
0 B
1
2
B>
H0B
mXX
j=1
0,j (✓j(B ) 1)
˙B (Z, t, T) = K>
1 B
1
2
B>
H1B
mXX
j=1
1,j (✓j(B ) 1)
(3.77)
with Z = Re(Z)+i Im(Z) 2 Cn
and the terminal conditions A (Z, T, T) = 0
and B (Z, T, T) = iZ.
12That is their explicit dependencies in the pricing formulas are a(T, T + ¯⌧; [T,T +¯⌧])
and b(T, T + ¯⌧; [T,T +¯⌧]).
13Refer to (Duffie et al., 2000, Prop. 1).
14Here x>H1x 2 Cn is the complex vector whose k-th component is
Pn
i,j=1 xi(H
(k)
1 )ijxj and H
(k)
1 2 Rn⇥n.
87

Proof. This proposition is a particular case of (Dufﬁe et al., 2000, Prop.
1).
The ODE for A is integrable once the Riccati equations for B has
been solved. These last often do not have a closed-form analytical so-
lution, mostly because model parameters in K1 and H1 may be time-
dependent or in case of co-jumps between volatility factors, modeled
through the jump MGF ✓j(Bi, , Bk, ) which eventually couples the two
Riccati for Bi, , and Bk, , making them not separately integrable.
88

Chapter 4
The Heston++ model:
empirical analysis
In our empirical analysis we jointly fit S&P500 options - together with
VIX futures and options - using the Heston++ model. We study the con-
tribution of the various features of the model analyzing their impact on
the pricing performance over a sample period of two years.
Our study endorses the results from literature concerning the need
of a multi-factor specification of the volatility dynamics (Andersen et al.,
2002; Bates, 2012; Egloff et al., 2010; Kaeck and Alexander, 2012; Menc´ıa
and Sentana, 2013) and of a discontinuous returns process (Bakshi et al.,
1997; Bates, 1996; Eraker, 2004; Pan, 2002). We therefore choose as a
benchmark model for our analysis the 2-SVJ model of Section 3.1.2,
which features two Heston stochastic volatility factors and normal jumps
in the returns dynamics. We then analyze the different contributions of
jumps in volatility inserting two sources of exponential jumps in one of
the two volatility factors. We first add them separately as an idiosyn-
cratic source of discontinuity (2-SVVJ model) and then correlated and
synchronized with jumps in price (2-SVCJ model). Then, we let the two
discontinuity components act together in the 2-SVCVJ model.
At the same time, we make a displacement transformation on the
volatility dynamics of each H model considered and analyze the per-
formance of the corresponding H + + model. In addition, we repeat
the analysis restricting the freedom of factor parameters imposing the
Feller condition (Andersen and Piterbarg, 2007; Cox et al., 1985; Duffie
89

and Kan, 1996).
Our analysis shows that the Heston++ model - calibrated consistently
on the three markets - works remarkably well, with an overall absolute
(relative) estimation error below 2.2% (4%). The key feature of the model
is a deterministic displacement of the instantaneous volatility, in addi-
tion to the usual multi-factor affine structure. Our empirical results show
a decisive improvement in the pricing performance over non-displaced
models, and also provide clear empirical support for the presence of
both price-volatility co-jumps and idiosyncratic jumps in the volatility
dynamics.
The chapter is structured as follows: Section 4.1 describes the finan-
cial data adopted for the empirical analysis and the descriptive statistics.
Section 4.2 discusses the results and Section 4.2.2 describes the impact of
restricting the volatility dynamics imposing the Feller condition. Section
4.3 concludes.
4.1 Empirical analysis
Our sample period spans two years, ranging from January 7, 2009 to De-
cember 29, 2010. The sampling frequency is weekly and the observation
day is Wednesday. In total, we have 104 weekly surfaces and term struc-
tures. Closing prices of S&P500 vanilla and VIX options are adopted,
together with settlement prices of VIX futures.
Commonly adopted exclusion filters are applied to data (A¨ıt-Sahalia
and Lo, 1998; Bakshi et al., 1997; Bardgett et al., 2013). We exclude option
quotes with negative bid-ask spreads, zero bids and filter out observa-
tions not satisfying standard no-arbitrage conditions.1
Potential liquid-
ity and asynchronicity biases are reduced considering only options with
maturity between one week and one year and excluding contracts not
traded on a given date. Following Bardgett et al. (2013), the analysis is
carried out only with liquid OTM options for the S&P500 market and
only with liquid call options for the VIX market. If a VIX ITM call lacks
of liquidity, we use the put-call parity in equation (2.12) to infer the liquid
price of the call from a more liquid VIX OTM put. 2
We compute mon-
eyness as the option exercise price divided by the current index level for
1For example, we eliminate VIX options on the basis of the appropriate parity relations
discussed in Section 2.1.3.
2We consider as liquid a contract, either option or futures, which has both positive Vol-
ume and Open Interests.
90

SPX options and as the ratio of the option strike and the VIX futures settle
price for options on VIX.3
We further exclude glaring outliers (for a total
of three market prices of VIX options) and eliminate SPX (VIX) maturity
slices made of less than 6 (3) options quoted.
The final sample is made of a total of 24,279 vanilla options (233 per day),
2,767 VIX options (27 per day), and 792 VIX futures (8 per day). OTM
vanilla (VIX call options) span on average 7 (5) maturity slices, ranging
from 1 (4) weeks-to-maturity to 12 (6) months and from 0.5 (0.4) to 1.4
(3.3) in the moneyness dimension. The term structure of VIX futures
ranges from roughly 7 days to 10 months. Vanilla options range in the
entire filtered time-frame from one week to the year, whereas VIX op-
tions range from 4 weeks to 6 months.
Summary statistics for S&P500 index options, are presented in Table
1 and sample characteristics of VIX derivatives are presented in Table 2.
The complementarity of SPX and VIX options markets reflects in
the opposite relative liquidities of calls and puts. Put (call) options on
S&P500 (VIX) index are more heavily traded than calls (puts), account-
ing for 59% (68%) of the total observations, with OTM puts (calls) more
than double than OTM calls (puts). As discussed in Section 2.1.3, a pos-
sible explanation for this dichotomy is the fact that both vanilla puts and
VIX call options provide insurance from equity market downturns.
Implied volatilities (IVs) of VIX options are generally higher than
those for SPX options, the latter averaging around 23% (respectively
32%) in the case of calls (resp. puts) quotes, while the first averaging
around 76% (resp. 70%).
The opposite sign of the skewness of S&P500 and VIX distributions,
translates in the opposite slopes of IV skews. They decrease with mon-
eyness for S&P500 options, ranging on average from about 30% to 20%
going from ITM to OTM calls (respectively from about 36% to 21% going
from OTM to ITM puts). The opposite is instead observed in the VIX op-
tions market, with average IV skews ranging from levels of 66% to 82%
going from ITM to OTM calls (respectively from levels of 61% to 85%
going from OTM to ITM puts).
Moreover, based on our sample of data, the ATM term structure of
S&P500 IVs does not display a clear trend, ranging roughly between 23%
for options of maturities below 45 days and 26% for those expiring in
more than 90 days. Nevertheless, this is not the case for VIX options,
3Therefore, OTM calls (puts), either vanilla or VIX options, are options of moneyness
> 1 (< 1).
91

where we observe a downward trend of about 20 volatility points on
average (going from approximately an ATM IV of 82% for options in the
nearest maturity bucket, to approximately 60% for options in the farthest
one), a fact compatible with a volatility-of-volatility decreasing with time
to maturity, which is in turn consistent with the mean-reverting nature
of volatility.
Tables 1 and 2 provide further details about the implied volatility sur-
faces of the two markets along the moneyness-maturity dimensions.
4.2 Calibration results
For each day in sample, we jointly calibrate each H and H++ model de-
scribed in the Section 3.1 to daily SPX and VIX option market surfaces
and VIX futures term structure. Joint calibration is performed minimiz-
ing for each date in sample the following normalized sum of squared
relative errors
L =
NSP XX
i=1
IV MKT
i,SP X IV mdl
i,SP X
IV MKT
i,SP X
!2
+
NSP X
NF ut
NF utX
j=1
FMKT
j Fmdl
j
FMKT
j
!2
+
NSP X
NV IX
NV IXX
k=1
IV MKT
k,V IX IV mdl
k,V IX
IV MKT
k,V IX
!2
(4.1)
where NSP X (NV IX) are the number of S&P500 (VIX) options quotes ob-
served in a given date, IV MKT
(IV mdl
) the corresponding market (mo-
del) implied volatilities and FMKT
(Fmdl
) the market (model) VIX fu-
tures prices term structure, made of NF ut points. The use of relative er-
rors is suggested by the different range of implied volatility values of SPX
and VIX options and normalizing factors NSP X /NV IX and NSP X /NF ut
adjust for the difference in the number of quotes, which would otherwise
severely penalize the ﬁt of the term structure of VIX futures. All H and
H++ models are nested with respect to the metrics in (4.1).
92

Table1:SamplecharacteristicsofS&P500options.Thetablereportstheaverageprices,bid-askspreads(BA),BlackandScholes(1973)implied
volatilities(IV),bid-askimpliedvolatilityspreads(IVBA),tradingvolume,openinterests(OI),thetotalnumberof(andinpercentageofthetotal)
observations(Obs)foreachmoneyness-maturitycategoryofcall(PanelA)andput(PanelB)optionsonS&P500index.Thesampleperiodisfrom
January7,2009toDecember29,2010andthesamplingfrequencyisweekly(Wednesdays).Maturityisdefinedasthenumberofdaystoexpiration.
Moneynessisdefinedastheratiooftheoptionexercisepricetothecurrentindexlevel.ITM(OTM),ATMandOTM(ITM)categoriesforcalls(puts)
aredefinedbyMoneyness<0.95,0.951.05,and>1.05,respectively.
MaturityMoneyness
PanelA:CallsPanelB:Puts
ITMATMOTMAllOTMATMITMAll
<45Days
Price90.8424.612.0230.122.9924.5481.0815.77
BA2.831.810.591.640.721.902.931.27
IV30.4322.9220.1423.4836.3523.3920.1930.92
IVBA7.372.171.913.042.902.327.403.04
Volume351.363378.772445.772557.842300.303394.44108.862498.61
OI19734.0725392.3417186.7721987.4623523.1022979.029930.2522350.05
Obs863250713734743423223165177065
Obs(%ofTOT)6.6319.2610.5536.4322.3512.232.7337.32
4590Days
Price114.5836.675.6429.278.5139.45112.7421.76
BA2.912.321.091.791.282.443.081.68
IV30.3723.9719.9322.6535.7924.7821.2132.13
IVBA2.991.471.651.702.311.533.862.17
Volume279.631914.14923.791282.751314.922479.65215.411577.75
OI20349.7116199.8010566.5513897.4818797.4917432.3411966.5818107.94
Obs459203422954788471018623246896
Obs(%ofTOT)3.5315.6217.6336.7824.889.831.7136.42
>90Days
Price149.6664.5217.6948.9224.2471.85168.2446.99
BA3.453.061.962.512.103.113.482.45
IV30.2025.3521.1623.6434.7725.9022.2431.67
IVBA1.691.131.231.251.561.162.251.52
Volume264.541313.80884.30959.231270.311794.78457.171330.29
OI23226.7624169.1324861.1324436.2032406.1822356.6320350.2429061.33
Obs403119018953488339611813954972
Obs(%ofTOT)3.109.1414.5626.7917.946.242.0926.26
All
Price110.9037.188.8534.8410.9540.15117.2326.15
BAspread3.002.251.271.931.312.363.141.73
IV30.3623.8020.4023.2235.7024.4321.1231.56
IVBAspread4.881.701.572.072.311.794.822.32
Volume311.992430.181285.981660.611640.632724.07248.101856.39
OI20713.8621875.8117069.8419668.2624164.1520914.6513794.0622567.39
Obs17255731556313019123385359123618933
Obs(%ofTOT)13.2544.0242.73100.0065.1728.316.53100.00
93

Table2:SamplecharacteristicsofVIXoptionsandVIXfutures.Thetablereportstheaverageprices,bid-askspreads(BA),Black(1976)implied
volatilities(IV),bid-askimpliedvolatilityspreads(IVBA),tradingvolume,openinterests(OI),thetotalnumberof(andinpercentageofthetotal)
observations(Obs)foreachmoneyness-maturitycategoryofcall(PanelA)andput(PanelC)optionsonVIXindex.PanelBreportsVIXfutures
settleprices,tradingvolume,openinterestsandobservationsforeachmaturitybucket.ThesampleperiodisfromJanuary7,2009toDecember
29,2010andthesamplingfrequencyisweekly(Wednesdays).Maturityisdefinedasthenumberofdaystoexpiration.Moneynessforanoptionof
maturityTisdefinedastheratiooftheoptionexercisepricetothecurrentVIXfuturespriceexpiringatT.ITM(OTM),ATMandOTM(ITM)for
calls(puts)aredefinedbyMoneyness<0.95,0.951.05,and>1.05,respectively.
MaturityMoneyness
PanelA:CallsPanelB:PanelC:Puts
ITMATMOTMAllFuturesOTMATMITMAll
<45Days
Price6.762.550.622.6428.240.622.408.634.71
BA0.360.190.110.190.110.180.340.23
IV93.1482.50109.75101.7774.3182.42100.9388.29
IVBA41.026.819.6418.537.926.7924.4515.47
Volume748.474256.803918.953026.725008.296154.985738.871133.003720.95
OI10959.0827865.8046552.4533978.5324226.5742227.5365398.4327405.7639504.99
Obs2098142271214415481208443
Obs(%)6.682.5913.4822.7418.1810.295.4113.8929.59
4590Days
Price7.033.511.032.7729.711.033.5110.554.37
BA0.400.260.160.230.150.260.390.24
IV66.6568.5585.9379.3863.8971.0780.6870.23
IVBA15.815.216.398.595.455.3312.357.55
Volume616.661705.281521.561317.621698.151874.181761.46252.971356.71
OI9259.1217392.9919378.6016679.6211052.1224158.3726138.525647.1818781.01
Obs1887949676314421566125406
Obs(%)6.002.5215.8424.3718.1814.364.418.3527.12
>90Days
Price7.604.121.683.7029.991.363.919.403.45
BA0.530.390.270.360.240.380.500.32
IV54.8459.3868.8763.7055.2760.9965.0458.26
IVBA13.975.605.558.075.685.748.246.22
Volume167.82372.16437.14349.97279.97634.48874.23131.62572.71
OI2588.514694.074182.873758.772629.106236.555368.431337.815081.06
Obs4951709911656504403112133648
Obs(%)15.815.4331.6552.8963.6426.927.488.8843.29
All
Price7.283.591.283.2329.621.123.349.374.07
BAspread0.460.310.210.290.190.290.400.27
IV66.3067.2582.3476.1861.4770.2685.2670.39
IVBAspread20.705.816.6710.586.065.9616.589.32
Volume398.471644.801488.581194.481397.512080.972621.70611.141716.98
OI5955.6913421.7217497.2013779.568087.3718407.2829435.0614129.2418983.52
Obs892330190931317927722594661497
Obs(%)28.4910.5460.97100.00100.0051.5717.3031.13100.00
94

We compare the pricing performance of each model separately on
each market in terms of the absolute errors
RMSESP X =
v
u
u
t 1
NSP X
NSP XX
i=1
IV MKT
i,SP X IV mdl
i,SP X
2
RMSEF ut =
v
u
u
t 1
NF ut
NF utX
j=1
FMKT
j Fmdl
j
2
(4.2)
RMSEV IX =
v
u
u
t 1
NV IX
NV IXX
k=1
⇣
IV MKT
k,V IX IV mdl
k,V IX
⌘2
and relative errors
RMSRESP X =
v
u
u
t 1
NSP X
NSP XX
i=1
IV MKT
i,SP X IV mdl
i,SP X
IV MKT
i,SP X
!2
RMSREF ut =
v
u
u
t 1
NF ut
NF utX
j=1
FMKT
j Fmdl
j
FMKT
j
!2
(4.3)
RMSREV IX =
v
u
u
t 1
NV IX
NV IXX
k=1
IV MKT
k,V IX IV mdl
k,V IX
IV MKT
k,V IX
!2
Moreover, we evaluate the overall calibration performance with the ag-
gregate errors
RMSEAll =
v
u
u
t 1
N
NX
i
QMKT
i Qmdl
i
2
RMSREAll =
v
u
u
t 1
N
NX
i
✓
QMKT
i Qmdl
i
QMKT
i
◆2
(4.4)
where N = NSP X + NF ut + NV IX and we have denoted synthetically
with QMKT
(Qmdl
) the market (model) quotes of the SPX (VIX) implied
volatilities IV MKT
(IV mdl
) and VIX futures prices FMKT
(Fmdl
). In
RMSEAll deﬁnition we have divided by 100 each VIX futures price F
in order to make it comparable with the implied volatility levels IVSP X
and IVV IX.
Figures from 12 to 18 show, in chronological order, the calibration
results and calibrated parameters of the 2-SVCVJ++ and of the nested 2-
SVCVJ models for some selected dates and different market situations.
95

Figures 14 and 16 provide examples of days in which the VIX futures
term structure displays a hump. These figures show, quite clearly, that
taking advantage of the added flexibility provided by the deterministic
shift t in fitting the term structure of VIX futures, the 2-SVCVJ++ mo-
del (solid red line) is able to calibrate the vanilla and VIX options jointly
without particular difficulty. The 2-SVCVJ model (dashed blue line),
missing such a flexibility, cannot reproduce the prices of the two market
even with the high (17) number of parameters employed. Moreover, the
fit of the two surfaces comes at expenses of the fit of the term structure
of VIX futures, where the non-displaced model is not able to reproduce
the hump, with a relative error RMSREF ut more than 5 (9) times the
corresponding error of the displaced 2-SVCVJ++ model on July 8 (re-
spectively September 2), 2009, which is remarkably low: 0.21% (resp.
0.42%).
Figures 12 and 18 show two rather common and different market sit-
uations. The first date, March 4, 2009 (respectively the second one, Au-
gust 11, 2010), displays a decreasing (resp. roughly increasing) VIX fu-
tures term structure. Also in this case, though with two jump sources,
the 2-SVCVJ model has some difficulty in reproducing adequately the
VIX options skew. To make this clear, consider the term structure of the
K = 40 strike in the VIX option surface of figure 18. We see that the
level of the surface goes from the 180 volatility points (vps) at the nearest
maturity of 7 days, to the roughly 60 vps at the longest maturity of 161
days. Nevertheless, the amplitude of the skew, which is of roughly 80
vps at 7 days, is still considerable at the longest maturity (approximately
25 vps). This phenomenon requires a model which is able to recreate the
positive sloping skew of the VIX implied surface, not only for the short-
est maturities, but for the entire term structure. The increased degrees
of freedom introduced by the displacement t help a lot also in these
situations, with the 2-SVCVJ++ making a relative error RMSREV IX of
2.55% (resp. 3.26%) on the backwarding (resp. contango) volatility market
of March 4, 2009 (resp. August 11, 2010), which is almost 3.9 (resp. 3.4)
times lower than the corresponding RMSREV IX made by the undis-
placed model 2-SVCVJ.
Tables 3 reports the summary statistics on the root mean squared er-
rors for the H and H++ models averaged over the three markets. Tables
4 and 5 report the same summary statistics dissected on the three mar-
kets. Table 6 reports average parameter estimates together with their
in-sample standard deviation.
Our results clearly show that the addition of the deterministic shift is
96

S&P500 Options implied volatility surface
600 800
40
50
60
70
80
90
Strike
Vol(%)
17 days
Calls
Puts
600 800
30
40
50
60
70
80
90
Strike
Vol(%)
27 days
2−SVCVJ
2−SVCVJ++
400 600 800
30
40
50
60
70
80
90
Strike
Vol(%)
45 days
500 1000
30
40
50
60
70
Strike
Vol(%)
73 days
500 1000
30
40
50
60
Strike
Vol(%)
108 days
500 1000
25
30
35
40
45
50
Strike
Vol(%)
199 days
500 1000
25
30
35
40
45
50
55
Strike
Vol(%)
290 days
VIX Futures term structure
0 50 100 150 200 250
36
37
38
39
40
41
42
43
44
45
Tenor (days)
SettlePrice(US$)
Data
2−SVCVJ
2−SVCVJ++
VIX Options implied volatility surface
40 50 60 70 80
80
90
100
110
120
130
140
150
Strike
Vol(%)
14 days
Data
30 40 50 60 70 80
60
70
80
90
100
110
Strike
Vol(%)
42 days
2−SVCVJ
2−SVCVJ++
30 40 50 60 70 80
50
60
70
80
90
Strike
Vol(%)
77 days
40 45 50 55
60
65
70
75
Strike
Vol(%)
105 days
Figure 12: Fit results on March 4, 2009. This ﬁgure reports market and model im-
plied volatilities for S&P500 (plot at the top) and VIX (plot at the bottom) options,
together with the term structure of VIX futures (plot in the middle) on March 4, 2009
obtained calibrating jointly on the three markets the 2-SVCVJ (blue dashed line) and
2-SVCVJ++ (red line). Maturities and tenors are expressed in days and volatilities are
in % points and VIX futures settle prices are in US$. Relative errors 2-SVCVJ++ (2-
SVCVJ) model: RMSRESP X = 2.04% (2.74%), RMSREF ut = 0.53% (1.31%),
RMSREV IX = 2.55% (9.83%). Absolute errors 2-SVCVJ++ (2-SVCVJ) model:
RMSESP X = 0.95% (1.30%), RMSEF ut = 0.20 US$ (0.51 US$), RMSEV IX =
2.74% (7.62%).
97

March 4, 2009 2-SVCVJ 2-SVCVJ++
↵1 3.3240 3.0912p
1 (%) 31.9752 24.0894
⇤1 1.0679 1.1152
⇢1 0.8431 0.9690
1,0 (%) 42.9752 34.5792
↵2 93.7102 43.2533p
2 (%) 17.1913 25.2210
⇤2 46.1993 8.7081
⇢2 0.5685 0.5891
2,0 (%) 30.6615 34.8718
0.0012 0.0016
E[cx] 4.3743 2.2030p
V ar[cx] 0.5652 0.6053
µco, 18.6006 68.4667
corr(cx, c ) 0.8477 0.8799
0.0051 0.0025
µid, 17.3522 57.6487
0 50 100 150 200 250 300
0
1
2
3
4
5
6
7
8
x 10
−3
T (d ays)
Iφ(0,T)
Figure 13: Calibrated parameters on March 4, 2009 of 2-SVCVJ and 2-SVCVJ++
models and I (0, T) displacement integrals of 2-SVCVJ++ model. Fit results are
shown in Figure 12.
98

800 1000
25
30
35
40
45
50
55
Strike
Vol(%)
10 days
Calls
Puts
600 800 1000
20
30
40
50
60
Strike
Vol(%)
45 days
2−SVCVJ
2−SVCVJ++
500 1000
20
30
40
50
60
Strike
Vol(%)
73 days
500 1000
25
30
35
40
45
50
55
60
Strike
Vol(%)
84 days
600 800 1000 1200
20
25
30
35
40
45
50
Strike
Vol(%)
101 days
500 1000
20
25
30
35
40
45
50
55
Strike
Vol(%)
164 days
500 1000
20
25
30
35
40
45
Strike
Vol(%)
255 days
800 1000 1200
20
25
30
35
Strike
Vol(%)
346 days
0 20 40 60 80 100 120 140 160 180 200
30.5
31
31.5
32
32.5
Tenor (days)
SettlePrice(US$)
Data
2−SVCVJ
2−SVCVJ++
30 40 50
70
80
90
100
110
Strike
Vol(%)
42 days
Data
40 50 60 70
75
80
85
90
95
100
105
Strike
Vol(%)
70 days
2−SVCVJ
2−SVCVJ++
40 60 80
65
70
75
80
85
90
95
Strike
Vol(%)
105 days
30 40 50 60 70
60
65
70
75
80
85
90
Strike
Vol(%)
133 days
30 40 50 60 70
60
65
70
75
80
85
90
Strike
Vol(%)
161 days
Figure 14: Fit results on July 8, 2009. This ﬁgure reports market and model im-
plied volatilities for S&P500 (plot at the top) and VIX (plot at the bottom) options,
together with the term structure of VIX futures (plot in the middle) on July 8, 2009
2-SVCVJ++ (red line). Maturities and tenors are expressed in days and volatilities are
in % points and VIX futures settle prices are in US$. Relative errors 2-SVCVJ++ (2-
SVCVJ) model: RMSRESP X = 1.77% (2.29%), RMSREF ut = 0.21% (1.11%),
1.55% (1.73%).
99

July 8, 2009 2-SVCVJ 2-SVCVJ++
↵1 2.1364 1.8702p
1 (%) 10.4533 9.1898
⇤1 0.3900 0.4164
⇢1 0.8850 0.9054
1,0 (%) 23.0944 22.2989
↵2 6.3082 6.5529p
2 (%) 27.1845 26.7377
⇤2 2.3147 2.4890
⇢2 0.9194 0.9265
2,0 (%) 12.7423 10.4210
0.4065 0.5185
E[cx] 0.0732 0.0778p
V ar[cx] 0.1637 0.1577
µco, 0.0019 0.0006
corr(cx, c ) 0.0357 0.0044
0.0009 0.0006
µid, 124.5221 147.0826
0 50 100 150 200 250 300 350
−0.5
0
0.5
1
1.5
2
2.5
x 10
−3
T (d ays)
Iφ(0,T)
Figure 15: Calibrated parameters on July 8, 2009 of 2-SVCVJ and 2-SVCVJ++ mod-
els and I (0, T) displacement integrals of 2-SVCVJ++ model. Fit results are shown
in Figure 14.
100

800 1000
20
25
30
35
40
45
50
55
Strike
Vol(%)
17 days
Calls
Puts
900 1000 1100
20
25
30
35
Strike
Vol(%)
28 days
2−SVCVJ
2−SVCVJ++
600 800 1000 1200
20
30
40
50
60
Strike
Vol(%)
45 days
600 800 1000 1200
20
25
30
35
40
45
50
55
Strike
Vol(%)
80 days
600 800 1000 1200
20
25
30
35
40
45
50
55
Strike
Vol(%)
108 days
500 1000
20
25
30
35
40
45
50
Strike
Vol(%)
199 days
500 1000
20
25
30
35
40
45
50
Strike
Vol(%)
290 days
0 20 40 60 80 100 120 140 160 180 200
29
29.5
30
30.5
31
31.5
32
32.5
33
Tenor (days)
SettlePrice(US$)
Data
2−SVCVJ
2−SVCVJ++
34 36 38 40 42 44
90
100
110
120
130
Strike
Vol(%)
14 days
Data
30 40 50
60
65
70
75
80
85
90
95
Strike
Vol(%)
49 days
2−SVCVJ
2−SVCVJ++
30 40 50 60 70 80
60
70
80
90
100
Strike
Vol(%)
77 days
20 30 40 50
50
55
60
65
70
75
80
85
Strike
Vol(%)
105 days
Figure 16: Fit results on September 2, 2009. This ﬁgure reports market and model
implied volatilities for S&P500 (plot at the top) and VIX (plot at the bottom) options,
together with the term structure of VIX futures (plot in the middle) on September
2, 2009 obtained calibrating jointly on the three markets the 2-SVCVJ (blue dashed
line) and 2-SVCVJ++ (red line). Maturities and tenors are expressed in days and
volatilities are in % points and VIX futures settle prices are in US$. Relative errors
2-SVCVJ++ (2-SVCVJ) model: RMSRESP X = 2.74% (5.65%), RMSREF ut =
0.42% (3.85%), RMSREV IX = 2.31% (6.11%). Absolute errors 2-SVCVJ++ (2-
SVCVJ) model: RMSESP X = 0.91% (1.56%), RMSEF ut = 0.13 US$ (1.18 US$),
RMSEV IX = 2.01% (4.87%).
101

September 2, 2009 2-SVCVJ 2-SVCVJ++
↵1 11.7166 0.8281p
1 (%) 23.3745 1.3579
⇤1 2.7121 0.3948
⇢1 0.5227 0.9446
1,0 (%) 0.0000 21.4092
↵2 2.5723 8.5742p
2 (%) 0.0336 23.5188
⇤2 0.4933 2.6570
⇢2 1.0000 0.7593
2,0 (%) 25.2973 7.4480
0.0080 0.0384
E[cx] 2.3407 0.5350p
V ar[cx] 0.4612 0.6800
µco, 10.1579 0.0002
corr(cx, c ) 0.9971 0.0851
0.0000 0.0243
µid, 1.0000 0.0001
0 50 100 150 200 250 300
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 10
−3
T (d ays)
Iφ(0,T)
Figure 17: Calibrated parameters on September 2, 2009 of 2-SVCVJ and 2-
SVCVJ++ models and I (0, T) displacement integrals of 2-SVCVJ++ model. Fit re-
sults are shown in Figure 16.
102

1000 1100 1200
15
20
25
30
35
40
Strike
Vol(%)
10 days
Calls
Puts
800 1000 1200
15
20
25
30
35
40
45
50
Strike
Vol(%)
38 days
2−SVCVJ
2−SVCVJ++
1000 1100 1200
20
25
30
35
Strike
Vol(%)
50 days
800 1000 1200
15
20
25
30
35
40
45
50
Strike
Vol(%)
66 days
600 800 1000 1200
20
25
30
35
40
45
50
55
Strike
Vol(%)
101 days
600 800 100012001400
20
30
40
50
Strike
Vol(%)
129 days
600 800 1000 1200
20
25
30
35
40
45
50
Strike
Vol(%)
220 days
6008001000120014001600
15
20
25
30
35
40
45
50
Strike
Vol(%)
311 days
0 50 100 150 200
26
27
28
29
30
31
32
Tenor (days)
SettlePrice(US$)
Data
2−SVCVJ
2−SVCVJ++
25 30 35 40
100
120
140
160
180
Strike
Vol(%)
7 days
Data
20 40 60 80
80
100
120
140
Strike
Vol(%)
35 days
2−SVCVJ
2−SVCVJ++
30 40 50 60 70
60
70
80
90
100
Strike
Vol(%)
70 days
30 40 50 60 70
60
65
70
75
80
85
90
Strike
Vol(%)
98 days
20 30 40
50
55
60
65
70
Strike
Vol(%)
133 days
40 60 80
40
50
60
70
80
Strike
Vol(%)
161 days
Figure 18: Fit results on August 11, 2010. This ﬁgure reports market and model
together with the term structure of VIX futures (plot in the middle) on August 11, 2010
2-SVCVJ++ (red line). Maturities and tenors are expressed in days and volatilities
are in % points and VIX futures settle prices are in US$. Relative errors 2-SVCVJ++
(2-SVCVJ) model: RMSRESP X = 2.88% (5.60%), RMSREF ut = 1.17% (2.21%),
2.76% (8.39%).
103

August 11, 2010 2-SVCVJ 2-SVCVJ++
↵1 0.0369 0.1349p
1 (%) 13.8529 44.9580
⇤1 0.3882 0.5256
⇢1 0.6825 0.9648
1,0 (%) 16.4115 16.4905
↵2 12.0826 15.1604p
2 (%) 26.8442 22.7877
⇤2 2.4498 3.6664
⇢2 0.9913 1.0000
2,0 (%) 4.3691 7.4445
0.0444 0.0013
E[cx] 0.2200 1.1115p
V ar[cx] 0.0270 1.9305
µco, 0.0003 36.4290
corr(cx, c ) 0.9993 0.7037
0.0010 36.7457
µid, 171.2908 0.0000
0 50 100 150 200 250 300
0
1
2
3
4
5
6
x 10
−3
T (d ays)
Iφ(0,T)
Figure 19: Calibrated parameters on August 11, 2010 of 2-SVCVJ and 2-SVCVJ++
models and I (0, T) displacement integrals of 2-SVCVJ++ model. Fit results are
shown in Figure 18.
104

Table 3: Calibration errors (in %). This table reports the sample average (max in
sample) of the Root Mean Squared Error (Panel A) and Root Mean Squared Relative
Error (Panel B) of all the H and H++ models calibrated jointly to S&P500 options,
VIX futures and VIX options market data. The sample period is from January 7, 2009
to December 29, 2010 and the sampling frequency is weekly (Wednesdays). For each
date in sample, the fit is performed minimizing the distance L in equation 4.1. Here
we report the absolute (relative) errors on (S&P500 and VIX options) implied volatil-
ity surfaces RMSESP X and RMSEV IX (RMSRESP X and RMSREV IX ) in per-
centage points and errors on the VIX futures term structures in US$. Performance
measures are defined in equations (4.2) to (4.3). Overall pricing errors RMSEAll and
RMSREAll are expressed in percentage points and defined in equation (4.4).
2-SVJ 2-SVJ++ 2-SVCJ 2-SVCJ++ 2-SVVJ 2-SVVJ++ 2-SVCVJ 2-SVCVJ++
Panel A: RMSE
RMSESP X 1.17 0.99 1.04 0.86 0.99 0.77 0.90 0.65
(6.01) (3.75) (4.11) (2.42) (4.28) (3.15) (4.28) (1.64)
RMSEF ut 0.70 0.49 0.59 0.34 0.59 0.31 0.53 0.22
(3.49) (1.85) (1.62) (1.32) (1.66) (1.19) (1.50) (1.07)
RMSEV IX 5.73 3.82 4.12 2.45 4.06 2.32 3.39 1.64
(27.91) (17.58) (17.66) (9.03) (15.55) (8.76) (14.70) (4.03)
RMSEAll 2.20 1.56 1.70 1.16 1.64 1.07 1.42 0.82
(8.80) (4.84) (5.44) (3.14) (7.12) (3.97) (4.57) (2.11)
Panel B: RMSRE
RMSRESP X 4.06 3.30 3.55 2.73 3.42 2.51 3.07 2.02
(16.79) (9.29) (10.93) (6.04) (11.31) (8.25) (11.31) (3.95)
RMSREF ut 2.32 1.61 2.01 1.13 1.98 1.02 1.81 0.74
(9.11) (5.01) (6.48) (3.73) (6.14) (2.92) (6.13) (2.60)
RMSREV IX 7.38 4.66 5.69 3.12 5.59 2.88 4.78 2.04
(28.32) (16.50) (25.14) (13.11) (23.66) (12.98) (23.56) (4.34)
RMSREAll 4.63 3.51 3.91 2.80 3.77 2.56 3.34 2.01
(15.75) (9.90) (10.54) (6.15) (10.70) (7.94) (10.70) (3.94)
crucial for the joint fit of the three markets. The 2-SVCVJ, which is the
richer non-displaced model considered, performs very well on average
with a sample mean relative error of 3.1% on SPX vanilla options, 1.8%
on VIX futures and 4.8% on VIX options, as, shown in 3 (Panel A, 7th
column). Nevertheless, as shown in Figures 14 and 16, it often fails in
reproducing a humped VIX futures term structure and, as confirmed by
Table 4 (Panel A), it tends to perform poorly at longer tenors. Moreover,
as shown by Figures 12 and 18, the change of the slope of call options on
VIX observed at low strikes and the skew term-structure is sometimes
hardly fitted and, as confirmed by Table 5 (Panel C), pricing errors tends
105

Table 4: Calibration RMSE (in %) on VIX futures by Tenor category. This table
reports the sample average of the Root Mean Squared Relative Error for different
Tenor categories of futures on VIX for all the H (Panel A) and H++ (Panel B) models.
Refer to main text and Table 3 for calibration details. Here we report the relative
errors on VIX futures term structures RMSEF ut, as deﬁned in the second of (4.3),
conditioned to the Tenor category considered, measured in days. Errors are expressed
in percentage points and the sample average is weighted by the number of daily
observations in each tenor category. Overall errors are reported in Table 3.
Tenor (days)
Panel A: H models Panel B: H + + models
< 45 45 90 > 90 < 45 45 90 > 90
2-SVJ 0.39 0.25 1.49 2-SVJ++ 0.29 0.17 1.04
2-SVCJ 0.38 0.24 1.19 2-SVCJ++ 0.19 0.13 0.70
2-SVVJ 0.36 0.22 1.22 2-SVVJ++ 0.17 0.13 0.64
2-SVCVJ 0.35 0.21 1.07 2-SVCVJ++ 0.12 0.13 0.44
Observations Observations (% of TOT = 792)
144 144 504 18.18 18.18 63.64
to concentrate at low values of moneyness and at intermediate maturi-
ties.
As shown in Table 5 (Panel A), the calibration error on vanilla options
on S&P500 is remarkably low and tends to increase, in absolute terms, at
short and long maturities and at higher strikes.
With the simple introduction of the displacement t, which is costless
from a computational perspective, the overall errors of 2-SVCVJ model,
mentioned above, collapse to the 2-SVCVJ++ model errors: 2.0%, 0.7%,
and 2.0% (respectively on SPX vanilla, VIX futures and VIX options),
which is roughly half of the average relative error without the extension
(see Table 3, Panel A, last two columns). It is particularly striking that the
maximum error in this case becomes 3.9%, 2.6% and 4.3%, which is com-
parable with the average error obtained without displacement (on VIX
options, the maximum error with displacement is less than the average
error without displacement). From Table 3, the mean (maximum) over-
all relative pricing error RMSREAll is 3.34% (10.70%) for the 2-SVCVJ
model and 2.01% (3.94%) for the displaced 2-SVCVJ++ model.
From Table 6, calibrated parameters are overall in line with typical
values found in the literature (Bates, 2000; Christoffersen et al., 2009;
Dufﬁe et al., 2000) for two-factor stochastic volatility models. We ob-
serve a fast mean-reverting factor 2
2,t, coupled with a slow factor 2
1,t.
Considering the 2-SVCVJ++ model, the fast (respectively slow) factor
shows a half-life log(2)/↵ of approximately 5 weeks (resp. 5 months) and
106

Table 5: Calibration RMSE (in %) on SPX and VIX options by Moneyness - Ma-
turity category. This table reports the sample average of the Root Mean Squared
Relative Error for different Moneyness and time-to-Maturity categories of options on
SPX (respectively VIX) for all the H models in Panel A (resp. C) and H++ models
in Panel B (resp. D). Refer to main text and Table 3 for calibration details. Here we
report the relative errors on VIX implied volatility surfaces RMSEV IX , as deﬁned
in the third of (4.3), conditioned to the Moneyness - Maturity category considered.
Time to Maturity is measured in days and Moneyness for an option of maturity T is
deﬁned as the ratio of the option exercise price to the current index level for S&P500
optionts and of the exercise price to the current VIX futures price expiring at T for
VIX options. For each category, errors are expressed in percentage points and the
sample average is weighted by the number of daily observations in each category.
Maturity Moneyness of SPX options
< 0.95 0.95 1.05 > 1.05 All < 0.95 0.95 1.05 > 1.05 All
< 45 Days
2-SVJ 3.92 4.59 5.21 4.57 2-SVJ++ 3.31 3.07 3.96 3.51
2-SVCJ 3.69 3.99 4.63 4.12 2-SVCJ++ 2.98 2.62 3.36 3.04
2-SVVJ 3.33 4.04 4.67 3.95 2-SVVJ++ 2.54 2.45 3.50 2.78
2-SVCVJ 3.00 3.68 4.34 3.60 2-SVCVJ++ 2.23 2.12 3.06 2.42
45 90 Days
2-SVJ 2.42 2.16 3.21 2.71 2-SVJ++ 2.17 1.75 2.50 2.26
2-SVCJ 2.11 1.84 2.64 2.31 2-SVCJ++ 1.98 1.24 1.96 1.90
2-SVVJ 2.02 1.83 2.70 2.25 2-SVVJ++ 1.73 1.27 2.13 1.81
2-SVCVJ 1.88 1.68 2.27 2.01 2-SVCVJ++ 1.62 0.90 1.64 1.53
> 90 Days
2-SVJ 2.61 3.99 6.39 4.40 2-SVJ++ 2.51 3.54 5.20 3.80
2-SVCJ 2.42 3.50 5.34 3.82 2-SVCJ++ 2.28 2.67 3.89 3.02
2-SVVJ 2.14 3.43 5.30 3.66 2-SVVJ++ 1.92 2.50 3.66 2.73
2-SVCVJ 2.11 3.16 4.48 3.27 2-SVCVJ++ 1.65 1.43 2.40 1.93
All Days
2-SVJ 3.21 3.91 5.22 2-SVJ++ 2.81 2.98 4.16
2-SVCJ 2.96 3.37 4.40 2-SVCJ++ 2.54 2.39 3.25
2-SVVJ 2.69 3.41 4.40 2-SVVJ++ 2.18 2.24 3.22
2-SVCVJ 2.49 3.10 3.83 2-SVCVJ++ 1.93 1.67 2.40
Observations Observations (% of TOT)
< 45 Days 4232 2642 1373 8247 < 45 Days 17.43 10.88 5.66 33.97
45 90 Days 4704 2368 2292 9364 45 90 Days 19.37 9.75 9.44 38.57
> 90 Days 3369 1418 1881 6668 > 90 Days 13.88 5.84 7.75 27.46
All Days 12305 6428 5546 24279 All Days 50.68 26.48 22.84 100.00
Maturity Moneyness of VIX options
Panel C: H models Panel D: H + + models
< 0.95 0.95 1.05 > 1.05 All < 0.95 0.95 1.05 > 1.05 All
< 45 Days
2-SVJ 10.66 9.50 7.54 9.13 2-SVJ++ 4.95 6.73 5.43 5.88
2-SVCJ 7.59 5.74 4.47 5.92 2-SVCJ++ 2.79 3.77 3.07 3.25
2-SVVJ 8.45 6.54 4.30 6.25 2-SVVJ++ 2.23 3.31 3.29 3.24
2-SVCVJ 6.20 4.95 3.32 4.67 2-SVCVJ++ 1.61 2.27 1.98 2.08
45 90 Days
2-SVJ 11.10 6.60 4.71 7.46 2-SVJ++ 4.99 5.00 3.51 4.37
2-SVCJ 9.32 4.51 3.93 6.25 2-SVCJ++ 2.88 2.72 2.96 3.11
2-SVVJ 9.02 4.56 3.54 5.79 2-SVVJ++ 2.43 2.70 2.66 2.77
2-SVCVJ 8.31 3.60 3.01 5.19 2-SVCVJ++ 1.80 1.89 2.08 2.11
> 90 Days
2-SVJ 8.89 4.50 4.26 6.07 2-SVJ++ 3.64 3.03 3.31 3.53
2-SVCJ 7.61 3.86 3.54 5.15 2-SVCJ++ 2.72 2.55 2.65 2.78
2-SVVJ 7.70 3.67 3.10 4.96 2-SVVJ++ 2.33 2.40 2.45 2.52
2-SVCVJ 7.25 3.66 2.99 4.73 2-SVCVJ++ 1.64 2.13 2.01 2.00
All Days
2-SVJ 10.78 6.83 5.45 2-SVJ++ 4.82 4.67 4.11
2-SVCJ 8.88 4.87 4.03 2-SVCJ++ 3.09 3.06 2.96
2-SVVJ 8.93 4.94 3.64 2-SVVJ++ 2.53 2.85 2.82
2-SVCVJ 8.02 4.31 3.20 2-SVCVJ++ 1.81 2.26 2.10
< 45 Days 135 59 390 584 < 45 Days 4.88 2.13 14.09 21.11
45 90 Days 190 57 477 724 45 90 Days 6.87 2.06 17.24 26.17
> 90 Days 384 137 938 1459 > 90 Days 13.88 4.95 33.90 52.73
All Days 709 253 1805 2767 All Days 25.62 9.14 65.23 100.00
107

Table 6: Calibrated parameters. This table reports the sample median (median ab-
solute deviation) of joint SPX, VIX futures and VIX options calibrated parameters
for all the H and H++ models considered in the empirical analysis. The sample pe-
riod is from January 7, 2009 to December 29, 2010 and the sampling frequency is
weekly (Wednesdays). Panel A (B) reports 1st (2nd) volatility factor diffusive pa-
rameters. Panel C reports intensity and unconditional mean and standard deviation
of normal jumps in price, where E[cx] = µx and V ar[cx] = 2
x under 2-SVJ, 2-
SVVJ models (respectively µx +⇢J µco, and 2
x +⇢2
J µ2
co, under 2-SVCJ, 2-SVCVJ)
and analogously under the corresponding displaced speciﬁcations. Panel D reports
the correlated co-jumps parameters. The unconditional correlation between jump
sizes is corr(cx, c ) = ⇢J µco, /
p
V ar[cx] under models 2-SVCJ, 2-SVCVJ and cor-
responding displaced speciﬁcations. Panel E reports the idiosyncratic jumps param-
eters.
Panel A: 1st
Factor
↵1 2.714 2.444 2.262 2.097 2.140 2.084 1.967 1.676
(1.564) (1.544) (1.347) (1.150) (1.279) (1.310) (1.334) (1.070)p
1 (%) 21.419 20.878 19.157 19.275 20.399 20.353 17.819 18.219
(5.460) (5.629) (6.282) (6.300) (8.181) (7.495) (9.162) (6.079)
⇤1 0.637 0.554 0.481 0.491 0.433 0.492 0.445 0.504
(0.341) (0.225) (0.227) (0.159) (0.204) (0.173) (0.219) (0.115)
⇢1 0.871 0.884 0.876 0.891 0.879 0.891 0.865 0.964
(0.122) (0.105) (0.121) (0.095) (0.117) (0.102) (0.121) (0.036)
1,0 (%) 16.679 16.691 16.977 16.484 16.307 16.047 16.250 16.376
(6.885) (5.927) (4.845) (4.615) (4.349) (4.850) (4.677) (4.837)
Panel B: 2nd
Factor
↵2 7.740 6.583 8.058 6.998 8.240 7.346 8.451 6.488
(3.005) (2.035) (3.190) (2.577) (4.221) (3.458) (3.420) (2.477)p
2 (%) 20.642 20.795 21.658 21.536 22.181 21.084 22.950 21.531
(2.707) (2.602) (3.401) (2.826) (4.456) (4.258) (4.308) (3.158)
⇤2 2.207 2.194 2.196 2.219 1.992 2.156 2.050 2.115
(1.036) (0.615) (0.870) (0.728) (0.778) (0.606) (0.738) (0.576)
⇢2 0.939 0.997 0.996 1.000 0.995 1.000 0.997 1.000
(0.061) (0.003) (0.004) (0.000) (0.005) (0.000) (0.003) (0.000)
2,0 (%) 10.085 7.461 9.024 8.387 7.895 7.634 8.683 7.984
(6.680) (5.627) (6.912) (6.050) (5.583) (4.176) (6.309) (4.640)
Panel C: Price jumps
0.038 0.040 0.040 0.045 0.041 0.079 0.079 0.064
(0.035) (0.037) (0.034) (0.040) (0.038) (0.061) (0.053) (0.055)
E[cx] 0.377 0.362 0.398 0.371 0.265 0.262 0.240 0.280
(0.281) (0.256) (0.257) (0.227) (0.189) (0.176) (0.151) (0.183)p
V ar[cx] 0.520 0.512 0.554 0.521 0.269 0.245 0.318 0.413
(0.291) (0.282) (0.347) (0.335) (0.184) (0.151) (0.215) (0.255)
Panel D: CO-jumps
µco, - - 0.153 0.090 - - 0.039 0.065
- - (0.153) (0.090) - - (0.039) (0.065)
corr(cx, c ) - - 0.428 0.341 - - 0.363 0.520
- - (0.428) (0.346) - - (0.366) (0.458)
Panel E: Idiosyncratic jumps
0
- - - - 0.003 0.002 0.002 0.013
- - - - (0.003) (0.002) (0.002) (0.012)
µid, - - - - 5.510 10.091 1.213 0.052
- - - - (5.500) (10.084) (1.213) (0.052)
108

−15 −10 −5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
Panel A: SPX Options (H models)
Relative Implied Volatility error (%)
Normalizedfrequency 2−SVJ
2−SVVJ
2−SVCJ
2−SVCVJ
−15 −10 −5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
Panel B: SPX Options (H++ models)
Normalizedfrequency
2−SVJ++
2−SVVJ++
2−SVCJ++
2−SVCVJ++
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Panel C: VIX Futures (H models)
Relative Settle Price error (%)
Normalizedfrequency
2−SVJ
2−SVVJ
2−SVCJ
2−SVCVJ
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Panel D: VIX Futures (H++ models)
Normalizedfrequency
2−SVJ++
2−SVVJ++
2−SVCJ++
2−SVCVJ++
−25 −20 −15 −10 −5 0 5 10 15 20 25
0
0.05
0.1
0.15
0.2
Panel E: VIX Options (H models)
Normalizedfrequency
2−SVJ
2−SVVJ
2−SVCJ
2−SVCVJ
−25 −20 −15 −10 −5 0 5 10 15 20 25
0
0.05
0.1
0.15
0.2
Panel F: VIX Options (H++ models)
Normalizedfrequency
2−SVJ++
2−SVVJ++
2−SVCJ++
2−SVCVJ++
Figure 20: Relative error distribution. This ﬁgure reports the relative pricing error
for all calibrated H and H + + models, computed for each of the 24279 S&P500 im-
plied volatilities, each of the 792 VIX Futures settle prices, and each of the 2767 VIX
implied volatilities distributed along the 104 Wednesdays in the sample period, from
January 7, 2009 to December 29, 2010. Refer to main text and Table 3 for calibration
details. In Panel A (B) we plot the error distribution of H (H++) models on SPX im-
plied volatilities. In Panel C (D) we plot the error distribution of H (H++) models
on VIX Futures settle prices. In Panel E (F) we plot the error distribution of H (H++)
models on VIX implied volatilities. All errors are expressed in percentage points.
109

−15 −10 −5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
Panel A: SPX Options (H models)
Normalizedfrequency
2−SVJ
2−SVVJ
2−SVCJ
2−SVCVJ
−15 −10 −5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
Panel B: SPX Options (H++ models)
Normalizedfrequency
2−SVJ++
2−SVVJ++
2−SVCJ++
2−SVCVJ++
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Panel C: VIX Futures (H models)
Normalizedfrequency
2−SVJ
2−SVVJ
2−SVCJ
2−SVCVJ
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Panel D: VIX Futures (H++ models)
Normalizedfrequency
2−SVJ++
2−SVVJ++
2−SVCJ++
2−SVCVJ++
−25 −20 −15 −10 −5 0 5 10 15 20 25
0
0.05
0.1
0.15
0.2
Panel E: VIX Options (H models)
Normalizedfrequency
2−SVJ
2−SVVJ
2−SVCJ
2−SVCVJ
−25 −20 −15 −10 −5 0 5 10 15 20 25
0
0.05
0.1
0.15
0.2
Panel F: VIX Options (H++ models)
Normalizedfrequency
2−SVJ++
2−SVVJ++
2−SVCJ++
2−SVCVJ++
Figure 21: Relative error distribution with Feller condition imposed. This ﬁgure
reports the relative pricing error for all calibrated H and H + + models, as in ﬁgure
20, but with the Feller condition 2↵i i ⇤2
i imposed on both stochastic volatility
factors ( 2
i,t i = 1, 2). Refer to Section 4.2.2 and Table 8 for calibration details. All
errors are expressed in percentage points.
110

contributes with roughly 18 vps4
(resp. 22 vps) to the long-term volatil-
ity level
p
. The low values of jump-intensities confirm that jumps are
rare events. The average number of jumps per year 252 ⇥ is estimated
around 16 for co-jumps and less (approximately 3) in the case of idiosyn-
cratic jumps. These numbers are respectively slightly above5
(resp. be-
low) the 8/9 (resp. 13) per annum estimated by Bandi and Renò, 2015
with an extensive econometric analysis. Co-jumps (respectively idiosyn-
cratic jumps) contribute to an increase of approximately 5% (resp. 2%)
of the long-term volatility level, which is approximately evaluated asp
µco, /↵1 (resp.
p
0µid, /↵1) for the case of co-jumps (resp. idiosyn-
cratic jumps).
Figure 20 shows, visually, the distribution of relative signed pricing
errors
QMKT
i Qmdl
i
QMKT
i
(4.5)
with Qi definitions depending on the considered market (as described
for equation 4.4), on all the 24, 279 S&P500 options, 2, 767 VIX options
and 792 VIX Futures implied volatilities observed in the 104 days con-
sidered in the sample. The advantage of the extension over the tradi-
tional specification is large and clearly displayed. If the 2-SVCVJ model
is used for consistent calibration of the three markets, the 10.7% (1.8%) of
S&P500 options implied volatilities (respectively the 2.8% (0.3%) of VIX
futures settle prices and 21.4% (6.2%) of VIX options implied volatilies)
are priced with a relative error greater than 5% (10%).
The displacement increases remarkably the pricing performance, es-
pecially in reproducing the term structure of VIX futures and the VIX
implied volatility surface. Indeed, when the 2-SVCVJ++ model is used,
only the 3.3% (0.4%) of S&P500 options implied volatilities (respectively
the 0.1% (0.0%) of VIX futures settle prices and 3.5% (0.1%) of VIX op-
tions implied volatilities) are priced with a relative error greater than 5%
(10%).
4.2.1 Impact of the short-term
Using vanilla options data on DAX, EuroStoxx50 and FTSE indexes,
Da Fonseca and Grasselli (2011) analyze the theoretical properties and
calibration performance of several competitive option pricing models,
4Volatility points.
5But inside the 95% confidence interval.
111

600 800
40
50
60
70
80
90
Strike
Vol(%)
17 days
Calls
Puts
600 800
30
40
50
60
70
80
90
Strike
Vol(%)
27 days
2−SVCVJ
2−SVCVJ++
400 600 800
30
40
50
60
70
80
90
Strike
Vol(%)
45 days
500 1000
30
40
50
60
70
Strike
Vol(%)
73 days
500 1000
30
40
50
60
Strike
Vol(%)
108 days
500 1000
25
30
35
40
45
50
Strike
Vol(%)
199 days
500 1000
25
30
35
40
45
50
55
Strike
Vol(%)
290 days
0 50 100 150 200 250
36
37
38
39
40
41
42
43
44
45
Tenor (days)
SettlePrice(US$)
Data
2−SVCVJ
2−SVCVJ++
40 50 60 70 80
80
90
100
110
120
130
140
150
Strike
Vol(%)
14 days
Data
30 40 50 60 70 80
60
70
80
90
100
110
Strike
Vol(%)
42 days
2−SVCVJ
2−SVCVJ++
30 40 50 60 70 80
50
60
70
80
90
Strike
Vol(%)
77 days
40 45 50 55
60
65
70
75
Strike
Vol(%)
105 days
Figure 22: Impact of the short-term: fit results on March 4, 2009. This figure reports
market and model implied volatilities for S&P500 (plot at the top) and VIX (plot at the
bottom) options, together with the term structure of VIX futures (plot in the middle)
on March 4, 2009 obtained calibrating jointly on the three markets the 2-SVCVJ (blue
dashed line) and 2-SVCVJ++ (red dashed line). Maturities and tenors are expressed
in days and volatilities are in % points and VIX futures settle prices are in US$. Su-
perimposed (continuous lines) is the corresponding fit obtained with same models
but excluding from the calibrating sample all contracts with maturity below 3 weeks.
112

March 4, 2009 2-SVCVJ 2-SVCVJ++
minimum term 1 week 3 weeks 1 week 3 weeks
↵1 3.3240 4.5873 3.0912 3.2527p
1 (%) 31.9752 31.1138 24.0894 26.4659
⇤1 1.0679 1.1687 1.1152 1.0705
⇢1 0.8431 0.7720 0.9690 0.9601
1,0 (%) 42.9752 44.0075 34.5792 33.7362
↵2 93.7102 76.7253 43.2533 36.0825p
2 (%) 17.1913 18.0654 25.2210 25.2725
⇤2 46.1993 44.2530 8.7081 10.4823
⇢2 0.5685 0.7326 0.5891 0.6765
1,0 (%) 30.6615 24.4879 34.8718 34.1177
0.0012 0.0013 0.0016 0.0015
E[cx] 4.3743 5.9784 2.2030 0.9576p
V ar[cx] 0.5652 2.6655 0.6053 0.4291
µco, 18.6006 23.7498 68.4667 62.3184
corr(cx, c ) 0.8477 0.7592 0.8799 0.8443
0
0.0051 0.0068 0.0025 0.0015
µid, 17.3522 17.4651 57.6487 56.0874
0 50 100 150 200 250 300
0
1
2
3
4
5
6
7
8
9
10
x 10
−3
T (days)
Iφ(0,T)
Figure 23: Impact of the short-term: calibrated parameters on March 4, 2009 of 2-
SVCVJ and 2-SVCVJ++ models and I (0, T) displacement integrals of 2-SVCVJ++
model: dashed (respectively continuous) line if short-term contracts are (resp. are
not) included. Fit results are shown in Figure 22.
113

800 1000
25
30
35
40
45
50
55
Strike
Vol(%)
10 days
Calls
Puts
600 800 1000
20
30
40
50
60
Strike
Vol(%)
45 days
2−SVCVJ
2−SVCVJ++
500 1000
20
30
40
50
60
Strike
Vol(%)
73 days
500 1000
25
30
35
40
45
50
55
60
Strike
Vol(%)
84 days
600 800 1000 1200
20
25
30
35
40
45
50
Strike
Vol(%)
101 days
500 1000
20
25
30
35
40
45
50
55
Strike
Vol(%)
164 days
500 1000
20
25
30
35
40
45
Strike
Vol(%)
255 days
800 1000 1200
20
25
30
35
Strike
Vol(%)
346 days
0 20 40 60 80 100 120 140 160 180 200
30.5
31
31.5
32
32.5
Tenor (days)
SettlePrice(US$)
Data
2−SVCVJ
2−SVCVJ++
30 40 50
70
80
90
100
110
Strike
Vol(%)
42 days
Data
40 50 60 70
75
80
85
90
95
100
105
Strike
Vol(%)
70 days
2−SVCVJ
2−SVCVJ++
40 60 80
60
65
70
75
80
85
90
95
Strike
Vol(%)
105 days
30 40 50 60 70
60
65
70
75
80
85
90
Strike
Vol(%)
133 days
30 40 50 60 70
60
65
70
75
80
85
90
Strike
Vol(%)
161 days
Figure 24: Impact of the short-term: fit results on July 8, 2009. This figure reports
market and model implied volatilities for S&P500 (plot at the top) and VIX (plot at the
bottom) options, together with the term structure of VIX futures (plot in the middle)
on March 4, 2009 obtained calibrating jointly on the three markets the 2-SVCVJ (blue
dashed line) and 2-SVCVJ++ (red dashed line). Maturities and tenors are expressed
in days and volatilities are in % points and VIX futures settle prices are in US$. Su-
perimposed (continuous lines) is the corresponding fit obtained with same models
but excluding from the calibrating sample all contracts with maturity below 3 weeks.
114

July 8, 2009 2-SVCVJ 2-SVCVJ++
↵1 2.1364 1.9274 1.8702 1.8702p
1 (%) 10.4533 7.7945 9.1898 9.1898
⇤1 0.3900 0.3741 0.4164 0.4164
⇢1 0.8850 0.8886 0.9054 0.9054
1,0 (%) 23.0944 22.8217 22.2989 22.2989
↵2 6.3082 7.2736 6.5529 6.5529p
2 (%) 27.1845 27.4401 26.7377 26.7377
⇤2 2.3147 2.5024 2.4890 2.4890
⇢2 0.9194 0.8755 0.9265 0.9265
1,0 (%) 12.7423 10.4250 10.4210 10.4210
0.4065 0.3682 0.5185 0.5185
E[cx] 0.0732 0.0902 0.0778 0.0778p
V ar[cx] 0.1637 0.1662 0.1577 0.1577
µco, 0.0019 0.0001 0.0006 0.0006
corr(cx, c ) 0.0357 0.3155 0.0044 0.0044
0
0.0009 0.0009 0.0006 0.5185
µid, 124.5221 109.0204 147.0826 147.0826
0 50 100 150 200 250 300 350
−0.5
0
0.5
1
1.5
2
2.5
x 10
−3
T (days)
Iφ(0,T)
Figure 25: Impact of the short-term: calibrated parameters on July 8, 2009 of 2-
115

800 1000
20
25
30
35
40
45
50
55
Strike
Vol(%)
17 days
Calls
Puts
900 1000 1100
20
25
30
35
Strike
Vol(%)
28 days
2−SVCVJ
2−SVCVJ++
600 800 1000 1200
20
30
40
50
60
Strike
Vol(%)
45 days
600 800 1000 1200
20
25
30
35
40
45
50
55
Strike
Vol(%)
80 days
600 800 1000 1200
20
25
30
35
40
45
50
55
Strike
Vol(%)
108 days
500 1000
20
25
30
35
40
45
50
Strike
Vol(%)
199 days
500 1000
20
25
30
35
40
45
50
Strike
Vol(%)
290 days
0 20 40 60 80 100 120 140 160 180 200
29
29.5
30
30.5
31
31.5
32
32.5
33
Tenor (days)
SettlePrice(US$)
Data
2−SVCVJ
2−SVCVJ++
34 36 38 40 42 44
90
100
110
120
130
Strike
Vol(%)
14 days
Data
30 40 50
60
65
70
75
80
85
90
95
Strike
Vol(%)
49 days
2−SVCVJ
2−SVCVJ++
30 40 50 60 70 80
60
70
80
90
100
Strike
Vol(%)
77 days
20 30 40 50
50
55
60
65
70
75
80
85
Strike
Vol(%)
105 days
Figure 26: Impact of the short-term: fit results on September 2, 2009. This figure
reports market and model implied volatilities for S&P500 (plot at the top) and VIX
(plot at the bottom) options, together with the term structure of VIX futures (plot in
the middle) on March 4, 2009 obtained calibrating jointly on the three markets the
2-SVCVJ (blue dashed line) and 2-SVCVJ++ (red dashed line). Maturities and tenors
are expressed in days and volatilities are in % points and VIX futures settle prices are
in US$. Superimposed (continuous lines) is the corresponding fit obtained with same
models but excluding from the calibrating sample all contracts with maturity below
3 weeks.
116

September 2, 2009 2-SVCVJ 2-SVCVJ++
↵1 11.7166 11.7166 0.8281 1.0315p
1 (%) 23.3745 23.3752 1.3579 1.5546
⇤1 2.7121 2.7121 0.3948 0.3961
⇢1 0.5227 0.5227 0.9446 0.8938
1,0 (%) 0.0000 0.1068 21.4092 22.2138
↵2 2.5723 2.5723 8.5742 8.3877p
2 (%) 0.0336 0.0336 23.5188 23.6298
⇤2 0.4933 0.4933 2.6570 2.6278
⇢2 1.0000 1.0000 0.7593 0.8246
1,0 (%) 25.2973 25.2983 7.4480 6.7962
0.0080 0.0080 0.0384 0.0374
E[cx] 2.3407 2.3407 0.5350 0.4986p
V ar[cx] 0.4612 0.4612 0.6800 0.6866
µco, 10.1579 10.1579 0.0002 0.0002
corr(cx, c ) 0.9971 0.9971 0.0851 0.0723
0
0.0000 0.0000 0.0243 0.0374
µid, 1.0000 1.0000 0.0001 0.0001
0 50 100 150 200 250 300
0
1
2
3
4
5
x 10
−3
T (days)
Iφ(0,T)
Figure 27: Impact of the short-term: calibrated parameters on September 2,
2009 of 2-SVCVJ and 2-SVCVJ++ models and I (0, T) displacement integrals of 2-
SVCVJ++ model: dashed (respectively continuous) line if short-term contracts are
(resp. are not) included. Fit results are shown in Figure 24.
117

1000 1100 1200
15
20
25
30
35
40
Strike
Vol(%)
10 days
Calls
Puts
800 1000 1200
15
20
25
30
35
40
45
50
Strike
Vol(%)
38 days
2−SVCVJ
2−SVCVJ++
1000 1100 1200
20
25
30
35
Strike
Vol(%)
50 days
800 1000 1200
15
20
25
30
35
40
45
50
Strike
Vol(%)
66 days
600 800 1000 1200
20
25
30
35
40
45
50
55
Strike
Vol(%)
101 days
600 800 100012001400
20
30
40
50
Strike
Vol(%)
129 days
600 800 1000 1200
20
25
30
35
40
45
50
Strike
Vol(%)
220 days
6008001000120014001600
15
20
25
30
35
40
45
50
Strike
Vol(%)
311 days
0 50 100 150 200
26
27
28
29
30
31
32
Tenor (days)
SettlePrice(US$)
Data
2−SVCVJ
2−SVCVJ++
25 30 35 40
100
120
140
160
180
Strike
Vol(%)
7 days
Data
20 40 60 80
80
100
120
140
Strike
Vol(%)
35 days
2−SVCVJ
2−SVCVJ++
30 40 50 60 70
60
70
80
90
100
Strike
Vol(%)
70 days
30 40 50 60 70
60
65
70
75
80
85
90
Strike
Vol(%)
98 days
20 30 40
50
55
60
65
70
Strike
Vol(%)
133 days
40 60 80
40
50
60
70
80
Strike
Vol(%)
161 days
Figure 28: Impact of the short-term: fit results on August 11, 2010. This figure
reports market and model implied volatilities for S&P500 (plot at the top) and VIX
(plot at the bottom) options, together with the term structure of VIX futures (plot in
the middle) on March 4, 2009 obtained calibrating jointly on the three markets the
2-SVCVJ (blue dashed line) and 2-SVCVJ++ (red dashed line). Maturities and tenors
are expressed in days and volatilities are in % points and VIX futures settle prices are
in US$. Superimposed (continuous lines) is the corresponding fit obtained with same
models but excluding from the calibrating sample all contracts with maturity below
3 weeks.
118

Augugst 11, 2010 2-SVCVJ 2-SVCVJ++
↵1 0.0369 0.0190 0.1349 0.1349p
1 (%) 13.8529 29.0256 44.9580 44.9580
⇤1 0.3882 0.4012 0.5256 0.5256
⇢1 0.6825 0.6891 0.9648 0.9648
1,0 (%) 16.4115 16.6534 16.4905 16.4905
↵2 12.0826 13.0322 15.1604 15.1604p
2 (%) 26.8442 26.6842 22.7877 22.7877
⇤2 2.4498 2.6060 3.6664 3.6664
⇢2 0.9913 0.9913 1.0000 1.0000
1,0 (%) 4.3691 0.4348 7.4445 7.4445
0.0444 0.0107 0.0013 0.0013
E[cx] 0.2200 0.2258 1.1115 1.1115p
V ar[cx] 0.0270 0.1759 1.9305 1.9305
µco, 0.0003 0.0968 36.4290 36.4290
corr(cx, c ) 0.9993 0.8893 0.7037 0.7037
0
0.0010 0.0011 36.7457 0.0013
µid, 171.2908 161.2064 0.0000 0.0000
0 50 100 150 200 250 300
0
1
2
3
4
5
6
x 10
−3
T (days)
Iφ(0,T)
Figure 29: Impact of the short-term: calibrated parameters on August 11, 2010 of 2-
119

Table 7: Impact of the short term: calibrated parameters. This table reports the sam-
ple median (median absolute deviation) of joint SPX, VIX futures and VIX options cal-
ibrated parameters for the 2-SVCVJ and 2-SVCVJ++ models considered in the empir-
ical analysis. The columns corresponding to the minimum term of 1 week are the last
two columns of Table 6 and the others report the calibrated parameters of the same
models calibrated excluding short-term contracts (less than 3 weeks to expiration)
from the calibration sample. The sample period is from January 7, 2009 to December
29, 2010 and the sampling frequency is weekly (Wednesdays). Panel A (B) reports
1st (2nd) volatility factor diffusive parameters. Panel C reports intensity and uncon-
ditional mean and standard deviation of normal jumps in price, where E[cx] = µx
and V ar[cx] = 2
x under 2-SVJ, 2-SVVJ models (respectively µx + ⇢J µco, and
2
x + ⇢2
J µ2
co, under 2-SVCJ, 2-SVCVJ) and analogously under the corresponding
displaced speciﬁcations. Panel D reports the correlated co-jumps parameters. The
unconditional correlation between jump sizes is corr(cx, c ) = ⇢J µco, /
p
V ar[cx]
under models 2-SVCJ, 2-SVCVJ and corresponding displaced speciﬁcations. Panel
E reports the idiosyncratic jumps parameters.
2-SVCVJ 2-SVCVJ++
Panel A: 1st
Factor
↵1 1.9674 1.9705 1.6757 1.6774
(1.3339) (1.3713) (1.0702) (0.9609)p
1 (%) 17.8186 17.8029 18.2188 18.8151
(9.1623) (9.5317) (6.0793) (6.8431)
⇤1 0.4445 0.4443 0.5040 0.4977
(0.2186) (0.2163) (0.1151) (0.1150)
⇢1 0.8651 0.8681 0.9641 0.9616
(0.1208) (0.1269) (0.0359) (0.0384)
1,0 (%) 16.2501 16.3388 16.3763 16.1583
(4.6771) (4.7309) (4.8365) (4.7190)
Panel B: 2nd
Factor
↵2 8.4510 8.3306 6.4882 6.4963
(3.4202) (3.2965) (2.4770) (2.3888)p
2 (%) 22.9496 22.8742 21.5312 21.5067
(4.3081) (4.3159) (3.1581) (3.1416)
⇤2 2.0495 2.0237 2.1152 2.0869
(0.7383) (0.6456) (0.5764) (0.5803)
⇢2 0.9972 0.9967 1.0000 1.0000
(0.0028) (0.0033) (0.0000) (0.0000)
1,0 (%) 8.6832 8.7519 7.9841 8.3909
(6.3088) (6.7075) (4.6395) (4.5115)
0.0791 0.0759 0.0644 0.0658
(0.0532) (0.0524) (0.0546) (0.0542)
E[cx] 0.2404 0.2449 0.2795 0.2728
(0.1508) (0.1497) (0.1834) (0.1797)p
V ar[cx] 0.3176 0.3152 0.4135 0.3889
(0.2155) (0.2162) (0.2546) (0.2291)
Panel D: CO-jumps
µco, 0.0393 0.0578 0.0650 0.0600
(0.0393) (0.0578) (0.0650) (0.0600)
corr(cx, c ) 0.3633 0.5991 0.5202 0.5490
(0.3660) (0.4009) (0.4578) (0.4458)
0
0.0021 0.0023 0.0125 0.0111
(0.0021) (0.0023) (0.0123) (0.0110)
µid, 1.2128 1.1341 0.0515 0.0211
(1.2126) (1.1340) (0.0515) (0.0211)
120

focusing on the SV Heston (1993) model, the 2-SV Christoffersen et al.
(2009) model and both single asset and multi-asset Wishart specifica-
tions: the Wishart Multidimensional Stochasti Volatility model (WMSV
hereafter, introduced by Da Fonseca et al. (2008)) and the Wishart Affine
Stochastic Correlation model (WASC hereafter, introduced by Da Fon-
seca et al. (2007)). We refer to Bru (1991) for a theoretical introduction
to Wishart processes and to Gourieroux and Sufana (2004, 2010); Gru-
ber et al. (2010); Leippold and Trojani (2008) for their application to
derivative pricing. Da Fonseca and Grasselli (2011) analyze the differ-
ent structural properties of the SV, 2-SV and WMSV models6
in terms of
the degrees of freedom relevant in describing the empirical features of
the vanilla options surface and the model reaction to its changes (level
and skew risks). As a setup for the comparison of the different models,
they consider the model implied leverage correlation7
and compare the
short-term volatility-of-volatility expansions of the call price and implied
volatility (Lewis, 2000), providing clear relations between the model im-
plied skew and the parameters. Their analysis in particular confirms that
multi factor models are needed to replicate a stochastic skew, as it is usu-
ally observed in market data. Furthermore, the WMSV model features
and additional degree of freedom8
w.r.t. the 2-SV model, which directly
affects the skew of the surface, though it leaves unaltered the level of the
surface. Our analysis, conducted calibrating multi-factor jump-diffusion
models on vanilla, VIX futures and VIX options data, qualitatively con-
firms their calibration results for the 2-SV model, which can be obtained
from our SVCVJ model switching off jumps (as detailed in Section 3.1.2).
In particular, as detailed in Table 6 and reported on a daily basis in Fig-
ures from 13 to 19, we can see a two-regime property in our 2-factor mod-
els, with a fast factor, associated with the short-term smile, featuring a
high volatility of volatility9
. Nevertheless, our calibrated risk-neutral
dynamics for the fast factor often degenerates to perfect anti-correlation.
Interestingly, authors observe that the addition of jumps would lead to a
lack of sensitivity of the skew term structure of the vanilla surface w.r.t.
correlation parameters ⇢1 and ⇢2. In our setting, the instantaneous lever-
6In the present analysis we consider only the single-asset case of the analysis in Da Fon-
seca and Grasselli (2011).
7Which is the correlation between asset returns and the stochastic volatility. This quan-
tity mainly drives the slope of the implied volatility surface (the skew), as it is clearly re-
lated to the skewness of returns distribution.
8The, possibly more than one, non-diagonal elements of the ⌃ state matrix.
9Which is in turns compatible with a more convex short-term smile of vanilla options.
121

age correlation of the simplest displaced model that we consider in this
thesis (the 2-SVJ++ model)
corr
✓
dSt
St
, d( 2
1,t + 2
2,t)
◆
=
⇢1⇤1
2
1,t + ⇢2⇤2
2
2,t
q
2
1,t + 2
2,t + ( 2
x + µ2
x) + t
q
⇤1
2
1,t + ⇤2
2
2,t
dt
(4.6)
would suggest that this could be the case, as part of the skew is jump-
induced. It is therefore interesting to test whether this observation can
be extended to the present analysis in which three distinct market data
sources are used to calibrate the models and if the presence of the dis-
placement has an impact. We therefore get inspiration from their Sec-
tion 2 and compare the calibration already performed with a new one in
which all contracts with expirations less than 3 weeks are excluded. This
analysis has been carried on for the 2-SVCVJ and 2-SVCVJ++ models
and calibration results, as well as calibrated parameters (compared with
those obtained including the short-term contracts) are displayed in Fig-
ures from 22 to 29 for the same days displayed before.
At least to the extent of the present analysis, from Table 7 and for
the 4 days displayed,10
we cannot see any evident difference, in terms of
calibrated parameters (neither of the undisplaced, nor of the displaced
models), as a consequence of the exclusion of the short term contracts.
In particular, we can still see a clear two-regime property of the 2-factor
calibrated models. We argue that this is in line with the value added by
VIX derivatives on the specification of the model. Even if the addition
of jumps introduces a mixing effect between the role of the correlation
parameters and jump parameters, which would make less clear the ef-
fect of the previous on the skew of the vanilla surface (as the leverage
correlation depends also on jumps), the introduction of volatility deriva-
tives in the calibration sample helps to identify the latter (and in turns
preserving the specification of the former): positive jumps in volatility,
which we model partly correlated with those in price and partly idiosyn-
cratic, mainly contribute to enhance the right skewness of the volatility
distribution (which translates into the positive slopes of the VIX options
surface).
10The whole handbook with fit and calibrated paramters for both the complete analysis
and this new analysis is available upon request.
122

1000 1200
20
30
40
50
60
Strike
Vol(%)
17 days
Calls
Puts
1000 1200
20
25
30
35
40
45
50
Strike
Vol(%)
28 days
Feller
NO Feller
800 1000 1200
20
30
40
50
60
70
Strike
Vol(%)
45 days
600 800 1000 1200
20
30
40
50
60
Strike
Vol(%)
80 days
600 800 100012001400
20
30
40
50
60
Strike
Vol(%)
108 days
600 800 100012001400
15
20
25
30
35
40
45
50
Strike
Vol(%)
199 days
600 800 1000
25
30
35
40
45
50
Strike
Vol(%)
290 days
0 50 100 150 200
29.5
30
30.5
31
31.5
32
32.5
Tenor (days)
SettlePrice(US$)
Data
Feller
NO Feller
30 40 50 60 70
80
100
120
140
160
180
200
Strike
Vol(%)
14 days
Data
20 40 60 80
60
80
100
120
140
Strike
Vol(%)
49 days
Feller
NO Feller
30 40 50 60 70
70
80
90
100
110
Strike
Vol(%)
77 days
30 40 50 60 70
60
70
80
90
Strike
Vol(%)
105 days
30 40 50 60
60
65
70
75
80
85
Strike
Vol(%)
140 days
30 40 50 60
55
60
65
70
75
80
Strike
Vol(%)
168 days
Figure 30: Impact of the Feller condition. This ﬁgure reports market and model
together with the term structure of VIX futures (plot in the middle) on June 06, 2010
obtained calibrating jointly on the three markets the 2-SVCVJ++ model with Feller
condition imposed (blue line) 2↵i i ⇤2
i on both stochastic volatility factors ( 2
i,t
i = 1, 2) and with NO Feller condition imposed (red line). Maturities and tenors are
expressed in days and volatilities are in % points and VIX futures settle prices are
in US$. Relative errors without (with) Feller condition imposed: RMSRESP X =
2.62% (7.81%), RMSREF ut = 0.80% (1.84%), RMSREV IX = 2.18% (9.77%).
Absolute errors without (with) Feller condition imposed: RMSESP X = 0.94%
(2.33%), RMSEF ut = 0.25 US$ (0.58 US$), RMSEV IX = 2.47% (10.78%).
12

4.2.2 Analysis with Feller condition imposed
As customary in the empirical S&P500 and VIX options pricing literature
(Bardgett et al., 2013; Branger et al., 2014; Chen and Poon, 2013), a Feller
condition is usually imposed on the volatility factors dynamics which re-
stricts the mutual range of variability of drift and vol-of-vol parameters.
The analysis of the preceding Section has been carried on without impos-
ing such condition. As discussed in Pacati et al. (2015a), assuming a log-
arithmic generating process for volatility - which is increasingly found
to provide an accurate description of the true volatility dynamics (An-
dersen et al., 2002; Bandi and Ren`o, 2015) - a square root diffusion which
approximates the statistical properties of the generating process violates
the Feller condition. To empirically assess the impact of the Feller con-
dition on the present analysis, we have repeated the same calibration of
the previous Section imposing
⌫ =
2↵i i
⇤2
i
1 i = 1, 2 (4.7)
separately on each volatility factor, as discussed in (Andersen and Piter-
barg, 2007; Dufﬁe and Kan, 1996). Overall, the H and H++, with or
without Feller condition imposed (which, considering the positivity of
drift and vol-of-vol parameters, corresponds respectively to the condi-
tions ⌫ 1 and ⌫ > 0), satisfy the following consistency conditions with
respect to the metric induced by the loss function L of equation (4.1). In
words:
1. each H++ model is better than the corresponding H model;
2. each H or H++ model with ⌫ > 0 is better than the same model with ⌫
1.
Table 8 (which corresponds to Table 3) reports the summary statistics
on the root mean squared errors for the H and H++ models averaged
over the three markets, while Tables 9 and 11 , report the same summary
statistics dissected on the three markets (Tables 9 and 11 are the analo-
gous of Tables 4 and 5, respectively).
Figure 21 shows visually the distribution of the signed relative er-
rors in equation (4.5) when the calibrations are performed imposing the
Feller condition. Considering the 2-SVCVJ++ model, the 17.9% (4.7%) of
S&P500 options implied volatilities (respectively the 1.9% (0.1%) of VIX
futures settle prices and 26.0% (5.8%) of VIX options implied volatilies)
124

are priced with a relative error greater than 5% (10%), which are values
comparable with those of the undisplaced 2-SVCVJ model with Feller
condition not imposed. Moreover, by visual inspection of the flattening
of the error distribution of the 2-SVJ model, especially in the VIX options
market, we see that the imposition of the Feller condition penalizes more
the models which do not have jumps in volatility. A possible explanation
could be the following: the Feller condition acts primarily as a binding on
the vol-of-vol parameters ⇤i, that become constrained to be smaller thanp
2↵i i. Then, if the model does not have another channel to increase the
skewness of the volatility/VIX distribution - such as jumps in volatility -
which is needed to reproduce the positive sloping smile of VIX options,
it ends up to be more affected by such restriction w.r.t. a model, like the
2-SVVJ, 2-SVCJ and 2-SVCVJ which features discontinuous volatility
dynamics.
Figure 30 shows a visual comparison between a typical calibration
performed with 2-SVCVJ++ model when the Feller condition is imposed
(blue line) and when it is not. It suggests that the restriction imposed pre-
vents the model from capturing the convexity of the skew of VIX options
- while still reproducing its positive slope - and from fitting long-term
futures.
The visual results of figure 30 are confirmed comparing tables 9
(Panel B) and 11 (Panel D) with their no-Feller counterparts 4 and 5 (same
Panels), where we see that the greatest increase in absolute pricing errors
of 2-SVCVJ++ model when the Feller condition is imposed is on futures
of long tenors, passing from 0.44% to 0.98% and, overall, on VIX op-
tions of short maturities, where the average absolute error increases from
roughly 2.1 vps11
to more than 6 vps, and high strikes, where it increases
from roughly 2 to 5 volatility points.
11Volatility points.
125

Table 8: Calibration errors (in %) with Feller condition imposed. This table reports
the sample average (max in sample) of the Root Mean Squared Error (Panel A) and
Root Mean Squared Relative Error (Panel B) of all the H and H++ models calibrated
jointly to S&P500 options, VIX futures and VIX options market data with the Feller
condition imposed 2↵i i ⇤2
i separately for i = 1, 2. The sample period is from
January 7, 2009 to December 29, 2010 and the sampling frequency is weekly (Wednes-
days). For each date in sample, the fit is performed minimizing the distance L in
equation (4.1). Here we report the absolute (relative) errors on (S&P500 and VIX op-
tions) implied volatility surfaces RMSESP X and RMSEV IX (RMSRESP X and
RMSREV IX ) in percentage points and errors on the VIX futures term structures in
US$. Performance measures are defined in equations (4.2) to (4.3). Overall pricing
errors RMSEAll and RMSREAll are expressed in percentage points and defined in
equation (4.4).
Panel A: RMSE
RMSESP X 2.19 2.12 1.93 1.81 1.62 1.46 1.40 1.21
(9.61) (5.95) (7.94) (6.17) (8.37) (4.83) (7.90) (4.41)
RMSEF ut 0.88 0.72 0.85 0.65 0.80 0.57 0.75 0.42
(3.52) (2.45) (3.39) (2.34) (3.03) (2.68) (2.03) (2.14)
RMSEV IX 16.07 15.21 13.27 12.03 6.20 4.56 5.77 4.02
(33.93) (33.73) (34.00) (33.88) (27.78) (14.85) (16.38) (12.02)
RMSEAll 5.57 5.28 4.64 4.21 2.57 2.05 2.33 1.76
(15.12) (13.18) (13.21) (13.31) (8.91) (6.36) (8.69) (5.88)
Panel B: RMSRE
RMSRESP X 7.09 6.56 6.30 5.71 5.65 4.95 4.89 4.03
(28.37) (12.40) (22.20) (11.99) (23.35) (12.90) (22.09) (11.71)
RMSREF ut 2.93 2.40 2.85 2.18 2.64 1.86 2.47 1.38
(9.18) (6.88) (8.85) (6.57) (8.29) (7.01) (5.79) (5.56)
RMSREV IX 17.87 16.23 14.79 12.70 7.96 5.24 7.32 4.42
(35.47) (27.94) (29.88) (27.94) (25.44) (14.00) (25.39) (12.05)
RMSREAll 8.84 8.08 7.65 6.75 5.96 4.97 5.23 4.05
(29.13) (14.68) (21.00) (14.73) (21.98) (12.61) (20.92) (11.63)
126

Table 9: Calibration RMSE (in %) on VIX futures by Tenor category with Feller
condition imposed. This table reports the sample average of the Root Mean Squared
Relative Error for different Tenor categories of futures on VIX for all the H (Panel
A) and H++ (Panel B) models. Refer to Section 4.2.2 and Table 8 for calibration de-
tails. Here we report the relative errors on VIX futures term structures RMSEF ut,
as deﬁned in the second of (4.3), conditioned to the Tenor category considered, mea-
sured in days. Errors are expressed in percentage points and the sample average is
weighted by the number of daily observations in each tenor category. Overall errors
are reported in Table 8.
Tenor (days)
< 45 45 90 > 90 < 45 45 90 > 90
2-SVJ 0.51 0.38 1.76 2-SVJ++ 0.40 0.28 1.53
2-SVCJ 0.45 0.34 1.79 2-SVCJ++ 0.32 0.23 1.46
2-SVVJ 0.33 0.28 1.80 2-SVVJ++ 0.19 0.18 1.35
2-SVCVJ 0.32 0.25 1.67 2-SVCVJ++ 0.16 0.14 0.98
Observations Observations (% of TOT = 792)
144 144 504 18.18 18.18 63.64
Figure 31: Scatter plot of the 2-SVCVJ++ mean-reversion parameters of 2
2,t:
log10(↵2) Vs log10(
p
2) obtained in the daily calibration imposing the Feller con-
dition 2↵i i ⇤2
i on both stochastic volatility factors ( 2
i,t i = 1, 2). ↵2 is the rate of
mean-reversion and
p
2 is the long-term volatility level.
−7 −6 −5 −4 −3 −2 −1 0
−2
0
2
4
6
8
10
12
14
log 1 0 (
√
β2)
log10(α2)
127

Table 10: Calibrated parameters with Feller condition imposed. This table reports the sample median
(median absolute deviation) of joint SPX, VIX futures and VIX options calibrated parameters for all the H
and H++ models considered in the empirical analysis when the Feller condition 2↵i i ⇤2
i is imposed
on both stochastic volatility factors ( 2
i,t i = 1, 2). The sample period is from January 7, 2009 to Decem-
ber 29, 2010 and the sampling frequency is weekly (Wednesdays). Panel A (B) reports 1st (2nd) volatility
factor diffusive parameters. Panel C reports intensity and unconditional mean and standard deviation of
normal jumps in price, where E[cx] = µx and V ar[cx] = 2
x under 2-SVJ, 2-SVVJ models (respectively
µx + ⇢J µco, and 2
x + ⇢2
J µ2
co, under 2-SVCJ, 2-SVCVJ) and analogously under the corresponding dis-
placed speciﬁcations. Panel D reports the correlated co-jumps parameters. The unconditional correlation
between jump sizes is corr(cx, c ) = ⇢J µco, /
p
V ar[cx] under models 2-SVCJ, 2-SVCVJ and correspond-
ing displaced speciﬁcations. Panel E reports the idiosyncratic jumps parameters.
Panel A: 1st
Factor
↵1 5.662 5.413 5.408 5.230 5.163 4.579 5.002 4.411
(0.960) (1.050) (1.224) (1.100) (1.300) (1.160) (1.357) (0.962)p
1 (%) 31.322 31.882 29.244 29.480 26.030 26.683 25.971 26.240
(3.169) (3.085) (3.167) (3.272) (3.969) (4.332) (4.686) (3.929)
⇤1 1.069 1.039 0.950 0.942 0.791 0.756 0.775 0.752
(0.172) (0.184) (0.175) (0.170) (0.240) (0.185) (0.231) (0.158)
⇢1 0.861 0.900 0.880 0.906 0.973 0.995 0.987 0.996
(0.083) (0.082) (0.097) (0.076) (0.027) (0.005) (0.013) (0.004)
1,0 (%) 20.727 19.009 20.318 19.185 14.592 14.698 14.612 16.912
(6.616) (5.879) (6.944) (6.834) (6.440) (6.398) (7.167) (7.055)
Panel B: 2nd
Factor
↵2 36408.4 2412.0 10561.0 19777.9 276.4 819.4 58.7 241.9
(36399.8) (24103.0) (10557.9) (19773.3) (276.4) (818.8) (54.7) (238.5)p
2 (%) 0.404 0.511 0.656 0.702 2.056 1.427 3.106 1.679
(0.376) (0.425) (0.638) (0.636) (2.004) (1.379) (2.948) (1.540)
⇤2 0.313 0.365 0.418 0.399 0.405 0.435 0.578 0.683
(0.285) (0.360) (0.383) (0.354) (0.401) (0.416) (0.405) (0.577)
⇢2 0.393 0.353 0.290 0.225 0.702 0.573 0.554 0.393
(0.544) (0.478) (0.677) (0.638) (0.298) (0.427) (0.446) (0.607)
2,0 (%) 34.976 51.811 12.342 16.251 21.087 21.308 14.692 13.913
(34.535) (50.943) (12.159) (16.116) (20.523) (18.232) (13.370) (12.189)
0.006 0.006 0.013 0.011 0.011 0.012 0.040 0.046
(0.005) (0.005) (0.009) (0.010) (0.009) (0.010) (0.028) (0.028)
E[cx] 0.404 0.295 0.242 0.214 0.912 0.973 0.364 0.380
(0.438) (0.331) (0.286) (0.228) (0.544) (0.481) (0.200) (0.198)p
V ar[cx] 0.258 0.224 0.271 0.231 0.414 0.396 0.353 0.297
(0.204) (0.171) (0.190) (0.151) (0.271) (0.239) (0.213) (0.146)
Panel D: CO-jumps
µco, - - 0.000 0.000 - - 0.601 0.674
- - (0.000) (0.000) - - (0.601) (0.602)
corr(cx, c ) - - 0.101 0.019 - - 0.774 0.878
- - (0.253) (0.236) - - (0.226) (0.122)
- - - - 0.110 0.131 0.082 0.134
- - - - (0.066) (0.076) (0.081) (0.112)
µid, - - - - 0.931 0.687 0.554 0.443
- - - - (0.521) (0.340) (0.390) (0.266)
128

From inspection of Panel B of tables 5 and 11, the increasing trend of
the errors is similar also on SPX options, passing from roughly 2 to 4.7
vps on the long-maturity bucket and from 2.4 to more than 6 vps on the
high strikes bucket.
With this restriction imposed, table 8 shows that the average (max-
imum) relative pricing error of 2-SVCVJ++ increase to 4% (11.7%) on
S&P500 options, 4.4% (12%) on VIX options and 1.4% (5.6%) on VIX fu-
tures, while for the 2-SVCVJ model we obtained 4.9% (22.1%) on S&P500
options, 7.3% (25.4%) on VIX options and 2.5% (5.8%) on VIX futures.
Overall, when the Feller condition is imposed, the mean (maximum)
overall relative pricing error RMSREAll grows up to 5.23% (20.92%) for
the 2-SVCVJ model and 4.05% (11.63%) for the 2-SVCVJ++ model.
Table 10 shows the calibrated parameters of all H and H++ models
when the Feller condition is imposed on each factor 2
i,t. Considering the
2-SVCVJ++ model, calibrated parameters still show the different role
played by the two volatility factors, with 2
1,t still representing the slow
mean-reverting factor, with a half life of almost 2 months. Neverthe-
less, while with Feller condition not imposed the two factors contribute
to a comparable fraction of the long-term volatility level (table 6, last
column), when the condition is imposed, the long-term level is driven
almost exclusively by the slow factor. Figure 31 shows a scatter plot of
the order of magnitude of the daily calibrated rate of mean reversion ↵2
with respect to the calibrated
p
2 for the fast factor 2
2,t. As it is clear,
insensately high values of ↵2 are coupled with so small values of 2 that
- as a consequence - the factor12
results to be simply unspeciﬁed. Con-
sistently, the restriction imposed by the Feller condition on the vol-of-vol
parameters ⇤i 
p
2↵i i, induces the jump parameters to compensate
for it, showing an increase in their mean value of roughly one order of
magnitue µco, = 0.67, µid, = 0.44, whereas from table 6 they would
have been µco, = 0.07, µid, = 0.05 if the Feller condition would have
not been imposed. Moreover, the long-term level of the 2-SVCVJ++
models
eff = 1 + 2 +
µco,
↵1
+ 0 µid,
↵1
(4.8)
results to be rather similar with or without the imposition of the Feller
condition:
p
eff is respectively 31.6% and 29.0%; a fact that is in line
with the intuition that models with jumps in volatility are able, to some
extent, to generate the necessary volatility-of-volatility - as required to
12Which, strictly speaking, would have a half-life of roughly one day.
129

Table 11: Calibration RMSE (in %) on SPX and VIX options by Moneyness - Matu-
rity category with Feller condition imposed This table reports the sample average of
the Root Mean Squared Relative Error for different Moneyness and time-to-Maturity
categories of options on SPX (respectively VIX) for all the H models in Panel A (resp.
C) and H++ models in Panel B (resp. D). Refer to Section 4.2.2 and Table 8 for cali-
bration details. Here we report the relative errors on VIX implied volatility surfaces
RMSEV IX , as deﬁned in the third of (4.3), conditioned to the Moneyness - Matu-
rity category considered. Time to Maturity is measured in days and Moneyness for an
option of maturity T is deﬁned as the ratio of the option exercise price to the current
index level for S&P500 optionts and of the exercise price to the current VIX futures
price expiring at T for VIX options.For each category, errors are expressed in percent-
age points and the sample average is weighted by the number of daily observations
in each category. Overall errors are reported in Table 8.
Maturity Moneyness of SPX options
< 0.95 0.95 1.05 > 1.05 All < 0.95 0.95 1.05 > 1.05 All
< 45 Days
2-SVJ 8.11 7.91 9.51 8.69 2-SVJ++ 8.38 6.51 7.91 7.99
2-SVCJ 6.99 7.51 8.55 7.75 2-SVCJ++ 6.93 5.95 7.19 6.90
2-SVVJ 5.58 6.04 7.71 6.37 2-SVVJ++ 5.25 4.60 5.97 5.38
2-SVCVJ 4.53 5.46 7.39 5.60 2-SVCVJ++ 4.17 3.87 5.43 4.44
45 90 Days
2-SVJ 4.32 5.27 6.08 5.25 2-SVJ++ 4.27 5.03 5.37 4.89
2-SVCJ 3.82 4.60 5.34 4.59 2-SVCJ++ 3.60 4.36 4.68 4.21
2-SVVJ 2.87 3.20 5.37 3.84 2-SVVJ++ 2.72 2.39 4.59 3.35
2-SVCVJ 2.58 2.85 4.50 3.35 2-SVCVJ++ 2.29 2.06 3.78 2.79
> 90 Days
2-SVJ 3.79 7.33 9.53 6.73 2-SVJ++ 3.71 7.06 8.78 6.34
2-SVCJ 3.47 6.33 8.23 5.91 2-SVCJ++ 3.29 6.11 7.79 5.63
2-SVVJ 2.63 4.69 10.12 6.16 2-SVVJ++ 2.38 3.95 9.74 5.79
2-SVCVJ 2.52 4.31 8.18 5.22 2-SVCVJ++ 2.18 3.45 7.56 4.68
All Days
2-SVJ 5.90 7.20 8.57 2-SVJ++ 6.00 6.34 7.56
2-SVCJ 5.17 6.61 7.65 2-SVCJ++ 5.04 5.68 6.79
2-SVVJ 4.08 5.12 8.31 2-SVVJ++ 3.84 4.00 7.36
2-SVCVJ 3.46 4.68 7.19 2-SVCVJ++ 3.13 3.41 6.02
< 45 Days 4232 2642 1373 8247 < 45 Days 17.43 10.88 5.66 33.97
45 90 Days 4704 2368 2292 9364 45 90 Days 19.37 9.75 9.44 38.57
> 90 Days 3369 1418 1881 6668 > 90 Days 13.88 5.84 7.75 27.46
All Days 12305 6428 5546 24279 All Days 50.68 26.48 22.84 100.00
Maturity Moneyness of VIX options
Panel C: H models Panel D: H + + models
< 0.95 0.95 1.05 > 1.05 All < 0.95 0.95 1.05 > 1.05 All
< 45 Days
2-SVJ 14.26 16.24 21.44 20.39 2-SVJ++ 8.57 12.00 20.97 18.62
2-SVCJ 13.06 13.30 18.21 17.63 2-SVCJ++ 7.82 9.83 17.16 15.35
2-SVVJ 9.96 7.77 8.00 9.12 2-SVVJ++ 3.95 5.47 7.20 6.64
2-SVCVJ 8.76 7.19 7.90 8.57 2-SVCVJ++ 3.49 4.67 6.64 6.06
45 90 Days
2-SVJ 12.87 8.31 20.74 18.73 2-SVJ++ 6.17 7.00 20.29 16.97
2-SVCJ 11.78 7.30 15.73 14.83 2-SVCJ++ 5.33 5.72 15.17 12.81
2-SVVJ 10.72 5.65 6.06 8.02 2-SVVJ++ 3.39 3.43 5.29 4.96
2-SVCVJ 9.95 4.83 5.73 7.47 2-SVCVJ++ 2.75 2.75 4.87 4.39
> 90 Days
2-SVJ 14.70 7.10 18.19 17.02 2-SVJ++ 9.54 6.31 17.71 15.37
2-SVCJ 13.14 5.48 13.63 13.56 2-SVCJ++ 7.32 4.92 13.06 11.51
2-SVVJ 10.45 4.35 5.49 7.34 2-SVVJ++ 4.01 3.27 4.79 4.61
2-SVCVJ 10.20 4.44 5.14 7.07 2-SVCVJ++ 3.14 3.15 4.07 3.90
All Days
2-SVJ 15.21 10.63 19.95 2-SVJ++ 9.32 8.36 19.48
2-SVCJ 13.81 8.62 15.56 2-SVCJ++ 7.51 6.74 14.85
2-SVVJ 11.38 5.95 6.45 2-SVVJ++ 4.19 4.10 5.69
2-SVCVJ 10.71 5.61 6.17 2-SVCVJ++ 3.36 3.65 5.08
Observations Observations ( of TOT)
< 45 Days 135 59 390 584 < 45 Days 4.88 2.13 14.09 21.11
45 90 Days 190 57 477 724 45 90 Days 6.87 2.06 17.24 26.17
> 90 Days 384 137 938 1459 > 90 Days 13.88 4.95 33.90 52.73
All Days 709 253 1805 2767 All Days 25.62 9.14 65.23 100.00
130

reproduce the positive sloping skew of VIX options - leveraging on an
increase of the contribution of jumps.
4.3 Conclusions
Our empirical results show a decisive improvement in the pricing per-
formance over non-displaced models, and also provide strong empirical
support for the presence of both price-volatility co-jumps and idiosyn-
cratic jumps in the volatility dynamics.
The displacement contributes to dramatically increase the fit of the
term structure of VIX futures, even when it displays humps. Moreover,
the addition of the rich jump structure of the Heston++ model makes
it able to capture the positive sloping smile of the VIX options surface
and its term structure. Based on our results, the maximum errors of the
2-SVCVJ++ model are comparable with the average errors of the non-
displaced 2-SVCVJ model.
The imposition of the Feller condition penalizes more the models
with a poorer volatility specification, while models featuring jumps are
able, to some extent, to compensate for the restrictions imposed on the
vol-of-vol parameters leveraging on an increased contribution of jumps.
Nevertheless, despite capturing the positive sloping skew of VIX op-
tions, the 2-SVCVJ++ model with Feller condition imposed seems un-
able to reproduce the correct convexity of the smile.
The pricing errors of displaced models with Feller condition imposed
are roughly comparable with those of non-displaced models without
Feller condition imposed. Overall, the imposition of the Feller condition
does not compromise the superiority of the t-displaced models over
those non-displaced.
A model which consistently prices both equity and volatility mar-
ket is a reasonable starting point in order to infer both equity and vari-
ance risk-premia from the data. In a possible research agenda we would
leverage on the enhanced ability of displaced models in capturing the
risk-neutral dynamics of the S&P500 and VIX indexes in order to try to
infer their true dynamics. This route goes through the definition of a
suitable change of measure between the risk-neutral and physical prob-
ability measure in this displaced jump-diffusion setup (Broadie et al.,
2007; Pan, 2002). A proper methodology has to be designed in order to
filter out unobserved latent variables, such as the volatility process and
jumps. In this respect, standard Kalman filter-based methodologies al-
131

ready employed to estimate equity and variance risk-premia (Bates, 2000;
Gruber et al., 2015) cannot be directly employed, due to the presence of
non-normal innovations in the latent processes. Therefore, more refined
non-standard filtering techniques will be required, such as the Auxiliary
Particle filter introduced by Pitt and Shephard (1999) and already suc-
cessfully employed for risk-premia estimation by Bardgett et al. (2013).
Future developments could lead toward the investigation and the
deeper understanding of the meaning and role of the displacement t,
which seems to play a crucial role in the option pricing context. In par-
ticular, from a mathematical point of view, could be interesting to inter-
pret displaced models as a kind of affine approximation of an unknown
non-affine process. Moreover, from a financial point of view could be
interesting to investigate whether, and to what extent, the displacement
deterministic function can be interpreted as an additional volatility state
vector.
132

Appendix A
Mathematical proofs and
addenda
A.1 Conditional characteristic functions of H
models
As the 2-SVCVJ is an affine model, ordinary calculations following
Duffie et al. (2000) lead to characteristic functions which are exponen-
tially affine in the state variables. For the logarithmic price and volatility
factors we obtain, respectively:
log f
2-SVCVJ
x (z; ⌧) = i(xt + (r q)⌧)z +
X
k=1,2
⇣
A
x
k(z; ⌧) + B
x
k (z; ⌧)
2
k,t
⌘
+ C
x
co(z; ⌧) + C
x
id(z; ⌧)
log f
2-SVCVJ
(z1, z2; ⌧) =
X
k=1,2
⇣
Ak (zk; ⌧) + Bk (zk; ⌧)
2
k,t
⌘
+ Cco(z1; ⌧) + Cid(z1; ⌧)
(A.1)
where coefficients satisfy the following sets of ODEs:
@Ax
k(z; ⌧)
@⌧
= ↵k kBx
k (z; ⌧)
@Bx
k (z; ⌧)
@⌧
=
1
2
⇤2
k (Bx
k (z; ⌧))2
(↵k iz⇢k⇤k) Bx
k (z; ⌧)
1
2
z(i + z)
@Cx
co(z; ⌧)
@⌧
=
⇣
✓co
(z, iBx
1 (z, ⌧)) 1 i¯µz
⌘
@Cx
id(z; ⌧)
@⌧
= 0
⇣
✓id
( iBx
1 (z, ⌧)) 1
⌘
133

with null initial conditions at ⌧ = 0, and
@Ak (zk; ⌧)
@⌧
= ↵k kBk (zk; ⌧)
@Bk (zk; ⌧)
@⌧
=
1
2
⇤2
k (Bk (zk; ⌧))2
↵kBk (zk; ⌧)
@Cco(z1; ⌧)
@⌧
=
⇣
✓co
(0, iB1 (z1, ⌧)) 1
⌘
@Cid(z1; ⌧)
@⌧
= 0
⇣
✓id
( iB1 (z1, ⌧)) 1
⌘
(A.2)
with initial conditions Ak (zk; 0) = Cco(z1; 0) = Cid(z1; 0) = 0 and Bk (zk; 0) =
izk. Explicit solutions can be found. For the f2-SVCVJ
x coefﬁcients, we have:
Ax
k(z; ⌧) =
↵k k
⇤2
k

(ck dk)⌧ 2 log
✓
1 gke dk⌧
1 gk
◆
Bx
k (z, ⌧) =
ck dk
⇤2
k
1 e dk⌧
1 gke dk⌧
Cx
co(z; ⌧) = ⌧
⇣
⇥co
(z; ⌧) 1 i¯µz
⌘
⇥co
(z; ⌧) = exp
⇢
iµxzx
1
2
2
xz2
x
⇥
1
Gco

1
2
⌧
µco,
⇤2
1
1
G+
co
log
✓
Gco g1G+
coe d1⌧
(1 g1)(1 iz⇢J µco, )
◆
Cx
id(z; ⌧) = 0
⌧
⇣
⇥id
(z; ⌧) 1
⌘
⇥id
(z; ⌧) =
1
Gid

1
2
⌧
µid,
⇤2
1
1
G+
id
log
✓
Gid g1G+
ide d1⌧
1 g1
◆
where we have deﬁned the auxiliary parameters:
ck = ↵k iz⇢k⇤k
dk =
q
c2
k + z(i + z)⇤2
k
gk =
ck dk
ck + dk
G±
co = 1 iz⇢J µco,
µco,
⇤2
1
(c1 ± d1)
G±
id = 1
µid,
⇤2
1
(c1 ± d1)
134

For the f2-SVCVJ
coefficients, we have:
Ak (zk; ⌧) =
2↵k k
⇤2
k
log
✓
1 izk
⇤2
k
2↵k
1 e ↵k⌧
◆
Bk (zk; ⌧) =
izke ↵k⌧
1 izk
⇤2
k
2↵k
(1 e ↵k⌧ )
Cco(z1; ⌧) = ⇥(z1; ⌧, µco, )
Cid(z1; ⌧) = 0
⇥(z1; ⌧, µid, )
⇥(z1; ⌧, µ) =
2µ
⇤2
1 2↵1µ
log
✓
1
iz1
1 iz1µ
⇤2
1 2↵1µ
2↵1
1 e ↵1⌧
◆
(A.3)
Characteristic functions of the other nested H models can be obtained applying
the appropriate simplifications to the corresponding expressions just presented
for the 2-SVCVJ model, as discussed in section (3.1.1), see Lian and Zhu (2013)
and Kokholm et al. (2015) for the case of the SVCJ model of Duffie et al. (2000)
and Chen and Poon (2013) for the case of the 2-SVCJ model with two volatil-
ity factors with correlated co-jumps between the first one and the price process.
We present here the expressions for the nested models adopted in the empiri-
cal analysis. For ease of exposition we begin with the results for the two factor
continuous 2-SV model of Christoffersen et al. (2009):
log f2-SV
x (z; ⌧) = i(xt + (r q)⌧)z +
X
k=1,2
⇣
Ax
k(z; ⌧) + Bx
k (z; ⌧) 2
k,t
⌘
log f2-SV
(z1, z2; ⌧) =
X
k=1,2
⇣
Ak (zk; ⌧) + Bk (zk; ⌧) 2
k,t
⌘ (A.4)
For the 2-SVJ model, with log-normal jumps in price only we obtain:
log f2-SVJ
x (z; ⌧) = log f2-SV
x (z; ⌧) + Cx
co(z; ⌧)|µco, =0
log f2-SVJ
(z1, z2; ⌧) = log f2-SV
(z1, z2; ⌧)
(A.5)
For the 2-SVVJ model, with log-normal jumps in price and idiosyncratic jumps
in 2
1,t we obtain:
log f2-SVVJ
x (z; ⌧) = log f2-SVJ
x (z; ⌧) + Cx
id(z; ⌧)
log f2-SVVJ
(z1, z2; ⌧) = log f2-SV
(z1, z2; ⌧) + Cid(z1; ⌧)
(A.6)
For the 2-SVCJ model, with correlated co-jumps in price and 2
1,t we obtain:
log f2-SVCJ
x (z; ⌧) = log f2-SV
x (z; ⌧) + Cx
co(z; ⌧)
log f2-SVCJ
(z1, z2; ⌧) = log f2-SV
(z1, z2; ⌧) + Cco(z1; ⌧)
(A.7)
Relations (3.11) are easily derived since each H++ model is an affine model nest-
ing the corresponding undisplaced H model.
135

A.2 Proof of Proposition 4: CH++
SPX (K, t, T)
The pricing formula is easily obtained from the first of (3.11) and from a straight-
forward application of results of Lewis (2000, 2001).
A.3 Proof of Proposition 5: V IXH++
t
Applying It¯o’s Lemma to the process log(St+¯⌧ /Ft,t+¯⌧ ), under the dynamics of
the 2-SVCVJ++ in (3.3), the VIX definition in (3.13) may be rewritten as
✓
V IXt
100
◆2
=
1
¯⌧
X
k=1,2
EQ
Z t+¯⌧
t
2
k,sds Ft +2 EQ
h
ecx
1 cx
i
+
1
¯⌧
I (t, t+ ¯⌧)
(A.8)
where we have also used the fact that t is a deterministic function. The inte-
grated volatilities and the co-jumps contribution can be computed in closed form
(see for example Lin (2007) and Duan and Yeh (2010) for similar computations)
EQ
Z t+¯⌧
t
2
1,sds Ft =
1 e ¯⌧↵1
↵1
2
1,t +
↵1 1 + µco, + 0
µid,
↵1
✓
¯⌧
1 e ¯⌧↵1
↵1
◆
EQ
Z t+¯⌧
t
2
2,sds Ft =
1 e ¯⌧↵2
↵2
2
2,t + 2
✓
¯⌧
1 e ¯⌧↵2
↵2
◆
EQ
h
ecx
1 cx
i
= ¯µ (µx + ⇢J µco, )
(A.9)
and therefore we have that the coefficients of affinity in (3.15) are
ak(¯⌧) =
1 e ¯⌧↵k
↵k
, k = 1, 2
b1(¯⌧) =
↵1 1 + µco, + 0
µid,
↵1
⇣
¯⌧ a1(¯⌧)
⌘
+ 2
h
¯µ (µx + ⇢J µco, )
i
b2(¯⌧) = 2
⇣
¯⌧ a2(¯⌧)
⌘
(A.10)
Relation (3.14) readily comes from the nesting of 2-SVCVJ model into 2-
SVCVJ++ if t ⌘ 0.
A.4 Proof of Proposition 6: FH++
V IX (t, T) and
CH++
V IX (K, t, T)
The payoffs of a VIX futures contract settled at time T and of a call option on
VIX of strike K maturing at T are linear functions of the VIX index value at
136

settle V IXT , respectively V IXT and (V IXT K)+
. As stated in Proposition
5, under H + + models, V IXT is non-linearly related to the value of volatility
factor processes at time T, whose conditional characteristic function is known in
closed form as shown in Lemma 2. To overcome this issue we rewrite the payoffs
as non-linear functions of the squared index
wF (V IX02
T )
100
=
q
V IX02
T
wC (V IX02
T )
100
=
✓q
V IX02
T K0
◆+ (A.11)
where V IX0
t = V IXt/100 and K0
= K/100 are, respectively, the index and strike
values expressed in percentage points. Fourier transforms for these payoffs are
available in closed form
ˆwF (z)
100
=
p
⇡
2
1
( iz)3/2
ˆwC (z)
100
=
p
⇡
2
1 erf(K0
p
iz)
( iz)3/2
(A.12)
and are single-valued regular functions in the upper half of the complex plane
Sw = {z 2 C : Im(z) > 0} (A.13)
Denote with f2-SVCVJ++
V IX02 the time t conditional characteristic function
EQ
h
eizV IX02
T Ft
i
of the squared index process V IX02
t at time T under
the 2-SVCVJ++ model. From Proposition 5 (with ⌧ = T t)
f2-SVCVJ++
V IX02 (z; ⌧) = eizI (T,T +¯⌧)/¯⌧
f2-SVCVJ
V IX02 (z; ⌧)
= eiz(
P
k=1,2 bk(¯⌧)+I (T,T +¯⌧))/¯⌧
f2-SVCVJ
(za1(¯⌧)/¯⌧, za2(¯⌧)/¯⌧; ⌧)
(A.14)
Following the approach of Lewis (2000, 2001), the value at time t of the call option
on VIX under the 2-SVCVJ++ model is given by
C2-SVCVJ++
V IX (K, t, T) = e r⌧
EQ ⇥
(V IXT K)+
Ft
⇤
= e r⌧
EQ ⇥
wC (V IX02
T ) Ft
⇤
=
e r⌧
2⇡
Z i Im(z)+1
i Im(z) 1
f2-SVCVJ++
V IX02 ( z; ⌧) ˆwC (z)dz
(A.15)
and similarly for futures
F2-SVCVJ++
V IX (t, T) = EQ
[V IXT | Ft]
= EQ ⇥
wF (V IX02
T ) Ft
⇤
=
1
2⇡
Z i Im(z)+1
i Im(z) 1
f2-SVCVJ++
V IX02 ( z; ⌧) ˆwF (z)dz
(A.16)
137

from which the results in Proposition 6 follow since the real (imaginary) part is an
even (odd) function of Re(z). For both claims, the integrands are well behaved
functions as long as z 2 S⇤
V IX02 Sw where f2-SVCVJ++
V IX02 (z; ⌧) is regular in the strip
SV IX02 and S⇤
V IX02 is the conjugate strip, obtained via reflection with respect to
the real z axis. The characteristic functions f2-SVCVJ++
V IX02 (z; ⌧) verifies
f2-SVCVJ++
V IX02 ( z; ⌧) = EQ
h
e izV IX02
T Ft
i
 EQ
h
e izV IX02
T Ft
i
= f2-SVCVJ++
V IX02 ( i Im(z); ⌧)
(A.17)
and therefore, considering the relation in (A.14), determining the strip of regu-
larity S⇤
V IX02 corresponds to analyze the stability of the solutions of the system
ODEs in equation (A.67) for zk = i Im(z)ak(¯⌧)/¯⌧ and k = 1, 2. Similar argu-
ments have been considered in Andersen and Piterbarg (2007); Lee et al. (2004);
Lord and Kahl (2010) in studying the regularity of the log-price characteristic
function fx(z; ⌧) of Heston-like stochastic volatility models. From the second of
the (A.3), the solution Bk ( i Im(z)ak(¯⌧)/¯⌧; ⌧) is regular as long as its denomi-
nator is not equal to zero, requiring:
Im(z) < ⇣
Bk
c (⌧) =
¯⌧
ak(¯⌧)
1
⇤2
k
2↵k
(1 e ↵k⌧ )
(A.18)
which, in addition, guarantees the regularity of Ak ( i Im(z)ak(¯⌧)/¯⌧; ⌧),
given in the first of (A.3). Idiosyncratic and correlated co-jumps solutions
Cco( i Im(z)a1(¯⌧)/¯⌧; ⌧) and Cid( i Im(z)a1(¯⌧)/¯⌧; ⌧) are regular as long as the
argument of the logarithms is not equal to zero, that requires, respectively:
Im(z) < ⇣
Cco
c (⌧) =
¯⌧
a1(¯⌧)
min
0
@ 1
µco,
,
1
⇤2
1
2↵1
(1 e ↵1⌧ ) + µco, e ↵1⌧
1
A (A.19)
and
Im(z) < ⇣
Cid
c (⌧) =
¯⌧
a1(¯⌧)
min
0
@ 1
µid,
,
1
⇤2
1
2↵1
(1 e ↵1⌧ ) + µid, e ↵1⌧
1
A (A.20)
We notice that, since µco, , µid, > 0, we have that min
⇣
⇣
Cco
c (⌧), ⇣
Cid
c (⌧)
⌘
<
⇣
B1
c (⌧) , and therefore ⇣c(⌧) is given by
⇣c(⌧) = min
⇣
⇣
Cco
c (⌧), ⇣
Cid
c (⌧), ⇣
B2
c (⌧)
⌘
(A.21)
138

A.5 Proof of proposition 9: EQ
hR T
t Xsds Ft
i
We derive the expression for functions Ax(⌧) and Bx(⌧) in (3.46)
EQ
Z T
t
Xsds Ft = Ax(⌧) + Bx(⌧)Xt (A.22)
performing the functional derivative of the expression for the expected diffusive
quadratic variation in (3.41), whose relevant term we report here for ease of the
reader,1
EQ
Z T
t
>
s Xsds Ft = Ac(t, T; [t,T ]) + B>
c (t, T; [t,T ])Xt (A.23)
w.r.t. the multiplicative displacement t. Without any pretensions to be rigorous,
we first introduce the concept of first variation and derivative of a functional.
Consider a functional F of the function f(x)
F[f] =
Z x1
x0
If (x)dx (A.24)
where the integrand If is assumed to depend on f(x) and possibly on its deriva-
tives and primitives. We will call the functional derivative of F w.r.t. f(x) the
function of x
F
f(x)
(A.25)
such that the first variation F = F[f + f] F[f] of F is (see (Courant and
Hilbert, 1953, pp. 186) and (Gelfand et al., 2000, pp. 11))
F =
Z x1
x0
F
f(x)
f(x)dx (A.26)
where the variation f(x) is an arbitrary sufficiently regular test function.2
We
interpret the expression in (A.23) as a functional Ft[ ] of the multiplicative dis-
placement function t : R+ ! Rn
Ft[ ] = EQ
Z T
t
>
s Xsds Ft + 1>
Z T
t
sds (A.29)
1We disregard the contribution of the -term in the expression (3.41) of
EQ[ [log S]c
t,T Ft].
2 In the physics literature, often dealing with functional derivatives of observables (func-
tionals F[f]) of fields (functions f(x)) defined on the entire space-time R4, the definition
employed is slightly different, with the variation f(x) inside (A.26) expressed formally in
terms of the Dirac delta ✏ (y x), and therefore in the scalar case the variation of F would
be
F =
Z
F
f(x)
✏ (x y)dx = ✏
F
f(y)
(A.27)
139

whose variation is
Ft = EQ
Z T
t
>
s Xsds Ft =
Z T
t
>
s EQ
[Xs| Ft] ds (A.30)
The last equality is an instance of Fubini theorem and therefore (A.30) holds as
long as
EQ
Z T
t
| >
s Xs|ds Ft < 1 (A.31)
but, since the variation is arbitrary small,
| >
s Xs| =
nX
i=1
i,s
2
i,s 
nX
i=1
| i,s
2
i,s| 
nX
i=1
2
i,s (A.32)
expression (A.30) holds a fortiori if we can interchange the expectation of the
volatility factor state vector with its integral
EQ
Z T
t
Xsds Ft =
Z T
t
EQ
[Xs| Ft] ds (A.33)
Moreover, the variation s has been taken outside of the expectation in (A.30)
since it is deterministic. For what was said before, the functional derivative of
Ft[ ] w.r.t. s, is the conditional expected value of the volatility state vector at
time s t
Ft
s
= EQ
[Xs| Ft] (A.34)
moreover if we assume (A.33), the expected integrated volatility factor state vec-
tor is the integral of Ft
(s)
EQ
Z T
t
Xsds Ft =
Z T
t
Ft
s
ds (A.35)
If we now interpret consistently Ac(t, T; t) and Bc(t, T; t), deﬁned in (3.42),
and reported here for ease of the reader,
Ac(t, T; [t,T ]) =
Z T
t
B>
c (s, T; [s,T ])ds K0 +
mXX
j=1
0,jr✓j(0)
!
Bc(t, T; [t,T ]) =
Z T
t
e(K>
1 +
PmX
j=1 1,j r✓>
j (0))(s t)
(s)ds
(A.36)
with the functional derivative retrieved in the limit of vanishing ✏ as:
F
f(x)
= lim
✏!0
F[f + ✏ (x y)] F[f]
✏
(A.28)
since the delta is symmetric. Good (non technical) introductions can be found in Parisi
(1988) and Greiner and Reinhardt (1996).
140

as functionals of t, denoted respectively as Ac,t[ ] and Bc,t[ ], the linear re-
lation (3.41) allows us to easily compute the functional derivative of Ft[ ] w.r.t.
s in terms of their own functional derivatives3
Ft
(s)
=
Ac,t
s
+
✓
Bc,t
s
◆>
Xt (A.37)
where4
Ac,t
s
=
"✓
Bc,t
s
◆>
Idn
#
K1 +
mXX
j=1
r✓j(0) >
1,j
! 1
K0 +
mXX
j=1
0,jr✓j(0)
!
Bc,t
s
= e(K>
1 +
PmX
j=1 1,j r✓>
j (0))(s t)
(A.40)
and we conclude observing that their integrals between time t and T
Z T
t
Ac,t
s
ds = [Bx(⌧) Idn⌧] K1 +
mXX
j=1
r✓j(0) >
1,j
! 1
K0 +
mXX
j=1
0,jr✓j(0)
!
Z T
t
✓
Bc,t
s
◆>
ds =
h
e(K1+
PmX
j=1 r✓j (0) >
1,j )⌧
Idn
i
K1 +
mXX
j=1
r✓j(0) >
1,j
! 1
(A.41)
are therefore the functions Ax(⌧) and Bx(⌧) of Proposition 9.
3Observe that
(B>
c,tXt)
s
=
B>
c,t
s
Xt =
✓
Bc,t
s
◆>
Xt
4 In deriving
Ac,t
s
it could be useful the following easy application of the Fubini theo-
rem to perform an interchange of the order of the integrals:
Z T
t
✓Z T
s
G(u, s) (u)du
◆
ds =
Z T
t
✓Z u
t
G(u, s) (u)ds
◆
du
=
Z T
t
✓Z u
t
G(u, s)ds
◆
(u)du
(A.38)
which corresponds to two distinct parametrizations of the triangular region Tt,T
Tt,T = (u, s) 2 R2
: s  u  T and t  s  T
= (u, s) 2 R2
: t  u  T and u  s  T
(A.39)
141

A.6 Proof of proposition 11: FV IX(t, T) and
CV IX(K, t, T) under the displaced affine
framework
By definition of conditional CF (VIX’ = VIX/100) and from the results in Propo-
sition 10, we have z = Re(z) + i Im(z) 2 C
fV IX02 (z; Xt, t, T) = EQ
h
eizV IX02
T Ft
i
= eiz(a+1>
I (T,T +¯⌧))/¯⌧
EQ
h
eizb>
XT /¯⌧
Ft
i
= eiz(a+1>
I (T,T +¯⌧))/¯⌧
f
✓
z
b
¯⌧
; Xt, t, T
◆
(A.42)
where f (Z; Xt, t, T) = EQ
h
eiZ>
XT Ft
i
with Z = Re(Z)+i Im(Z) 2 Cn
, is the
risk-neutral conditional characteristic function of Xt 2 Rn
. The results in Lewis
(2000) and Lewis (2001), based on the regularity theorem for CF of Lukacs (1970),
ensure us that fV IX02 is a regular function in the strip
z 2 C : | Im(z)| < ⇣c(t, T) (A.43)
that will in general depend on the model considered, through f (as discussed in
Appendix A.4 for the specific case of the Heston++ model). Recalling the rewrit-
ten payoffs
wF (V IX02
T ) = 100 ⇥
q
V IX02
T
wC (V IX02
T ) = 100 ⇥
✓q
V IX02
T K0
◆+ (A.44)
and their Fourier transforms
ˆwF (z) = 100 ⇥
p
⇡
2
1
( iz)3/2
ˆwC (z) = 100 ⇥
p
⇡
2
1 erf(K0
p
iz)
( iz)3/2
(A.45)
that are single-valued regular functions in the upper half of the complex plane
Im(z) > 0, we can apply the definition of arbitrage-free pricing. and compute
the VIX derivative prices by Fourier inversion of their payoffs. For futures on
142

VIX we have:
FV IX (t, T) = EQ
[V IXT | Ft]
= EQ
[wF (V IX02
T ) | Ft]
= EQ
"
1
2⇡
Z i Im(z)+1
i Im(z) 1
e izV IX02
T ˆwF (z)dz Ft
#
=
1
2⇡
Z i Im(z)+1
i Im(z) 1
EQ
h
e izV IX02
T | Ft
i
ˆwF (z)dz
=
1
2⇡
Z i Im(z)+1
i Im(z) 1
fV IX02 ( z; Xt, t, T) ˆwF (z)dz
(A.46)
where we have used Fubini Theorem to move the expectation inside the integral.
Considering that the real (imaginary) part of the complex integrand is an even
(odd) function of Re(z), can be rewritten as
FV IX (t, T) =
1
⇡
Z 1
0
Re
h
fV IX02 ( z; Xt, t, T) ˆwF (z)
i
d Re(z) (A.47)
with 0 < Im(z) < ⇣c, given in (A.43). Substituting fV IX02 expression in (A.42)
and ˆwF (z) given in (A.45), we get the ﬁrst of (3.72). Analogously, for call options
on VIX,
CV IX (K, t, T) = e r⌧
EQ
[(V IXT K)+
| Ft]
= e r⌧
EQ
[wC (V IX02
T ) | Ft]
= e r⌧
EQ
"
1
2⇡
Z i Im(z)+1
i Im(z) 1
e izV IX02
T ˆwC (z)dz Ft
#
=
e r⌧
2⇡
Z i Im(z)+1
i Im(z) 1
EQ
h
e izV IX02
T | Ft
i
ˆwC (z)dz
=
e r⌧
2⇡
Z i Im(z)+1
i Im(z) 1
fV IX02 ( z; Xt, t, T) ˆwC (z)dz
=
e r⌧
⇡
Z 1
0
Re
h
fV IX02 ( z; Xt, t, T) ˆwC (z)
i
d Re(z)
(A.48)
with 0 < Im(z) < ⇣c. Substituting fV IX02 expression in (A.42) and ˆwC (z) given
in (A.45), we get the second of (3.72). Similarly, for a put option
wP (V IX02
T ) = 100 ⇥ max
✓q
V IX02
T K0
◆
(A.49)
143

with Fourier transform
ˆwP (z) =
Z 1
1
eizV IX02
T wP (V IX02
T )dV IX02
T
= 100 ⇥
✓
iK0
z
p
⇡
2
erf(K0
p
iz)
( iz)3/2
◆ (A.50)
Therefore, a put option on VIX can be priced either by put-call parity in (2.12),
given call and futures prices in (3.72), or directly
PV IX (K, t, T) =
e r⌧
⇡
Z 1
0
Re
h
fV IX02 ( z; Xt, t, T) ˆwP (z)
i
d Re(z) (A.51)
with 0 < Im(z) < ⇣c.
A.7 Affinity conservation under displacement
transformation of instantaneous volatility
From inspection of VIX derivatives pricing formulas in Propositions 6 or 11, it
is clear that VIX futures and options prices depend strongly on the risk-neutral
statistical properties of the stochastic volatility process
Xt = ( 2
1,t, ..., 2
n,t)>
2 Rn
, (A.52)
and only say, indirectly (through the VIX affinity coefficients), on the dynamics of
the underlying price process St. Moreover, by direct inspection of the (a, b) co-
efficients in Proposition 10, it is clear that they do not depend on the correlation
between the diffusive dynamics of St and Xt. VIX derivative prices do directly
depend on the statistical properties of the volatility factors and the only relevant
process to be affine in order for their price to be computable in closed-form is the
stochastic volatility process Xt.
This means that to price volatility derivatives, one can either compute the condi-
tional PDF of Xt: pQ
(XT |Xt), or more in general can express the pricing formu-
las, as in Proposition 11, in terms of the conditional CF
f (Z; Xt, t, T) = EQ
h
eiZ>
XT
Ft
i
(A.53)
which, as we have seen in Proposition 12, is computable in closed form under
our present affine framework for Xt.
The price of equity derivatives instead, depends on the risk-neutral distribution
of St, to which will in general contribute the dynamics of Xt. In other words, to
compute the no-arbitrage price of a contingent claim on St, one has to consider
either the transition PDF pQ
S(ST |St), or the conditional CF
fS(z; St, Xt, t, T) = EQ
h
eizST
Ft
i
(A.54)
144

that will in general be a function of the volatility factors too. The transform anal-
ysis of Duffie, Pan and Singleton Duffie et al. (2000) ensures us that the function
fS(z; St, Xt) can be computed in closed-form (and in the usual exponential affine
form), provided that the complete process
(Xt, St)>
= ( 2
1,t, ..., 2
n,t, St)>
2 Rn+1
(A.55)
is an affine process, according to the affine dependence structure described in
(Duffie et al., 2000, Sec 2.2). As will be shown in what follows, if we consider a
( t, t)-displaced AJD model, in order for the affinity structure of the complete
process (Xt, St)>
to hold, binds have to be imposed on the risk-neutral correla-
tion structure between the price process St and those stochastic volatility factors
Xi,t that are displaced. In other terms, if the i-th stochastic volatility factor is
displaced, that is
Vc,i( 2
i,t) = i,t
2
i,t + i,t (A.56)
the instantaneous correlation between dWS
i,t and dWX
i,t cannot be chosen arbi-
trarily in order for the process (Xt, St)>
to be affine. We do not make here any
general statement and prefer to investigate deeper on this point with a couple of
examples of models that fit in the present framework.
Example 1. Pacati, Renò and Santilli (2014) In Pacati et al. (2014), the authors
consider a jump-diffusion model, labelled 2fj++, in which the price process St of a non-
dividend paying underlying follows the risk-neutral dynamics (refer to main text for
details)
dSt = rStdt + St
q
2
1,t + tdWS
1,t + 2,tdWS
2,t + kJ StdNt
d 2
i,t = ↵i( i
2
i,t)dt + ⇤i i,tdWi,t (i = 1, 2)
log(1 + kJ ) ⇠ N
✓
log(1 + ¯kJ )
1
2
2
J , 2
J
◆
(A.57)
The contribution to the spot variance Vc( 2
1,t, 2
2,t) of the first stochastic volatility factor
is displaced by a non-negative deterministic function t 0
Vc( 2
1,t, 2
2,t) = Vc,1( 2
1,t) + Vc,2( 2
2,t)
Vc,1( 2
1,t) = 2
1,t + t
Vc,2( 2
2,t) = 2
2,t
(A.58)
As they pointed out, the unique functional form of the instantaneous correlation between
dWS
1,t and dWX
1,t which guarantees the linearity of the pricing PDE for a contingent
claim on St is:
corr(dWS
1,t, dW1,t) = ⇢
s
2
1,t
2
1,t + t
dt (A.59)
where ⇢ 2 [ 1, 1] is an additional constant.
145

In the second example that we consider we make explicit the correspondence
between a model for (Xt, St)>
with a linear backward Fokker-Planck equation
for fS(z; St, Xt), that is the vanilla pricing PDE, and the affinity propriety in the
sense of (Duffie et al., 2000, Sec 2.2). This correspondence is an identity and the
linearity of the PDE / affinity holds provided a particular form for the correlation
structure is imposed.
Example 2. Christoffersen, Heston and Jacobs (2009) ( t, t)-displaced Consider
a filtered probability space (⌦, F, {Ft}t 0 , Q), satisfying usual assumptions. Under
the risk-neutral measure Q, we specify the evolution of the logarithmic price of the un-
derlying S&P500 index xt = log St as follows
dxt =
h
r q
1
2
t
2
1,t + t + 2
2,t
i
dt +
q
t
2
1,t + tdWS
1,t + 2,tdWS
2,t
d 2
i,t = ↵i( i
2
i,t)dt + ⇤i tdWi,t (i = 1, 2)
(A.60)
where r is the short rate, q is the continuously compounded dividend yield rate, and in
which the risk-neutral dynamics of the index is driven by continuous shocks, modeled
by the Wiener processes WS
i,t, i = 1, 2. The first volatility factor is displaced by two
sufficiently regular deterministic functions t and t which verify the conditions (3.27)
of our setting
t 0 and 0 = 0
t 0 and 0 = 1
(A.61)
and ↵i, i, ⇤i are non-negative constants.5
The corresponding dynamics of the index St
is, by It¯o’s lemma:
dSt
St
= (r q)dt +
q
t
2
1,t + tdWS
1,t + 2,tdWS
2,t (A.62)
This model is a ( t, t)-displaced version of the two-factor model of Christoffersen, He-
ston and Jacobs Christoffersen et al. (2009), which we will call 2-SV⇥+. The only non-
zero correlations imposed are
corr(dWS
1,t, dW1,t) = ⇢1(t)dt (A.63)
corr(dWS
2,t, dW2,t) = ⇢2dt (A.64)
with |⇢1(t)|  1 but left otherwise unspecified and ⇢2 2 [ 1, 1] an additional constant.
Consider first the stochastic volatility process Xt = ( 2
1,t, 2
2,t)>
alone. It’s easy to check
that this process fits in our affine framework, is unaffected by the ( t, t)-displacements
and its distributional properties can be described by means of the conditional CF
f2-SV⇥+
(z1, z2; 2
1,t, 2
2,t, ⌧) = EQ
h
eiz1
2
1,T +iz2
2
2,T Ft
i
(A.65)
5In the present context the Feller condition 2↵i i ⇤2
i , i = 1, 2 is not relevant and we
do not consider it further.
146

which, from Proposition 12, takes the following exponential affine form
log f2-SV⇥+
(z1, z2; 2
1,t, 2
2,t, ⌧) =
X
i=1,2
⇣
Ai (zi; ⌧) + Bi (zi; ⌧) 2
i,t
⌘
(A.66)
where coefficients satisfy the following set of ODEs:
@Ai (zi; ⌧)
@⌧
= ↵i iBi (zi; ⌧)
@Bi (zi; ⌧)
@⌧
=
1
2
⇤2
i (Bi (zi; ⌧))2
↵iBi (zi; ⌧)
(A.67)
with initial conditions Ai (zi; 0) = 0 and Bi (zi; 0) = izi. Explicit solutions can be
found:
Ai (zi; ⌧) =
2↵i i
⇤2
i
log
✓
1 izi
⇤2
i
2↵i
1 e ↵i⌧
◆
Bi (zi; ⌧) =
izie ↵i⌧
1 izi
⇤2
i
2↵i
(1 e ↵i⌧ )
(A.68)
We can conclude that the price of VIX derivatives does not require any specification
of the correlations ⇢1(t), ⇢2, since it does not depend on them. By direct inspection of
Proposition 11, the price of a futures or option written on VIX, depends on the dynamics
of St only thorough the affinity coefficients of VIX scaled squared
✓
V IX2-SV⇥+
t
100
◆2
=
1
¯⌧
X
i=1,2
ai(¯⌧) + bi(¯⌧) 2
i,t +
Z t+¯⌧
t
sds
!
(A.69)
which for this 2-SV⇥+ model take the form (i = 1, 2)
ai(¯⌧) = i
⇣
¯⌧ bi(¯⌧)
⌘
bi(¯⌧) =
1 e ¯⌧↵i
↵i
(A.70)
but not on any correlation between dWS
i,t and dWj,t.
We now go back to the complete specification of the 2-SV⇥+ model for ( 2
1,t, 2
2,t, xt)>
and analyze the role of correlation function ⇢1(t) and begin with the affine approach
introduced by Duffie, Pan and Singleton in Duffie et al. (2000). Borrowing from their
notation, we rewrite the model in 2-SV⇥+ model in matricial form as
d
0
@
2
1,t
2
2,t
xt
1
A = µdt + dW (A.71)
147

where dW = (dW(1)
, dW(2)
, dW(3)
, dW(4)
)>
2 R4
is a 4-dimensional standard
Wiener process and the drift is the 3-dimensional vector µ = (µ 1 , µ 2 , µx)>
2 R3
µ i = ↵l( l
2
l,t) (i = 1, 2)
µx = r q
1
2
t
2
1,t + t + 2
2,t
(A.72)
and it’s easy to see that µ is an affine function of the complete process ( 1,t, 2,t, xt)>
µ = K0 + K1
0
@
2
1,t
2
2,t
xt
1
A
=
0
@
↵1 1
↵2 2
r q 1
2 t
1
A +
0
@
↵1 0 0
0 ↵2 0
1
2 t
1
2
0
1
A
0
@
2
1,t
2
2,t
xt
1
A
The volatility matrix 2 R3⇥4
is given by the following matrix
=
0
B
@
⇤1 1,t 0 0 0
0 0 ⇤2 2,t 0
⇢1(t)
q
t
2
1,t + t
p
1 ⇢2
1(t)
q
t
2
1,t + t ⇢2 2,t
p
1 ⇢2
2 2,t
1
C
A
(A.73)
The complete process is affine in the sense of (Duffie et al., 2000, Sec. 2.2) provided
the variance-covariance matrix >
2 R4⇥4
can be written as an affine function of
( 1,t, 2,t, xt)>
>
= H0 + H1 ·
0
@
2
1,t
2
2,t
xt
1
A
= H0 + H
(1)
1
2
1,t + H
(2)
1
2
1,t + H
(3)
1 xt
for some real symmetric 3 ⇥ 3 matrices H0 and H
(i)
1 , i = 1, 2, 3. It’s easy to realize
that, for a general form of ⇢1(t), there are no such matrices. Let us impose the following
functional form on the correlation:
corr(dWS
1,t, dW1,t) = ⇢1(t)dt = ⇢1
s
t
2
1,t
t
2
1,t + t
dt (A.74)
with ⇢1 2 [ 1, 1] an additional constant. With this correlation structure imposed we
148

easily find that (H
(3)
1 = 03⇥3)
>
=
0
@
⇤2
1
2
1,t 0 ⇢1⇤1
p
t
2
1,t
0 ⇤2
2
2
2,t ⇢2⇤2
2
2,t
⇢1⇤1
p
t
2
1,t ⇢2⇤2
2
2,t t
2
1,t + t + 2
2,t
1
A
= H0 + H
(1)
1
2
1,t + H
(2)
1
2
1,t
=
0
@
0 0 0
0 0 0
0 0 t
1
A +
0
@
⇤2
1 0 ⇢1⇤1
p
t
0 0 0
⇢1⇤1
p
t 0 t
1
A 2
1,t +
0
@
0 0 0
0 ⇤2
2 ⇢2⇤2
0 ⇢2⇤2 1
1
A 2
2,t
and therefore the 2-SV⇥+ model, equipped with the correlation structure in (A.74), is
an affine model in the sense of (Duffie et al., 2000, Sec. 2.2). We conclude this affine
approach noting that with this ad-hoc form for the correlation, the diffusion matrix
can be written in the extended canonical form of Collin-Dufresne and Goldstein (2002)
and Cheridito et al. (2010)
3⇥4 = ⌃3⇥4
p
V 4⇥4
=
0
B
@
⇤1 0 0 0
0 0 ⇤2 0
⇢1
p
t
q
1 ⇢2
1,t ⇢2
q
1 ⇢2
2
1
C
A
0
B
B
B
B
B
B
B
@
q
2
1,t 0 0 0
0
r
t
1 ⇢2
1,t
+ t
2
1,t 0 0
0 0
q
2
2,t 0
0 0 0
q
2
2,t
1
C
C
C
C
C
C
C
A
and thus satisfies their sufficient condition for affinity.
Let us now step back to the unspecified correlation ⇢1(t) in (A.63) and follow the standard
PDE approach for pricing derivatives. Consider the conditional CF of the complete
process ( 2
1,t, 2
2,t, xt)>
2 R3
f2-SV⇥+
x (z, z1, z2; 2
1,t, 2
2,t, xt, t, T) = EQ
h
eizxT +iz1
2
1,T +iz2
2
2,T Ft
i
(A.75)
The Feynmann-Kaˇc theorem states that fx is a solution of the following boundary value
problem6
(Bjork, 1998, Chap. 5)
@tfx + µ>
rfx +
1
2
Tr
h
>
Hx
i
= 0
fx (z, z1, z2; 2
1,T , 2
2,T , xT , T, T) = eizxT +iz1
2
1,T +iz2
2
2,T
(A.76)
6@tfx is for @fx
@t
, rfx 2 R3 denotes the gradient of fx w.r.t ( 2
1,t, 2
2,t, xt)> 2 R3,
Hx 2 R3⇥3 is the Hessian matrix of fx and Tr[·] the trace operator.
149

where µ 2 R3
and 2 R3⇥4
have been defined in (A.73) and (A.73), respectively.
From the dynamics in (A.60), the PDE for fx may be written explicitly as follows
(f := fx , Vi := 2
i,t)7
0 = @tf +

r q
1
2
( tV1 + t + V2) @xf +
1
2
( tV1 + t + V2)@2
xxf
+
X
k=1,2

↵k( k Vk)@kf +
1
2
⇤2
kVk@2
kkf
+ ⇢1(t)
p
tV1 + t⇤1
p
V1@2
x1f + ⇢2⇤2V2@2
x2f
It’s easy to realize that, for a general form of ⇢1(t), the PDE is not analytically tractable,
due to the non-linear dependence w.r.t. V1 := 2
1,t which prevents us from applying a
separation argument. Let us impose the correlation in (A.74)
corr(dWS
1,t, dW1,t) = ⇢1(t)dt = ⇢1
s
tV1
tV1 + t
dt (A.77)
With this correlation structure, we obtain a linearization of the PDE
0 = @tf +

r q
1
2
( tV1 + t + V2) @xf +
1
2
( tV1 + t + V2)@2
xxf
+
X
k=1,2

↵k( k Vk)@kf +
1
2
⇤2
kVk@2
kkf + ⇢k⇤kVk@2
xkf
(A.78)
If we look for a solution of (A.78) with z = 0, we are in fact looking for a solution
verifying
fx (0, z1, z2; 2
1,t, 2
2,t, xt, t, T) = EQ
h
eiz1
2
1,T +iz2
2
2,T Ft
i
= f (z1, z2; 2
1,t, 2
2,t, ⌧)
that is the conditional CF of the volatility process Xt = ( 2
1,t, 2
2,t)>
. Since the dynam-
ics of Xt does not depends on xt, the PDE (A.78) satisfied by f , simplifies to (f := f)8
@tf +
X
k=1,2

↵k( k Vk)@kf +
1
2
⇤2
kVk@2
kkf = 0
f (z1, z2; 2
1,T , 2
2,T , 0) = eiz1
2
1,T +iz2
2
2,T
(A.79)
7For ease of notation, in the PDE we will also write @xfx for @fx
@xt
, @kfx for @fx
@Vk
,
@2
xxfx for @2
fx
@x2
t
, @2
ijfx for @2
fx
@Vi@Vj
, @2
xifx for @2
fx
@xt@Vi
.
8One can easily realize that the PDE in (A.78), with the terminal condition f(T) =
eiz1
2
1,T +iz2
2
2,T can be verified by a function independent from xt, that is verifying
@xf = @2
xxf = @2
xkf = 0.
150

and if we substitute the educated guess of equation A.66
f (z1, z2; 2
1,t, 2
2,t, ⌧) = e
P
k=1,2 Ak (zk;⌧)+Bk (zk;⌧) 2
k,t (A.80)
it’s a simple check of internal consistency to verify that the coefficients Ak and Bk will
satisfy the set of ODEs in (A.67).
We now go back to the full linear PDE in (A.78) and look for a solution with z1 = z2 = 0,
that is we look for the conditional CF of the log-price, needed in pricing equity derivatives
Lewis (2000, 2001)
fx (z, 0, 0; 2
1,t, 2
2,t, xt, t, T) = EQ
h
eizxT
Ft
i
= fx(z, 2
1,t, 2
2,t, t, T)
Since the dynamics of xt = log St depends on the dynamics of the volatility factors Xt,
the choice z1 = z2 = 0 will only modifies the terminal condition, otherwise leaving the
PDE in (A.78) unchanged (fx := f)
@tf +

r q
1
2
( tV1 + t + V2) @xf +
1
2
( tV1 + t + V2)@2
xxf+
X
k=1,2

↵k( k Vk)@kf +
1
2
⇤2
kVk@2
kkf + ⇢k⇤kVk@2
xkf = 0
fx(z; 2
1,T , 2
2,T , xT , T, T) = eizxT
(A.81)
Now we substitute in (A.77) the educated guess
log fx(z; V1, V2, xt, t, T ) = i(xt+(r q)⌧)z+
X
k=1,2
⇣
A
x
k(z, t, T ) + B
x
k (z, t, T )
2
k,t
⌘ 1
2
z(i+z)I (t, T )
(A.82)
where I (t, T) =
R T
t sds, it’s easy to show that the coefficients Ax
k(z; t, T) and
Bx
k (z; t, T) solve the following set of ODEs
@tAx
k = ↵k kBx
k
@tBx
1 =
1
2
⇤2
1 (Bx
1 )2
+ (↵1 iz⇢1⇤1) Bx
1 +
1
2
z(i + z) t
@tBx
2 =
1
2
⇤2
2 (Bx
2 )2
+ (↵2 iz⇢2⇤2) Bx
2 +
1
2
z(i + z)
(A.83)
with null initial conditions at t = T. For generic t the Riccati equation for Bx
1 (and
thus Ax
1 ) does not have a closed-form solution, but can be easily integrate numerically,
whereas the others can be given explicitly:
Ax
1 (z; ⌧) =
↵1 1
⇤2
1

(c1 d1)⌧ 2 log
✓
1 g1e d1⌧
1 g1
◆
Bx
1 (z, ⌧) =
c1 d1
⇤2
1
1 e d1⌧
1 g1e d1⌧
(A.84)
151

where we have defined the auxiliary parameters:
ck = ↵k iz⇢k⇤k
dk =
q
c2
k + z(i + z)⇤2
k
gk =
ck dk
ck + dk
(A.85)
This examples suggest that ( t, t)-displaced affine models of the volatil-
ity factor process Xt are in general subjected to restrictions in their correlation
structure (such as those in equations A.59 and A.74) in order for the affinity to be
extended to the complete (Xt, St)>
process. Moreover, this last example shows
the problems arising from the presence of the displacement functions in two dif-
ferent perspectives. On one side the restriction on the correlation structure allows
the variance-covariance matrix of the complete process to be an affine function
of (Xt, log St)>
, as required by the affinity definition in Duffie et al. (2000). On
the other side, the ad-hoc correlation structure leads to a separable equity pricing
PDE (for the log-price CF fx), therefore easily numerically or even analytically
integrable, but it does not affect the separable VIX derivatives pricing PDE (for
the factor process CF f ), consistently with the fact that Xt is affine despite the
non-affinity of the complete process (Xt, St)>
.
152

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