Smiling Twice: The Heston++ Model

Smiling twice: The Heston++ model
C. Pacati1 G. Pompa2 R. Renò3
1Dipartimento di Economia Politica e Statistica
Università di Siena, Italy
2,IMT School for Advanced Studies Lucca, Italy
3Dipartimento di Scienze Economiche
Università degli Studi di Verona, Italy
XVII Workshop on Quantitative Finance, Pisa 2016
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model

The problem
There is growing demand, and correspondingly a liquid market, for trading
volatility derivatives and managing volatility risk.
SPX and VIX derivatives both provide informations on the same volatility
process, a model which is able to price one market, but not the other, is
inherently misspecified.
There is need of a pricing framework for consistent pricing both equity
derivatives and volatility derivatives;
Affine models are unable to reproduce VIX Futures and Options features;
Non-affine models are often analitically intractable and computationally heavy.
We tackle the problem of jointly fit the IV surface of SPX index
options, together with the term structure of VIX futures and the
surface of VIX options, leveraging on an affinity-preserving
deterministic shift extension of the volatility process.

VIX: the Fear Index
Since 1993, VIX reﬂects the 30-day expected risk-neutral
S&P500 index volatility.
Leverage effect: inverse relationship SPX-VIX (2004-2016)
Jan04 Jan06 Jan08 Jan10 Jan12 Jan14 Jan16
0
250
500
750
1000
1250
1500
1750
2000
S&P500
0
10
20
30
40
50
60
70
80
90
VIXDailyClose

VIX: the Fear Index
Since 1993, VIX reﬂects the 30-day expected risk-neutral
S&P500 index volatility.
Positively skewed and leptokurtic (2004-2016)
0 10 20 30 40 50 60 70 80 90
0
0.02
0.04
0.06
0.08
0.1
0.12
VIX Daily Close
EmpiricalPDF

VIX Futures: the term structure of Fear
Traded since 2004, convey market visions on volatility of
S&P500 (2004-2014).
0
1
2
3
4
5
6
7
Nov−14
Feb−13
May−11
Jul−09
Oct−07
Jan−06
Mar−04
0
10
20
30
40
50
60
70
DateTenor (months)

VIX Futures: the term structure of Fear
Traded since 2004, convey market visions on volatility of
S&P500 (2004-2014).
Humped term structure (June 29, 2009)

VIX Options: the cross-section of Fear
Traded since 2006, provide insurance from equity market
downturns: S&P500 vanilla below (June 29, 2009)

VIX Options: the cross-section of Fear
Traded since 2006, provide insurance from equity market
downturns: VIX options below (June 29, 2009).

Modeling VIX and VIX derivatives: literature review
Standalone approach: volatility is directly modeled,
separated from the underlying stock price process. Whaley
1993 (GBM), Grünbichler and Longstaff 1996 (SQR),
Detemple and Osakwe 2000 (LOU), Mencia and Sentana
2013 (CTLOUSV), Goard and Mazur 2013 (3/2).

Modeling VIX and VIX derivatives: literature review
Standalone approach: volatility is directly modeled,
separated from the underlying stock price process. Whaley
1993 (GBM), Grünbichler and Longstaff 1996 (SQR),
Detemple and Osakwe 2000 (LOU), Mencia and Sentana
2013 (CTLOUSV), Goard and Mazur 2013 (3/2).
Consistent approach: VIX is derived from the speciﬁcation
of SPX dynamics. Sepp 2008 (SVVJ), Lian Zhu 2013
(SVCJ), Lo et al. 2013 (2-SVCJ), Bardgett et al. 2013
(2-SMRSVCJ), Branger et al. 2014 (2-SVSVJ), Baldeaux
and Badran 2014 (3/2J), Pacati et al. 2015 (2-SVCVJ++).

The 2-SVCVJ++ model: motivation
++ extension of an afﬁne processes:
Brigo and Mercurio (2001): CIR++ ﬁts observed term
structure of forward rates.
Pacati, Renò and Santilli (2014): Heston++ reproduces
ATM term structure of FX options.
Two sources of jumps:
CO- market downturns correlated with volatility spikes
(Todorov and Tauchen, 2011, Bandi and Renò, 2015).
Idiosyncratic- direct channel for right skewness of volatility.

The 2-SVCVJ++ model



dSt
St−
= (r − q − λ¯µ) dt + σ2
1,t + φt dWS
1,t + σ2,t dWS
2,t + (ecx
− 1)dNt
dσ2
1,t = α1(β1 − σ2
1,t )dt + Λ1σ1,t dWσ
1,t + cσdNt + cσdNt
dσ2
2,t = α2(β2 − σ2
2,t )dt + Λ2σ2,t dWσ
2,t
under Q, where φ0 = 0, φt ≥ 0, cx ∼ N µx + ρJcσ, δ2
x |cσ,
cσ ∼ E(µco,σ) and cσ ∼ E(µid,σ). The model is afﬁne provided
corr(dWS
1,t , dWσ
1,t ) = ρ1
σ2
1,t
σ2
1,t + φt
corr(dWS
2,t , dWσ
2,t ) = ρ2

The 2-SVCVJ++ model: nested models
Taxonomy of the H++ models used in the empirical analysis.
All models have two factors.
jumps in price volatility volatility displacement
(idiosyncratic) (co-jumps) φt
2-SVJ
2-SVJ++
2-SVCJ
2-SVCJ++
2-SVVJ
2-SVVJ++
2-SVCVJ
2-SVCVJ++

The 2-SVCVJ++ model: afﬁnity
Lemma (Conditional Characteristic Functions)
Under the H++ models, the conditional characteristic function of
returns fH++
x (z) = EQ
eizxT Ft and of the two stochastic volatility
factors fH++
σ (z1, z2) = EQ
eiz1σ2
1,T +iz2σ2
2,T Ft are given by:
fH++
x (z; xt , σ2
1,t , σ2
2,t , t, T, φ) = fH
x (z; xt , σ2
1,t , σ2
2,t , τ)e−1
2
z(i+z)Iφ(t,T)
fH++
σ (z1, z2; σ2
1,t , σ2
2,t , τ) = fH
σ (z1, z2; σ2
1,t , σ2
2,t , τ)
where τ = T − t, z, z1, z2 ∈ C and Iφ(t, T) =
T
t φsds.

The 2-SVCVJ++ model: S&P500 Options
Proposition (Price of SPX Options)
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 τ = T − t, is given by (Lewis 2000, 2001)
CH++
SPX
(K, t, T)
= St e−qτ
−
1
π
St Ke− 1
2
(r+q)τ
∞
0
Re eiuk
fH
x u −
i
2
e
− u2
+ 1
4
Iφ(t,T)
u2 + 1
4
du
where k = log St
K + (r − q)τ.

The 2-SVCVJ++ model: VIX Index
Proposition (VIX Index)
Under the H++ models,
VIXH++
t,¯τ
100
2
=
VIXH
t,¯τ
100
2
+
1
¯τ
Iφ(t, t + ¯τ)
where ¯τ = 30 days, VIXH
t,¯τ is the corresponding quotation under
H models, which is an afﬁne function of the volatility factors σ2
1,t
and σ2
2,t
VIXH
t,¯τ
100
2
=
1
¯τ


k=1,2
ak (¯τ)σ2
k,t + bk (¯τ)


where Iφ(t, t + ¯τ) =
t+¯τ
t φsds.

The 2-SVCVJ++ model: VIX Futures and Options
Proposition (Price of VIX derivatives)
Under H++ models, the time t value of a futures on VIXt,¯τ settled at time T and the
arbitrage-free price at time t of a call option on VIXt,¯τ , with strike price K and time to
maturity τ = T − t are given respectively by
FH++
VIX
(t, T)
100
=
1
2
√
π
∞
0
Re


f
H
σ −z
a1(¯τ)
¯τ
, −z
a2(¯τ)
¯τ
e
−iz k=1,2 bk ( ¯τ)+Iφ(T,T+ ¯τ) / ¯τ
(−iz)3/2


 d Re(z)
and
CH++
VIX
(K, t, T)
100
=
e−rτ
2
√
π
∞
0
Re f
H
σ −z
a1(¯τ)
¯τ
, −z
a2(¯τ)
¯τ
×
e
−iz k=1,2 bk ( ¯τ)+Iφ(T,T+ ¯τ) / ¯τ
1 − erf(K/100
√
−iz)
(−iz)3/2


 d Re(z)
where z = Re(z) + i Im(z) ∈ C, 0 < Im(z) < ζc(τ) and erf(z) = 2√
π
z
0 e−s2
ds.

SPX Vanilla (September 2, 2009)
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

VIX Futures (September 2, 2009)
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++

VIX Options (September 2, 2009)
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

Calibration Errors (in %)
2-SVJ 2-SVJ++ 2-SVCJ 2-SVCJ++ 2-SVVJ 2-SVVJ++ 2-SVCVJ 2-SVCVJ++
Panel A: RMSE
RMSESPX 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)
RMSEFut 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)
RMSEVIX 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
RMSRESPX 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)
RMSREFut 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)
RMSREVIX 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)

Agenda
Infer the true dynamics of S&P500: change of measure
between P and Q measure, non-standard Kalman filtering
of latent variables, pricing kernel and variance risk-premia
estimation.
Understanding the meaning of φt :
1 Is it an affine approximation of some non-affine (true)
model?
2 Is it (and to what extent) an additionally volatility state
vector?

Thanks for your attention!

References I
Chicago Board Options Exchange. The CBOE volatility index-VIX. White Paper
(2009).
Grünbichler, A., and Longstaff, F. A. Valuing futures and options on volatility.
Journal of Banking & Finance 20 (6), 985-1001.
Mencía, J. and Sentana, E. Valuation of VIX derivatives. Journal of Financial
Economics 108 (2), 367-391.
Bardgett, C., Gourier, E., and Leipold, M. Inferring volatility dynamics and risk
premia from the S&P 500 and VIX markets. Working paper.
Cont, R., and Kokholm, T. A consistent pricing model for index options and
volatility derivatives. Mathematical Finance 23.2 (2013): 248-274.
Sepp, A. Pricing options on realized variance in the Heston model with jumps in
returns and volatility. Journal of Computational Finance 11 (4), 33Ð70.
Sepp, A. VIX option pricing in a jump-diffusion model. Risk (April), 84-89.
Pacati, C., Renò, R. and Santilli, M. (2014). Heston Model: shifting on the
volatility surface. Risk (November), 54-59.
Brigo, D-, and Mercurio, F. A deterministic-shift extension of analytically-tractable
and time-homogeneous short-rate models. Finance & Stochastics 5, 369-388.

References II
Bakshi, G., and Madan, D. Spanning and derivative-security valuation. Journal of
Financial Economics 55.2 (2000): 205-238.
Schoutens, W. Levy processes in Finance: Pricing Financial Derivatives. Wiley,
2003.
Duffie, D., Pan, J., and Singleton, K. Transform analysis and asset pricing for
affine jump-diffusions. Econometrica 68.6 (2000): 1343-1376.
Zhu, S.-P., and Lian, G.-H. An analytical formula for VIX futures and its
applications. Journal of Futures Markets 32.2 (2012): 166-190.
Lian, G.-H., and Zhu S.-P. Pricing VIX options with stochastic volatility and
random jumps. Decisions in Economics and Finance 36.1 (2013): 71-88.
Heston, S. L. A closed-form solution for options with stochastic volatility with
applications to bond and currency options. Review of financial studies 6.2 (1993):
327-343.
Christoffersen, P., Heston, S., and Jacobs, K.. The shape and term structure of
the index option smirk: Why multifactor stochastic volatility models work so well.
Bates, D. S. Jumps and stochastic volatility: Exchange rate processes implicit in
deutsche mark options. Review of financial studies 9.1 (1996): 69-107.

References III
Branger, N., and Völkert, C. The ﬁne structure of variance: Consistent pricing of
VIX derivatives. Working paper.

Smiling Twice: The Heston++ Model

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