Hierarchical Deterministic Quadrature Methods for Option Pricing under the Rough Bergomi Model

Hierarchical Deterministic Quadrature Methods
for Option Pricing under the Rough Bergomi Model
Chiheb Ben Hammouda
Christian Bayer Raúl Tempone
SIAM Conference on Financial Mathematics and Engineering
(FM21)
June 1-4, 2021

1 Option Pricing under the Rough Bergomi Model: Motivation &
Challenges
2 Our Approach Based on Hierarchical Deterministic Quadrature
Methods (Bayer, Ben Hammouda, and Tempone 2020a)
3 Numerical Experiments and Results
4 Conclusions and Future Work
0

Rough Volatility 1
1
Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum. “Volatility is rough”.
In: Quantitative Finance 18.6 (2018), pp. 933–949 1

The Rough Bergomi Model
(Bayer, Friz, and Gatheral 2016)
This model, under a pricing measure, is given by
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
dSt =
√
vtStdZt,
vt = ξ0(t)exp(η̃
WH
t − 1
2
η2
t2H
),
Zt ∶= ρW1
t + ρ̄W⊥
t ≡ ρW1
+
√
1 − ρ2W⊥
,
(1)
(W1
,W⊥
): two independent standard Brownian motions
̃
WH
is Riemann-Liouville process, defined by
̃
WH
t = ∫
t
0
KH
(t − s)dW1
s , t ≥ 0,
KH
(t − s) =
√
2H(t − s)H−1/2
, ∀ 0 ≤ s ≤ t. (2)
H ∈ (0,1/2] controls the roughness of paths, ρ ∈ [−1,1] and η > 0.
t ↦ ξ0(t): forward variance curve at time 0.
B Model allows for accurate fit to market prices for different
ranges of maturities, with few parameters. 2

Model Challenges
Numerically:
▸ The model is non-Markovian and non-affine ⇒ Standard numerical
methods (PDEs, Transform methods) seem inapplicable.
▸ Prevalent pricing method for vanilla options is Monte Carlo (MC)
/ still computationally expensive.
▸ Discretization methods with poor strong convergence error (rate of
order H) (Neuenkirch and Shalaiko 2016) ⇒ Variance reduction
methods, such as multilevel Monte Carlo, are inefficient.
▸ Improvements to the available numerical methods, based on deep
learning.
Theoretically:
▸ Hard to analyse the available simulation methods.2
Due to
▸ Non Markovianity ⇒ infinite dimensional state.
▸ Singularity in the Kernel, KH
(.), in (2).
2
(Bayer, Hall, and Tempone 2020) provides a preliminary weak error analysis but
not for the standard schemes (Exact and Hybrid schemes). 3

Option Pricing Challenges
European option price: V (S,0) = e−rT
EQ[g(ST )]
The integration problem is challenging
Issue 1: Time-discretization of the rough Bergomi process (large N
(number of time steps)) ⇒ S takes values in a high-dimensional
space ⇒ / Curse of dimensionality3
when using standard
deterministic quadrature methods.
Issue 2: The payoff function g is typically not smooth ⇒ low
regularity⇒ / Deterministic quadrature methods suffer from a
slow convergence.
3
Curse of dimensionality: An exponential growth of the work (number of
function evaluations) in terms of the dimension of the integration problem.

Challenges
4

Methodology 4
We design efficient fast hierarchical pricing methods, for options whose
underlyings follow the rough Bergomi model, based on
1 Analytic smoothing to uncover the available regularity.
2 Approximating the resulting integral of the smoothed payoff using
deterministic quadrature methods
▸ Adaptive sparse grids quadrature (ASGQ).
▸ Quasi Monte Carlo (QMC).
3 Combining our methods with hierarchical representations
▸ Brownian bridges as a Wiener path generation method ⇒ ↘ the
effective dimension of the problem.
▸ Richardson Extrapolation (Condition: stable weak error
expansion in ∆t to extrapolate) ⇒ Faster convergence of the weak
error ⇒ ↘ number of time steps (smaller input dimension).
4
Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone. “Hierarchical
adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough
Bergomi model”. In: Quantitative Finance 20.9 (2020), pp. 1457–1473
5

Conditional Expectation for Analytic Smoothing
CrB (T,K) = E [(ST − K)
+
]
= E[E[(ST − K)+
∣ σ(W1
(t),t ≤ T)]]
= E [CBS (S0 = exp(ρ∫
T
0
√
vtdW1
t −
1
2
ρ2
∫
T
0
vtdt),k = K, σ2
= (1 − ρ2
)∫
T
0
vtdt)]
≈ ∫
R2N
CBS (GrB(w(1)
,w(2)
))ρN (w(1)
)ρN (w(2)
)dw(1)
dw(2)
∶= CN
rB. (3)
Idea: (3) obtained by using the orthogonal decomposition of St into
S1
t = E{ρ∫
t
0
√
vsdW1
s }, S2
t = E{
√
1 − ρ2
∫
t
0
√
vsdW⊥
s },
and then apply conditional log-normality.
Notation:
CBS(S0,k,σ2
): the Black-Scholes call price, for initial spot price S0, strike price k, and
volatility σ2
.
GrB maps 2N independent standard Gaussian random inputs to the parameters fed to
Black-Scholes formula. GrB depends on the simulation scheme.
ρN : the multivariate Gaussian density, N: number of time steps.
For a continuous (semi)martingale Z, E(Z)t = exp(Zt − Z0 − 1
2
[Z,Z]0,t), where [Z,Z]0,t is
the quadratic variation of Z.
6

Simulation of the Rough Bergomi Dynamics
Goal: Simulate jointly (W1
t , ̃
WH
t ∶ 0 ≤ t ≤ T), resulting in W1
t1
,...,WtN
and ̃
WH
t1
,..., ̃
WH
tN
along a given grid t1 < ⋅⋅⋅ < tN
1 Covariance based approach (Bayer, Friz, and Gatheral 2016)
▸ 2N-dimensional Gaussian random as input vector.
▸ Based on Cholesky decomposition of the covariance matrix of the
(2N)-dimensional Gaussian random vector
W1
t1
,...,W1
tN
, ̃
WH
t1
,..., ̃
WtN
.
▸ Exact method but slow
▸ Work: O (N2
).
2 The hybrid scheme (Bennedsen, Lunde, and Pakkanen 2017)
▸ 2N-dimensional Gaussian random as input vector.
▸ Based on Euler discretization
▸ Approximates the kernel function in (2) by a power function near
zero and by a step function elsewhere.
▸ Accurate scheme that is much faster than the Covariance based
approach.
▸ Work: O (N) up to logarithmic factors.
7

On the Choice of the Simulation Scheme
Figure 2.1: The convergence of the relative weak error EB, using MC with
6 × 106
samples, for example parameters:
H = 0.07, K = 1,S0 = 1, T = 1, ρ = −0.9, η = 1.9, ξ0 = 0.0552. The upper and
lower bounds are 95% confidence intervals.
10−2 10−1
Δt
10−2
10−1
∣
E
[
g
∣
X
Δ
t
)
−
g
∣
X
)]
∣
weak_error
Lb
Ub
rate=Δ1.02
rate=Δ1.00
(a) With the hybrid scheme
10−2 10−1
Δt
10−3
10−2
10−1
∣
E
[
g
∣
X
Δ
t
)
−
g
∣
X
)]
∣
weak_error
Lb
Ub
rate=Δ0.76
rate=Δ1.00
(b) With the exact scheme
B The weak error behavior in Figure 2.1 was observed consistently
for different sets of parameters.
8

Sparse Grids Quadrature (I)
Aim: Approximate E[F], with F ∶ Rd
→ R.
Notation:
▸ a multi-index β ∈ Nd
+.
▸ Fβ ∶= Qm(β)
[F] a quadrature operator based on a Cartesian quadrature grid
(Qm(β)
∶= ⊗d
j=1 Qm(βj )
); m(βj) quadrature points along dimension yj; m(.) ∶ N+ → N+
an increasing function).
Issue: Approximating E[F] with Fβ is not an appropriate option due to the
well-known curse of dimensionality.
Alternative Idea: A quadrature estimate of E[F] is MI`
[F]
E[F] = ∑
β∈Nd
+
∆[Fβ] ≈ ∑
β∈I`
∆[Fβ] ∶= MI`
[F],
where I` ⊂ Nd
+ is a properly chosen index set.
▸ The mixed (first-order tensor) difference operators
∆[Fβ] =
d
⊗
i=1
∆iFβ ∶= ∑
α∈{0,1}d
(−1)∑
d
i=1 αi
Fβ−α. (4)
▸ The first-order difference operators
∆iFβ = {
Fβ − Fβ−ei , if βi > 1
Fβ if βi = 1
, (5)
where ei denotes the ith d-dimensional unit vector
9

Sparse Grids Quadrature (II)
E[F] ≈ MI`
[F] = ∑
β∈I`
∆[Fβ],
Product approach: I` = {∣∣ β ∣∣∞≤ `; β ∈ Nd
+} ⇒ EQ(M) = O (M−r/d
)
(for functions with bounded total derivatives up to order r).
Regular sparse grids:
I` = {∣∣ β ∣∣1≤ ` + d − 1; β ∈ Nd
+} ⇒ EQ(M) = O (M−s
(log M)(d−1)(s+1)
)
(for functions with bounded mixed derivatives up to order s).
Adaptive sparse grids quadrature (ASGQ): I` = IASGQ
(defined in next slides)
⇒ EQ(M) = O (M−sw
) (for functions with bounded weighted mixed derivatives
up to order sw).
B Notation: M: number of quadrature points; EQ: quadrature error.
Figure 2.2: Left: product
grids ∆β1 ⊗∆β2 for
1 ≤ β1,β2 ≤ 3.
Right: the corresponding
sparse grids construction.
10

ASGQ in Practice
E[F] ≈ MIASGQ [F] = ∑
β∈IASGQ
∆[Fβ],
The construction of IASGQ
is done by profit thresholding
IASGQ
(T) = {β ∈ Nd
+ ∶ Pβ ≥ T}.
Profit of a hierarchical surplus Pβ =
∣∆Eβ∣
∆Wβ
.
Error contribution: ∆Eβ = ∣MI∪{β} − MI∣.
Work contribution: ∆Wβ = Work[MI∪{β}] − Work[MI].
Figure 2.3: A posteriori,
adaptive construction as in
(Beck et al. 2012; Haji-Ali
et al. 2016): Given an index
set Ik, compute the profits
of the neighbor indices and
select the most profitable
one
11

ASGQ in Practice
β∈IASGQ
∆[Fβ],
IASGQ
(T) = {β ∈ Nd
+ ∶ Pβ ≥ T}.
∣∆Eβ∣
∆Wβ
.
one
11

Randomized QMC
A (rank-1) lattice rule (Sloan 1985; Nuyens 2014) with n points
Qn(F) ∶=
1
n
n−1
∑
k=0
F (
kz mod n
n
),
where z = (z1,...,zd) ∈ Nd
(the generated vector).
A randomly shifted lattice rule (for practical error estimates)
Qn,q(F) =
1
q
q−1
∑
i=0
Q(i)
n (F) =
1
q
q−1
∑
i=0
(
1
n
n−1
∑
k=0
F (
kz + ∆(i)
mod n
n
)), (6)
where {∆(i)
}q
i=1: independent random shifts, and MQMC
= q × n.
12

Wiener Path Generation Methods
{ti}N
i=0: Grid of time steps, {Bti }N
i=0: Brownian motion increments
Random Walk
▸ Proceeds incrementally, given Bti ,
Bti+1 = Bti +
√
∆tZi, Zi ∼ N(0,1).
▸ All components of Z = (Z1,...,ZN ) have the same scale of importance (in
terms of variance): isotropic.
Hierarchical Brownian Bridge
▸ Given a past value Bti and a future value Btk
, the value Btj (with
ti < tj < tk) can be generated according to (ρ = j−i
k−i
)
Btj = (1 − ρ)Bti + ρBtk
+ Zj
√
ρ(1 − ρ)(k − i)∆t, Zj ∼ N(0,1). (7)
▸ The most important values (capture a large part of the total variance) are
the first components of Z = (Z1,...,ZN ).
▸ ↘ the effective dimension (# important dimensions) by ↗ anisotropy
between different directions ⇒ Faster convergence of deterministic
quadrature methods. 13

Richardson Extrapolation (Talay and Tubaro 1990)
Conjecture
Let us denote by (Xt)0≤t≤T a stochastic process following the rBergomi dynamics,
and by (X̂∆t
ti
)0≤ti≤T its approximation using the hybrid scheme with a time step ∆t.
Then, for sufficiently small ∆t, and a suitable smooth function f, we assume that
E[f(X̂∆t
T )] = E[f(XT )] + c∆t + O (∆t2
). (8)
⇒ 2E[f( ̂
X
∆t/2
T )] − E[f( ̂
X∆t
T )] = E[f(XT )] + O (∆t2
).
General Formulation
{∆tJ = ∆t02−J
}J≥0: grid sizes, KR: level of Richardson extrapolation, I(J,KR):
approximation of E[f(XT )] by terms up to level KR
I(J,KR) =
2KR
I(J,KR − 1) − I(J − 1,KR − 1)
2KR − 1
, J = 1,2,...,KR = 1,2,... (9)
Advantage
Applying level KR of Richardson extrapolation dramatically reduces the bias ⇒↘
the number of time steps N needed to achieve a certain error tolerance ⇒↘ the
total dimension of the integration problem.
14

Challenges
14

Numerical Experiments
Table 1: Reference solution of call option price under the rough Bergomi model, for
different parameter constellations. The numbers between parentheses correspond to
the statistical errors estimates.
Parameters Reference solution
H = 0.07,K = 1,S0 = 1,T = 1,ρ = −0.9,η = 1.9,ξ0 = 0.0552 0.0791
(5.6e−05)
H = 0.02,K = 1,S0 = 1,T = 1,ρ = −0.7,η = 0.4,ξ0 = 0.1 0.1246
(9.0e−05)
H = 0.02,K = 0.8,S0 = 1,T = 1,ρ = −0.7,η = 0.4,ξ0 = 0.1 0.2412
(5.4e−05)
H = 0.02,K = 1.2,S0 = 1,T = 1,ρ = −0.7,η = 0.4,ξ0 = 0.1 0.0570
(8.0e−05)
Set 1 is the closest to the empirical findings (Gatheral, Jaisson, and
Rosenbaum 2018), suggesting that H ≈ 0.1. The choice ν = 1.9 and
ρ = −0.9 is justified by (Bayer, Friz, and Gatheral 2016).
For the remaining three sets, we test the potential of our method for a
very rough case, where variance reduction methods are inefficient.
15

Error Comparison
Etot: the total error of approximating the expectation in (3).
When using ASGQ estimator, QN
Etot ≤ ∣CrB − CN
rB∣ + ∣CN
rB − QN ∣ ≤ EB(N) + EQ(TOLASGQ,N),
where EQ is the quadrature error, EB is the bias, TOLASGQ is a user
selected tolerance for ASGQ method.
When using randomized QMC or MC estimator, Q
MC (QMC)
N
Etot ≤ ∣CrB − CN
rB∣ + ∣CN
rB − Q
MC (QMC)
N ∣ ≤ EB(N) + ES(M,N),
where ES is the statistical error, M is the number of samples used
for MC or randomized QMC method.
MQMC
and MMC
, are chosen so that ES,QMC(MQMC
) and
ES,MC(MMC
) satisfy
ES,QMC(MQMC
) = ES,MC(MMC
) = EB(N) =
Etot
2
.
16

Relative Errors and Computational Gains
Table 2: Computational gains achieved by ASGQ and QMC over the MC
method to meet a certain error tolerance. The ratios (ASGQ/MC) and
(QMC/MC) are referred to CPU time ratios in %, and are computed for the
best configuration with Richardson extrapolation for each method.
Parameters Relative CPU time ratio CPU time ratio
Set error (ASGQ/MC) in % (QMC/MC) in %
Set 1 1% 7% 10%
Set 2 0.2% 5% 1%
Set 3 0.4% 4% 5%
Set 4 2% 20% 10%
17

Computational Work of the MC Method
with Different Configurations
Figure 3.1: Computational work of the MC method with the different
configurations in terms of Richardson extrapolation’s level. Case of parameter
set 1 in Table 1.
10−2 10−1 100
Relative Error
10−3
10−1
101
103
105
CPU
time
MC
slope= -3.33
MC+Rich(level 1)
slope= -2.51
MC+Rich(level 2)

Computational Work of the QMC Method
Figure 3.2: Computational work of the QMC method with the different
set 1 in Table 1.
10−2 10−1 100
Relative Error
10−2
10−1
100
101
102
103
CPU
time
QMC
lope= -1.98
QMC+Rich(level 1)
slope= -1.57
QMC+Rich(level 2)

Computational Work of the ASGQ Method
Figure 3.3: Computational work of the ASGQ method with the different
set 1 in Table 1.
10−2 10−1 100
Relative Error
10−2
10−1
100
101
102
103
CPU
time
ASGQ
slo e= -5.18
ASGQ+Rich(level 1)
slope= -1.79
ASGQ+Rich(level 2)
slope= -1.27

Computational Work of the Different Methods
with their Best Configurations
Figure 3.4: Computational work comparison of the different methods with the
best configurations, for the case of parameter set 1 in Table 1.
10−2 10−1 100
Relative Error
10−3
10−2
10−1
100
101
102
103
104
CPU
time
MC+Rich(level 1)
slo e= -2.51
QMC+Rich(level 1)
slope= -1.57
ASGQ+Rich(level 2)
slope= -1.27

Challenges
21

Conclusions and Contributions
1 Proposed novel fast option pricers, for options whose underlyings
follow the rough Bergomi model, based on
▸ Conditional expectations for analytic smoothing.
▸ Hierarchical deterministic quadrature methods.
2 For different parameter constellations, our proposed methods
significantly outperform the MC approach, which is the prevalent
method in this context.
3 We propose an original way to overcome the high dimensionality
of the integration domain, by
▸ Reducing the total input dimension using Richardson extrapolation.
▸ Combining the Brownian bridge construction with ASGQ or QMC.
22

Conclusions and Contributions
4 Our proposed methodology shows a robust performance with
respect to the values of the Hurst parameter H.
5 Our approach is applicable to a wide class of stochastic volatility
models, in particular rough volatility models.
6 More details can be found in
Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Hierarchical adaptive sparse grids and quasi-Monte Carlo for
option pricing under the rough Bergomi model”. In: Quantitative
Finance 20.9 (2020), pp. 1457–1473.
23

Future Work and Remarks
1 For cases where analytic smoothing is not possible, one can use
similar numerical smoothing techniques as proposed in (Bayer,
Ben Hammouda, and Tempone 2020b).
2 Accelerating our novel approach can be achieved by using
▸ Better versions of the ASGQ and QMC methods.
▸ Alternative simulation schemes for rough volatility models.
B Using the fast scheme based on Donsker type approximation
(Horvath, Jacquier, and Muguruza 2017) is not helpful since the
observed weak error rate is of order H for small values of H.
24

References I
“Hierarchical adaptive sparse grids and quasi-Monte Carlo for
option pricing under the rough Bergomi model”. In: Quantitative
Finance 20.9 (2020), pp. 1457–1473.
“Numerical smoothing and hierarchical approximations for
efficient option pricing and density estimation”. In: arXiv
preprint arXiv:2003.05708 (2020).
Christian Bayer, Peter Friz, and Jim Gatheral. “Pricing under
rough volatility”. In: Quantitative Finance 16.6 (2016),
pp. 887–904.
Christian Bayer, Eric Joseph Hall, and Raúl Tempone. “Weak
error rates for option pricing under the rough Bergomi model”.
In: arXiv preprint arXiv:2009.01219 (2020).
25

References II
Joakim Beck et al. “On the optimal polynomial approximation of
stochastic PDEs by Galerkin and collocation methods”. In:
Mathematical Models and Methods in Applied Sciences 22.09
(2012), p. 1250023.
Mikkel Bennedsen, Asger Lunde, and Mikko S Pakkanen.
“Hybrid scheme for Brownian semistationary processes”. In:
Finance and Stochastics 21.4 (2017), pp. 931–965.
Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum.
“Volatility is rough”. In: Quantitative Finance 18.6 (2018),
pp. 933–949.
Abdul-Lateef Haji-Ali et al. “Multi-index stochastic collocation
for random PDEs”. In: Computer Methods in Applied Mechanics
and Engineering (2016).
26

References III
Blanka Horvath, Antoine Jacquier, and Aitor Muguruza.
“Functional central limit theorems for rough volatility”. In:
Available at SSRN 3078743 (2017).
Andreas Neuenkirch and Taras Shalaiko. “The Order Barrier for
Strong Approximation of Rough Volatility Models”. In: arXiv
preprint arXiv:1606.03854 (2016).
Dirk Nuyens. The construction of good lattice rules and
polynomial lattice rules. 2014.
Ian H Sloan. “Lattice methods for multiple integration”. In:
Journal of Computational and Applied Mathematics 12 (1985),
pp. 131–143.
Denis Talay and Luciano Tubaro. “Expansion of the global error
for numerical schemes solving stochastic differential equations”.
In: Stochastic analysis and applications 8.4 (1990), pp. 483–509.
27

Thank you for your attention
27

Hierarchical Deterministic Quadrature Methods for Option Pricing under the Rough Bergomi Model

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