Efficient Fourier Pricing of Multi-Asset Options: Quasi-Monte Carlo & Domain Transformation Approach

Efficient Fourier Pricing of Multi-Asset Options:
Quasi-Monte Carlo & Domain Transformation Approach
Chiheb Ben Hammouda
Christian
Bayer
Michael
Samet
Antonis
Papapantoleon
Raúl
Tempone
The International Conference on Computational Finance, CWI
Amsterdam, April 3, 2024

1 Motivation, Challenges and Framework
2 Quasi-Monte Carlo with Effective Domain transformation for Fast
Fourier Pricing
3 Numerical Experiments and Results
4 Conclusion
0

Setting
Pricing multi-asset options: compute E[P(XT )]
P(⋅): payoff function (typically non-smooth), e.g., (K: the strike price)
▸ Basket put P(x) = max(∑
d
i=1 ciexi
− K,0), s.t. ci > 0,∑
d
i=1 ci = 1;
▸ Rainbow (E.g., Call on min):
P(x) = max(min(ex1
,...,exd
) − K,0)
▸ Cash-or-nothing put: P(x) = ∏
d
i=1 1[0,Ki](exi
).
XT is a d-dimensional (d ≥ 1) vector of log-asset prices at time T,
following a certain multivariate stochastic model with an affine
structure (e.g., Lévy models).
x1
4
2
0
2
4
x
2
4
2
0
2
4
P(x
1
,
x
2
)
0.0
0.2
0.4
0.6
0.8
(a) Basket put
x1
0.0
0.2
0.4
0.6
0.8
1.0
x
2
0.0
0.2
0.4
0.6
0.8
1.0
P
(
x
1
,
x
2
)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
(b) Call on min
x1
0
1
2
3
4
5
6
7
x
2
0
1
2
3
4
5
6
7
P(x
1
,
x
2
)
0.0
0.2
0.4
0.6
0.8
1.0
(c) Cash-or-nothing
put
Figure 1.1: Payoff functions illustration

Setting
Pricing multi-asset options: compute E[P(XT )]
P(⋅) is a payoff function (typically non-smooth), e.g., (K: the strike price)
▸ Basket put P(x) = max(∑
d
i=1 ciexi
− K,0), s.t. ci > 0,∑
d
i=1 ci = 1;
▸ Rainbow (E.g., Call on min): P(x) = max(min(ex1
,...,exd
) − K,0)
▸ Cash-or-nothing put: P(x) = ∏
d
i=1 1[0,Ki](exi
).
XT is a d-dimensional vector of log-asset prices at time T, following a certain
multivariate stochastic model with an affine structure (e.g., Lévy models).
Challenges
1 Monte Carlo (MC) method (prevalent choice) has a rate of convergence independent of
the problem’s dimension and regularity of the payoff but can be very slow.
2 P(⋅) is non-smooth ⇒ deteriorates convergence of deterministic quadrature.
3 The curse of dimensionality and other issues ⇒ Most proposed Fourier pricing
approaches efficient for only 1D and 2D options (Carr et al. 1999; Lewis 2001; Fang
et al. 2009; Hurd et al. 2010; Ruijter et al. 2012),. . . .
Aim: Empower Fourier-based pricing methods of multi-asset options
1 C. Ben Hammouda et al. “Optimal Damping with Hierarchical Adaptive Quadrature for
Efficient Fourier Pricing of Multi-Asset Options in Lévy Models”. In: Journal of
Computational Finance 27.3 (2024), pp. 43–86. (Michael’s talk)
2 C. Ben Hammouda et al. “Quasi-Monte Carlo for Efficient Fourier Pricing of Multi-Asset
Options”. In: arXiv preprint arXiv:2403.02832 (2024). (Today’s talk)

Numerical Integration Methods: Sampling in [0,1]2
E[P(X(T))] = ∫Rd P(x)ρXT
(x)dx ≈ ∑
M
m=1 ωmP (xm).
Monte Carlo (MC)
0 0.2 0.4 0.6 0.8 1
u1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u2
Tensor Product Quadrature
0 0.2 0.4 0.6 0.8 1
u1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u2
Quasi-Monte Carlo (QMC)
0 0.2 0.4 0.6 0.8 1
u1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u2
Adaptive Sparse Grids Quadrature
0 0.2 0.4 0.6 0.8 1
u1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u2

Fast Convergence: When Regularity Meets
Structured Sampling
Monte Carlo (MC)
(-) Slow convergence:
O(M− 1
2 ).
(+) Rate independent of
dimension and regularity
of the integrand.
Tensor Product Quadrature
Convergence: O(M− r
d )
(Davis et al. 2007).
r > 0 being the order of
bounded total
derivatives of the
integrand.
Quasi-Monte Carlo (QMC)
Optimal Convergence: O(M−1
)
(Dick et al. 2013).
Requires the integrability of
first mixed partial derivatives
of the integrand.
Worst Case Convergence:
O(M−1/2
).
Adaptive Sparse Grids Quadrature
Convergence: O(M− p
2 ) (Chen
2018).
p > 1 is related to the order of
bounded weighted mixed
(partial) derivatives of the
integrand.

Challenge 1: Original problem is non smooth (low regularity)
x1
4
2
0
2
4
x
2
4
2
0
2
4
P(x
1
,
x
2
)
0.0
0.2
0.4
0.6
0.8
(a) Basket Put
x1
0.0
0.2
0.4
0.6
0.8
1.0
x
2
0.0
0.2
0.4
0.6
0.8
1.0
P
(
x
1
,
x
2
)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
(b) Call on min
x1
0
1
2
3
4
5
6
7
x
2
0
1
2
3
4
5
6
7
P(x
1
,
x
2
)
0.0
0.2
0.4
0.6
0.8
1.0
(c)
Cash-or-nothing
Solution: Uncover the available hidden regularity in the
problem
1 Analytic smoothing (Bayer et al. 2018; Ben Hammouda et al.
2020): taking conditional expectations over subset of integration
variables. / Good choice not always trivial.
2 Numerical smoothing (Ben Hammouda et al. 2022):
/ Additional computational work! Attractive when explicit smoothing or
Fourier mapping not possible.
3 Mapping the problem to the Fourier space (Today’s talk)
(Ben Hammouda et al. 2024b; Ben Hammouda et al. 2024c).
" Fourier transform of the density function (characteristic function)
available/cheap to compute.

Better Regularity in the Fourier Space
x1
4
2
0
2
4
x
2
4
2
0
2
4
P(x
1
,
x
2
)
0.0
0.2
0.4
0.6
0.8
(a) Payoff: Basket put
u1
20
10
0
10
20
u
2
20
10
0
10
20
|P(u
1
,
u
2
)|
1e
9
0.0
0.5
1.0
1.5
2.0
2.5
(b) Fourier Transform
x1
0
1
2
3
4
5
6
7
x
2
0
1
2
3
4
5
6
7
P(x
1
,
x
2
)
0.0
0.2
0.4
0.6
0.8
1.0
(a) Payoff:
Cash-or-nothing
u1
15
10
5
0
5
10
15
u
2
15
10
5
0
5
10
15
|P(u
1
,
u
2
)|
0.002
0.000
0.002
0.004
0.006
x1
2 1 0 1 2 3 4 5
x
2
2
1
0
1
2
3
4
5
P(x
1
,
x
2
)
0
20
40
60
80
100
120
140
(a) Payoff: Call on min
u1
30
20
10
0
10
20
30
u
2
30
20
10
0
10
20
30
|P(u
1
,
u
2
)|
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
5

Fourier Pricing Formula in d Dimensions
Notation
Θm,Θp: the model and payoff parameters, respectively;
̂
P(⋅): the Fourier transform of the payoff P(⋅);
XT : vector of log-asset prices at time T, with extended characteristic function ΦXT
(⋅);
R: vector of damping parameters ensuring integrability;
δP : strip of regularity of ̂
P(⋅); δX: strip of regularity of ΦXT
(⋅),
Assumption 1.1
1 x ↦ P(x) is continuous on Rd
(Can be replaced by more regularity assumptions on
the model).
2 δP ∶= {R ∈ Rd
∶ x ↦ e−⟨R,x⟩
P(x) ∈ L1
bc(Rd
) and y ↦ ̂
P(y + iR) ∈ L1
(Rd
)} ≠ ∅.
3 δX ∶= {R ∈ Rd
∶ y ↦∣ ΦXT
(y + iR) ∣< ∞,∀ y ∈ Rd
} ≠ ∅.
Proposition (Ben Hammouda et al. 2024b)
Under Assumptions 1, 2 and 3, and for R ∈ δV ∶= δP ∩ δX, the value of the option price on
d stocks is
V (Θm,Θp) = e−rT
E[P(XT)] (1)
= ∫
Rd
(2π)−d
e−rT
R(ΦXT
(y + iR) ̂
P(y + iR))
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∶=g(y;R,Θm,Θp)
dy.
6

Challenge 2: The choice of the damping parameters
Damping parameters, R, ensure integrability and control the
regularity of the integrand.
Figure 1.6: Example of a strip of analyticity of the integrand of a 2D call
on min option under VG model. Parameters:
θ = (−0.3,−0.3),ν = 0.5,Σ = I2.
-30.0 -25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0
R1
30
25
20
15
10
5
0
5
10
R
2
V
X
P
(a) σ = (0.2, 0.2)
-30.0 -25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0
R1
30
25
20
15
10
5
0
5
10
R
2
V
X
P
(b) σ = (0.2, 0.5)
7

Challenge 2: The choice of the damping parameters
Damping parameters, R, ensure integrability and control the
regularity of the integrand.
Solution: (Ben Hammouda et al. 2024b) and Michael’s talk
Based on contour integration error estimates:
Parametric smoothing of the Fourier integrand via an (generic)
optimization rule for the choice of damping parameters.
Near-Optimal Damping Rule (Ben Hammouda et al. 2024b)
We propose an optimization rule for choosing the damping parameters
R∗
∶= R∗
(Θm,Θp) = arg min
R∈δV
∥g(u;R,Θm,Θp)∥∞
= arg min
R∈δV
g(0Rd ;R,Θm,Θp). (2)
where R∗
∶= (R∗
1,...,R∗
d) denotes the optimal damping parameters.

Challenge 3: Curse of dimensionality
1 Most of the existing Fourier approaches face hurdles in
high-dimensional settings due to the tensor product (TP) structure of
the commonly employed numerical quadrature techniques.
2 Complexity of (standard) TP quadrature to solve (1) ↗ exponentially
with the number of underlying assets (Recall Convergence: O(M− r
d )).
2 3 4 5 6 7 8
dimension
10−1
100
101
102
103
104
Runtime
TP
TP
Figure 1.7: Call on min option under Normal Inverse Gaussian model: Runtime (in sec)
versus dimension for TP for a relative error TOL = 10−2
.
Solution: Effective treatment of the high dimensionality
1 (Ben Hammouda et al. 2024b): Sparsification and dimension-adaptivity
techniques to accelerate convergence (Michael’s talk).
2 (Ben Hammouda et al. 2024c): Quasi-Monte Carlo (QMC) with efficient
domain transformation (Today’s talk).

Fourier Pricing
4 Conclusion
9

Quasi-Monte Carlo (QMC):
Need for Domain Transformation
Recall: our Fourier integrand is:
g (y;R) = (2π)−d
e−rT
R(ΦXT
(y + iR) ̂
P(y + iR)), y ∈ Rd
, R ∈ δV ∶= δP ∩ δX
Our Fourier integrand is in Rd
BUT QMC constructions are restricted
to the generation of low-discrepancy point sets on [0,1]d
.
⇒ Need to transform the integration domain
Using an inverse cumulative distribution function, we obtain the value of
the option price on d stocks:
V (Θm,Θp) = ∫
Rd
g(y)dy = ∫
[0,1]d
g ○ Ψ−1
(u;Λ)
ψ ○ Ψ−1(u;Λ)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
=∶g̃(u;Λ)
du.
▸ ψ(⋅;Λ): a probability density function (PDF) with parameters Λ.
▸ Ψ(⋅;Λ): the corresponding cumulative distribution function (CDF).
10

Randomized Quasi-Monte Carlo (RQMC)
The transformed integration problem reads now:
V (Θm,Θp) = ∫
[0,1]d
g ○ Ψ−1
(u;Λ)
ψ ○ Ψ−1(u;Λ)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
=∶g̃(u;Λ)
du. (3)
Once the choice of ψ(⋅;Λ) (respectively Ψ−1
(⋅;Λ)) is determined,
the RQMC estimator of (3) can be expressed as follows:
QRQMC
N,s [g̃] ∶=
1
S
S
∑
i=1
1
N
N
∑
n=1
g̃ (u(s)
n ;Λ), (4)
▸ {un}N
n=1 is the sequence of deterministic QMC points
▸ For n = 1,...,N, {u
(s)
n }S
s=1: obtained by an appropriate
randomization of {un}N
n=1, such that {u
(s)
n }S
s=1
i.i.d
∼ U([0,1]d
).
Why Randomization?
▸ Practical error estimates based on the central limit theorem.
10

Challenge 4: Deterioration of QMC convergence if ψ
or/and Λ are badly chosen
Observe: The denominator of g̃(u) =
g○Ψ−1(u;Λ)
ψ○Ψ−1(u;Λ)
decays to 0 as
uj → 0,1 for j = 1,...,d.
The transformed integrand may have singularities near the
boundary of [0,1]d
⇒ Deterioration of QMC convergence.
−20 −15 −10 −5 0 5 10 15 20
u
0.0
0.2
0.4
0.6
0.8
g
(
u
)
(a) Original Fourier integrand (1)
for call option under GBM
0.0 0.2 0.4 0.6 0.8 1.0
u
0
10
20
30
40
50
60
70
̃
g
(
u
)
̃ ̃
σ = 1.0
̃ ̃
σ = 5.0
̃ ̃
σ = 9.0
(b) Domain transformation for
the integrand (1)
Questions
Q1: Which density to choose? Q2: How to choose its parameters?
11

How to choose ψ(⋅;Λ) (respectively Ψ−1(⋅;Λ) ) and
and its parameters, Λ?
For u ∈ [0,1]d
,R ∈ δV , the transformed Fourier integrand reads:
g̃(u) =
g ○ Ψ−1
(u;Λ)
ψ ○ Ψ−1(u;Λ)
=
e−rT
(2π)d
R
⎡
⎢
⎢
⎢
⎣
̂
P(Ψ−1
(u) + iR)
ΦXT
(Ψ−1
(u) + iR)
ψ (Ψ−1(u))
⎤
⎥
⎥
⎥
⎦
.
⇒ Sufficient to design the domain transformation to control the growth
at the boundaries of the term
ΦXT
(Ψ−1(u)+iR)
ψ(Ψ−1(u))
(Conservative choice).
The payoff Fourier transforms ( ̂
P(⋅)) decay at a polynomial rate.
PDFs of the pricing models (light and semi-heavy tailed models), if
they exist, are much smoother than the payoff ⇒ the decay of their
Fourier transforms (charactersitic functions) is faster the one of the
payoff Fourier transform (Trefethen 1996; Cont et al. 2003).

Model-dependent Domain Transformation
Solution (Ben Hammouda et al. 2024c): Effective Domain Transformation
1 Choose the density ψ(⋅;Λ) to asymptotically follow the same functional form of the
characteristic function.
Table 1: Extended characteristic function: ΦXT
(z) = exp(iz′
X0)exp(iz′
µT)ϕXT
(z), and choice of ψ(⋅).
ϕXT
(z),z ∈ Cd
, I[z] ∈ δX ψ(y;Λ),y ∈ Rd
Gaussian (Λ = Σ̃):
GBM model: exp(−T
2
z′
Σz)
(2π)− d
2 (det(Σ̃))− 1
2 exp(−1
2
(y′
Σ̃
−1
y))
Generalized Student’s t (Λ = (ν̃,Σ̃)):
VG model: (1 − iνz′
θ + 1
2
νz′
Σz)
−T /ν Γ( ν̃+d
2
)(det(Σ̃))− 1
2
Γ( ν̃
2
)ν̃
d
2 π
d
2
(1 + 1
ν̃
(y′
Σ̃y))
− ν̃+d
2
NIG model: Laplace (Λ = Σ̃) and (v = 2−d
2
):
exp(δT (
√
α2 − β′
∆β −
√
α2 − (β + iz)′∆(β + iz)))
(2π)− d
2 (det(Σ̃))− 1
2 (y′
Σ̃
−1
y
2
)
v
2
Kv (
√
2y′Σ̃
−1
y)
Notation:
Σ: Covariance matrix for the Geometric Brownian Motion (GBM) model.
ν,θ,σ,Σ: Variance Gamma (VG) model parameters.
α,β,δ,∆: Normal Inverse Gaussian (NIG) model parameters.
µ is the martingale correction term.
Kv(⋅): the modified Bessel function of the second kind.
13

Model-dependent Domain Transformation:
Case of Independent Assets
Using independence: Observe
ϕXT
(Ψ−1
(u)+iR)
ψ(Ψ−1(u))
= ∏
d
j=1
ϕ
X
j
T
(Ψ−1
(uj )+iRj )
ψj (Ψ−1(uj ))
1 Choose the density ψ(⋅;Λ) in the change of variable to asymptotically follow the same
functional form of the extended characteristic function.
2 Select the parameters Λ to control the growth of the integrand near the boundary of
[0,1]d
i.e limuj →0,1 g̃(uj) < ∞,j = 1,...,d.
Table 2: Choice of ψ(u;Λ) ∶= ∏
d
j=1 ψj(uj;Λ) and conditions on Λ for GBM, (ii) VG and (iii) NIG.
See (Ben Hammouda et al. 2024c) for the derivation.
Model ψj(yj;Λ) Growth condition on Λ
GBM
1
√
2σ̃j
2
exp(−
y2
j
2σ̃j
2 ) (Gaussian) σ̃j ≥ 1
√
T σj
VG
Γ( ν̃+1
2
)
√
ν̃πσ̃j Γ( ν̃
2
)
(1 +
y2
j
ν̃σ̃j
2 )
−(ν̃+1)/2
(t-Student) ν̃ ≤ 2T
ν
− 1,
σ̃j = (
νσ2
j ν̃
2
)
T
ν−2T
(ν̃)
ν
4T −2ν
NIG
exp(−
∣yj ∣
σ̃j
)
2σ̃j
(Laplace) σ̃j ≥ 1
δT
" In case of equality conditions, the integrand still decays at the speed of the payoff transform.

Should Correlation Be Considered
in the Domain Transformation?
104
NxS
10−4
10−3
10−2
10−1
Relative
Statistical
Error
ρ= −0.7
N−0.99
ρ=0
N−1.48
ρ=0.7
N−0.69
Figure 2.2: Two-dimensional call on the minimum option under the GBM model:
Effect of the correlation parameter, ρ, on the convergence of RQMC.
For the domain transformation, we set σ̃j = 1
√
T σj
= 5, j = 1, 2. N: number of QMC
points; S = 32: number of digital shifts.
15

Case of Correlated Assets
Challenge 5: Numerical Evaluation of the inverse CDF Ψ−1
(⋅)
1 We can not evaluate the inverse CDF componentwise using the univariate inverse
CDF as in the independent case (Ψ−1
d (u) ≠ (Ψ−1
1 (u1),...,Ψ−1
1 (ud))).
2 The inverse CDF is not given in closed-form for most multivariate distributions,
and its numerical approximation is generally computationally expensive.
Observe: For GBM model: If Z ∼ N(0,Id) ⇒ X = L̃Z ∼ N(0,Σ̃) (L̃: Cholesky factor
of Σ̃) ⇒ we have Ψ−1
nor,d(u;Σ̃) = L̃Ψ−1
nor,d(u;Id) = L̃(Ψ−1
nor,1(u1),...,Ψ−1
nor,1(ud))
Solution: Avoid the expensive computation of the inverse CDF
1 We consider multivariate transformation densities, ψ(⋅,Λ), which belong to the
class of normal mean-variance mixture distributions; i.e., for X ∼ ψ(⋅,Λ), we can
write X = µ + WZ, with Z ∼ Nd(0,Σ), and W ≥ 0, independent of Z.
2 We use the eigenvalue or Cholesky decomposition to eliminate the dependence
structure.

Illustration
GBM model : Using L̃Ψ−1
nor,d(u;Id) = L̃(Ψ−1
nor,1(u1),...,Ψ−1
nor,1(ud))
(L̃: Cholesky factor of Σ̃), we obtain
∫
Rd
g(y)dy = ∫
[0,1]d
g (L̃Ψ−1
nor,d(u;Id))
ψnor (L̃Ψ−1
nor,d(u;Id))
du,
VG model: Observe: If Z ∼ N(0,Σ̃),Y ∼ χ2
(ν̃) ⇒
X = Z ×
√
ν̃
√
Y
∼ td(ν̃,0,Σ̃), with Z,Y independent
⇒ we obtain (see Proposition 3.4 in (Ben Hammouda et al. 2024c))
∫
Rd
g(u)du = ∫
+∞
0
⎛
⎜
⎜
⎜
⎝
∫
[0,1]d
g (
L̃⋅Ψ−1
nor,d(u;Id)
√
y
)
ψstu (
L̃⋅Ψ−1
nor,d
(u;Id)
√
y
)
du
⎞
⎟
⎟
⎟
⎠
ρY (y)dy
▸ td(ν̃,0,Σ̃): generalized t-student distribution.
▸ ρY (⋅): density of χ2
(ν̃) (chi-squared) distribution.
▸ L̃: Cholesky factor of ν̃ × Σ̃
17

Case of Correlated Assets
1 Choose the density ψ(⋅;Λ) in the change of variable to asymptotically follow the
same functional form of the extended characteristic function.
2 Select the parameters Λ to control the growth of the integrand near the boundary
of [0,1]d
i.e limuj→0,1 g̃(uj) < ∞,j = 1,...,d.
Table 3: Choice of ψ(u;Λ) ∶= ∏
d
j=1 ψj(uj;Λ) and conditions on Λ for GBM, (ii) VG and (iii) NIG.
See (Ben Hammouda et al. 2024c) for the derivation.
Model ψ(y;Λ) Growth condition on Λ
GBM Gaussian: (2π)−d
2 (det(Σ̃))−1
2 exp(−1
2(y′
Σ̃
−1
y)) TΣ − Σ̃
−1
⪰ 0
VG Generalized Student’s t:
Γ(ν̃+d
2
)(det(Σ̃))− 1
2
Γ(ν̃
2
)ν̃
d
2 π
d
2
(1 + 1
ν̃
(y′
Σ̃y))
− ν̃+d
2
ν̃ = 2T
ν − d, and
Σ − Σ̃
−1
⪰ 0
or
ν̃ ≤ 2T
ν − d, and
Σ̃ = Σ−1
NIG Laplace (v = 2−d
2 ): (2π)− d
2 (det(Σ̃))− 1
2 (y′Σ̃
−1
y
2 )
v
2
Kv (
√
2y′Σ̃
−1
y) δ2
T2
∆ − 2Σ̃
−1
⪰ 0

Illustration: Case of Correlated Assets
104
NxS
10−4
10−3
10−2
10−1
Relative
Statistical
Error
̃
Σ=1
TΣ
N−1.69
̃
σj = 1
√Tσj
N−0.69
Figure 2.3: Two-dimensional call on the minimum option under the GBM model:
Effect of the correlation parameter, ρ, on the convergence of RQMC. N: number of
QMC points; S = 32: number of digital shifts.
19

Fourier Pricing
4 Conclusion
19

Effect of Domain Transformation on
RQMC Convergence
0.0 0.2 0.4 0.6 0.8 1.0
u
0
10
20
30
40
50
60
̃
g
(
u
)
̃ ̃
σ = 1.0
̃ ̃
σ = 5.0
̃ ̃
σ = 9.0
(a)
103 104
NxS
10−6
10−5
10−4
10−3
10−2
10−1
100
Relative
Statistical
Error
̃
σ=1.0
N−0.68
̃
σ=5.0
N−1.42
̃
σ=9.0
N−2.26
(b)
Figure 3.1: Call option under the NIG model: Effect of the parameter σ̃ of the
Laplace PDF on
(a) the shape of the transformed integrand g̃(u) and
(b) convergence of the relative statistical error of RQMC
N: number of QMC points; S = 32: number of digital shifts.
Boundary growth condition: σ̃ ≥ 1
T δ
= 5.
20

Effect of Domain Transformation on
RQMC Convergence
0.0 0.2 0.4 0.6 0.8 1.0
u
−5
0
5
10
15
20
25
30
35
̃
g
(
u
)
̃ ̃
ν = 3.0
̃ ̃
ν = 9.0
̃ ̃
ν = 15.0
(a)
103 104
NxS
10−8
10−7
10−6
10−5
10−4
10−3
10−2
Relative
Statistical
Error
̃
ν = 3.0
N−3.05
̃
ν = 9.0
N−1.35
̃
ν = 15.0
N−0.66
(b)
Figure 3.2: Call option under the VG model: Effect of the parameter ν̃ of the
t-student PDF on
(a) the shape of the transformed integrand g̃(u) and
(b) convergence of the RQMC error
N: number of QMC points; S = 32: number of digital shifts.
Boundary growth condition: ν̃ ≤ 2T
ν
− 1 = 9
21

RQMC In Fourier Space vs MC in Physical Space
Figure 3.3: Average runtime in seconds with respect to relative tolerance
levels TOL: Comparison of RQMC in the Fourier space (with optimal
damping parameters and appropriate domain transformation) and MC in the
physical space.
10−2
10−1
TOL
100
101
102
Runtime
MC
TOL−1.97
RQMC
TOL−0.98
(a) 6D-VG call on min
10−2
10−1
TOL
10−1
100
101
102
Runtime
MC
TOL−2.0
RQMC
TOL−1.13
(b) 6D-NIG call on min
22

Comparison of the Different Methods
Figure 3.4: Call on min option: Runtime (in sec) versus dimensions to reach
a relative error, TOL = 10−2
. RQMC in the Fourier space (with optimal
damping parameters and appropriate domain transformation), TP in the
Fourier space with optimal damping parameters, and MC in the physical
space.
2 3 4 5 6 7 8 9 10 12 15
dimension
10−1
101
103
105
107
109
1011
Runtime
RQMC TP
MC
MC
(a) NIG model with:
α = 15, βj = −3, δ = 0.2, ∆ = Id,
σ̃j =
√
2
δ2T 2
2 3 4 5 6 7 8 9 10 12 15
dimension
10−2
100
102
104
106
108
1010
Runtime
RQMC TP
MC
MC
(b) VG model with:
σj = 0.2, θj = −0.3, ν = 0.2, Σ = Id,
ν̃ = 2T
ν
− d, σ̃j = 1
σj
.
23

Fourier Pricing
4 Conclusion
23

Conclusion
1 We empower Fourier pricing methods of multi-asset options by
employing QMC with an appropriate domain transformation.
2 We desing a practical (model dependent) domain transformation
strategy that prevents singularities near boundaries, ensuring the
integrand retains its regularity for faster QMC convergence in the
Fourier space.
3 The designed QMC-based Fourier pricing approach outperforms
the MC (in physical domain) and tensor product quadrature (in
Fourier space) for pricing multi-asset options across up to 15
dimensions.
4 Accompanying code for the paper can be found here:
Git repository: Quasi-Monte-Carlo-for-Efficient-Fourier-Pricing-of-
Multi-Asset-Options
24

Related References
Thank you for your attention!
1 C. Ben Hammouda et al. “Quasi-Monte Carlo for Efficient Fourier
Pricing of Multi-Asset Options”. In: arXiv preprint
arXiv:2403.02832 (2024)
2 C. Ben Hammouda et al. “Optimal Damping with Hierarchical Adaptive
Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy
Models”. In: Journal of Computational Finance 27.3 (2024), pp. 43–86
3 C. Ben Hammouda et al. “Numerical smoothing with hierarchical adaptive
sparse grids and quasi-Monte Carlo methods for efficient option pricing”. In:
Quantitative Finance (2022), pp. 1–19
4 C. Ben Hammouda et al. “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
25

References I
[1] Christian Bayer, Markus Siebenmorgen, and Rául Tempone.
“Smoothing the payoff for efficient computation of basket option
pricing.”. In: Quantitative Finance 18.3 (2018), pp. 491–505.
[2] C. Ben Hammouda, C. Bayer, and R. 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.
[3] C. Ben Hammouda, C. Bayer, and R. Tempone. “Multilevel
Monte Carlo combined with numerical smoothing for robust and
efficient option pricing and density estimation”. In: arXiv
preprint arXiv:2003.05708, to appear in SIAM Journal on
Scientific Computing (2024).
26

References II
[4] C. Ben Hammouda, C. Bayer, and R. Tempone. “Numerical
smoothing with hierarchical adaptive sparse grids and
quasi-Monte Carlo methods for efficient option pricing”. In:
Quantitative Finance (2022), pp. 1–19.
[5] C. Ben Hammouda et al. “Optimal Damping with Hierarchical
Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset
Options in Lévy Models”. In: Journal of Computational Finance
27.3 (2024), pp. 43–86.
[6] C. Ben Hammouda et al. “Quasi-Monte Carlo for Efficient
Fourier Pricing of Multi-Asset Options”. In: arXiv preprint
arXiv:2403.02832 (2024).
[7] Peter Carr and Dilip Madan. “Option valuation using the fast
Fourier transform”. In: Journal of computational finance 2.4
(1999), pp. 61–73.
27

References III
[8] Peng Chen. “Sparse quadrature for high-dimensional integration
with Gaussian measure”. In: ESAIM: Mathematical Modelling
and Numerical Analysis 52.2 (2018), pp. 631–657.
[9] Rama Cont and Peter Tankov. Financial Modelling with Jump
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[10] Philip J Davis and Philip Rabinowitz. Methods of numerical
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[11] Josef Dick, Frances Y Kuo, and Ian H Sloan. “High-dimensional
integration: the quasi-Monte Carlo way”. In: Acta Numerica 22
(2013), pp. 133–288.
[12] Ernst Eberlein, Kathrin Glau, and Antonis Papapantoleon.
“Analysis of Fourier transform valuation formulas and
applications”. In: Applied Mathematical Finance 17.3 (2010),
pp. 211–240.
28

References IV
[13] Fang Fang and Cornelis W Oosterlee. “A novel pricing method
for European options based on Fourier-cosine series expansions”.
In: SIAM Journal on Scientific Computing 31.2 (2009),
pp. 826–848.
[14] Thomas R Hurd and Zhuowei Zhou. “A Fourier transform
method for spread option pricing”. In: SIAM Journal on
Financial Mathematics 1.1 (2010), pp. 142–157.
[15] Alan L Lewis. “A simple option formula for general
jump-diffusion and other exponential Lévy processes”. In:
Available at SSRN 282110 (2001).
[16] M. J. Ruijter and Cornelis W Oosterlee. “Two-dimensional
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options”. In: SIAM Journal on Scientific Computing 34.5 (2012),
B642–B671.
29

References V
[17] Lloyd Nicholas Trefethen. “Finite difference and spectral methods
for ordinary and partial differential equations”. In: (1996).
30

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