Talk_HU_Berlin_Chiheb_benhammouda.pdf

Smoothing Techniques and Hierarchical Approximations
for Efficient Option Pricing
Chiheb Ben Hammouda
Christian
Bayer
Michael
Samet
Center for Uncertainty
Quantification
Quantification Logo Lock-up
Antonis
Papapantoleon
Raúl
Tempone
Center for Uncertainty
Quantification
Cen
Qu
Center for Uncertainty Quantification Logo Loc
Mathematical Finance Seminar, Humboldt University, Berlin
October 27, 2022

Related Manuscripts to the Talk
Part I
▸ Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Numerical Smoothing with Hierarchical Adaptive Sparse Grids
and Quasi-Monte Carlo Methods for Efficient Option Pricing”. In:
arXiv preprint arXiv:2111.01874 (2021). Accepted in Quantitative
Finance Journal (2022).
▸ Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone.
“Multilevel Monte Carlo combined with numerical smoothing for
robust and efficient option pricing and density estimation”. In:
arXiv preprint arXiv:2003.05708 (2022).
Part II
▸ Christian Bayer, Chiheb Ben Hammouda, Antonis Papapantoleon,
Michael Samet, and Raúl Tempone. “Optimal Damping with
Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of
Multi-Asset Options in Lévy Models”. In: arXiv preprint
arXiv:2203.08196 (2022).
1

1 Motivation and Framework
2 Numerical Smoothing for Efficient Option Pricing
The Numerical Smoothing Idea
The Smoothness Theorem
Error Discussion
Numerical Experiments and Results
Conclusions
3 Optimal Damping with Hierarchical Adaptive Quadrature for
Efficient Fourier Pricing of Multi-Asset Options in Lévy Models
Problem Setting and Motivation
Optimal Damping Rule
Conclusions
1

Option Pricing as a High-Dimensional, non-Smooth
Numerical Integration Problem
Option pricing often correspond to
high-dimensional integration problems
with non-smooth integrands.
▸ High-dimensional:
(i) Time-discretization of an SDE;
(ii) A large number of underlying
assets;
(iii) Path dependence on the whole
trajectory of the underlying.
▸ Non-smooth: E.g., digital options:
1ST >K, rainbow options:
max(min(S1,...,Sd) − K,0), . . .
Methods like quasi Monte Carlo
(QMC) or (adaptive) sparse grids
quadrature (ASGQ) critically rely on
smoothness of integrand and dimension
of the problem.
Monte Carlo (MC) does not care (in
terms of convergence rates) BUT
multilevel Monte Carlo (MLMC) does.
Figure 1.1: Integrand of two-dimensional
European basket option (Black–Scholes
model)

Goal: Approximate efficiently E[g(X(T))]
How Does Regularity Affect
QMC, ASGQ, and MLMC Performance?
2

Randomized QMC (rQMC)
A (rank-1) lattice rule (Sloan 1985; Nuyens 2014) with n points
Qn(g) ∶=
1
n
n−1
∑
k=0
(g ○ F−1
)(
kz mod n
n
),
where z = (z1,...,zd) ∈ Nd
(the generating vector) and F−1
(⋅) is the inverse of
the standard normal cumulative distribution function.
A randomly shifted lattice rule
Qn,q(g) =
1
q
q−1
∑
i=0
Q(i)
n (g) =
1
q
q−1
∑
i=0
(
1
n
n−1
∑
k=0
(g ○ F−1
)(
kz + ∆(i)
mod n
n
)),
where {∆(i)
}q−1
i=0 : independent random shifts, and MrQMC = q × n.
3

How Does Regularity Affect QMC Performance?
(Niederreiter 1992): convergence rate of O (M
−1
2
−δ
rQMC), where 0 ≤ δ ≤ 1
2
is related to the degree of regularity of the integrand g.
(Dick, Kuo, and Sloan 2013): convergence rate of O (M−1
rQMC) can be
observed for the lattice rule if the integrand g belongs to Wd,γ
equipped with the norm
∣∣g∣∣2
Wd,γ
= ∑
α⊆{1∶d}
1
γα
∫
[0,1]∣α∣
(∫
[0,1]d−∣α∣
∂∣α∣
∂yα
g(y)dy−α)
2
dyα,
where yα ∶= (yj)j∈α and y−α ∶= (yj)j∈{1∶d}∖α.
(Nuyens and Suzuki 2021): Higher mixed smoothness and periodicity
requirements can lead to higher order convergence rates, when using
scaled lattice rules.
Notation
Wd,γ: a d-dimensional weighted Sobolev space of functions with
square-integrable mixed first derivatives.
γ ∶= {γα > 0 ∶ α ⊆ {1,2,...,d}} being a given collection of weights, and
d being the dimension of the problem. 4

Quadrature Methods (I)
Naive Quadrature for E[g]: " Curse of dimension
Qβ
d [g] ∶=
d
⊗
i=1
Qm(βi)
[g] ≡
m(β1)
∑
k1=1
⋯
m(βd)
∑
kd=1
wβ1
k1
⋯wβd
kd
g (xβ1
k1
,...,xβd
kd
)
Notation:
{xβi
ki
}
m(βi)
ki=1
,{ωβi
ki
}
m(βi)
ki=1
are respectively the sets of quadrature points and
corresponding quadrature weights.
β = (βi)d
i=1 ∈ Nd
is a multi-index, and m ∶ N → N is a level-to-nodes function mapping
a given level to the number of quadrature points.
Better Quadrature estimate E[g]:
QI
d [g] = ∑
β∈I
∆Qβ
d [g], with the multivariate detail: ∆Qβ
d [g] =
d
⊗
i=1
∆iQβ
d [g]
and univariate detail: (with ei denotes the ith d-dimensional unit vector.)
∆iQβ
d [g] ∶= {
(Qβ
d − Qβ′
d )[g], with β′
= β − ei, if βi > 1
Qβ
d [g], if βi = 1,
Illustration:
∆Qβ
2 = ∆2∆1Q
(β1,β2)
2 = Q
(β1,β2)
2 − Q
(β1,β2−1)
2 − Q
(β1−1,β2)
2 + Q
(β1−1,β2−1)
2
∑∣∣β∣∣∞≤2 ∆Qβ
2 [g] = ∆Q
(1,1)
2 [g] + ∆Q
(2,1)
2 [g] + ∆Q
(1,2)
2 [g] + ∆Q
(2,2)
2 [g] = Q
(2,2)
2 [g]
5

Quadrature Methods (II)
E[g] ≈ QI
d [g] = ∑
β∈I
∆Qβ
d [g]
Tensor Product (TP) approach: I ∶= I` = {∣∣ β ∣∣∞≤ `; β ∈ Nd
+}.
Regular sparse grids: I ∶ I` = {∣∣ β ∣∣1≤ ` + d − 1; β ∈ Nd
+}
Adaptive sparse grids quadrature (ASGQ): The construction of I = IASGQ
is
done a posteriori and adaptively by profit thresholding
IASGQ
= {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 1.2: Left are product grids ∆β1 ⊗ ∆β2
for 1 ≤ β1,β2 ≤ 3. Right is the corresponding
sparse grids construction.
Figure 1.3: Illustration of ASG grid
construction.
6

How Does Regularity Affect ASGQ Performance?
Product approach: EQ(M) = O (M−r/d
) (for functions with
bounded total derivatives up to order r).
Adaptive sparse grids quadrature (ASGQ): From the analysis in
(Chen 2018; Ernst, Sprungk, and Tamellini 2018)
EQ(M) = O (M−p/2
) (where1
p is independent from the problem
dimension, and is related to the order up to which the weighted
mixed derivatives are bounded).
Notation: M: number of quadrature points; EQ: quadrature error.
1
characterizes the relation between the regularity of the integrand and the
anisotropic property of the integrand with respect to different dimensions. 7

ASGQ Quadrature Error Convergence
101
102
103
104
MASGQ
10-3
10
-2
10
-1
100
101
Relative
Quadrature
Error
ASGQ without smoothing
ASGQ with numerical smoothing
(a)
101
102
103
104
MASGQ
10-3
10
-2
10-1
100
101
Relative
Quadrature
Error
(b)
Figure 1.4: Digital option under the Heston model: Comparison of the relative
quadrature error convergence for the ASGQ method with/out numerical
smoothing. (a) Without Richardson extrapolation (N = 8), (b) with
Richardson extrapolation (Nfine level = 8).
8

Multilevel Monte Carlo (MLMC)
(Heinrich 2001; Kebaier et al. 2005; Giles 2008b)
Aim: Improve MC complexity, when estimating E[g(X(T))].
Setting:
▸ A hierarchy of nested meshes of [0,T], indexed by {`}L
`=0.
▸ ∆t` = K−`
∆t0: The time steps size for levels ` ≥ 1; K>1, K ∈ N.
▸ X` ∶= X∆t`
: The approximate process generated using a step size of ∆t`.
MLMC idea
E[g(XL(T))] = E[g(X0(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
+
L
∑
`=1
E[g(X`(T)) − g(X`−1(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
(1)
Var[g(X0(T))] ≫ Var[g(X`(T)) − g(X`−1(T))] ↘ as ` ↗
M0 ≫ M` ↘ as ` ↗
MLMC estimator: ̂
Q ∶=
L
∑
`=0
̂
Q`,
̂
Q0 ∶=
1
M0
M0
∑
m0=1
g(X0,[m0](T)); ̂
Q` ∶=
1
M`
M`
∑
m`=1
(g(X`,[m`](T)) − g(X`−1,[m`](T))), 1 ≤ ` ≤ L
9

How Does Regularity Affect MLMC Complexity?
Complexity analysis for MLMC
MLMC Complexity (Cliffe, Giles,
Scheichl, and Teckentrup 2011)
O (TOL
−2−max(0, γ−β
α
)
log (TOL)2×1{β=γ}
)
(2)
i) Weak rate:
∣E[g (X`(T)) − g (X(T))]∣ ≤ c12−α`
ii) Variance decay rate:
Var[g (X`(T)) − g (X`−1(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∶=V`
≤ c22−β`
iii) Work growth rate: W` ≤ c32γ`
(W`:
expected cost)
For Euler-Maruyama (γ = 1):
▸ If g is Lipschitz ⇒ V` ≃ ∆t` due to strong rate 1/2, that is β = γ and MLMC complexity
O (TOL−2
);
▸ Otherwise (without any smoothing or adaptivity techniques):
β < γ ⇒ worst-case complexity, O (TOL− 5
2 ).
10

How Does Regularity Affect MLMC Robustness?
/ For non-lipschitz payoffs (without any smoothing or adaptivity
techniques):
The Kurtosis, κ` ∶=
E[(Y`−E[Y`])4
]
(Var[Y`])2 is of O(∆t
−1/2
` ) for Euler-Maruyama.
Large kurtosis problem: discussed previously in (Ben Hammouda, Moraes,
and Tempone 2017; Ben Hammouda, Ben Rached, and Tempone 2020) ⇒
/ Expensive cost for reliable/robust estimates of sample statistics.
Why is large kurtosis bad?
σS2(Y`) =
Var[Y`]
√
M`
√
(κ` − 1) +
2
M` − 1
; " M` ≫ κ`.
Why are accurate variance estimates, V` = Var[Y`], important?
M∗
` ∝
√
V`W−1
`
L
∑
`=0
√
V`W`.
Notation
Y` ∶= g(X`(T)) − g(X`−1(T))
σS2(Y`): Standard deviation of the sample variance of Y`;
M∗
` : Optimal number of samples per level; W`: Cost per sample path.
11

Error Discussion
Conclusions
Conclusions
11

Framework
Approximate efficiently E[g(X(T))]
Given a (smooth) φ ∶ Rd
→ R, the payoff g ∶ Rd
→ R has either
jumps or kinks:
▸ Hockey-stick functions: g(x) = max(φ(x),0) (put or call payoffs,
. . . ).
▸ Indicator functions: g(x) = 1(φ(x)≥0) (digital option, financial
Greeks, probabilities, . . . ).
▸ Dirac Delta functions: g(x) = δ(φ(x)=0) (density estimation, financial
Greeks, . . . ).
The process X is approximated (via a discretization scheme) by
X, E.g., (for illustration)
▸ Stochastic volatility model: E.g., the Heston model
dXt = µXtdt +
√
vtXtdWX
t
dvt = κ(θ − vt)dt + ξ
√
vtdWv
t ,
(WX
t ,Wv
t ): correlated Wiener processes with correlation ρ.
12

Error Discussion
Conclusions
Conclusions
12

Numerical Smoothing Steps
Guiding example:
g ∶ Rd
→ R payoff function, X
∆t
T (∆t = T
N ) Euler discretization of
d-dimensional SDE , E.g., dX
(i)
t = ai(Xt)dt + ∑d
j=1 bij(Xt)dW
(j)
t ,
where {W(j)
}d
j=1 are standard Brownian motions.
X
∆t
T = X
∆t
T (∆W
(1)
1 ,...,∆W
(1)
N ,...,∆W
(d)
1 ,...,∆W
(d)
N )
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∶=∆W
.
E[g(X
∆t
T )] =?
1 Identify hierarchical representation of integration variables ⇒ locate the
discontinuity in a smaller dimensional space and create more anisotropy
(a) X
∆t
T (∆W) ≡ X
∆t
T (Z), Z = (Zi)dN
i=1 ∼ N(0,IdN ): s.t. “Z1 ∶= (Z
(1)
1 ,...,Z
(d)
1 )
(coarse rdvs) substantially contribute even for ∆t → 0”.
E.g., Brownian bridges of ∆W / Haar wavelet construction of W.
(b) Design of a sub-optimal smoothing direction (A: rotation matrix which is
easy to construct and whose structure depends on the payoff g.)
Y = AZ1.
E.g., for an arithmetic basket option with payoff max(∑
d
i=1 ciXi(T) − K,0), a
suitable A is a rotation matrix, with the first row leading to Y1 = ∑
d
i=1 Z
(i)
1
up to rescaling without any constraint for the remaining rows (using the
Gram-Schmidt procedure). 13

2
E[g(X(T))] ≈ E[g(X
∆t
(T))]
= ∫
Rd×N
G(z)ρd×N (z)dz
(1)
1 ...dz
(1)
N ...dz
(d)
1 ...dz
(d)
N
= ∫
RdN−1
I(y−1,z
(1)
−1 ,...,z
(d)
−1 )ρd−1(y−1)dy−1ρdN−d(z
(1)
−1 ,...,z
(d)
−1 )dz
(1)
−1 ...dz
(d)
−1
= E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] ≈ E[I(Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )], (3)
" How to obtain I(⋅) and I(⋅)? ⇒ See next slide.
Notation
G ∶= g ○ Φ ○ (ψ(1)
,...,ψ(d)
) maps N × d Gaussian random inputs to g(X
∆t
(T));
where
▸ ψ(j)
∶ (Z
(j)
1 ,...,Z
(j)
N ) ↦ (B
(j)
1 ,...,B
(j)
N ) denotes the mapping of the Brownian
bridge/Haar wavelet construction.
▸ Φ ∶ (∆t,B) ↦ X
∆t
(T) denotes the mapping of the time-stepping scheme.
x−j denotes the vector of length d − 1 denoting all the variables other than xj in
x ∈ Rd
.
ρd×N (z) = 1
(2π)d×N/2 e−1
2
zT z
.
14

3
I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) = ∫
R
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
= ∫
y∗
1
−∞
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1 + ∫
+∞
y∗
1
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
≈ I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) ∶=
Mlag
∑
k=0
ηkG(ζk (y∗
1),y−1,z
(1)
−1 ,...,z
(d)
−1 ), (4)
where
▸ y∗
1 (y−1,z
(1)
−1 ,...,z
(d)
−1 ): the exact discontinuity location s.t
φ(X
∆t
(T)) = P(y∗
1 ;y−1,z
(1)
−1 ,...,z
(d)
−1 ) = 0. (5)
▸ y∗
1(y−1,z
(1)
−1 ,...,z
(d)
−1 ): the approximated discontinuity location via root finding
to solve (5);
▸ MLag: the number of Laguerre quadrature points ζk ∈ R, and corresponding
weights ηk;
4 Compute the remaining (dN −1)-integral in (3) by ASGQ or QMC or MLMC.
15

Extending Numerical Smoothing for
Multiple Discontinuities
Multiple Discontinuities: Due to the payoff structure/use of Richardson extrapolation.
R different ordered multiple roots, e.g., {y∗
i }R
i=1, the smoothed integrand is
I (y−1,z
(1)
−1 ,...,z
(d)
−1 ) = ∫
y∗
1
−∞
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1 + ∫
+∞
y∗
R
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1
+
R−1
∑
i=1
∫
y∗
i+1
y∗
i
G(y1,y−1,z
(1)
−1 ,...,z
(d)
−1 )ρ1(y1)dy1,
and its approximation I is given by
I(y−1,z
(1)
−1 ,...,z
(d)
−1 ) ∶=
MLag,1
∑
k=0
ηLag
k G(ζLag
k,1 (y∗
1),y−1,z
(1)
−1 ,...,z
(d)
−1 )
+
MLag,R
∑
k=0
ηLag
k G(ζLag
k,R (y∗
R),y−1,z
(1)
−1 ,...,z
(d)
−1 )
+
R−1
∑
i=1
⎛
⎝
MLeg,i
∑
k=0
ηLeg
k G(ζLeg
k,i (y∗
i ,y∗
i+1),y−1,z
(1)
−1 ,...,z
(d)
−1 )
⎞
⎠
,
{y∗
i }R
i=1: the approximated discontinuities locations; MLag,1 and MLag,R: the number
of Laguerre quadrature points ζLag
.,. ∈ R with corresponding weights ηLag
. ; {MLeg,i}R−1
i=1 :
the number of Legendre quadrature points ζLeg
.,. with corresponding weights ηLeg
. .
I can be approximated further depending on (i) the decay of G × ρ1 in the semi-infinite
domains and (ii) how close the roots are to each other.

Extending Numerical Smoothing for
Density Estimation
Goal: Approximate the density ρX at u, for a stochastic process X
ρX(u) = E[δ(X − u)], δ is the Dirac delta function.
" Without any smoothing techniques (regularization, kernel
density,. . . ) MC and MLMC fail due to the infinite variance caused
by the singularity of the function δ.
Strategy in (Bayer, Hammouda, and Tempone 2022): Conditioning
with respect to the Brownian bridge
ρX(u) =
1
√
2π
E[exp(−(Y ∗
1 (u))
2
/2)∣
dY ∗
1
du
(u)∣]
≈
1
√
2π
E
⎡
⎢
⎢
⎢
⎣
exp(−(Y
∗
1(u))
2
/2)
R
R
R
R
R
R
R
R
R
R
R
dY
∗
1
du
(u)
R
R
R
R
R
R
R
R
R
R
R
⎤
⎥
⎥
⎥
⎦
, (6)
Y ∗
1 (u;Z−1): the exact discontinuity; Y
∗
1(u;Z−1): the approximated
discontinuity.

Why not Kernel Density Techniques
in Multiple Dimensions?
Similar to approaches based on parametric regularization as in (Giles, Nagapetyan,
and Ritter 2015).
This class of approaches has a pointwise error that increases exponentially with
respect to the dimension of the state vector X.
For a d-dimensional problem, a kernel density estimator with a bandwidth matrix,
H = diag(h,...,h)
MSE ≈ c1M−1
h−d
+ c2h4
. (7)
M is the number of samples, and c1 and c2 are constants.
Our approach in high dimension: For u ∈ Rd
ρX(u) = E[δ(X − u)] = E[ρd (Y∗
(u))∣det(J(u))∣]
≈ E[ρd (Y
∗
(u))∣det(J(u))∣], (8)
where
▸ Y∗
(u;⋅): the exact discontinuity; Y
∗
(u;⋅): the approximated discontinuity.
▸ J is the Jacobian matrix, with Jij =
∂y∗
i
∂uj
; ρd(⋅) is the multivariate Gaussian density.
Thanks to the exact conditional expectation with respect to the Brownian bridge
⇒ the smoothing error in our approach is insensitive to the dimension of the
problem.
18

Error Discussion
Conclusions
Conclusions
18

Notations
The Haar basis functions, ψn,k, of L2
([0,1]) with support
[2−n
k,2−n
(k + 1)]:
ψ−1(t) ∶= 1[0,1](t); ψn,k(t) ∶= 2n/2
ψ (2n
t − k), n ∈ N0, k = 0,...,2n
− 1,
where ψ(⋅) is the Haar mother wavelet:
ψ(t) ∶=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
1, 0 ≤ t < 1
2
,
−1, 1
2
≤ t < 1,
0, else,
Grid Dn ∶= {tn
` ∣ ` = 0,...,2n+1
} with tn
` ∶= `
2n+1 T.
For i.i.d. standard normal rdvs Z−1, Zn,k, n ∈ N0, k = 0,...,2n
− 1,
we define the (truncated) standard Brownian motion
WN
t ∶= Z−1Ψ−1(t) +
N
∑
n=0
2n
−1
∑
k=0
Zn,kΨn,k(t).
with Ψ−1(⋅) and Ψn,k(⋅) are the antiderivatives of the Haar basis
functions. 19

Notations
We define the corresponding increments for any function or
process F as follows:
∆N
` F ∶= F(tN
`+1) − F(tN
` ).
The solution of the Euler scheme of dXt = b(Xt)dWt, X0 = x ∈ R,
along DN
as
XN
`+1 ∶= XN
` + b(XN
` )∆N
` W, ` = 0,...,2N
− 1, (9)
with XN
0 ∶= X0 = x; for convenience, we also define XN
T ∶= XN
2N .
We define the deterministic function HN
∶ R2N+1−1
→ R as
HN
(zN
) ∶= EZ−1
[g (XN
T (Z−1,zN
))], (10)
where ZN
∶= (Zn,k)n=0,...,N, k=0,...2n−1.
20

Theorem 2.1 ((Bayer, Ben Hammouda, and Tempone 2022))
Assume that XN
T , defined by (9), satisfies Assumptions 3.1 and 3.2.a
Then, for any p ∈ N and indices n1,...,np and k1,...,kp (satisfying
0 ≤ kj < 2nj ), the function HN
defined in (10) satisfies the following
(with constants independent of nj,kj)b
∂p
HN
∂zn1,k1 ⋯∂znp,kp
(zN
) = Ô
P (2− ∑
p
j=1 nj/2
).c
In particular, HN
is of class C∞
.
a
We showed in (Bayer, Ben Hammouda, and Tempone 2022) that
Assumptions 3.1 and 3.2 are valid for many pricing models.
b
The constants increase in p and N.
c
For sequences of rdvs FN , we write that FN = Ô
P(1) if there is a rdv C
with finite moments of all orders such that for all N, we have ∣FN ∣ ≤ C a.s.
21

Sketch of the Proof
1 We consider a mollified version gδ of g and the corresponding function
HN
δ (defined by replacing g with gδ in (10)).
2 We prove that we can interchange the integration and differentiation,
which implies
∂HN
δ (zN
)
∂zn,k
= E [g′
δ (XN
T (Z−1,zN
))
∂XN
T (Z−1,zN
)
∂zn,k
].
3 Multiplying and dividing by
∂XN
T (Z−1,zN )
∂y (where y ∶= z−1) and
replacing the expectation by an integral w.r.t. the standard normal
density, we obtain
∂HN
δ (zN
)
∂zn,k
= ∫
R
∂gδ (XN
T (y,zN
))
∂y
(
∂XN
T
∂y
(y,zN
))
−1
∂XN
T
∂zn,k
(y,zN
)
1
√
2π
e− y2
2 dy.
(11)
4 We show that integration by parts is possible, and then we can discard
the mollified version to obtain the smoothness of HN
because
∂HN
(zN
)
∂zn,k
= −∫
R
g (XN
T (y,zN
))
∂
∂y
⎡
⎢
⎢
⎢
⎢
⎣
(
∂XN
T
∂y
(y,zN
))
−1
∂XN
T
∂zn,k
(y,zN
)
1
√
2π
e− y2
2
⎤
⎥
⎥
⎥
⎥
⎦
dy.
5 The proof relies on successively applying the technique above of
dividing by
∂XN
T
∂y and then integrating by parts.
22

Error Discussion
Conclusions
Conclusions
22

Error Discussion for ASGQ
QASGQ
N : the ASGQ estimator
E[g(X(T)] − QASGQ
N = E[g(X(T))] − E[g(X
∆t
(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error I: bias or weak error of O(∆t)
+ E[I (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − E[I (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error II: numerical smoothing error of O(M
−s/2
Lag
)+O(TOLNewton)
+ E[I (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − QASGQ
N
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error III: ASGQ error of O(M
−p/2
ASGQ
)
, (12)
Notations
MASGQ: the number of quadrature points used by the ASGQ estimator, and p > 0.
y∗
1: the approximated location of the non smoothness obtained by Newton iteration
⇒ ∣y∗
1 − y∗
1∣ = TOLNewton
MLag is the number of points used by the Laguerre quadrature for the one
dimensional pre-integration step.
s > 0: For the parts of the domain separated by the discontinuity location,
derivatives of G with respect to y1 are bounded up to order s.
23

Work and Complexity Discussion for ASGQ
An optimal performance of ASGQ is given by
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
min
(MASGQ,MLag,TOLNewton)
WorkASGQ ∝ MASGQ × MLag × ∆t−1
s.t. Etotal,ASGQ = TOL.
(13)
Etotal, ASGQ ∶= E[g(X(T)] − QASGQ
N
= O (∆t) + O (M
−p/2
ASGQ) + O (M
−s/2
Lag ) + O (TOLNewton).
We show in (Bayer, Ben Hammouda, and Tempone 2022) that
under certain conditions for the regularity parameters s and p
(p,s ≫ 1)
▸ WorkASGQ = O (TOL−1
) (for the best case) compared to
WorkMC = O (TOL−3
) (for the best case of MC).
24

Error Discussion for rQMC
̂
QrQMC
: the rQMC estimator
E[g(X(T)] − ̂
QrQMC
= E[g(X(T))] − E[g(X
∆t
(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error I: bias or weak error of O(∆t)
+ E[I (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − E[I (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
−s/2
Lag
)+O(TOLNewton)
+ E[I (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − ̂
QrQMC
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error III: rQMC error of O(M−1+δ
rQMC
), δ>0
, (14)
Notations
MrQMC: the number of rQMC points.
y∗
1: the approximated location of the non smoothness obtained by Newton iteration
⇒ ∣y∗
1 − y∗
1∣ = TOLNewton
MLag is the number of points used by the Laguerre quadrature for the one
dimensional pre-integration step.
s > 0: For the parts of the domain separated by the discontinuity location,
derivatives of G with respect to y1 are bounded up to order s.
25

Error Discussion for MLMC
̂
QMLMC
: the MLMC estimator
E[g(X(T)] − ̂
QMLMC
= E[g(X(T))] − E[g(X
∆tL
(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error I: bias or weak error of O(∆tL)
+ E[IL (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − E[IL (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
−s/2
Lag,L
)+O(TOLNewton,L)
+ E[IL (Y−1,Z
(1)
−1 ,...,Z
(d)
−1 )] − ̂
QMLMC
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
Error III: MLMC statistical error of O
⎛
⎝
√
∑L
`=L0
√
MLag,`+log(TOL−1
Newton,`
)
⎞
⎠
.
(15)
" The level ` approximation of I in ̂
QMLMC
, I` ∶= I`(y`
−1,z
(1),`
−1 ,...,z
(d),`
−1 ), is
computed with: step size of ∆t`; MLag,` Laguerre quadrature points;
TOLNewton,` as the tolerance of the Newton method at level `
26

Multilevel Monte Carlo with Numerical Smoothing:
Variance Decay, Complexity and Robustness
Notation: I`:= I`(y`
−1,z
(1),`
−1 ,...,z
(d),`
−1 ): the level ` approximation of I, computed with
step size ∆t`; MLag,` Laguerre points; TOLNewton,` as the tolerance of Newton at level `.
Theorem 2.2 ((Bayer, Hammouda, and Tempone 2022))
Under Assmuptions 3.1 and 3.2 and further regularity assumptions for the drift and
diffusion, V` ∶= Var[I` − I`−1] = O (∆t1−
` ) for any 0, compared with O (∆t
1/2
` ) for
MLMC without smoothing.
Corollary 2.3 ((Bayer, Hammouda, and Tempone 2022))
Under Assmuptions 3.1 and 3.2 and further regularity assumptions for the drift and
diffusion, the complexity of MLMC combined with numerical smoothing using Euler is
O (TOL−2−
) for any 0, compared with O (TOL−2.5
) for MLMC without smoothing.
Corollary 2.4 ((Bayer, Hammouda, and Tempone 2022))
Let κ` be the kurtosis of the random variable Y` ∶= I` − I`−1, then under assmuptions 3.1
and 3.2 and further regularity assumptions for the drift and diffusion, we obtain
κ` = O (1) compared with O (∆t
−1/2
` ) for MLMC without smoothing.

Error Discussion
Conclusions
Conclusions
27

ASGQ Quadrature Error Convergence
101
102
103
104
MASGQ
10-3
10
-2
10
-1
100
101
Relative
Quadrature
Error
(a)
101
102
103
104
MASGQ
10-3
10
-2
10-1
100
101
Relative
Quadrature
Error
(b)
Figure 2.1: Digital option under the Heston model: Comparison of the relative
quadrature error convergence for the ASGQ method with/out numerical
smoothing. (a) Without Richardson extrapolation (N = 8), (b) with
Richardson extrapolation (Nfine level = 8).
28

QMC Error Convergence
102
103
104
QMC samples
10-3
10
-2
10-1
95%
Statistical
error
QMC without smoothing
slope=-0.52
QMC with numerical smoothing
slope=-0.85
(a)
102
103
104
105
QMC samples
10-5
10
-4
10
-3
10-2
10-1
95%
Statistical
error
QMC without smoothing
slope=-0.64
QMC with numerical smoothing
slope=-0.92
(b)
Figure 2.2: Comparison of the 95% statistical error convergence for rQMC
with/out numerical smoothing. (a) Digital option under Heston, (b) digital
option under GBM.
29

Errors in the Numerical Smoothing
101
102
MLag
10-5
10
-4
10
-3
10-2
10
-1
Relative
numerical
smoothing
error
Relative numerical smoothing error
slope=-4.02
(a)
10-10
10-8
10-6
10-4
10-2
100
TOLNewton
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Relative
numerical
smoothing
error
10
-4
Relative numerical smoothing error
(b)
Figure 2.3: The relative numerical smoothing error for a fixed number of
ASGQ points MASGQ = 103
plotted against (a) different values of MLag with a
fixed Newton tolerance TOLNewton = 10−10
, (b) different values of TOLNewton
with a fixed number of Laguerre quadrature points MLag = 128.
30

Comparing ASGQ with MC
In (Bayer, Ben Hammouda, and Tempone 2022):
Consider digital/call/basket options in Heston or discretized GBM models.
ASGQ with numerical smoothing 10 - 100× faster in dim. around 20 than MC.
10
-3
10
-2
10
-1
Total Relative Error
10
-1
100
101
10
2
103
Computational
Work
MC (without smoothing, without Richardson extrapolation)
MC (without smoothing, with Richardson extrapolation)
ASGQ (with smoothing, without Richardson extrapolation)
ASGQ (with smoothing, with Richardson extrapolation)
Figure 2.4: Digital option under Heston: Computational work comparison for ASGQ with
numerical smoothing and MC with the different configurations in terms of the level of
Richardson extrapolation. 31

Error Discussion
Conclusions
Conclusions
31

Conclusions
Contributions in the context deterministic quadrature methods
1 A novel numerical smoothing technique, combined with ASGQ/QMC,
Hierarchical Brownian Bridge and Richardson extrapolation.
2 Our approach outperforms substantially the MC method for moderate/high
dimensions and for dynamics where discretization is needed.
3 We provide a regularity analysis for the smoothed integrand in the time stepping
setting, and a related error discussion of our approach.
4 We illustrate that traditional schemes of the Heston dynamics deteriorate the
regularity, and we use an alternative smooth scheme to simulate the volatility
process based on the sum of Ornstein-Uhlenbeck (OU) processes.
5 More analysis and numerical examples can be found in:
Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone. “Numerical
Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo
Methods for Efficient Option Pricing”. In: arXiv preprint arXiv:2111.01874
(2021). Accepted in Quantitative Finance Journal (2022).
Contributions in the context MLMC methods
1 A numerical smoothing approach that can be combined with MLMC estimator for
efficient option pricing and density estimation.
2 Compared to the case without smoothing
☀ We significantly reduce the kurtosis at the deep levels of MLMC which improves
the robustness of the estimator.
☀ We improve the strong convergence rate ⇒ improvement of MLMC complexity
from O (TOL−2.5
) to O (TOL−2
).
Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone. “Multilevel Monte
Carlo combined with numerical smoothing for robust and efficient option pricing
and density estimation”. In: arXiv preprint arXiv:2003.05708 (2022). 32

Error Discussion
Conclusions
Conclusions
32

Problem Setting
Using the inverse generalized Fourier transform theorem, the value of the
option price on d stocks can be expressed as
V (Θm,Θp) = e−rT
E[P(XT)] = e−rT
E[(2π)−d
R(∫
Rd+iR
ei⟨u,XT ⟩ ̂
P(u)du)], R ∈ δP
= (2π)−d
e−rT
R(∫
Rd+iR
ΦXT
(u) ̂
P(u)du), R ∈ δV ∶= δP ∩ δX
∶= ∫
Rd
g (u;R,Θm,Θp)du (16)
Notations
ΦXT
(u): the joint characteristic function of XT ∶= (X1
T ,...,Xd
T );
Xi
T ∶= log (Si
T ), {Si
T }
d
i=1
are the prices of the underlying assets at time T.
̂
P(⋅): the conjugate of the Fourier transform of the payoff P(⋅).
R: vector of damping parameters ensuring L1
integrability of g(⋅).
δV ∶= δP ∩ δX: the strip of regularity of g(⋅); δP : strip of regularity of ̂
P(⋅);
δX: strip of regularity of ΦXT
(⋅).
Θm,Θp are respectively the model and payoff parameters.
i is the unit complex number; R(⋅): real part of the complex number .
33

Motivation
The integrand in the frequency space has often higher regularity
than in the physical space.
For the Fourier-based pricing approach: two key aspects affecting
the numerical complexity of the numerical quadrature methods:
1 The choice of the damping parameters that ensure integrability and
control the regularity of the integrand.
No precise analysis of the effect of the damping parameters on
the convergence speed or guidance on choosing them to improve the
numerical performance, particularly in the multivariate setting.
2 The effective treatment of the high dimensionality.
Methodology
1 Smoothing the Fourier integrand via an optimized choice of
damping parameters.
2 Sparsification and dimension-adaptivity techniques to accelerate
the convergence of the quadrature in moderate/high dimensions.

Framework
Basket put:
P(XT ) = max(K −
d
∑
i=1
eXi
T ,0);
̂
P(z) = K1−i ∑
d
j=1 zj
∏
d
j=1 Γ(−izj)
Γ(−i∑
d
j=1 zj + 2)
, z ∈ Cd
,I[z] ∈ δP
Call on min:
P(XT ) = max(min(eX1
T ,...,eXd
T ) − K,0);
̂
P(z) =
K1−i ∑
d
j=1 zj
(i(∑
d
j=1 zj) − 1)∏
d
j=1 (izj)
, z ∈ Cd
,I[z] ∈ δP
Table 1: Strip of regularity of ̂
P(⋅): δP (Eberlein, Glau, and Papapantoleon 2010).
Payoff δP
Basket put {R ∈ Rd
, Ri 0 ∀i ∈ {1,...,d}}
Call on min {R ∈ Rd
, Ri 0 ∀i ∈ {1,...,d},∑
d
i=1 Ri −1}
Notation: Γ(⋅): complex Gamma function, I(⋅): imaginary part of the
argument. 35

Framework: Pricing Models
Table 2: ΦXT
(⋅) for the different models. ⟨.,.⟩ is the standard dot product on Rd
.
Model ΦXT
(z)
GBM exp(i⟨⋅,X0⟩) × exp(i⟨z,r1Rd − 1
2
diag(Σ)⟩T − T
2
⟨z,Σz⟩),I[z] ∈ δX.
VG exp(i⟨⋅,X0⟩) × exp(i⟨z,r1Rd + µV G⟩T)(1 − iν⟨θ,z⟩ + 1
2
ν⟨z,Σz⟩)
−1/ν
,
I[z] ∈ δX
NIG exp(i⟨⋅,X0⟩)×
exp(i⟨z,r1Rd + µV G⟩T)(1 − iν⟨θ,z⟩ + 1
2
ν⟨z,Σz⟩)
−1/ν
,I[z] ∈ δX
Table 3: Strip of regularity of ΦXT
(⋅): δX (Eberlein, Glau, and Papapantoleon 2010).
Model δX
GBM Rd
VG {R ∈ Rd
,(1 + ν⟨θ,R⟩ − 1
2
ν⟨R,ΣR⟩) 0}
NIG {R ∈ Rd
,(α2
− ⟨(β − R),∆(β − R)⟩) 0}
Notation:
Σ: Covariance matrix related to the Geometric Brownian Motion (GBM)
model.
ν,θ,σ,Σ: Variance Gamma (VG) model parameters.
α,β,δ,∆: Normal Inverse Gaussian (NIG) model parameters.
µV G,i = 1
ν log (1 − 1
2σ2
i ν − θiν), i = 1,...,d
µNIG,i = −δ (
√
α2 − β2
i −
√
α2 − (βi + 1)2
), i = 1,...,d
36

Strip of Regularity: 2D Illustration
Figure 3.1: Strip of regularity of characteristic functions, δX,
(left) VG: σ = (0.2,0.8), θ = (−0.3,0),ν = 0.257
(right) NIG: α = 10,β = (−3,0),δ = 0.2,∆ = I2.
−8 −6 −4 −2 0 2 4 6
R1
−4
−3
−2
−1
0
1
2
3
4
R
2
−15 −10 −5 0 5 10
R1
−10.0
−7.5
−5.0
−2.5
0.0
2.5
5.0
7.5
10.0
R
2
There is no guidance in the litterature on how to choose the
damping parameters, R, to improve error convergence of quadrature
methods in the Fourier space.
37

Effect of the Damping Parameters: 2D Illustration
u1
-1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0
u2
-1.0
-0.75
-0.5
-0.25
0.0
0.25
0.5
0.75
1.0
g(u
1
,
u
2
)
0.0
200.0
400.0
600.0
800.0
u1
-1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0
u2
-1.0
-0.75
-0.5
-0.25
0.0
0.25
0.5
0.75
1.0
g(u
1
,
u
2
)
2.0
4.0
6.0
8.0
10.0
u1
-1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0
u2
-1.0
-0.75
-0.5
-0.25
0.0
0.25
0.5
0.75
1.0
g(u
1
,
u
2
)
3.0
4.0
5.0
6.0
u1
-1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0
u2
-1.0
-0.75
-0.5
-0.25
0.0
0.25
0.5
0.75
1.0
g(u
1
,
u
2
)
-5000.0
0.0
5000.0
10000.0
15000.0
20000.0
Figure 3.2: Effect of the damping parameters on the shape of the integrand in case of
basket put on 2 stocks under VG: σ = (0.4,0.4), θ = (−0.3,−0.3),ν = 0.257 (top left)
R = (0.1,0.1) (top right) R = (1,1), (bottom left) R = (2,2), (bottom right)
R = (3.3,3.3). 38

Error Discussion
Conclusions
Conclusions
38

Optimal Damping Rule: 1D Motivation
Figure 3.3: (left) Shape of the integrand w.r.t the damping parameter, R.
(right) Convergence of relative quadrature error w.r.t. number of quadrature
points, using Gauss-Laguerre quadrature for the European put option under
VG: S0 = K = 100,r = 0,T = 1,σ = 0.4,θ = −0.3,ν = 0.257.
0 1 2 3 4 5
u
0
5
10
15
20
g(u)
R = 1
R = 2
R = 3
R = 4
101
Number of quadrature points
10−5
10−4
10−3
10−2
10−1
Relative
error
R = 1
R = 2
R = 3
R = 4

Optimal Damping Rule: Characterization
The analysis of the quadrature error can be performed through two representations:
1 Error estimates based on high-order derivatives for a smooth function g:
▸ (-) High-order derivatives are usually challenging to estimate and control.
▸ (-) Will result in a complex rule for optimally choosing the damping parameters.
2 Error estimates valid for functions that can be extended holomorphically into the
complex plane
▸ (+) Corresponds to the case in Eq (16).
▸ (+) Will result in a simple rule for optimally choosing the damping parameters.
Error Estimate Based on Contour Integration
∣EQN
[g]∣ = ∣
1
2πi
∮
C
KN (z)g(z)dz∣ ≤
1
2π
sup
z∈C
∣g(z)∣∮
C
∣KN (z)∥dz∣, (17)
with KN (z) =
HN (z)
πN (z)
, HN (z) = ∫
b
a
ρ(x)
πN (z)
z − x
dx
Extending (17) to the multiple dimensions is straightforward using tensorization.
The upper bound on sup
z∈C
∣f(z)∣ is independent of the quadrature method.
Notation:
The quadrature error: EQN
[g] ∶= ∫
b
a g(x)ρ(x)dx − ∑N
k=1 g (xk)wk; (a,b can be infinite)
C: a contour containing the interval [a,b] within which g(z) has no singularities.
πN (z): the roots of the orthogonal polynomial related to the considered quadrature.
40

Optimal Damping Heuristic Rule
We propose a heuristic optimization rule for choosing the damping
parameters
R∗
∶= R∗
(Θm,Θp) = arg min
R∈δV
∥g(u;R,Θm,Θp)∥∞, (18)
where R∗
∶= (R∗
1,...,R∗
d) denotes the optimal damping parameters.
For the considered models, the integrand attains its maximum at
the origin point u = 0Rd ; thus solving (18) is reformulated to a
simpler optimization problem
R∗
= arg min
R∈δV
g(0Rd ;R,Θm,Θp).
We denote by R the numerical approximation of R∗
using
interior-point method.

Error Discussion
Conclusions
Conclusions
41

Effect of the Optimal Damping Rule on ASGQ
Figure 3.4: GBM model with anisotropic parameters: Convergence of the
relative quadrature error w.r.t. number of the ASGQ points for different
damping parameter values. (left) Basket put, (right) Call on min
100
101
102
103
104
10-4
10-3
10-2
10-1
100
101
102
Relative
Error
100
101
102
103
10-4
10-3
10-2
10-1
100
101
Relative
Error

Reference values computed by MC method using M = 109
samples.
The used parameters are based on the literature on model
calibration (Healy 2021). 42

Effect of the Optimal Damping Rule on ASGQ
Figure 3.5: VG model with anisotropic parameters: Convergence of the
relative quadrature error w.r.t. number of the ASGQ points for different
damping parameter values. (left) Basket put, (right) call on min
100
101
102
103
104
10-3
10-2
10-1
100
Relative
Error
100
101
102
103
10-4
10-3
10-2
10-1
100
Relative
Error

samples.
calibration (Aguilar 2020). 43

Comparison of Fourier approach against MC
Table 4: Comparison of the Fourier approach against the MC method for the
European basket put and call on min under the VG model.
Example d Relative Error CPU Time Ratio
Basket put under VG (isotropic) 4 3 × 10−4
4.7%
Basket put under VG
(anisotropic)
4 4 × 10−4
5.2%
Call on min under VG (isotropic) 4 6 × 10−4
0.57%
Call on min under VG
(anisotropic)
4 9 × 10−4
0.56%
Basket put under VG (isotropic) 6 8 × 10−4
42.6%
Basket put under VG
(anisotropic)
6 5 × 10−3
11%
Call on min under VG (isotropic) 6 2 × 10−3
23%
Call on min under VG
(anisotropic)
6 3 × 10−3
1.3%
CPU times provided in Table 4 are obtained by averaging the CPU time of
10 runs. CPU Time Ratio = CP U(Quadrature)+CP U(Optimization)
CP U(MC)
× 100.
samples.
calibration (Aguilar 2020).
44

Error Discussion
Conclusions
Conclusions
44

Conclusions
1 We propose a rule for the choice of the damping parameters which
accelerates the convergence of the numerical quadrature for
Fourier-based option pricing.
2 We used adaptivity and sparsification techniques to alleviate the
curse of dimensionality.
3 Our Fourier appraoch combined with the optimal damping rule
and adaptive sparse grids quadrature achieves substantial
computational gains compared to the MC method for multi-asset
options under GBM and Lévy models.
Christian Bayer, Chiheb Ben Hammouda, Antonis Papapantoleon,
Michael Samet, and Raúl Tempone. “Optimal Damping with
Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of
Multi-Asset Options in Lévy Models”. In: arXiv preprint
arXiv:2203.08196 (2022)
45

Related References
Thank you for your attention!
[1] C. Bayer, C. Ben Hammouda, R. Tempone. Numerical Smoothing with
Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for
Efficient Option Pricing, arXiv:2111.01874 (2021). Accepted in
Quantitative Finance (2022).
[2] C. Bayer, C. Ben Hammouda, R. Tempone. Multilevel Monte Carlo
combined with numerical smoothing for robust and efficient option pricing
and density estimation, arXiv:2003.05708 (2022).
[3] C. Bayer, C. Ben Hammouda, A. Papapantoleon, M. Samet, R. Tempone.
Optimal Damping with Hierarchical Adaptive Quadrature for Efficient
Fourier Pricing of Multi-Asset Options in Lévy Models, arXiv:2203.08196
(2022).
[4] C. Bayer, C. Ben Hammouda, R. Tempone. Hierarchical adaptive sparse
grids and quasi-Monte Carlo for option pricing under the rough Bergomi
model, Quantitative Finance, 2020.
46

Assumptions
Assumption 3.1
There are positive rdvs Cp with finite moments of all orders such that
∀N ∈ N, ∀`1,...,`p ∈ {0,...,2N
− 1} ∶
R
R
R
R
R
R
R
R
R
R
R
R
∂p
XN
T
∂XN
`1
⋯∂XN
`p
R
R
R
R
R
R
R
R
R
R
R
R
≤ Cp a.s.
this means that
∂pXN
T
∂XN
`1
⋯∂XN
`p
= Ô
P(1).
Assumption 3.1 is natural because it is fulfilled if the diffusion
coefficient b(⋅) is smooth; this situation is valid for many option pricing
models.

Assumptions
Assumption 3.2
For any p ∈ N we obtaina
(
∂XN
T
∂y
(Z−1,ZN
))
−p
= Ô
P(1).
a
y ∶= z−1
In (Bayer, Ben Hammouda, and Tempone 2022), we show
sufficient conditions where this assumption is valid. For instance,
Assumption 3.2 is valid for
▸ one-dimensional models with a linear or constant diffusion.
▸ multivariate models with a linear drift and constant diffusion,
including the multivariate lognormal model (see (Bayer,
Siebenmorgen, and Tempone 2018)).
There may be some cases in which Assumption 3.2 is not fulfilled,
e.g., XT = W2
T . However, our method works well in such cases
since we have g(XT ) = G(y2
1).

References I
[1] Jean-Philippe Aguilar. “Some pricing tools for the Variance
Gamma model”. In: International Journal of Theoretical and
Applied Finance 23.04 (2020), p. 2050025.
[2] 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.
“Numerical Smoothing with Hierarchical Adaptive Sparse Grids
and Quasi-Monte Carlo Methods for Efficient Option Pricing”.
In: arXiv preprint arXiv:2111.01874 (2021). Accepted in
Quantitative Finance Journal (2022).
49

References II
“Multilevel Monte Carlo combined with numerical smoothing for
robust and efficient option pricing and density estimation”. In:
[5] 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.
[6] Christian Bayer et al. “Optimal Damping with Hierarchical
Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset
Options in Lévy Models”. In: arXiv preprint arXiv:2203.08196
(2022).
[7] Chiheb Ben Hammouda, Nadhir Ben Rached, and Raúl Tempone.
“Importance sampling for a robust and efficient multilevel Monte
Carlo estimator for stochastic reaction networks”. In: Statistics
and Computing 30.6 (2020), pp. 1665–1689.
50

References III
[8] Chiheb Ben Hammouda, Alvaro Moraes, and Raúl Tempone.
“Multilevel hybrid split-step implicit tau-leap”. In: Numerical
Algorithms 74.2 (2017), pp. 527–560.
[9] Peng Chen. “Sparse quadrature for high-dimensional integration
with Gaussian measure”. In: ESAIM: Mathematical Modelling
and Numerical Analysis 52.2 (2018), pp. 631–657.
[10] K Andrew Cliffe et al. “Multilevel Monte Carlo methods and
applications to elliptic PDEs with random coefficients”. In:
Computing and Visualization in Science 14.1 (2011), p. 3.
[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.
51

References IV
[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.
[13] Oliver G Ernst, Bjorn Sprungk, and Lorenzo Tamellini.
“Convergence of sparse collocation for functions of countably
many Gaussian random variables (with application to elliptic
PDEs)”. In: SIAM Journal on Numerical Analysis 56.2 (2018),
pp. 877–905.
[14] Thomas Gerstner and Michael Griebel. “Numerical integration
using sparse grids”. In: Numerical algorithms 18.3 (1998),
pp. 209–232.
[15] Alexander D Gilbert, Frances Y Kuo, and Ian H Sloan.
“Preintegration is not smoothing when monotonicity fails”. In:
52

References V
[16] Michael B Giles. “Improved multilevel Monte Carlo convergence
using the Milstein scheme”. In: Monte Carlo and Quasi-Monte
Carlo Methods 2006. Springer, 2008, pp. 343–358.
[17] Michael B Giles. “Multilevel Monte Carlo path simulation”. In:
Operations Research 56.3 (2008), pp. 607–617.
[18] Michael B Giles, Kristian Debrabant, and Andreas Rößler.
“Numerical analysis of multilevel Monte Carlo path simulation
using the Milstein discretisation”. In: arXiv preprint
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[19] Michael B Giles and Abdul-Lateef Haji-Ali. “Multilevel Path
Branching for Digital Options”. In: arXiv preprint
arXiv:2209.03017 (2022).
53

References VI
[20] Michael B Giles, Tigran Nagapetyan, and Klaus Ritter.
“Multilevel Monte Carlo approximation of distribution functions
and densities”. In: SIAM/ASA Journal on Uncertainty
Quantification 3.1 (2015), pp. 267–295.
[21] Andreas Griewank et al. “High dimensional integration of kinks
and jumps—smoothing by preintegration”. In: Journal of
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[22] Abdul-Lateef Haji-Ali, Jonathan Spence, and Aretha Teckentrup.
“Adaptive Multilevel Monte Carlo for Probabilities”. In: arXiv
preprint arXiv:2107.09148 (2021).
[23] Jherek Healy. “The Pricing of Vanilla Options with Cash
Dividends as a Classic Vanilla Basket Option Problem”. In:
54

References VII
[24] Stefan Heinrich. “Multilevel monte carlo methods”. In:
International Conference on Large-Scale Scientific Computing.
Springer. 2001, pp. 58–67.
[25] Ahmed Kebaier et al. “Statistical Romberg extrapolation: a new
variance reduction method and applications to option pricing”.
In: The Annals of Applied Probability 15.4 (2005), pp. 2681–2705.
[26] J Lars Kirkby. “Efficient option pricing by frame duality with the
fast Fourier transform”. In: SIAM Journal on Financial
Mathematics 6.1 (2015), pp. 713–747.
[27] Harald Niederreiter. Random number generation and
quasi-Monte Carlo methods. Vol. 63. Siam, 1992.
[28] Dirk Nuyens. The construction of good lattice rules and
polynomial lattice rules. 2014.
55

References VIII
[29] Dirk Nuyens and Yuya Suzuki. “Scaled lattice rules for
integration on Rd
achieving higher-order convergence with error
analysis in terms of orthogonal projections onto periodic spaces”.
In: arXiv preprint arXiv:2108.12639 (2021).
[30] Ian H Sloan. “Lattice methods for multiple integration”. In:
Journal of Computational and Applied Mathematics 12 (1985),
pp. 131–143.
56

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