MCQMC_talk_Chiheb_Ben_hammouda.pdf

Quasi-Monte Carlo & Multilevel Monte Carlo Methods
Combined with Numerical Smoothing
for Efficient Option Pricing and Density Estimation
Chiheb Ben Hammouda
Christian Bayer Raúl Tempone
Center for Uncertainty
Quantification
Center for Un
Quantification
nter for Uncertainty Quantification Logo Lock-up
15th International Conference on Monte Carlo and Quasi-Monte
Carlo Methods in Scientific Computing, Linz, Austria
July 20, 2022

Related Manuscripts to the Talk
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), to appear 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)
1

1 Motivation and Framework
2 The Numerical Smoothing Idea
3 The Smoothness Theorem
4 Error Discussion for rQMC and MLMC combined with Numerical
Smoothing
5 Numerical Experiments and Results
6 Conclusions

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., call options
(ST − K)+
, digital options 1ST >K, . . .
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)

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, distribution functions, . . . ).
▸ 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)
▸ One/multi-dimensional geometric Brownian motion (GBM) process.
▸ Multi-dimensional 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.
3

How Does Regularity Affect
QMC and MLMC Performance?
3

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 (f) =
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.
4

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. 5

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
6

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. Otherwise β < γ ⇒ worst-case complexity.
Higher order schemes, E.g., the Milstein scheme, may lead to better complexities
even for non-Lipschitz observables (Giles, Debrabant, and Rößler 2013; Giles 2015).
However,
▸ For moderate/high-dimensional dynamics, coupling issues may arise and the scheme
becomes computationally expensive.
▸ Deterioration of the robustness of the MLMC estimator because the kurtosis explodes
as ∆t` decreases: O (∆t−1
` ) compared with O (∆t
−1/2
` ) for Euler-Maruyama without
smoothing (Giles, Nagapetyan, and Ritter 2015) and O (1) in (Bayer, Ben Hammouda,
and Tempone 2022) (see next slides).
7

How Does Regularity Affect MLMC Robustness?
/ For non-lipschitz payoffs: The kurtosis, κ` ∶=
E[(Y`−E[Y`])4
]
(Var[Y`])2 is of
O(∆t
−1/2
` ) for Euler scheme and O(∆t−1
` ) for the Milstein scheme.
Large kurtosis problem: (Giles 2015; 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.
8

Alternative Smoothing Techniques
for QMC and ASGQ
Bias-free mollification (E.g., (Bayer, Siebenmorgen, and Tempone
2018; Bayer, Ben Hammouda, and Tempone 2020)) by taking
conditional expectations or exact integration over subset of
integration variables;
(-) Not always possible.
Mollification: E.g., by convolution with Gaussian kernel or
manually;
(-) Additional error that may scale exponentially with the
dimension.
Mapping the problem to the frequency space (E.g., (Bayer,
Ben Hammouda, Papapantoleon, Samet, and Tempone 2022)):
Better regularity compared to the physical space;
(-) Applicable when the Fourier transform of the density function is
available and cheap to compute.
Preintegration techniques (E.g., (Griewank, Kuo, Leövey, and Sloan
2018; Gilbert, Kuo, and Sloan 2021)).

Alternative Smoothing Techniques
for MLMC
Implicit smoothing based on conditional expectation combined with
the Milstein scheme: (E.g, (Giles, Debrabant, and Rößler 2013));
(-) Not always possible and drawbacks of Milstein scheme.
Parametric smoothing: carefully constructing a regularized version
of the observable (E.g, (Giles, Nagapetyan, and Ritter 2015));
Malliavin calculus integration by parts: (E.g, (Altmayer and
Neuenkirch 2015)): splitting the payoff function into a smooth
component (treated by standard MLMC) and a compactly
supported discontinuous part (treated via Malliavin MLMC).
Adaptive sampling: (E.g, (Haji-Ali, Spence, and Teckentrup 2021))

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 )
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∶=∆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, 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).
11

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
.
12

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.
▸ y∗
1(y−1,z
(1)
−1 ,...,z
(d)
−1 ): the approximated discontinuity location via root finding;
▸ 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.
" Our approach is different from previous MLMC techniques which smooth out
only at the final step with respect to ∆W ⇒ smoothing effect vanishes as ∆t → 0.
13

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
⎤
⎥
⎥
⎥
⎦
, (5)
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
. (6)
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))∣], (7)
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.
16

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. 17

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, (8)
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
))], (9)
where ZN
∶= (Zn,k)n=0,...,N, k=0,...2n−1.
18

The Smoothness Theorem
Theorem 3.1 ((Bayer, Ben Hammouda, and Tempone 2022))
Assume that XN
T , defined by (8), satisfies Assumptions 6.1 and 6.2.a
Then, for any p ∈ N and indices n1,...,np and k1,...,kp (satisfying
0 ≤ kj < 2nj ), the function HN
defined in (9) 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 6.1 and 6.2 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.
19

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 (9)).
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.
(10)
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.
20

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¶
Error II: numerical smoothing error of O(M
−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
, (11)
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.
21

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¶
Error II: numerical smoothing error of O(M
−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,`
)
⎞
⎠
.
(12)
" 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 `
22

Multilevel Monte Carlo with Numerical Smoothing:
Variance Decay, Complexity and Robustness
Notation: I`:= I`(y`
−1,z
(1),`
−1 ,...,z
(d),`
−1 ): the level ` Euler approximation of I, computed
with: step size of ∆t`; MLag,` Laguerre quadrature points; TOLNewton,` as the tolerance of
the Newton method at level `.
Corollary 4.1 ((Bayer, Hammouda, and Tempone 2022))
Under Assmuptions 6.1 and 6.2, V` ∶= Var[I` − I`−1] = O (∆t`) compared with O (∆t
1/2
` )
for MLMC without smoothing.
Under Assmuptions 6.1 and 6.2, the complexity of MLMC combined with numerical
smoothing using Euler discretization is O (TOL−2
(log(TOL))2
) compared with
O (TOL−2.5
) for MLMC without smoothing.
Let κ` be the kurtosis of the random variable Y` ∶= I` − I`−1, then under Assmuptions 6.1
and 6.2, we obtain κ` = O (1) compared with O (∆t
−1/2
` ) for MLMC without smoothing.

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 5.1: Comparison of the 95% statistical error convergence for rQMC
with/out numerical smoothing. (a) Digital option under Heston, (b) digital
option under GBM.
24

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 5.2: Call option under GBM with N = 4: The relative numerical
smoothing error 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.
25

Digital Option under the Heston Model:
MLMC Without Smoothing
0 1 2 3 4 5 6
-8
-6
-4
-2
0 1 2 3 4 5 6
-10
-5
0
0 1 2 3 4 5 6
2
4
6
0 1 2 3 4 5 6
50
100
150
200
kurtosis
Figure 5.3: Digital option under Heston: Convergence plots for MLMC
without smoothing.
26

MLMC With Numerical Smoothing
0 1 2 3 4 5 6 7
-15
-10
-5
0 1 2 3 4 5 6 7
-10
-5
0
0 1 2 3 4 5 6 7
2
4
6
8
0 1 2 3 4 5 6 7
8
9
10
11
12
kurtosis
Figure 5.4: Digital option under Heston: Convergence plots for MLMC with
numerical smoothing.
27

Numerical Complexity Comparison
10-4
10-3
10-2
10-1
TOL
1e-04
1e-02
10
2
10
3
E[W]
MLMC without smoothing
TOL
-2.5
MLMC+ Numerical smoothing
TOL-2
log(TOL)2
Figure 5.5: Digital option under Heston: Comparison of the numerical
complexity of i) standard MLMC, and ii) MLMC with numerical smoothing.
Numerical computational cost improved from O(TOL−5/2
) without smoothing
to O(TOL−2
) with numerical smoothing.
28

Density estimation in the Heston model:
MLMC With Numerical Smoothing
0 2 4 6 7
-10
-5
0
0 2 4 6 7
-10
-5
0
0 2 4 6 7
2
4
6
8
0 2 4 6 7
10
15
20
kurtosis
Figure 5.6: Density of Heston: Convergence plots for MLMC with numerical
smoothing for computing the density ρX(T )(u) at u = 1, where X follows the
Heston dynamics.
29

Conclusions
1 We propose a numerical smoothing approach that can be combined with
QMC and MLMC (also ASGQ) estimators for efficient option pricing and
density estimation.
2 The numerical smoothing improves the regularty structure of the problem
⇒ optimal convergence rates for QMC.
3 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 variance decay rate ⇒ improvement of MLMC complexity
from O (TOL−2.5
) to O (TOL−2
log(TOL)2
)
" without the need to use higher order schemes such as Milstein scheme.
4 When estimating densities: Compared to the smoothing strategy used in
(Giles, Nagapetyan, and Ritter 2015), the pointwise error of our numerical
smoothing approach does not increases exponentially with respect to the
dimension of state vector.
5 Our approach is simple and generic, and can be extended:
▸ Many model dynamics and payoff structures;
▸ Computing financial Greeks;
▸ Computing distribution functions;
▸ Risk estimation.
30

Related References
[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), to appear in
Quantitative Finance.
[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.
Thank you for your attention!
31

Assumptions
Assumption 6.1
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 6.1 is natural because it is fulfilled if the diffusion
coefficient b(⋅) is smooth; this situation is valid for many option pricing
models.

Assumptions
Assumption 6.2
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 6.2 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 6.2 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] Martin Altmayer and Andreas Neuenkirch. “Multilevel Monte
Carlo quadrature of discontinuous payoffs in the generalized
Heston model using Malliavin integration by parts”. English. In:
SIAM Journal on Financial Mathematics : SIFIN 6.1 (2015).
Online-Ressource, pp. 22–52. doi: 10.1137/130933629. url:
https://madoc.bib.uni-mannheim.de/37052/.
[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), to appear in
Quantitative Finance Journal (2022).
34

References II
“Multilevel Monte Carlo combined with numerical smoothing for
robust and efficient option pricing and density estimation”. In:
arXiv preprint arXiv:2003.05708 (2022).
[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 L/’evy 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.
35

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] 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.
[10] 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.
[11] Alexander D Gilbert, Frances Y Kuo, and Ian H Sloan.
“Preintegration is not smoothing when monotonicity fails”. In:
arXiv preprint arXiv:2112.11621 (2021).
36

References IV
[12] 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.
[13] Michael B Giles. “Multilevel Monte Carlo methods”. In: Acta
Numerica 24 (2015), pp. 259–328.
[14] Michael B Giles. “Multilevel Monte Carlo path simulation”. In:
Operations Research 56.3 (2008), pp. 607–617.
[15] Michael B Giles, Kristian Debrabant, and Andreas Rößler.
“Numerical analysis of multilevel Monte Carlo path simulation
using the Milstein discretisation”. In: arXiv preprint
arXiv:1302.4676 (2013).
37

References V
[16] 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.
[17] Andreas Griewank et al. “High dimensional integration of kinks
and jumps—smoothing by preintegration”. In: Journal of
Computational and Applied Mathematics 344 (2018), pp. 259–274.
[18] Abdul-Lateef Haji-Ali, Jonathan Spence, and Aretha Teckentrup.
“Adaptive Multilevel Monte Carlo for Probabilities”. In: arXiv
preprint arXiv:2107.09148 (2021).
[19] Stefan Heinrich. “Multilevel monte carlo methods”. In:
International Conference on Large-Scale Scientific Computing.
Springer. 2001, pp. 58–67.
38

References VI
[20] 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.
[21] Harald Niederreiter. Random number generation and
quasi-Monte Carlo methods. Vol. 63. Siam, 1992.
[22] Dirk Nuyens. The construction of good lattice rules and
polynomial lattice rules. 2014.
[23] 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).
[24] Ian H Sloan. “Lattice methods for multiple integration”. In:
Journal of Computational and Applied Mathematics 12 (1985),
pp. 131–143.
39

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