Presentation.pdf

Optimal Damping with Adaptive Quadrature
for Efficient Fourier Pricing of Multi-Asset Options
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
SIAM Conference on Financial Mathematics and Engineering
(FM23), Philadelphia, U.S.
June 6, 2023
0

1 Motivation and Framework
2 Optimal Damping with Hierarchichal Adaptive Quadrature
(ODHAQ)
3 Numerical Experiments and Results
4 Conclusions
0

Motivation
Aim: Pricing multi-asset options: compute E[P(XT)]
▸ P(⋅) is a payoff function (typically non-smooth).
▸ XT is a d-dimensional vector of asset prices at time T.
Standard Monte Carlo (MC)
▸ (-) Slow rate of convergence: O(M− 1
2 ).
▸ (+) Rate independent of the dimension and the regularity of the
integrand.
Deterministic quadrature methods:
▸ Tensor Product (TP): Rate of convergence O (M− r
d ) with r
being the order of bounded total derivatives of the integrand.
▸ Adaptive sparse grids quadrature (ASGQ): Rate of
convergence, O(M− p
2 ) (Chen 2018), where p > 1 is independent
from the problem dimension, and related to the order of bounded
weighted mixed (partial) derivatives of the integrand.
Better regularity of the integrand’s transform in the Fourier space
than the one in the physical space
⇒ Map the problem to the frequency space when applicable.
Notation: M: number of MC/quadrature points. 1

Popular Fourier Pricing Methods
1 The whole option price is Fourier transformed w.r.t. the strike
variable (Carr and Madan 1999).
▸ A damping factor w.r.t. the strike is introduced to ensure integrability.
▸ Challenging to generalize to the multidimensional setting.
▸ The derivation must be performed separately for each problem.
2 Using the Plancherel-Parseval Theorem and the generalized inverse
Fourier transform w.r.t the log-asset price (Lewis 2001).
▸ Highly modular pricing method.
▸ A damping parameter w.r.t. the stock price variable is introduced to
ensure integrability.
▸ Easy to extend to the multivariate case (Eberlein, Glau, and
Papapantoleon 2010).
3 Replacing the probability density function by its Fourier cosine
series expansion (Fang and Oosterlee 2009)
▸ Truncation parameter of the integration domain is introduced.
▸ Cosine series coefficients of the payoff are known analytically for most
1D payoff functions, otherwise approximated numerically.
2

Fourier Pricing Formula in d Dimensions
Proposition
Let
Θm,Θp be respectively the model and payoff parameters;
̂
P(⋅): the conjugate of the Fourier transform of the payoff P(⋅);
XT : vector of log-asset prices at time T, with joint characteristic function ΦXT
(⋅);
R: vector of damping parameters ensuring integrability;
δP : strip of regularity of ̂
P(⋅); δX: strip of regularity of ΦXT
(⋅),
then the value of the option price on d stocks can be expressed as
V (Θm,Θp) = (2π)−d
e−rT
R(∫
Rd
ΦXT
(u + iR) ̂
P(u + iR)du), R ∈ δV ∶= δP ∩ δX (1)
∶= ∫
Rd
g (u;R,Θm,Θp)du
Proof.
Using the inverse generalized Fourier transform and Fubini theorems:
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
E[ei⟨u,XT⟩
] ̂
P(u)du), R ∈ δV ∶= δP ∩ δX
= (2π)−d
e−rT
R(∫
Rd
ΦXT
(u + iR) ̂
P(u + iR)du).
3

Framework: Pricing Models
Table 1: ΦXT
(⋅) for the different models. ⟨.,.⟩ is the standard dot product on Rd
.
Model ΦXT
(z)
GBM exp(i⟨z,X0⟩) × exp(i⟨z,r1Rd − 1
2
diag(Σ)⟩T − T
2
⟨z,Σz⟩),I[z] ∈ δX.
VG exp(i⟨z,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⟨z,X0⟩)×
exp(i⟨z,r1Rd + µV G⟩T)(1 − iν⟨θ,z⟩ + 1
2
ν⟨z,Σz⟩)
−1/ν
,I[z] ∈ δX
Table 2: 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 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.
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
4

Strip of Regularity: 2D Illustration
Figure 1.1: Strip of regularity of characteristic functions, δX,
(left) VG: σ = (0.2,0.8), θ = (−0.3,0),ν = 0.257, Σ = I2
(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.
5

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 1.2: 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).
6

Aim
Design a computationally efficient Fourier method to solve (1).
Challenges
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.
2 The effective treatment of the high dimensionality.
" Curse of dimensionality: cost of standard quadrature
methods grows exponentially with the number of assets.
Solution (Bayer, Ben Hammouda, Papapantoleon, Samet,
and Tempone 2022)
Parametric smoothing of the Fourier integrand via an optimized
choice of damping parameters.
Sparsification and dimension-adaptivity techniques to accelerate
the convergence of the quadrature in moderate/high dimensions.

(ODHAQ)
4 Conclusions
7

Optimal Damping Rule: 1D Motivation
Figure 2.1: (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.
−4 −2 0 2 4
u
0
5
10
15
20
g(u)
R = 1
R = 3
R = 4
R = 2.29
101
N
10−3
10−2
10−1
ε
R
R = 1
R = 3
R = 4
R = 2.29

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.
▸ (-) Might result in an optimal but complex rule for 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 (1).
▸ (+) Will result in a simple rule for optimally choosing the damping parameters.
Theorem 2.1 (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∣, (2)
with KN (z) =
HN (z)
πN (z)
, HN (z) = ∫
b
a
ρ(x)
πN (z)
z − x
dx
Extending (2) to the multiple dimensions is straightforward using tensorization.
The upper bound on sup
z∈C
∣g(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.
9

Optimal Damping Rule
We propose an optimization rule for choosing the damping
parameters
R∗
∶= R∗
(Θm,Θp) = arg min
R∈δV
∥g(u;R,Θm,Θp)∥∞, (3)
where R∗
∶= (R∗
1,...,R∗
d) denotes the optimal damping parameters.
The integrand attains its maximum at the origin point u = 0Rd ;
thus solving (3) is reformulated to a simpler optimization problem
R∗
= arg min
R∈δV
g(0Rd ;R,Θm,Θp).
R: the numerical approximation of R∗
using interior-point
method.

Quadrature Methods
Naive Quadrature operator based on a Cartesian quadrature grid
∫
Rd
g(x)ρ(x)dx ≈
d
⊗
k=1
QNk
k [g] ∶=
N1
∑
i1=1
⋯
Nd
∑
id=1
wi1 ⋯wid
g (xi1 ,...,xid
)
" Caveat: Curse of dimension: i.e., total number of quadrature
points N = ∏d
k=1 Nk.
Solution:
1 Sparsification of the grid points to reduce computational work.
2 Dimension-adaptivity to detect important dimensions of the
integrand.
Notation:
{xik
,wik
}Nk
ik=1 are respectively the sets of quadrature points and
corresponding quadrature weights for the kth dimension, 1 ≤ k ≤ d.
QNk
k [.]: the univariate quadrature operator for the kth dimension.
11

Hierarchical Quadrature - 1D case
Let m(β) ∶ N+ → N+ a strictly increasing function with m(1) = 1;
β: level of quadrature
m(β): number of quadrature points used at level β, E.g.,
m(β) = 2β
− 1
Hierarchical construction: example for level 3 quadrature Qm(3)
[g]:
Qm(3)
[g] = Qm(1)
´¹¹¹¹¸¹¹¹¹¶
∆m(1)
[g] + (Qm(2)
− Qm(1)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∆m(2)
)[g] + (Qm(3)
− Qm(2)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
∆m(3)
)[g]
Clearly ∫
Rd
g(x)ρ(x)dx =
∞
∑
β=1
∆m(β)
(g).
12

Hierarchical Sparse grids: Construction
Let β = [β1,...,βd] ∈ Nd
+, m(β) ∶ N+ → N+ an increasing function,
1 1D quadrature operators: Q
m(βk)
k on m(βk) points, 1 ≤ k ≤ d.
2 Detail operator: ∆
m(βk)
k = Q
m(βk)
k − Q
m(βk−1)
k , Q
m(0)
k = 0.
3 Hierarchical surplus: ∆m(β)
= ⊗d
k=1 ∆
m(βk)
k .
4 Hierarchical sparse grid approximation: on an index set I ⊂ Nd
QI
d [g] = ∑
β∈I
∆m(β)
[g] (4)
13

Grids Construction
Tensor Product (TP) approach: I ∶= I` = {∣∣ β ∣∣∞≤ `; β ∈ Nd
+}.
Regular sparse grids (SG): I ∶ I` = {∣∣ β ∣∣1≤ ` + d − 1; β ∈ Nd
+}
Adaptive sparse grids (ASG): Adaptive and a posteriori
construction of I = IASGQ
by profit rule
IASGQ
= {β ∈ Nd
+ ∶ Pβ ≥ T}, with Pβ =
∣∆Eβ∣
∆Wβ
:
▸ ∆Eβ = ∣Q
I∪{β}
d [g] − QI
d [g]∣;
▸ ∆Wβ = Work[Q
I∪{β}
d [g]] − Work[QI
d [g]].
Figure 2.2: 2-D Illustration (Chen 2018): Admissible index sets I (top) and
corresponding quadrature points (bottom). Left: TP; middle: SG; right: ASG .
14

(ODHAQ)
4 Conclusions
14

Effect of the Optimal Damping Rule on ASGQ
Figure 3.1: Convergence of the relative quadrature error w.r.t. number of the
ASGQ points for different damping parameter values.
(left) 4-Assets Basket put under GBM, (right) 4-Assets Call on min under VG.
100
101
102
103
104
Number of quadrature points
10-4
10-3
10-2
10-1
100
101
102
Relative
Error
100
101
102
103
Number of quadrature points
10-4
10-3
10-2
10-1
100
Relative
Error
Reference values computed by MC method using M = 109
samples.
The used parameters are based on the literature on model
calibration (Aguilar 2020; Healy 2021).
15

Comparison of our Approach against COS Method
Figure 3.2: Convergence of the relative error w.r.t. N, number of
characteristic function evaluations
(left) 1D put option under GBM, (right) 2D basket put option under GBM.
100
101
102
N
10-10
10-5
100
Relative
Error
ODHAQ
COS
100
101
102
103
104
105
N
10-4
10-3
10-2
10-1
100
Relative
Error
ODHAQ
COS

Comparison of our Approach against COS Method
Figure 3.3: Convergence of the relative error w.r.t. N, number of
characteristic function evaluations
(left) 1D put option under VG, (right) 1D put option under NIG.
100
101
102
N
10-4
10-3
10-2
10-1
100
101
Relative
Error
ODHAQ
COS
100
101
N
10-4
10-3
10-2
10-1
100
101
Relative
Error
ODHAQ
COS

Comparison of Fourier approach against MC
Table 3: Comparison of our ODHAQ 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 4 4 × 10−4
5.2%
Call on min under VG 4 9 × 10−4
0.56%
Basket put under VG 6 5 × 10−3
11%
Call on min under VG 6 3 × 10−3
1.3%
CPU Time Ratio ∶= CP U(ODHAQ)+CP U(Optimization)
CP U(MC)
× 100.
Reference values computed by MC method using M = 109
samples.
The used parameters are based on the literature on model
calibration (Aguilar 2020).
18

(ODHAQ)
4 Conclusions
18

Conclusions and Extension
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 approach 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.
4 Extension: Combine domain transformation with quasi-Monte
Carlo in the Fourier space: to scale better with high dimensions
" See Michael’s talk tomorrow
19

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

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] C. 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).
[3] Peter Carr and Dilip Madan. “Option valuation using the fast
Fourier transform”. In: Journal of computational finance 2.4
(1999), pp. 61–73.
[4] Peng Chen. “Sparse quadrature for high-dimensional integration
with Gaussian measure”. In: ESAIM: Mathematical Modelling
and Numerical Analysis 52.2 (2018), pp. 631–657.
[5] Rama Cont and Peter Tankov. Financial Modelling with Jump
Processes. Chapman and Hall/CRC, 2003.
21

References II
[6] 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.
[7] 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.
[8] Jherek Healy. “The Pricing of Vanilla Options with Cash
Dividends as a Classic Vanilla Basket Option Problem”. In:
arXiv preprint arXiv:2106.12971 (2021).
[9] Alan L Lewis. “A simple option formula for general
jump-diffusion and other exponential Lévy processes”. In:
Available at SSRN 282110 (2001).
22

References III
[10] Wim Schoutens. Lévy Processes in Finance: Pricing Financial
Derivatives. Wiley Online Library, 2003.
23

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 4: 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. 24

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