Fourier_Pricing_ICCF_2022.pdf

Optimal Damping with Hierarchical Adaptive Quadrature
for Efficient Fourier Pricing of Multi-Asset Options in
Lévy Models
Center for Uncertain
Quantification
Michael Samet, michael.samet@kaust.edu.sa
Joint work with 1
Christian Bayer (WIAS Berlin)
Chiheb Ben Hammouda (RWTH Aachen)
Antonis Papapantoleon (TU Delft, FORTH)
Raúl Tempone (RWTH Aachen, KAUST)
International Conference on Computational Finance 2022, June 10, Wuppertal.
1
Bayer, Christian, et al. ”Optimal Damping with Hierarchical Adaptive Quadrature
for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models.” arXiv preprint,
2022.

Motivation
Aim: compute E[P(XT)], P(·) is a payo↵ functional (typically non-smooth),
XT is a vector of log of asset prices at final time T.
Standard Monte Carlo (MC) has slow rate of convergence, O(M
1
2 ), and
may not take advantage of the smoothness of the problem.
Adaptive sparse grids quadrature (ASGQ) has rate of convergence,
O(M
p
2 ), where p > 1 is independent from the problem dimension, and is
related to the order up to which the weighted mixed derivatives are
bounded. (Chen 2018).
Better regularity of the integrand’s transform in the Fourier space (when
applicable) than the integrand’s regularity in the physical space.
Challenges
1 For integrability purposes, several Fourier methods rely on damping
parameters whose values a↵ect considerably the performance of numerical
quadrature methods.
2 Curse of dimensionality: cost of tensor product (TP) quadrature grows
exponentially with the number of underlying assets.
M.Samet (KAUST) Efficient Fourier Pricing 2 / 26

Outline
1 Problem Setting and Pricing Framework
2 Fourier Pricing Combined with Optimal Damping and Adaptive Sparse
Grids Quadrature
3 Numerical Experiments and Results
4 Conclusion

Outline
Grids Quadrature
4 Conclusion

Problem Setting
Using the inverse generalized Fourier transform theorem, the value of the option price
on d stocks can be expressed as
V (⇥m, ⇥p) = e rT
E⇢XT
[P(XT)] = e rT
Z
Rd
P(x)⇢XT
(x)dx
= (2⇡) d
e rT
<
✓Z
Rd +iR
XT
(u) b
P(u)du
◆
, R 2 V
= (2⇡) d
e rT
<
✓Z
Rd
XT
(u + iR) b
P(u + iR)du
◆
:=
Z
Rd
g (u; R, ⇥m, ⇥p) du
XT
(u) is the joint characteristic function of XT := X1
T , . . . , Xd
T ,
Xi
T := log Si
T , i = 1, . . . , d, Si
T
d
i=1
are the prices of the underlying assets at
the maturity T. ⇢XT
is the risk neutral transition probability density function.
b
P(u) is the conjugate of the Fourier transform of the payo↵ function P(x).
R is a vector of damping parameter ensuring L1
integrability of g(.). V is the
strip of regularity of g(.).
i is the unit complex number, <(.) is a real part the argument. ⇥m, ⇥p are
respectively the model and payo↵ parameters.

Considered Payo↵s
Basket put:
Payo↵ (scaled):
P (XT ) = max 1
d
X
i=1
eXi
T , 0
!
Conjugate of the Fourier transform of the payo↵:
b
P(z) =
Qd
j=1 ( izj )
⇣
i
Pd
j=1 zj + 2
⌘, z 2 Cd
, =[z] 2 P
Call on min:
Payo↵ (scaled):
P (XT ) = max
⇣
min
⇣
eX1
T , . . . , eXd
T
⌘
1, 0
⌘
Conjugate of the Fourier transform of the payo↵:
b
P(z) =
1
⇣
i
⇣Pd
j=1 zj
⌘
1
⌘ Qd
j=1 (izj )
, z 2 Cd
, =[z] 2 P
Notation: (·): complex Gamma function, =(·): imaginary part of the argument, P : strip
of regularity of the conjugate of the payo↵ Fourier transform.

Pricing Models
Table: Discounted characteristic function of XT. XT
(·) = eih·,X0i
XT
(·), h., .i is the standard
dot product on Rd
, X is the strip of regularity of the characteristic function.
Model XT
(u)
GBM exp i
⌦
z, r1Rd
1
2 diag(⌃)
↵
T T
2 hz, ⌃zi , =[z] 2 X .
VG exp (i hz, µVG i T) 1 i⌫h✓, zi + 1
2 ⌫hz, ⌃zi
1/⌫
, =[z] 2 X .
NIG exp
⇣
i hz, µNIG i T + T
⇣p
↵2 h , i
p
↵2 h + iz, ( + iz)i
⌘⌘
,
=[z] 2 X .
, ⌃: Geometric Brownian Motion (GBM) parameters, controlling respectively the
volatility and the correlation between stocks.
⌫, ✓, , ⌃: Variance Gamma (VG) parameters, controlling respectively the peakedness,
skewness, volatility, and the correlation between stocks.
↵, , , : Normal Inverse Gaussian (NIG) parameters, controlling respectively the
peakedness, the skewness, the tail-heaviness, and the degree of correlation between
stocks.
µVG,i = r + 1
⌫ log 1 1
2
2
i ⌫ ✓i ⌫ , i = 1, . . . , d
µNIG,i = r
✓
p
↵2 2
i
q
↵2 ( i + 1)
2
◆
, i = 1, . . . , d

Strip of Regularity
To ensure the L1
integrability of g(·), a vector of damping parameters R 2 Rd
is
introduced, which must belong to V , where V = ¯P X .
Table: Strip of regularity of the conjugate of payo↵ transforms, P (Eberlein,
Glau, and Papapantoleon 2010).
Payo↵ P
Basket put {R 2 Rd
, Ri > 0 8i 2 {1, . . . , d}}
Call on min {R 2 Rd
, Ri < 0 8i 2 {1, . . . , d},
Pd
i=1 Ri < 1}
Table: Strip of regularity of the characteristic functions, X (Eberlein, Glau, and
Papapantoleon 2010).
Model X
GBM Rd
VG {R 2 Rd
, 1 + ⌫h✓, Ri 1
2 ⌫hR, ⌃Ri > 0}
NIG {R 2 Rd
, ↵2
h( R), ( R)i > 0}

Strip of Regularity: 2D Illustration
Figure: Strip of regularity of characteristic functions, X , (left) VG: = (0.2, 0.8),
✓ = ( 0.3, 0), ⌫ = 0.257 (right) NIG: ↵ = 10, = ( 3, 0), = 0.2, = I2.
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.

E↵ect of the Damping Parameters: 2D Illustration
Figure: E↵ect of the damping parameters on the shape of the integrand in case of basket
put on 2 stocks under VG: = (0.4, 0.4), ✓ = ( 0.3, 0.3), ⌫ = 0.257 (top left)
R = (0.2, 0.2) (top right) R = (1, 1), (bottom left) R = (2, 2), (bottom right) R = (3, 3).

Outline
Grids Quadrature
4 Conclusion

Methodology
1 Smoothing the Fourier integrand via a proposed heuristic
optimization rule.
2 Approximating the Fourier integral using adaptive sparse grid
quadrature (ASGQ), hence accelerating the convergence of the
numerical quadrature.

Optimal Damping Rule: 1D Motivation
We define the quadrature error as
EQN
[g] :=
Z b
a
g(x)⇢(x)dx
N
X
k=1
g (xk) wk
g : [a, b] ⇢ R 7! R (a, b can be infinite), ⇢(.) be a weight function, {xk}N
k=1,
{wk}N
k=1 are respectively quadrature nodes and quadrature weights, N is the number
of quadrature points.
Theorem: Derivative-free quadrature error estimate (Donaldson and Elliott
1972)
| EQN
[g] |=|
1
2⇡i
I
C
KN(z)g(z)dz |
1
2⇡
max
z2C
|g(z)|
I
C
|KN(z)kdz|
KN(z) =
HN(z)
⇡N(z)
, HN(z) =
Z b
a
⇢(x)
⇡N(z)
z x
dx
C is a contour containing the interval [a, b] within which g(z) has no singularities
(poles, fractional powers, branch points, discontinuities in a function or in any of its
derivatives.), ⇡N(z) are the roots of the orthogonal polynomial related to the
considered quadrature.

Optimal Damping Rule: 1D Motivation
Figure: (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, = 0.4, ✓ = 0.3, ⌫ = 0.257.

Optimal Damping Heuristic Rule
We propose a heuristic optimization rule for choosing the damping
parameters
R⇤
:= R⇤
(⇥m, ⇥p) = arg min
R2 V
kg(u; R, ⇥m, ⇥p)k1, (1)
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 (1) is reformulated to a simpler
optimization problem
R⇤
= arg min
R2 V
g(0Rd ; R, ⇥m, ⇥p).
We denote by R the numerical approximation of R⇤ using
interior-point method.

Quadrature Estimator
Qd [g] :=
d
O
i=1
Qm( i )
[g] ⌘
m( 1)
X
k1=1
· · ·
m( d )
X
kd =1
w 1
k1
· · · w d
kd
g
⇣
x 1
k1
, . . . , x d
kd
⌘
n
x i
ki
om( i )
ki =1
,
n
! i
ki
om( i )
ki =1
are respectively the sets of quadrature points and
corresponding quadrature weights.
= ( i )
d
i=1 2 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.
i Qd :=
(
Qd Q
0
d , with 0
= ei , when i > 0
Qd , otherwise
QI
d =
X
2I
Qd , Qd =
d
O
i=1
i
!
Qd

Quadrature Estimator
Table: Construction details for the quadrature methods. ` 2 N represents a given level.
T̄ 2 is a threshold value. r is the order of bounded total derivatives, p depends on the
order up to which weighted mixed derivatives are bounded (Chen 2018).
Quadrature
Method
m : N 7! N I(`) ⇢ Nd
Complexity
TP m( ) = IT P
(`) ⌘ { 2 Nd
+ : || ||1  `} O N
r
d
ASGQ m( ) = 2 1
+1,
> 1, m(1) = 1
IASGQ
⌘ 2 Nd
+ : P T̄ O N
p
2
New indices are selected iteratively, a-posteriori, based on the error versus cost-profit rule,
with a hierarchical surplus defined by P =
| E |
W
E = Q
IASGQ
[{ }
d [g] QIASGQ
d [g] (Error Contribution)
W = Work
h
Q
IASGQ
[{ }
d [g]
i
Work
h
QIASGQ
d [g]
i
(Work Contribution)

ASGQ Construction: 2D Illustration
Figure: A-posteriori construction of the index set for ASGQ (Bayer,
Ben Hammouda, and Tempone 2020).

Outline
Grids Quadrature
4 Conclusion

Setting for the Numerical Experiments
The relative quadrature error is defined as
Relative Error =
| QI
d [g] Reference Value |
Reference Value
,
where QI
d [g] is the quadrature estimator with Gauss-Laguerre measure used
to approximate the integral of g(.).
Reference values are computed by the MC method using M = 109
samples.
The used parameters are based on the literature on model calibration, GBM:
(Healy 2021), VG: (Aguilar 2020), NIG: (Kirkby 2015).
For illustration, we provide the numerical results under VG model. These
results are consistent with the results found under other pricing models,
namely, GBM and NIG, for di↵erent parameter sets and dimensions,
d 2 {2, 4, 6}.

Numerical Experiments and Results
Payo↵ Set Model Parameters
Basket
put
1 = (0.4, 0.4, 0.4, 0.4), ✓ = ( 0.3, 0.3, 0.3, 0.3), ⌫ = 0.257
Basket
put
2 = (0.2, 0.4, 0.6, 0.8), ✓ = ( 0.3, 0.2, 0.1, 0), ⌫ = 0.257
Call on
min
3 = (0.4, 0.4, 0.4, 0.4), ✓ = ( 0.3, 0.3, 0.3, 0.3), ⌫ = 0.257
Call on
min
4 = (0.2, 0.4, 0.6, 0.8), ✓ = ( 0.3, 0.2, 0.1, 0), ⌫ = 0.257
Basket
put
5
= (0.4, 0.4, 0.4, 0.4, 0.4, 0.4), ✓ = (0.3, 0.3, 0.3, 0.3, 0.3, 0.3),
⌫ = 0.257
Basket
put
6
= (0.2, 0.3, 0.4, 0.5, 0.6, 0.7), ✓ = ( 0.3, 0.2, 0.1, 0, 0.1, 0.2),
⌫ = 0.257
Call on
min
7
= (0.4, 0.4, 0.4, 0.4, 0.4, 0.4), ✓ = (0.3, 0.3, 0.3, 0.3, 0.3, 0.3),
⌫ = 0.257
Call on
min
8
= (0.2, 0.3, 0.4, 0.5, 0.6, 0.7), ✓ = ( 0.3, 0.2, 0.1, 0, 0.1, 0.2),
⌫ = 0.257
Table: Parameters setting for the numerical experiments under VG model

E↵ect of the ASGQ
Figure: Convergence of the relative quadrature error w.r.t. number of quadrature
points for TP and ASGQ methods for European 4-asset options under VG model,
when optimal damping parameters, R, are used (left) basket put: set 2 (right)
call on min: set 4
100
101
102
103
104
Number of quadrature points
10-4
10-3
10-2
10-1
100
Relative
Error
100
101
102
103
10-4
10-3
10-2
10-1
100
Relative
Error

E↵ect of the Optimal Damping Rule on ASGQ
Figure: Convergence of the relative quadrature errorr, w.r.t. number of
quadrature points, for the ASGQ method for various damping parameter values
under VG model (left) basket put: set 2 (right) call on min: set 4
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

Comparison of Fourier approach against MC
Table: Comparison of the Fourier approach against the MC method for the
European basket put and call on min under the VG model.
Payo↵ d Set Best
Quadra-
ture
Relative
Error
CPU
Time
Ratio
Basket put 4 1 ASGQ 2.6 ⇥ 10 4
4.7%
5.2%
Call on min 4 3 ASGQ 5.9 ⇥ 10 4
0.57%
0.56%
42.6%
11%
23%
1.3%
CPU times provided in Table 6 are obtained by averaging the CPU time of
10 runs. CPU Time Ratio = CPU(Quadrature)+CPU(Optimization)
CPU(MC) ⇥ 100.

Outline
Grids Quadrature
4 Conclusion

Conclusion
We proposed a heuristic rule for a choice of the damping parameters
which improves the convergence of the numerical quadrature.
We used adaptive sparse grids quadrature to approximate the Fourier
integral in order to alleviate the curse of dimensionality.
Our Fourier method combined with the optimal damping rule and
adaptive sparse grids quadrature achieves substantial computational
gains compared to the MC method for basket and rainbow options
under GBM and in Lévy models.
More details can be found in 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).

References I
[1] 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).
[2] Christian Bayer, Chiheb Ben Hammouda, and Raul Tempone
Numerical smoothing with hierarchical adaptive sparse grids and
quasi-Monte Carlo methods for efficient option pricing. arXiv preprint
arXiv:2111.01874, 2021.
[3] P Chen Sparse quadrature for high-dimensional integration with
Gaussian measure,ESAIM: Mathematical Modelling and Numerical
Analysis, 2018.

References II
[4] Ernst Eberlein, Kathrin Glau, and Antonis Papapantoleon Ernst
Eberlein, Kathrin Glau, and Antonis Papapantoleon. Analysis of
Fourier transform valuation formulas and applications, Applied
Mathematical Finance, 17(3):211–240, 2010.
[5] David Elliott and P.D. Tuan Asymptotic estimates of Fourier
coefficients,SIAM Journal on Mathematical Analysis, 5(1):1–10, 1974.
[6] Alan L. Lewis A simple option formula for general jump-di↵usion and
other exponential lévy processes,SSRN 282110, 2001.
[7] J. Lars. Kirkby,Efficient option pricing by frame duality with the fast
Fourier transform,SIAM Journal on Financial Mathematics,
6(1):713–747, 2015.
[8] Wim Schoutens,Lévy Processes in Finance: Pricing Financial
Derivatives,Wiley Online Library, 2003.

Thank you
for your attention!

Fourier_Pricing_ICCF_2022.pdf

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