European option valuation under the Bates PIDE in finance: A numerical implementation of the Gaussian scheme

Fazlollah Soleymani; Ali Akgül

doi:10.3934/dcdss.2020052

Article Contents

2020, Volume 13, Issue 3: 889-909. Doi: 10.3934/dcdss.2020052

This issue Previous Article Inclusion of fading memory to Banister model of changes in physical condition Next Article Existence results of Hilfer integro-differential equations with fractional order

European option valuation under the Bates PIDE in finance: A numerical implementation of the Gaussian scheme

Fazlollah Soleymani^1, and
Ali Akgül^2, ,

1.
Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137–66731, Iran
2.
Department of Mathematics, Art and Science Faculty, Siirt University, Siirt, Turkey

^* Corresponding author
^* Corresponding author

Received: April 30, 2018

Revised: June 30, 2018

Published: December 2019

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Abstract

Models at which not only the asset price but also the volatility are assumed to be stochastic have received a remarkable attention in financial markets. The objective of the current research is to design a numerical method for solving the stochastic volatility (SV) jump–diffusion model of Bates, at which the presence of a nonlocal integral makes the coding of numerical schemes intensive. A numerical implementation is furnished by gathering several different techniques such as the radial basis function (RBF) generated finite difference (FD) approach, which keeps the sparsity of the FD methods but gives rise to the higher accuracy of the RBF meshless methods. Computational experiments are worked out to reveal the efficacy of the new procedure.

Keywords:

Mathematics Subject Classification: Primary: 91B25; Secondary: 65M22.

Citation:

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Figure 1. Results based on GRBF–FDI in Problem 4.1. Top left: Numerical solution. Top right: List plot of the numerical solution indicating the non–uniform nodes' distribution. Bottom left: Contour plot of the solution. Bottom right: The sparsity pattern of the system of ODEs' coefficient matrix

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Figure 2. Results based on GRBF–FDI in Problem 4.2. Top left: Numerical solution. Top right: List plot of the numerical solution indicating the non–uniform nodes' distribution. Bottom left: Contour plot of the solution. Bottom right: The sparsity pattern of the system of ODEs' coefficient matrix

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Figure 3. Results based on GRBF–FDI in Problem 4.3. Top left: Numerical solution. Top right: List plot of the numerical solution indicating the non–uniform nodes' distribution. Bottom left: Contour plot of the solution. Bottom right: The sparsity pattern of the system of ODEs' coefficient matrix

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Table 1. Numerical reports of call vanilla option pricing for Problem 4.1

Scheme	$ m $	$ n $	$ N $	$ k $	$ u $	$ \varepsilon $	Time (s)
SFD–EM
	20	20	400	400	8.7001	$ 1.947\times10^{-1} $	0.91
	40	25	1000	2000	8.5974	$ 2.973\times10^{-1} $	2.67
	40	40	1600	2000	8.6739	$ 2.209\times10^{-1} $	5.39
	65	45	2925	4000	8.8609	$ 3.397\times10^{-2} $	15.77
	80	50	4000	10000	8.8745	$ 2.036\times10^{-2} $	33.95
HFM–DM
	10	10	100	250	8.3465	$ 5.483\times10^{-1} $	0.45
	15	15	225	250	8.6980	$ 1.968\times10^{-1} $	0.64
	25	20	500	400	8.8601	$ 3.473\times10^{-2} $	1.12
	30	30	900	600	8.8705	$ 2.431\times10^{-2} $	2.03
	50	30	1500	2000	8.8852	$ 9.624\times10^{-3} $	5.41
	80	30	2400	5000	8.8905	$ 4.320\times10^{-3} $	13.10
GRBF–FDI
	10	10	100	NR	8.4182	$ 4.766\times10^{-1} $	0.08
	15	15	225	NR	8.7578	$ 1.370\times10^{-1} $	0.13
	25	20	500	NR	8.8513	$ 4.355\times10^{-2} $	0.14
	30	30	900	NR	8.8659	$ 2.891\times10^{-2} $	0.18
	60	30	900	NR	8.8905	$ 4.321\times10^{-3} $	0.58
	80	30	2400	NR	8.8952	$ \bf{3.305\times10^{-4}} $	1.39

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Table 2. Numerical reports of put vanilla option pricing for Problem 4.2

Scheme	$ m $	$ n $	$ N $	$ k $	$ Re(\lambda_{\max}) $	$ \epsilon $	Time (s)
SFD–EM
	20	20	400	400	-516.30	$ 2.582\times10^{-2} $	0.15
	40	25	1000	2000	-2438.80	$ 2.368\times10^{-2} $	0.99
	40	40	1600	2000	-2512.03	$ 2.109\times10^{-2} $	1.55
	65	45	2925	4000	-7041.97	$ 4.009\times10^{-3} $	5.01
	80	50	4000	10000	-10932.60	$ 2.972\times10^{-3} $	19.43
HFM–DM
	10	10	100	250	-109.47	$ 5.948\times10^{-2} $	0.10
	15	15	225	250	-298.15	$ 2.459\times10^{-2} $	0.12
	25	20	500	500	-951.58	$ 3.982\times10^{-3} $	0.22
	30	30	900	1000	-1418.65	$ 3.015\times10^{-3} $	0.51
	80	30	2400	10000	-11134.30	$ 4.722\times10^{-4} $	11.12
GRBF–FDI
	10	10	100	NR	-238.16	$ 8.173\times10^{-2} $	0.09
	15	15	225	NR	-639.62	$ 3.579\times10^{-2} $	0.10
	25	20	500	NR	-2032.56	$ 1.210\times10^{-2} $	0.15
	30	30	900	NR	-3030.1	$ 8.521\times10^{-3} $	0.19
	60	30	1800	NR	-13211.4	$ 1.746\times10^{-3} $	0.57
	80	30	2400	NR	-24023.6	$ \bf{5.879\times10^{-4}} $	1.28

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Table 3. Parameter settings for the Bates model

Descriptions	Parameters	Values
Correlation between the Brownian motions	$ \rho $	-0.5
Rate of interest	$ r $	0.03
Dividend yield	$ q $	0
Variance volatility	$ \sigma $	0.25
Mean reversal rate	$ \kappa $	2
Mean level of variance	$ \theta $	0.04
Price of strike	$ E $	100
Rate of jump arrival	$ \lambda $	0.2
Time to expiry	$ T $	0.5
Jump size log–variance	$ \hat{\sigma} $	0.4
Jump size log–mean	$ \gamma $	-0.5

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Table 4. Numerical reports of put option pricing in Problem 4.3

Scheme	$ m $	$ n $	$ N $	$ k $	$ Re(\lambda_{\max}) $	$ \epsilon $	Time (s)
SFD–EM
	10	10	100	250	-104.46	$ 4.321\times10^{-1} $	0.10
	15	15	225	500	-276.31	$ 8.815\times10^{-2} $	0.23
	25	20	500	1000	-884.68	$ 8.121\times10^{-2} $	0.74
	30	30	900	2000	-1316.01	$ 1.498\times10^{-2} $	1.78
	45	30	1350	2000	-3209.72	$ 4.817\times10^{-3} $	3.77
	60	30	1800	5000	-1316.01	$ 1.834\times10^{-3} $	7.50
	80	30	2400	5000	-10982.70	$ 1.434\times10^{-3} $	13.92
GRBF–FDI
	10	10	100	NR	-158.68	$ 5.584\times10^{-2} $	0.16
	15	15	225	NR	-435.54	$ 2.018\times10^{-2} $	0.30
	25	20	500	NR	-1381.41	$ 6.922\times10^{-3} $	0.89
	30	30	900	NR	-2054.73	$ 5.998\times10^{-3} $	1.77
	45	30	1350	NR	-4874.35	$ 2.042\times10^{-3} $	4.26
	60	30	1800	NR	-8904.53	$ 1.055\times10^{-3} $	8.26
	80	30	2400	NR	-16165.60	$ \bf{5.754\times10^{-4}} $	15.68

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Table 5. Numerical results of the AMG–STS method for Problem 4.3

$m$	$n$	$\varepsilon$ at $s=90$	$\varepsilon$ at $s=100$	$\varepsilon$ at $s=110$	$\epsilon$
17	9	$1.081\times10^{0}$	$1.577\times10^{0}$	$1.968\times10^{-1}$	$1.512\times10^{-1}$
33	17	$2.808\times10^{-2}$	$5.205\times10^{-1}$	$1.389\times10^{-1}$	$4.948\times10^{-2}$
65	33	$4.783\times10^{-3}$	$1.253\times10^{-1}$	$2.845\times10^{-2}$	$1.166\times10^{-2}$
129	65	$7.383\times10^{-3}$	$3.098\times10^{-2}$	$5.255\times10^{-3}$	$2.834\times10^{-3}$
257	129	$1.700\times10^{-5}$	$7.781\times10^{-3}$	$3.455\times10^{-3}$	$8.313\times10^{-4}$

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