# American Institute of Mathematical Sciences

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

 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

Received  May 2018 Revised  July 2018 Published  March 2019

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.

Citation: Fazlollah Soleymani, Ali Akgül. European option valuation under the Bates PIDE in finance: A numerical implementation of the Gaussian scheme. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020052
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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
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
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
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
 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
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
 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
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
 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
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
 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
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 $1.700\times10^{-5}$ $7.781\times10^{-3}$ $3.455\times10^{-3}$ $8.313\times10^{-4}$
 $m$ $n$ $\varepsilon$ 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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