Numerical solution of systems of partial integral differential equations with application to pricing options

Abstract

We introduce and analyze a strongly stable numerical method designed to yield good performance under challenging conditions of irregular or mismatched initial data for solving systems of coupled partial integral differential equations (PIDEs). Spatial derivatives are approximated using second order central difference approximations by treating the mixed derivative terms in a special way. The integral operators are approximated using one and two–dimensional trapezoidal rule on an equidistant grid. Computational complexity of the method for solving large systems of PIDEs is discussed. A detailed treatment for the consistency, stability, and convergence of the proposed method is provided. Two asset American option under regime–switching with jump–diffusion model when solved using a penalty term, leads to a system of two dimensional PIDEs with mixed derivatives. This model involves double probability density function which brings more challenges to the numerical solution in already a complicated partial integral differential equation. The complexity of the dense jump probability generator, the nonlinear penalty term and the regime–switching terms are treated efficiently, while maintaining the stability and convergence of the method. The impact of the jump intensity and other parameters is shown in the graphs. Numerical experiments are performed to demonstrated efficiency, accuracy, and reliability of the proposed approach.

Partial differential integral equation model for pricing American option under multi state regime switching with jumps

“…We define the set, where denotes the exercise boundary and the payoff function is denoted by q ( S 1 , S 2 ). For any regime α = 1, 2, …, m o , and asset prices S 1 and S 2 , the option values V α ( S 1 , S 2 , t ) in the regime α at time t , satisfies the following free boundary value problems, (see References [26, 28] for details) …”

“…Detailed derivation of the spatial discretization matrix A is given in Yousuf et al [28]. Details of the integral operator approximation can be seen in Reference [26]. The differential and integral operators discretization yields the following initial‐value problem: where A is L ⋅ M × L ⋅ M block tridiagonal matrix and F ( u , t ) is the nonlinear forcing term obtained by the discretization of (2.10) for penalty method and (2.18) for rationality parameter approach.…”

“…Detailed derivation of the spatial discretization matrix A is given in Yousuf et al [28]. Details of the integral operator approximation can be seen in Reference [26]. The differential and integral operators discretization yields the following initial-value problem:…”

“…where 𝜕 𝛼 denotes the exercise boundary and the payoff function is denoted by q(S 1 , S 2 ). For any regime 𝛼 = 1, 2, … , m o , and asset prices S 1 and S 2 , the option values V 𝛼 (S 1 , S 2 , t) in the regime 𝛼 at time t, satisfies the following free boundary value problems, (see References [26,28] for details)…”

Numerical analysis of the European and American options with the SPH method

“…Several numerical studies have been done to predict the European/American options. In most cases, the numerical schemes used are based on the finite difference method: (Acevedo and Lelièvre, 2018;Akpan and Fatokun, 2015;Anwar and Andallah, 2018;Company et al, 2009;Dilloo and Tangman, 2017;Khodayari and Ranjbar, 2018;Kiyoumarsi, 2018;Koleva and Vulkov, 2016;Matus et al, 2017;Rao and Manisha, 2018;Vahdati et al, 2018;Yousuf, 2018;Phaochoo et al, 2016;Zhang et al, 2016).…”