Option pricing in the illiquid markets under the mixed fractional Brownian motion model

Abstract

This paper deals with the option pricing in the illiquid markets under the mixed fractional geometric Brownian motion model with jump process. We propose a general long memory financial model, where its featuring parameters are related to demand and supply by showing also the existence of some restrictions on them. Moreover, by using the delta Hedging strategy and replicating portfolio, we obtain an integro partial differential equation (PIDE) for the option price which is solved by the spectral numerical method with suitable diagonal functions and an infinite series. In particular, by using some operational matrices and Gauss–Hermite quadrature rule, we derive a linear system of algebraic equations solved by a standard collocation method. Moreover, we study the existence and uniqueness of the solution of PIDE and prove the convergence of the numerical scheme. The applicability and efficiency of the collocation method are shown on some nontrivial numerical examples.