Monte-Carlo method for option pricing in sub-diffusive arithmetic models

Abstract

This paper focuses on applying the Monte Carlo approach to option pricing in markets with illiquid assets. Anomalous sub-diffusion is a well-known model for describing such markets when relatively long periods without any trading are observed. For constructing sub-diffusive models we need to replace a calendar time t with some stochastic processes S(t), which is called inverse subordinator. The inverse subordinator S(t) means first hitting time and is based on subordinator processes. In this paper, we propose to use the gamma process as a subordinator for Bashelie sub-diffusion model. Using well-known properties for gamma and inverse gamma processes we find the covariance structure of fractional Bachelier model with FBM time-changed by gamma process and then explore the asymptotic behavior of it. Then we apply the Monte-Carlo method and propose a procedure of option pricing for the Bashelie sub-diffusion model. For this aim, we use iterative schemes for simulating N scenarios of stock prices for our models. Finally, we demonstrate numerical results.