Fractional Black-Scholes equations and their robust numerical simulations

Abstract

Conventional partial differential equations under the classical Black-Scholes approach have been extensively explored over the past few decades in solving option pricing problems. However, the underlying Efficient Market Hypothesis (EMH) of classical economic theory neglects the effects of memory in asset return series, though memory has long been observed in a number financial data. With advancements in computational methodologies, it has now become possible to model different real life physical phenomenons using complex approaches such as, fractional differential equations (FDEs). Fractional models are generalised models which based on literature have been found appropriate for explaining memory effects observed in a number of financial markets including the stock market. The use of fractional model has thus recently taken over the context of academic literatures and debates on financial modelling. Fractional models are usually of a non-linear and complex nature, which pose a considerable amount of computational and theoretical difficulties in deriving their analytical solutions. To the best of our knowledge, currently, there exist no tractable exact/analytical solution methods for solving fractional Black-Scholes equations, and as such, numerical solution methods become of a vital importance in understanding nature of solutions to such models. This thesis therefore, serves to derive some Generalised (fractional) Black-Scholes Partial Differential Equations (fBS-PDEs), as well as, propose their respective tractable, efficient and robust numerical simulation methods.