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.