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Oct 2009

Numerical Methods for fractional Black-Scholes Equations for Option Pricing

Fractional partial differential equations (fPDEs) are generalization of classical integer order partial differential equations (PDEs). The fractional Black-Scholes (fBS) equation is a fPDE, where the stock price is assumed to follows a Lévy process, rather than the Brownian motion assumed by the classical BS equations.

Our research will focus on using numerical methods for solving the fBS equations governing European and American Options. We will analyze the existence and uniqueness of the solutions, and then develop numerical schemes, eg. finite difference method, finite volume method, to solve the fBS equations. The fBS equation for American Option pricing, which is a linear complementarity problem (LCP), is equivalent to a variational inequality problem. Therefore a penalty method will be used for solving this LCP. We will analyze the stability and convergence of all these numerical methods. FBS equations for other exotic options pricing will also be investigated.

Option pricing has attracted a large amount of attentions from both mathematicians and engineers. One shortcomings of the standard BS model is that it underestimates the probability of large jumps of stock price in a short time. Since the fractional partial differential operator has the ‘non-local’ property, the fPDEs are increasingly used to model problems in finance, fluid flow, mechanics and other applications. However, the numerical methods of fBS equations and the convergence analysis of these methods have not yet been fully explored, so this is an important area for future research.

Assistance in statistics is available for Postgraduates students by research at the UWA Centre for Applied Statistics.