Valuation of the European -Style Put Options under Lé́vy Process via the Mellin Transform

Abstract

This paper considers the valuation of the European-style put options under Le ́vy process via the Mellin transform. We obtain the partial integro-differential equation for the price of the European put option with log-normal jump diffusion process. The Mellin transform with its derivatives and shifting properties were used to solve the partial integro-differential equation. We recover the integral equation for the price of a European put option consisting of single integral when the underlying asset governed by the log-normally distributed jumps by means of the Mellin transform inversion formula. With the special case of the log-normally distributed jumps, we show that our integral equation reduces to the infinite series solution of the Black-Scholes-like valuation formula for a European put option. We also show how to estimate the solution of the integral equation of complex function for the price of the European put option to a form that permits the use of numerical integration. The numerical experiment shows that the Mellin transform provides comparable results, agrees with the values of the Black-Scholes model and is a good alternative approach to Rannacher timestepping for the valuation of the European style put options. Hence the Mellin transform produces a powerful and flexible tool for handling log-normal jump diffusion process.