Distinguished Limits of Levy-Stable Processes, and Applications to Option Pricing

In this paper we derive analytic expressions for the value of European Put and Call options when the stock process follows an exponential Levy-Stable process. It is shown that the generalised Black-Scholes operator for the Levy-Stable case can be obtained as an asymptotic approximation of a process where the random variable follows a damped Levy process. Finally, it is also shown that option prices under the Levy-Stable case generate the volatility smile encountered in the financial markets when the Black-Scholes framework is employed.