Title: Tempered Double Fractional Diffusion Model for Option Pricing
Since 1973, the celebrated Black-Schole-Merton option pricing formula has dominated the minds of both researchers and practitioners alike until the crash of the stock market in 1987 when market practitioners and researchers realized that the Black-Scholes model wasn’t sufficient at capturing some of the wild jumps that were being observed in stock price time series.
The main purpose of this dissertation is to extend the ideas of H. Kleinert and J. Korbel , Jean-Philippe Aguilar and Jan Korbel to provide framework for an analytical solution to the option pricing problem by modelling the evolution of stock log-returns density via the (double) space-time fractional diffusion equation with a tempering in the space variable (or log-returns). Our novel contribution is that we temper the space variable (or log-return variable) so that we can regulate the tail behaviour of the distribution. The idea of tempering the tails of the log-returns distribution enables us to capture different risk profiles of the underlying asset return. We also formulate the corresponding optimization problem and provides numerical solution to the model parameters. Finally, we provide estimates for the implied volatilities under these models under the assumption that options are priced at-the-money on forward basis, that is, the spot price equals the present value of the strike price.
Advisor: Dr. Wojbor Woyczynski