Date of Award
Open Access Thesis
College of Arts and Sciences
The Black-Scholes model is commonly used to track the price of European options with respect to maturity in many financial markets. This model degenerates into a partial differential equation that relates the European-style option price to the underlying price and time of expiry. Black-Scholes assumes that underlying prices satisfy a geometric Brownian motion.
After the U.S. stock market crash of 1987, this assumption becomes inaccurate as it fails to represent the behavior of S&P 500 European vanilla option prices. Specifically, under the measure of moneyness, the volatility smirk does not flatten out and the resulting conditional return distribution does not converge to normality. Recent academic literature have proposed readjusted financial models to account for the shortcomings of Black-Scholes, none which successfully have combined infinite return moments and finite price moments.
To reduce the effects of these consequences and to incorporate the additional moment conditions, we assume that the underlying satisfy a Levy −stable motion. Under this assumption, we will derive the Finite Moment Log Stable (FMLS) model and its respective fractional partial differential equation counterpart. Then, we will solve the Black-Scholes equation under FMLS by using the standard finite difference method and a finite volume scheme that significantly reduces the computational and storage cost in comparison. Lastly, we will perform a numerical simulation of our methods by using recent financial data in the S&P 500 market acquired within a one-year time frame to compare the performance of these methods.
Cheung, W.(2015). Modeling, Simulation, and Applications of Fractional Partial Differential Equations. (Master's thesis). Retrieved from https://scholarcommons.sc.edu/etd/3732