Numerical techniques for solving the Black-Scholes equation

Abstract

Trabajo de Máster en Economía, Finanzas y Computación (2021-22). Tutor: Dr. D. Jared Lee Aurentz. The Black-Scholes model (named after Fischer Black and Myron Scholes) for option valuation is a model used in financial mathematics to theoretically estimate the value of a financial option, of the European option type. However, solving the Black-Scholes equation in higher dimensions requires numerical techniques. In this Master’s thesis, we propose a Chebyshev Pseudo spectral method and Euler Implicit method for pricing European call options and a comparative study of several possible configurations of these two methods. An option is a financial asset that offers the buyer the opportunity to buy or sell depending on the type of contract they hold. Each options contract will have a specific expiration date by which the holder must exercise their option, and it is either worthless or worth more than it was bought for. Black-Scholes partial differential equation presented in 1973, models the fair value of a European call option under certain market assumption. The terminal condition is derived from the difference between the stock price upon maturity and the option strike price, while the boundary conditions are derived from the put-call parity. We use the Chebyshev points as a set of points when discretizing Black-Scholes equation. Knowing that options has been priced with the use of finite differences it works as a comparison to the results of Chebyshev Pseudo- Spectral method. By approximating the initial condition with orthogonal Chebyshev polynomials and truncating the domain, the convergence rate increases significantly. In this context, numerical experiments confirm a considerable increase in efficiency, especially for large data sets. [1] [2] [2] [3] [4]