Valoració d'opcions financeres i equacions en derivades parcials : resolució de l'equació de Black-Scholes

Abstract

We are currently facing many di culties with the economic crisis that has destroyed both the nancial and labour markets. Because of this I have decided to enter the financial sector once I have completed my mathematics degree, and try to understand the complexities and learn the secrets of this sector. When thinking of the di fferent financial stocks and options you cannot ignore the Black-Scholes equation, which is for many the most important factor in economic development, and for which Robert C. Merton and Myron Scholes received the Nobel prize for economics in 1997. Here we derive and solve this equation. Our work is based on the valuation of fi nancial options, speci cally the Call and Put options. In order to value these options we use the Black Scholes parabolic partial di fferential equation. We derive this equation in three diff erent ways. The first uses instantaneous replication, the second the "risk price" of the market and the final method calculates the forward price. We present also three distinct methods to solve it. The first two are based on the resolution of the Heat Equation, as the Black-Scholes Equation can be transformed to this using an apropriate change of variables. The third derivation uses the Mellin transform. Finally we give a numerical solution using a finite di erence method. Obviously, before we derive the equation, we must first defi ne and explain the fi nancial and mathematical concepts needed to derive, solve and interpret the equation. From this we can also introduce the Heat Equation and the Fourier Transform. Finally to finish we present a C program which calculates the price of a Call and Put options.