Investigating the Black-Scholes partial differential equation: applications in finance

Abstract

The Black-Scholes stochastic differential equation is a well-known useful model in fi nance. In this thesis, we present the derivations, assumptions, existence, and uniqueness of its solutions and practical applications of the Black-Scholes model. We obtained the equa tion through derivation and then we discussed its well-posedness. Analytical solutions of the Black-Scholes equation were obtained using the Feynman-Kac formula. From the analytical solution, we have derived put and call formula for European options. For American options, we used finite difference approximation. As an application, we used historical data of dif ferent stocks from Yahoo Finance and tested the efficiency of the Black-Scholes equation. In addition, geometric Brownian motion is applied to historical data of different periods having different market behavior using the Montecarlo simulation to predict stock prices. The result shows the effectiveness of the Black-Scholes model in stable situations despite its assumption of constant market volatility. Since Montecarlo’s simulation relies on past prices, it provides a satisfactory approximation when market price behavior is stable, but its approximation diminishes in extremely volatile markets.Results show that the Black-Scholes equation can compute option prices until expiration in low-volatility markets with an efficient prediction. Future studies should give attention to develop an adjusted version of the model that is understandable while guaranteeing efficiency in high-volatility markets.