Application of stochastic differential equations to option pricing

Abstract

The financial world is a world of random things and unpredictable events. Along with the innovative development of diversity and complexity in modern financial market, there are more and more financial derivative emerged in the financial industry in order to gain higher yields as well as hedge the risk . As a result, to price the derivative , indeed the future uncertainty, become an interesting topic in the field of mathematical finance and financial quantitative analysis. In this thesis, I mainly focus on the application of stochastic differential equations to option pricing. Based on the arbitrage-free and risk-neutral assumption, I used the stochastic differential equations theory to solve the pricing problem for the European option of which underlying assets can be described by a geometric Brownian motion. The thesis explores the Black-Scholes model and forms an optimal control problem for the volatility that is an essential parameter in the Black-Scholes formula. Furthermore, the application of backward stochastic differential equations (BSDEs) has been discussed. I figured that BSDEs can model the pricing problem in a more clarifying and logical way. Also, based on the model discussed in the thesis, I provided a case study on pricing a Chinese option-like deposit product by using Mathematica, that shows the feasibility and applicability for the option pricing method based on stochastic differential equations.