Applications of Stochastic Calculus to Finance

Stochastic Calculus has been applied to the problem of pricing financial derivatives since 1973 when Black and Scholes published their famous paper The Pricing of Options and Corporate Liabilities in the Joumal of Political Economy. The purpose of this thesis is to show the mathematical principles underlying the methods applied to finance and to present a new model of the stock price process. As part of this paper, we present proofs of Ito\u27s Formula and Girsanov\u27s Theorem which are frequently used in financial applications. We demonstrate the application of these theorems to calculating the fair price of a European call option. There are two methods that result in the same price: the risk neutral valuation and the Black-Scholes partial differential equation. A new model of the stock price process is presented in Section 4. This model was inspired by the model of Cox and Ross published in 1976. We develop the model such that a martingale measure will exist for the present value of the stock price. We fit data to the traditional geometric Brownian motion model and the new model and compare the resulting prices. The data fit some stocks well, but in some cases the new model provided a better fit. The price of a European call is calculated for both models for several different stocks