Research On Some European Option Pricing Problems

In this paper we consider three problems on European option pricing: European option pricing with transaction costs and stochastic dividends; option pricing with stock return following Ornstein-Uhlenbeck process; option pricing on foreign currency under stochastic interest rate. Firstly, we consider pricing European option problems with transaction costs which are proportional to stocks'value and discrete stochastic dividends of stocks under an arbitrage-free framework and give an explicit expression of option pricing formula, which extends option pricing formula where underlying assets are forward contracts.(when both transaction costs and dividends become zero ,it reduces to the option pricing formula where underlying assets are forward contracts).The result indicates that option price increases as transaction costs of stocks increase and decreases as dividends of stocks increase. It demonstrates that corporate stocks and bonds are options based on corporate assets. And then we apply option theory to price stocks and bonds. The interest relationship between shareholders and loaners is analyzed and corporate security's effects on option price are discussed. Secondly, European option pricing problem is discussed when price of stocks follows geometric Brown motion and return of stocks follows Ornstein-Uhlenbeck process. Using Arbitrage-free principle, we give a partial differential equation with which option price is satisfied and using Fourier inverse transformation, the closed-form solution to European Call option pricing formula is given. Finally, we consider three risk resources: domestic term structure risk; foreign term structure risk; exchange rate risk. Using martingale method, we deal with option pricing problems on foreign currency, and obtain price expression of European call and put option under stochastic interest rate and their parity. We also consider hedging problems of options.