This dissertation studies the power options and reset option, and obtains the following results:①Pricing of Eurpean power options under fractional Ornstein-Uhlenback(O-U)process and stochastic rateConsidering the randomness and mean-reversion of interest rate and underlying asset, we incorporate an expanding Vasick model and fractional O-U process to study the pricing of European power options.

Pricing lookback options on the stocks driven by exponential Ornstein-Uhlenback process;

Under the hypothesis of stock price submitting to exponential Ornstein-Uhlenbeck process and considering the relation ship between the fluctuation of interest rate and the fluctuation of stock price, this paper focuses on analyzing the effect of the fluctuation of market interest rate on European option price, and then compares the obtained formula with Black-Scholes pricing formula by sample.

Through analysis and comparison,it has been found that both of the models can meet the same stochastic differential equations and the option is of the same price under the model,with Black-Scholes option pricing model given first,and then its pricing formula deduced by martingale approach,and finally option pricing model of O-U process introduced.

They are governed by the following stochastic differential equations:where O_t is an m-dimensional O-U process governed by the following SDE:H,b_1,b_0,C are m×d, d×d, d×1, d×d matrices, (B_t, W_t) is a (d+m) dimensional Brownian motion.

The first model is a simple one-factor model in which the logarithm of the spot price of the commodity is assumed to follow O—U process which has a mean reverting character.