European Option Pricing Under The Unknown Volatility And The Unknown Expected Rate Of Return

In recent years, with the improvement of people’s investment philosophy,their concern for derivative products have become more and more sophisticated.A lot of investors have participated in the derivatives market. Option which is an important part of the financial derivatives market, has been concerned by investors. Then, how to pricing option is important to academics and investors. In 1973, Black and Scholes got European call option pricing formula under the volatility is constant and no expected rates of return. However, a large number of empirical analysis shows that real financial market volatility is not constant. In order to better meet the real market situation, this paper gets European call option, cash or nothing call and asset or nothing call’s optional prices ’s closed form solution under conditions of incomplete information and the volatility and expected rate of return is unknown. We also do numerical analysis about the optional price of European call option.Specifically, assuming that the stock price tS satisfies here μ =α +βη is the excepted rate of return,α, β,σ >0 are constants,η is the standard normal random variable,tW is the Winner process, the density function of the random variable ξ is here a, b are constants, Γ(x) is the Gamma function, and,,tW η ξ are independent of each other.With the hypothesis of no arbitrage opportunity in the market, we construct the portfolio to get the formula of European call option, cash-or-nothing call and asset-or-nothing call through Ito formula and delta hedging strategy under the condition of complete information. But,there are part of variables which are unobserved in these formulas. Therefore, the operability is not practical. In fact, the investor’s information is not complete. In order to determine the price of an option further in the incomplete information situation, we assume that the investor use the minimum mean square error hedging strategy. And we obtain the optional price closed form solution of European call option, cash-or-nothing call and asset-or-nothing call. Finally, we do the empirical analysis with the data of the domestic and foreign stock market. And the result shows that the excepted rate of return μ and ξ ave certain influence on the option pricing and implied volatility smile.