Option Pricing Model And Its Generalization

The Black--Scholes Option Pricing Model has been a useful toolin proper pricing of option .It is on the following conditions :(1)The price of underlying assets derives in geometric Brown motion way. (2)In the term of validity, risk-free rate and financial capital payoff's volatility are constant. (3)There exists no frictions on markets i.e. there are no taxes and transaction costs. (4) Financial assets have no dividend or other income during the term of validity.(5)The options are European Options.However, there are some differences between these conditions and the real society. According to this, the writer makes some adaptation and generalization to that model mainly on the following aspects:(1) The writer made a detailed introduction and analysis to the Black-Scholes model and the Binomial Tree Option pricing model. On the basis of arbitrage-free principle.the binomial tree methods define a risk-neutral world by making use of-hedging skills. As a result , the expected yield of all security are risk-free .So afair price independent on every investor's risk aversion is given. The continuous model of option pricing -Black-Scholes model, which is on the basis of the consumption that the price of the underlying assets derivates in the geometric Brown motion,thewriter uses the A-hedging skills and ltd formula and gives the Black-Schcolesequation which the option price satisfies. By computing its terminal value, the model gives a fair price of European option - Black-Scholes formula independent on every investor' risk aversion. By introducing the two models, the paper show the right way for option price modeling and its reasonability.(2) This paper tries to build the basic equation of the European option pricing with transaction costs by simulating the option payoff using the portfolio i.e. build a portfolio including the option positions and the derivative security position on the basis of this option, and give the option a price when the payoff of this portfolio position is equal to the risk-free payoff (the A- hedging).The paper has tested it by binomial tree model.(3) In Black-Scholes pricing model, the payoff volatility of stock investment a and risk-free rate r are constant during the term of validity. In fact it is hard for them to remain the same in real society. The writer consideres cr and r as functions of time t, then the writer gets a pricing formula for European option. The paper also makes improvements to binomial tree model to made the results more satisfying.(4) When people are performing risk management to the appeared risk by using the option as a financial tool, the following ratios denoted by some Greek letters are very important. This paper introduces three important ratios Delta, Gamma, Theta, and their applications, and discussed the Phi value and Psi value of the changes of the option value relative to the changes of the interest rate and that of the exercises price . To some extent, it improves our abilities to use the option pricing model more exactly and make option transaction strategies against risks.