Option Pricing On The Models Of Stock Price Flucturation

The models of the stock price fluctuation is a mathematics model discribing the fluctuation of the stock price,It is all along the question financial scholars research over a long period of time, The models existing at present are mainly the model of Randonm Walk and the model of lognormal distribution etc.Economists analyse the two models by authentic proof,which indicates that this two models do not fully qualify the actual stock market.In view of the above- mentioned facts,at the time some scholar have studied a new model of the stock price that even conforms to the actual stock market- that is the model of lognormal distribution.Underlying the assumption that the stock price accords with the model of the stock price fluctuating sources,By comprehensivily applying the stochasitic differential theory and no-arbitriagc thcory,this paper,under the conditions that the risk-free rate r is constant or Ito stochasitic process, successively works out the option pricing about the stock price model with that the short-term profit function is piecewise lecture function arid that one with that the short-term profit function is Possion Jump Process,derivats counterpart partial differential equation of option pricing.The outcome states:1.When the short-term profit function is unusual flunctuating sources bring out a piecewise lecture function,this amendment on the lognormal distribution model does not improve the option price,because this partial differential equation of option pricing is the same one underlying the lognormal distribution model(see equation 2.14).2.When the risk-free rate r is stochastie interest rate,the partial differential equation of option pricing,underlying the above-mentioned stock price model,popularizes the Black-Scholes partial differential equation(see equation 3.19).3. In no-effective market,investors will face trading costs which cann't be neglected.Based on defining the trading costs,this paper set up a no-linearoption pricing model with discrete trading time, and discuss European option long position and option short position with trading cost(3.33).4.After changing the short-term profit function to Possion Jump Process, in the view of that the derivated partial differential equation of the option pricing which different from Black-Scholes partial differential equation still is that interest rate is constant(4.2), the model which does not accord with the real market under the assumption. At last,we derivat a new model of option pricing whoso profit rate is Possion Jump Process under stochastic interest rate(5.13),this model not only changes the form of the short-term profit function of the stock price model and avaids the simplization of the profit rate function the unusual flunction sources bring about,but also relaxes the basis assumption of Black-Scholes option pricing model and makes that the partial differential equation builds the foundation which even approaches the actual market.