Modified Black-Scholes Model On Stochastic Differential Equations

The first option pricing model which was produced by doctor Black and professor Scholes on The Journal of Political Economy in 1973 was the landmark of the modern finance.In the following developing and perfecting this model,Professor Merton did lots of preeminence work.For honor their crackajack work.The twenty-ninth Nobel price in economic sciences had been given to them.It represented that the option pricing model got enough affirmation from economics.The core technique of finance is availability pricing for the tools of finance.There is tightness relation between Black-Scholes model and the practicality of the finance market.The model with the direct practice on finance trading brought greatness influence and developed the finance market of the whole world fast.After the Black-Scholes model appeared,It has got common attention and recognition,Some scholars took deep checkout on the veracity of model.But at same time,A few economist oppugned the problem in the Black-Scholes model and modified it.In 1997,The American scholar Calai first time checked the Black-Scholes model with date coming from the Chicago Board option exchange.After this,Trippi,Chiras,Manuster and so on have made lots of exploring in this domain.Some other research were began with theory analyze bringing forward some problem in Black-Scholes model.They focused on the hypothesis of the model,many scholar think that the too strict hypothesis must affect the credibility of the Black-Scholes model, as follows:First of all,The parameter in the model are constant.But for example,The risk-free interest rate is variety with the macroeconomics and it is very difficult for a investor to borrow or lender money with the same risk-free interest rate.The variance rate of the return on stock being constant is contradiction to the fact.After some research,Black also agreed with the variance rate of the return on stock depending on the price of stock.As changing those parameters into some function about the price of stock and time.The Black-Scholes model is still applicable and the frame of the equation has not changed.Second,The stock pays no dividends or other distributions. This hypothesis restricted the application of the model.Many scholars considered that the time which dividends send and the number would effect the price of the options.It took no much time after the Black-Scholes model published, Professor Merton had solved this problem perfect.He succeed got the Black-Scholes model with dividends or other distribution.Last but not the least,There are no transactions costs in buying or selling the stock or the option.It was quiet knotty to modified this hypothesis.The most of method for it used a utility function and stochastic control to finish the option pricing,the work of Leland which was rigorous according to ideology of Black and Scholes in Lelang's framework.Transaction costs are compensated by systematic gains accumulated during the dynamic hedging process with volatility adjustment.Based on the means of Leland's,this paper use two option to get a new Black-Scholes equation and compensate the transaction costs by systematic gains accumulated with risk-free interest rate adjustment.We get a nice result for this improving with the real date of the price of option,the improve model in more accurate than the original Black-Scholes model.