| In 1973, Black and Scholes gave the standard option pricing equation, B-S Equation for short. Then, Merton gave its explicit solution. During the subse-quent several decades, the theory has made great progress and improvements, so that it has been closer to reality, such as underlying asset dividends being added, position'change resulting transaction costs, compound options, options under stochastic interest rate, exotic options, and so on. However, how to pre-cisely calculate their prices is a problem payed close attention to by many scholars and financial institutions.In this paper, we discuss the problem of option pricing with transaction costs under stochastic interest rate and CEVP. The so called CEVP is actually a spread of geometric brownian motion,which is only a special case of it. Six respects are considered:European call option, general case of interest rate and stock price equations,spread option,multi-asset option, generalized definition of transaction costs, and generalized definition of transaction costs under stochastic interest rate and CEVP.In the process of derivation of option prices, we take advantage of the method by constructing a riskless portfolio; we use the classic method "Leland'Method" to deal with transaction costs, that's, to consider transaction costs in a small time interval, otherwise, they may approach infinity; stochastic interest rate and transaction costs exist at the same time, so we can't get V's explicit expression which is in regard to S, r and t,however,numerical calculation is feasible.If transaction costs and stochastic interest rate do not exist, the equation becomes classic B-S equation. Predecessors only have dealt the case involv-ing CEVP and transaction costs without stochastic interest rate,here, we get the option'partial differential equation, where the option price is a function of stock price,interest rate and time. Because stock price and interest rate both have stochastic factors,it involves how to deal with more than one brownian mo-tion,which is not dealt with by predecessors. This is the first innovation in this paper.Then, we spread the conclusion to the case in which interest rate equation and stock price equation are generalized. And then, we consider the spread options, which is a simple multi-asset option, and its price equation is given. And,we have derived the equation of a multi-asset option whose price is determined by n options,which is influenced by in stochastic factors each.It's noteworthy that,in the process of dealing with position'change, we use Paul Wilmott'method which is first mentioned in the reference [3]. The higher order infinitesimal of Brownian Motion's differential is sacrificed, which is not to be let approach 0, and it is only a smaller constant.The main part constituting transaction cost is bigger, and it becomes infinite as transaction'frequency ap-proaches infinite.We may find that though Paul Wilmott'method is more simple, it is not absolutely precise because higher order infinitesimal of Brownian Motion's dif-ferential is sacrificed. In this paper, another definition of transaction cost is given witch is more general; what's more,it is let be a twice-differential function of position'change, so it's reasonable; Delta is related to the frequency of posi-tion'changes, which has practical sense. The ordinary case that the transaction costs are let to be|v| can be approached by a twice differentiable function of v. This is the second innovation in this paper.The third innovation is that stochastic interest rate is considered,giving the price equation with generalized definition of transaction costs form under stochas-tic interest rate and CEVP. Similarly, dealt does not include transaction fre-quency. And option is a function of stock price, time and interest rate.Finally, we give the implicit format finite difference method and programmes for the calculation of the price of options with transaction costs... |
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