Surveys On Options Pricing In Frictional Market

An option is one of basic financial derivatives, it is a successful example of the financial derivatives market innovation in the 20th century. It is only 30 years from 1973 that the options market has become an important part of the international financial market.French mathematician Louis Bachelier introduced the Option Pricing Model in his paper firstly in 1909, which laid the foundation for the modern theory of option pricing. The latest revolution of modern option pricing theory began in the 1970's. In1973, the famous American financial experts, professor of the University of Chicago Fischer Black and Myron Scholes established the European-style call option pricing formula : it is not dependent on the investor's preference and lead up all investors to a risk-neutral world.But in the real stock market, investors were faced with significant transaction costs which can not be ignored. For these reasons, I read some concerned literature as to Option Pricing with transaction costs, and analysis, generalize ,make up,come into being this paper.There are two main methods for Option Pricing with Transaction Costs : no-arbitrage principle and the utility function method.The Black-Scholes model was improved by Leland useing of the no-arbitrage principle in 1985. In the main the Leland assumptions are those in the Black-Scholes model but with the following exceptions:The portfolio is revised everyδtwhereδtis a finite and fixed ,small timestep;The random walk is given in discrete time by whereφis drawn from a standardized normal distribution.;Transaction costs are proportional to the value of the transaction in the underlying .thus ifνshares are bought or sold at a price S ,then the cost incurred is k |ν| S;The hedged portfolio has an expected return equal to that from a risk-free bank deposit.The Leland model is extended to arbitrary option payoffs or portfolios by Hoggard,Whalley and Wilmott in 1992 , and using of the no-arbitrage principle, they obtained the option pricing formula with transaction costs:Compared with the standard Black-Scholes equation, the volatility has made an adjustment. Let We can drop the modulus sign from the above formula . Corresponding , long positions should be valued with an adjusted :Short positions should be valued with an adjusted : Utility Based is another approach for the option pricing with transaction cost . In modern finance, it is customary to describe risk preferences by a utility function , and proved that the utility based theory is the best way to hedge the options with transaction costs. Expected utility theory maintains that individual behavior as if they were maximizing the expectation of some utility function of the possible outcimes. Hodges and Neuberger (HN) pioneered the option pricing and hedging approach. Davis , Panas and Zariphopoulou (DPZ) made improvements later. In the article, HN assume that an investor holds an investment portfolio and he has the opportunity to issue an optionand hedge the risk using the underlying. However, since rehedgingis costly, most define strategy in terms of a "loss function". The aim is to maximize the expected utility, which entails the investors specifying the "utility function". The case considered in mostdetail by HN and DPZ is of the exponential utility function. It has the nice property of constant risk aversion. Mathematically, such a problem is one of stochastic control and the differential equations involved are very similar to the Black-Scholes equation.HN introduced the general approach of Option Pricing with transaction costs. However, they carried out computations of the optimal hedging strategies and option prices in a market with only proportional transaction costs, without really presenting the continuous time model and the numerical procedure.The ideas of HN were modified by DPZ. Instead of valuing an option on its own, they embed the option valuation problem within a more general portfolio management approach. Considering the effect on a portfolio of adding the constraint that at a given date, expiry, the portfolio investment has an element of obligation due to the option contract, they only considered costs proportional to the value of the transaction( k|ν| S).In HN and DPZ the value of the option is given in terms of the solution of a three-dimensional free boundary problem.The variable in the problem are asset prices S ,time t , as always, and also D , the number of shares held in the hedged portfolio.Later the author introduced a special pricing method based on the utility function–– Option Pricing with the deposit rate isn't equal to the borrowing rate.Before, we summary the first case of market friction–– option pricing with transaction costs . Next, I discuss another case of market friction–– On-One 's-Own Option Pricing with tax, basing on the On-One 's-Own option pricing, I give the nature of their functions.Generally speaking, the option we are talking about are trading by the investors. The price of the underlying has an impact on the price of option, but not vice versa. When, however, a company issue its own options, the value of those options become part of the company's assets or liabilities, So any change in the option price would affect the market value of the company and thus the stock price. To distinguish this special situation from the standard case, We propose the term On-One 's-Own option (listed after the OOO option)to denote the situation described.Let X_t tdenote the market value of the company at time t ; W_t the hidden liabilities; S_t the market value of the company's assets;γtime t price of p European OOO call options with strike price K and expiration timeT ,which are sold by the company. Assuming p European OOO call options are sold by the company at timet₀,there is tax and tax rate is q . Well, we can prove that their pricesr_t^s satisfy the following equation :Subsequently,we can prove that its prices function meet the following theorem :Theorem : let p European OOO call options are sold by the company at timet₀,there is tax and tax rate is q . These options have a unique priceγsatisfy and Furthermore,...