Pricing Lookback Options On Two Different Markets Models

Option as an effective tool of keeping away financial risks and hedging has beenwidely used. In modern market, there exists many exotic options which are traded?exible, and also the profit was suited to the investors. One of these options includethe Lookback options whose payoff depends on the maximum or minimum priceof the assets during the life of this option, which leads to pricing is much morecomplex than that of the plain vanilla options.It is well known that an effective market model plays an important role inmaking decision, financial risk management and hedging, etc. Since the classi-cal Black-Scholes model has many deficiencies in describing the risk of marketsystem. So, many extensions were widely applied, such as the fractional Brown-ian motion model, Stochastic Volatility model and so on. Relating to the classicalBlack-Scholes model, such models are much more suitable to describe the feature ofthe actual market risk. Recently, a large number of empirical studies show that themarket share price changes has dependence in long term, and the distribution of thelogarithm of the share price income has the feature about"leptokurtic, fat tail"andso on. While the fractional Brownian motion model and the Stochastic Volatilitymodel can describe those features, so they become the hot model in present re-search. In this article, we consider the valuation of European lookback option in thefractional Brownian motion model and the Hull-White Stochastic Volatility model,respectively.In Chapter 1, a brief introduction of the significance of option pricing is pre-sented, the literature reviews on both the options and lookback options are shown.In Chapter 2, we consider the pricing of European lookback option under the fractional Black-Scholes model. As the payoff in the lookback options includingthe maximum or minimum value of the assets, it is difficult to obtain the maximumor minimum distribution function of the fractional Brownian motion model directly.Using the partial differential equation approach, the pricing of the lookback optionsis derived. First, the closed-form solution of the floating exercise price of the Eu-ropean lookback call (or put) option pricing is obtained. Second, some numericalexamples and the risk analysis are provided.In Chapter 3, assume that the underlying asset satisfying the Hull-White stochas-tic volatility model, using the Martingale approach, the nature of the conditional dis-tribution and the Taylor expansion of the explicit formula of the European lookbackcall option prices, we obtain the approximated solution of the ?oating exercise priceof the European lookback put option, and also given some numerical examples.In Chapter 4, Our main conclusion and the further research works are summa-rized.