Path-dependent Option Pricing Research

The optons pricing is the key points in modern financial theory, the simple is the problem of European options and American options. With the development of society, the options that are different from European options and American options called the singular options are appearing. And they are widly used because they can reduce investment risks or as a public incentive method in a programme. While the majority of such options are closely related to their experiences and path, so they are also called the path-dependent options. The path-dependent options are options whose payoff depends on the behavior of the price of the underlying between 0 and maturity (assumed to be fixed),rather than merely on the final price of the underlying. So, we must describe the properity of the path in the Black-scholes model , then we can get the formula of path-dependent options in Black-scholes option pricing methods. To some path-dependent options, we can define a new varible that is the function of the time. In this paper, we intend to study the compound options and reset options that all belongs to the path-dependent options. The compound options are used to simulate American options. For compound options, including having one date and one strike price and having many dates and strike price, they are separated to four kinds: the call option on a call option, the call option on a put option, the put option on a call option and the put option on a put option. It also includes the compound options about another options. It also divided real compound options and financial options by their underlying assets. The pricing of reset options is easier than compound options'. It is similar to European options and make the holder has more right to decide to invest or not. With the social development, especially the rapid development of computer technology ,complex computing is not a difficulty problems,so these options that are more complex than European Options have the greater practical value and prospects in future.In this paper the main conclusions are:(1), We study the pricing of compound options from easier to harder, we consider the compound options which has only one time point befor the maturity. In this model, the riskless rate r(t),the dividend rate and the volatilityσ(t) are all the function of time. We get the formular of the call option on a call option by risk neutral method. Then using the call-put paring for European option, we also get the formula of the call option on a put option. We find that their correlation is not. similar to the call-put parity for European option. (2),Considering the important thing effecting the price of the underting, we assume that the risk asset is driven by jump-diffusion process and the jump is described by the equivalent to Possion jump.We don't consider the dividend and the risk rateμ(t), the volatilityσ(t) are all the function of time, and the riskless rate r(t) may be. We can get the option on option pricing formalar in this model by applying martingale method and changing of numeraire or changing of probablility measure.(3), We study the reset optins with predetermined dates in jump-difusion models. By using the martingales method and choosing different numeraire or changing the probability measure, we give the formula of this kind of optins.Under the assumption that the risk asset pays no dividends and is driven by the Levy jump diffusion process, and the risk rate and the volatility are all the function of time. We obtain the generalization of reset options pricing formulae, and we get the solution of reset options with one reset time.