Research On Path-dependent Options Pricing Models And Methods

Options, as important derivative products，play a key role in the financial market.Options are important instrument to hedge, can avert the risk, and conduct financial investorshow to invest their money. With the development of the financial market in our country, andthe demand of controlling the risk in financial market, options get further development inChinese market, and the option market is improved with time. Path-dependent options areintroduced because of the demand of financial market. For the specificity of Path-depandentoptions, they can satisfy the demand of investors, therefore, they are becoming the maininstrument in finiancial market. How to model the option pricing model reasonly and how toprice the path-depandent options have become major focus of attention from corporate andregulatory sectors. Since the1970s, many scholars have considered the problem of pricingoptions and give the pricing method under the geometric Brownian motion environment.However, in recent years, some researches about assets’ returns have illustrated that thedistribution of asset log-return are not symmetrical and have the properties of heavy tail andaiguilles, and the volatility has the shape of smile, also there is volatility clustering. Theseproperties indicate that the traditional Brownian motion can not describe the randomphenomena well. Fortunately, the random process, namely tempered stable process, cancapture the random phenomena very well. Hence pricing options under GARCH model withtempered stable process and giving the price method under these pricing models haveimportant theoretical and practical significance for both determining the fair and reasonablevalues of the warrants and promoting the strong development of China’s financial derivativesmarket.Owing to fluctuations in the financial markets from time to time, due to the human errors,the parameters may have different values in the different commercial banks and financialinstitutions. So it is unreasonable to assume the parameters as constants. To solve thisproblem, we do research on the underingly asset price, and present option pricing under fuzzyenvironments. Morevoer, we apply tempered stable distributions to model the price ofunderingly assets, then, present the method to price path-depandent options. The main resultsand innovations in this thesis are listed as follows:First, under the jump diffusion model, this thesis deals with the problem of pricing thebarrier options and lookback options, and the method the price American barrier options andlookback options is also proposed. Previous studies dealing with option pricing usually assumed that the underlying assetfollowed the geometric Brownian motion. However, numerious empirical analysis shows thatthe distribution of underlying asset returns is skewed distributions. The process of underlyingasset is not continuous, and there are jumps. We first apply jump diffusion process to modelthe underlying assets, then we present to how to price the option under the models. Since thevalue of Lookback option and barrier option is related to the price of underlying asset, it isdifficult to get the analytical solution for the path-depandent options. Here we present leastsquare simulations methods to price options, based on the present methods. The method wepresent can compute the option price in shorter time, and it is more efficient.Second, with the facts of the return of asset in financial market, this thesis appliestempered stable distributions to model the asset price. The tempered stable process capturesthe skewness, heavy tail of the asset return. The GARCH model combined with temperedstable distributions model the asset, which also captures the volatility clustering. Therefore,they can describe the process of assets better.Although the GARCH model can descirble the volatility clustering, the distribution ofasset return is skew, and has heavy tail, so it is not reasonable to use normal distribution. Thisthesis use the tempered stable distributions as the innovation of GARCH model, and presentthe TS-GARCH model, which can decrible the process of asset returns. The K-S and A-Dstatistic show that the TS-GARCH models are close to the real financial market.Third, we use markov chain approximation to price American options under theTS-GARCH models, and prove the convergence of the markov chain method.Since the tempered stable distributions and GARCH models are used to describe thebehavior of the underlying asset, how to price option, especially American option under theTS-GARCH models is the critical problem. The classical methods are Monte Carlo simulationand tree method. We apply markov chain approximation to compute the option price by usingtrasiton probability matrix. It can solve the pricing problem of both of European options andAmerican options. Compared with Monte Carlo simulation, it is more efficient.Fourth, we do research on option pricing models under fuzzy environments, and presentfuzzy double exponential jump diffusion option pricing model. We also obtain the crisppossibilistic mean option pricing formula in fuzzy double enponential jump diffusion modelby using the crisp possibilistic mean values of the fuzzy numbers. The method to price optionin the fuzzy option pricing model is presented.Previous studies dealing with option pricing usually assumed that the underlying assetfollowed the geometric Brownian motion under the cirsp environment. Using the double exponential jump diffusion models to describe the behavior of the underlying asset, fuzzy theparameters of financial market, we introduce the fuzzy double exponential jump diffusionmodel. Futhermore, we obtain the crisp possibilistic mean option pricing formula in fuzzydouble enponential jump diffusion model by using the crisp possibilistic mean values of thefuzzy numbers. The empirical research shows that: the fuzzy double exponential jumpdiffusion model is more reasonable, and it can supply more valuable information of thefinancial market. Hence the introduction of fuzzy option pricing model can improve theoption pricing.