Pring Of Exotic Options On Fractional Jump-Diffusions

Financial mathematies is a new developing branch of seience, it is now being paid close attention to in the domain of international finance and applied mathematies. Uncertain pricing is one core of financial mathematies study, it involves the theories of modern finance such as asset pricing theory, investment combination theory and it involves theories of modem mathematics such as stochastic analyzing and optimizing theory too.Effective management of risk occupies the right evaluation of derivative securities.The critical thing is that the financial derivative scceurities exist reasonably and develop properly is how to value its fair price. Recently,in addition to known European options and American options,there appear many new variety which are changed,composed,derived by vanilla options in international financial market.So far, most researches in option pricing,including Black-Scholes formula,have been conducted in the approach of the linear classic financial theories.However,the linear methodlogy of CMT has limitations inherently as they are invalid to capture complicated"patern"in stock price. So,a new research trend,from the point of nonlinearity and evolution instead of in a linear view,emerges(stock price based on fractional Brownian motion in stend of Brownian motion). It shows that traditional finance theory based on the assumptions of normal return distribution, random walk, and independence, Brownian motion,cannot accurately characterize the price behavior; while with the hypothesis of fractal capital market, non-normality, fractional Brownian motion,and the long-term memory of the financial time series, The actual financial time series follow a skewed random walk; have fractals and exhibit long-term memory, the behavior of the actual stock price can be characterized well.This dissertation studies option pricing with assumptions that the stock price fluctuation follows a fractional Brownian motion and the capital market has fractal characteristics. It extends the jump-diffusions model and puts forward the fractional jump-diffusions model which based on fractional Brownian motion instead of Brownian motion, and then researches the pricing of two exotic options (compound options, exchange options) on fractional jump-diffusions.The main results of this dissertation are summarized as follows:①The pricing formulas for European exchange option are obtained using insurance actuary pricing methods, where the underlying asset follows a fractional jump-diffusion process with the time-dependent parameters (expected rate, volatility and risk-less rate). These pricing formulas generalize the corresponding European option and European exchange option pricing on jump-diffusions.②The pricing formulas for European compound options are obtained using insurance actuary pricing methods, where the underlying asset follows a fractional jump-diffusion process with the time-dependent parameters (expected rate, volatility and risk-less rate). These pricing formulas generalize the corresponding ones about European compound option pricing on jump-diffusions, which are obtained by Gukhal and Li, etc.