Several Applications Of Stochastic Differential Equation In Finance

With the fast development of mathematical finance in recent years, stochastic differ-ential equations have been widely used in finance. As important financial tools, option and stock have attracted more and more attention. In this thesis, we study the risk of option pricing and the hedging error under the assumption that the asset prices follow some differential equations. Then we discuss the feasibility of technical analysis in stock market, and test the fitness of current stock pricing models. The main contents of this thesis are listed in the following:In the first chapter, the origin and development of mathematical finance are first introduced. Then we introduce the conception of option, hedging and the classification of option. Finally, some preliminaries of this thesis are provided.In the second chapter, the hedging problem of options is considered under the geo-metric Brownian Motion model. We suppose the underlying stocks are dividend payed, then we explicitly compute the variance-optimal hedging strategy in discrete time for bi-nary options and European call option taking advantage of the Girsanov theorem and the predictable quadratic covariation process.In the third chapter, we simulate the ratio of hedging error to option pricing. Then we gain the upper and lower bound of hedging error by Ito's formula and illustrate the risk of traditional option pricing.In the fourth chapter, the option pricing and the hedging strategy are provided when the risky asset prices follow stochastic differential equations with delay. As has been pointed out, the current stock price is influenced by the past price. We consider such a kind of Black-Scholes model with delay, and give the variance-optimal hedging strategy. Then we study the option pricing problem. Also, we consider the pricing and hedging problem under a kind of random delay models.In the fifth chapter, we investigate the technical analysis in the stock market. Firstly, we introduce some popular technical analysis indicators such as BOLL, ROC and RSI. Since the empirical analysis indicate that the stock returns are long-range dependent and the widely discussed exponential Levy model is not long-range dependent, we consider the exponential fractional Brownian motion with drift as real market. Considering the fractional Brownian motion is neither the process with independent increments as Levy process nor the Markov process, we obtain our results using the knowledge of stochastic analysis and matrix theory. We then give the stationary property of statistics consist of these technical indicators. Also, we gain the convergence property by Birkhoff's ergodic theorem and derive the law of large numbers for frequencies of the stock prices falling out normal scope of these technical indicators. In addition, We obtain the rate of convergence. In the end, we test the the independence of the change of stock price using the data of daily close prices in American stock market as well as the high frequency stock prices in Chinese stock market.