| As an important part of stochastic calculus,backward stochastic differential equation has important applications in many fields,such as stochastic optimal control,mathematical finance,derivative pricing and so on.Compared with the stochastic differential equation,the backward stochastic differential equation deduces the strategy that can achieve the expected goal from the result,which is more in line with the law of financial market.This thesis uses backward stochastic differential equation to analyze the option price.Starting from the classical backward stochastic differential equation,the existence and uniqueness of the solution under the standard assumption are proved.Combined with the Black-Scholes model,it shows that the wealth process under the Black-Scholes model is a backward stochastic differential equation.The explicit expression of the Black-Scholes formula is derived by the methods of equivalent martingale measure and Feynman-Kac formula.With the rapid development of the financial market,there are more and more types of financial derivatives,and the stock price presents discontinuity.Consider using It(?)-L(?)vy process to describe the stock price.Under this assumption,the corresponding wealth process is a backward stochastic differential equation with jumps.In this thesis,the existence and uniqueness theorem of the solution of the backward stochastic differential equation with jumps is given.Based on Malliavin calculus,the differentiability of the solution to the backward stochastic differential equation with jumps is proved,and the expression of portfolio is obtained.Although the solution of backward stochastic differential equation with jumps exists under relevant assumptions,the explicit expression of the solution can be obtained only in special forms.In this thesis,Monte Carlo numerical simulation method is used to simulate the random motion path of the underlying asset price under the risk neutral probability measure,and the option price is simulated through the mean value of option return under all paths.7 pictures,2 tables,74 references. |
