The Euler-Maruyama Approximations For The Stochastic Volatility With Jumps Model

In this dissertation, we mainly investigate the numerical solution property of the stochastic volatility with jumps (SVJ) model, numerical simulation and some applica-tion on European and barrier options pricing.In the first chapter, we outline the background and the major work.In the second chapter, we state the Ito formula of the stochastic differential equa-tion (SDE) with jumps, the method of Euler-Maruyama approximations, and some inequalities.In the third chapter, we give equivalent form of the SVJ model and obtain the nonnegative solution, we then give and prove the moments of the model. We also prove the strong convergence of the numerical solution. It should be pointed out that for the mean-reverting square root process, Higham and Mao give the proof of the strong convergence of the solution. In this dissertation, these strong convergence will be given in the form of lemma.In the fourth chapter, we prove that the numerical approximation of the pay-off for the European put and up-and-out barrier options are convergent. It should be pointed out that our proof idea is inspired by Higham and Mao, but their stock price process is a continuous stochastic process, and ours is with jumps, so we need seeking new methods to solve it.In the fifth chapter, we first introduced the Monte Carlo method, then price of option was simulated.