Numerical Simulation For The Payoff Of European Option Based On Stochastic Delay Differential Equations

The thesis applies the practical significant and adaptive strong convergence numerical algorithm to simulate European option payoff for specific financial models. Because it is hard to acquire the explicit solution of stochastic delay differential equations in the studies of financial models, the development of numerical algorithm which can describe the operation law of asset prices properly and control the errors of financial quantity simulation is a great theoretical and practical significant task.Chapter one introduces the application background and development process of Black-Scholes model, reviews the development of the numerical algorithm of stochastic delay differential equations.For special option pricing models in finance, chapter two studies the application of Euler-Maruyama and Monte Carlo algorithm in simulating European option payoff. Firstly, the algorithm which determines the step-size and the number of sample path was deduced through controlling the global error of final time asset price. The thesis presents the step-size and sample path number in special situations. Secondly, the mean-square-error was analyzed in the simulation of European option payoff. Finally, the related numerical experiments are given according to the conclusion.Chapter three discusses the general autonomous stochastic delay differential equations in financial mathematics. This chapter studies the application of grid method (Multigrid Euler-Maruyama method and Multilevel Monte Carlo method) in simulating European option payoff under certain conditions. Firstly, it was proved that the appropriate sample estimator can be structured in certain conditions and the mean-square error of this estimator can be controlled in a certain range. Second, the thesis deduces the complexity of algorithm in the grid method. Finally, the conclusion is that grid method decreases the complexity of algorithm in comparison with the numerical algorithmin in chapter two under certain conditions.