Study On Algorithms For Partial Differential Equations Of Path-dependent Option Pricing

This thesis studies the algorithms for partial differential equation of pathdependent option pricing,including the algorithms of lookback option under regime-switching jump-diffusion,Asian option under regime-switching jumpdiffusion,vulnerable Asian option and other complex models path-dependent options.In the aspect of lookback option under regime-switching jump-diffusion model,this thesis uses the high-order methods to study partial integro differential equations of lookback option pricing.The value function of the lookback option under regime-switching jump-diffusion model is governed by a system of two-dimensional partial integro differential equations.In this thesis,the two-dimensional partial integro differential equations are recast into a onedimensional partial integro differential equations.Using the high-order methods to solve one-dimensional problems,at the same time of getting the option price,the calculate method is proposed to solve Greeks.Numerical examples are carried out to verify the accuracy and the high-order convergence rate of option price.In the aspect of Asian option under regime-switching jump-diffusion model,we use the high-order methods and the moving mesh methods to complete the solution and analyse convergence rate.In the part of the high-order methods,we obtain the option price and Greeks.Numerical examples are carried out to verify the accuracy and the high-order convergence rate of option price.In the part of the moving mesh methods,the two-dimensional partial integro differential equations are recast into a one-dimensional moving boundary problems,the partial Integro differential equations of moving boundary is solved,and then the convergence rate is proved by theory and numerical examples respectively.In terms of vulnerable geometric Asian option,the probability method such as conditional expectation theorem is used to obtain the exact solution.In this thesis,we use the conditional expectation theorem to transform the vulnerable geometric Asian option into a geometric Asian option with new terminal condition,then the exact solution of the geometric Asian option with new terminal condition is obtained by using the probability method,thus the pricing of vulnerable geometric Asian option is completed.In terms of vulnerable arithmetic Asian option,we use the conditional expectation theorem and the finite difference method to complete the option pricing.In this thesis,we use the conditional expectation theorem to transform the vulnerable arithmetic Asian option into a arithmetic Asian option with new terminal condition,owing to the arithmetic average Asian option has no exact solution,we use partial differential equation pricing method to complete the option pricing.In this thesis,we give the numerical solution of vulnerable arithmetic Asian option,combining the conditional expectation theorem with partial differential equation pricing method.Numerical examples are carried out to verify the accuracy of option price.