Noncompact Difference Schemes Based On Expectations And Its Application In Option Pricing

In the 1970s,Black and Scholes proposed the famous Black-Scholes option pricing model,which laid a theoretical foundation for the development of the global options market and greatly promoted the development of the intersection of mathematics and finance.This thesis studies the difference method of option pricing.Since the positiveness of option price is a basic requirement,we are committed to developing a difference method with positiveness.Based on the Feynman-Kac formula,the solution of the parabolic equation can be expressed as a conditional expectation,and we hope to obtain a numerical scheme with positiveness by approximating the conditional expectation.For multi-dimensional problems,a compact difference scheme with a discrete probability structure can only be constructed under certain conditions(such as diagonal dominance of the covariance matrix),so we give up the requirement of compactness.Firstly,for one-dimensional problems,we propose two types of noncompact explicit schemes with positiveness,and analyze the consistence,convergence and stability of the algorithms;on this basis,we propose a noncompact difference scheme for solving two-dimensional problems,numerical experiments and application to multi-asset option pricing verify that the new algorithm is effective.The establishment of positiveness of the algorithms in this thesis has no conditions required to the covariance matrix corresponding to the problem,and compared with the explicit algorithm in the compact format,the time step in our algorithm can be enlarged,and it only needs to be of the same order as the space step.Both theoretical analysis and numerical experimental results show that the noncompact difference scheme based on expectation proposed in this thesis can provide an unconditionally positive difference scheme for multi-asset option pricing problems,thereby providing accurate option prices for financial markets.