Study On Fast Operator Splitting Numerical Solution Of Multi-Asset Option Pricing

A multitude of financial derivatives have emerged in the market due to the rapid development of the economy,catering to the varying investment demands and risks.As one of these diversified portfolio methods,the research on the pricing of multi-asset option is of vital importance.Since Black and Scholes established the option pricing model in 1973,many scholars began to focous on how to provide accurate and efficient numerical solutions for option pricing problems based on the Black-Scholes equation.In addition,to make the model more realistic,many optimization models were proposed,such as stochastic volatility models,stochastic rate models,jump diffusion models,and more.Therefore,it is meaningful to develop quick and effective numerical schemes for pricing multi-asset options and optimizing models in financial engineering.Based on the above research background,this thesis studies the numerical methods of multi-asset option pricing problems under the assumption of the geometric Brownian motions and Heston stochastic volatility.The thesis gives new numerical schemes for multi-asset pricing problems based on orthogonal transformation of the Brownian motions,operator splitting method and pseudo-spectral method on local encrypted grids.The main contents and innovations of this thesis are as follows:1.We construct numerical schemes to solve the two-asset option pricing problems under the assumption that the underlying asset price processes are geometric Brownian motions.For the corresponding two-dimensional Black-Scholes equation,the mixed derivative terms are eliminated first,so as to avoid the asymmetric linear equations caused by discrete mixed derivative terms.Then we use the operator splitting method to split the problem into two one-dimensional problems without mixed derivative terms.The thesis introduces C-LRG(Chebyshev Locally Refined Grids)obtained by the transformation of Chebyshev interpolation points and the grid point to be solved.Combining the operator splitting method with the pseudo-spectral method differential matrix on locally refined grids,the fast OS-PS(Operator Splitting and Pseudo-spectral)method is proposed to solve the two-asset option pricing problems.The experiment results on the spread option and two-asset basket option demonstrate that the OS-PS scheme on C-LRG achieves quicker convergence to the precise solution compared to the finite difference method based on uniform meshing.Additionally,the result demonstrates a first-order convergence rate that aligns with the theoretical estimation regarding time.Finally,we compare the numerical solution errors of the OS-PS scheme on C-LRG and other locally refined grids,further illustrating the advantages of the C-LRG constructed in this thesis.2.We construct numerical schemes to solve the pricing problems of the two asset basket option under Heston stochastic volatility model.First,the partial differential equation of the stochastic volatility option pricing model is derived by constructing the risk-free asset portfolio and the delta-hedging technique.The variable coefficient partial differential equation without mixed derivative terms is derived through the correlation between random processes.The operator splitting method is used to divide the problem into three one-dimensional problems which can be quickly calculated.Then,we provide the finite difference scheme and the OS-PS scheme to solve the problem.The numerical experiment confirms that the OS-PS scheme on C-LRG still has higher solving efficiency compared to finite difference discretization when applied to the stochastic volatility model.Additionally,the results confirm that the numerical approach possesses first-order accuracy on time.