Numerical Methods For Option Pricing Problems Under The Heston Stochastic Volatility Model

The derivatives market is an integral component of the capital market.As one of the most active derivatives in the world,options and its pricing have received widespread attention in both academia and industry.The classic Black-Scholes model makes strong assumptions and it cannot reflect the features of the real market data.Then a variety of models have been developed to overcome its shortcomings.Among them,the Heston stochastic volatility model is one of the most popular models in practice and research.This paper mainly studies the asymptotic analysis and numerical solutions of pricing European options under the Heston stochastic volatility model.The European option pricing problem under the Heston stochastic volatility model has a closed-form solution,but it involves in complex integrals which are difficult to calculate.In this paper,we present the asymptotic solution to the problem in powers of the volatility of variance.We obtain some asymptotic properties of the option price and its derivatives with respect to the asset price,variance and time.The initial boundary value problem derived from the Heston model is defined in an infinite domain which needs to be handled carefully.To find a numerical solution,it is common to truncate the domain and apply the boundary condition at infinity directly,which can cause high inaccuracy.In this paper,we try to give an approximation of the original equation based on the asymptotic properties of the option price and derive an approximate artificial boundary condition for the Heston model.Then we make modifications and get another two artificial boundary conditions(ABCs)to improve the accuracy.Compared with the ABC corresponding to the Black-Scholes model,we analyze the asymptotic properties of the three proposed ABCs.We use the ABCs to transform the European option pricing problem in an unbounded domain into a reduced problem in a bounded domain and discuss the equivalence of the two problems.We also analyze the stability of the reduced problem and the existence and uniqueness of its solution.As for the numerical method,we use a typical finite difference scheme combined with a numerical integration method and a curve fitting approach to approximate the ABCs.We discuss the accuracy and stability of the numerical method.Numerical experiments turn out that compared with the commonly-used boundary condition at infinity in Heston's original paper,the proposed ABCs work remarkably better with higher accuracy and lower computational cost.They also show a big advantage when calculating the Greeks which play an important role in option trading.This paper also introduces an iterative splitting method to solve the European option pricing problem under the Heston model.Based on the idea of operator splitting,the method transforms the two-dimensional problem into quasi one-dimensional problems and a mixed method is used to improve the convergence of the iterative process.This method is intuitive and simple,and can be easily extended to other option pricing models.Numerical experiments show that the method can give more accurate results of the option price and the Greeks compared to the classic finite difference scheme.