Numerical Solution Research Of Options Pricing Under Uncertain Volatility

In the options with stock as underlying asset, volatility smile exists widely, which makesclassic constant volatility hypothesis no longer fit. As a result of researchers' different startingpoints and using tools, we got different conclusions about the motion law of volatility insteadof an unified or generally acknowledged conclusion. However, different volatility hypotheseshave a very noticeable influence on option pricing. In this paper, assuming that volatility liesin a certain confidence interval, we are going to price Asian Option for underlying asset pricewith jump-diffusion process, regardless of volatility structure.According to underlying asset price with jump-diffusion process, we assume that volatilityinterval σ∈[σmin,σmax], then we construct value equation of Asian Option, consideringdifferent risk preferences. Even under a very simple hypothesis, the analytical solution forAsian Option, which satisfies classic Partial Differential Equation, does not exist. Hence, thispaper uses numerical solution for Partial Differential Equation to solve the problem. Fordifferent parts of the equation, we will use different numerical calculation strategies asfollows: to reduce the complexity of equation using semi-Lagrangian method; to deal withintegral terms in the Partial Differential Equation using related discrete integration method; tosolve the America-Style part using common Penalty Function Method. To the restConvection-Diffusion part, out of theoretical proof need, we use ordinary Difference Methodand the chosen difference format is supposed to ensure the M-Matrix property for CoefficientMatrix. In time aspect, we have proved that both implicit scheme and mixed scheme areunconditionally stable, and we have given the stability condition for the C-N format. In orderto solve the related non-smooth non-linear equations, we have constructed an efficientiteration method, accompanied by several numerical experiments. Finally, we get the idealexperimental result which is quite consistent with theoretical analysis.Through the volatility interval hypothesis, we have got a possible interval for option value.This interval could provide decision basis for investors. Under the constant volatilityhypothesis, the corresponding option value can not reach the interval end point. Therefore,when ail possible volatility values used by researchers lie in the volatility interval, it alsomeans the option value for related calculation is supposed to lie in the corresponding priceinterval.