Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem (Advances in Applied Mathematics and Mechanics, 2021)
This work presents a comparison study of different numerical methods to solveBlack-Scholes-type partial differential equations (PDE) in the convection-dominated case,i.e., for European options, if the ratio of the risk-free interest rateand the squared volatility-known in fluid dynamics as Péclet number is high.For Asian options, additional similar problems arise when the "spatial" variable,the stock price, is close to zero. Here we focus on three methods:the exponentially fitted scheme, a modification of Wang's finite volume methodspecially designed for the Black-Scholes equation, and the Kurganov-Tadmor schemefor a general convection-diffusion equation, that is applied for the first timeto option pricing problems. Special emphasis is put on the Kurganov-Tadmor becauseits flexibility allows the simulation of a great variety of types of options andit exhibits quadratic convergence. For the reduction technique proposed by Wilmott,a put-call parity is presented based on the similarity reduction and the put-callparity expression for Asian options.Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.