Optimal and near-optimal advection—diffusion finite-difference schemes III. Black—Scholes equation

Optimal and near–optimal compact finite–difference schemes are presented and tested for the numerical solution of an extended Black–Scholes equation: ∂tc-rc+rs∂sc+12σ2s2∂s2c=-q-s∂s(s∂sg).Here c(s,t) is the expected value of the right to buy or sell an asset at some future date, s is the asset price, r(t) is the rate of increase available from alternative riskless investments, and σ(t) is the asset volatility. The terms on the right–hand side allow the applicability to be extended beyond the basic European options model. The compactness of the numerical scheme keeps any computer programming elementary. The required computational resources can be as small as 0.001 of conventional schemes.