Laplace Transform and finite difference methods for the Black-Scholes equation

Keywords: Black–Scholes equation Completely monotonic function Finite difference scheme Laplace Transform M-Matrix Positivity-preserving Post–Widder formula a b s t r a c t In this paper we explore discrete monitored barrier options in the Black–Scholes framework. The discontinuity arising at each monitoring data requires a careful numerical method to avoid spurious oscillations when low volatility is assumed. Here a technique mixing the Laplace Transform and the finite difference method to solve Black–Scholes PDE is considered. Equivalence between the Post–Widder inversion formula joint with finite difference and the standard finite difference technique is proved. The mixed method is positivity-preserving, satisfies the discrete maximum principle according to financial meaning of the involved PDE and converges to the underlying solution. In presence of low volatility, equivalence between methods allows some physical interpretations. One of the main concerns about financial options is what the exact values of the options are. In absence of evaluation formula for non-standard options, numerical technique is required. Usually the choice goes toward numerical methods with high order of accuracy (for instance in the finite difference method the Crank–Nicolson scheme) and no attention is paid to the fact how the financial provision of the contract can affect the reliability of the numerical solution. Special options, as discretely monitored barrier options are characterized by discontinuities that are renewed at each monitoring date. In presence of low volatility the Black–Scholes PDE becomes convection dominated. As a consequence, numerical diffusion or spurious oscillations may arise, so that special numerical techniques have to be employed. A viable route to circumvent discontinuity issues is considering an Integral Transforms method. If we assume that the volatility r of the underlying asset price S and the risk-free interest rate r of the market depends only on S, i.e., r ¼ rðSÞ; r ¼ rðSÞ, then the Laplace Transform becomes a useful tool. The Black–Scholes equation is solved by the Laplace Transform method for time t discretization. The resulting ordinary differential equation (ODE) is solved by a finite difference scheme and, as a final result, a Laplace Transform of the solution is obtained. Hereinafter we call that method a 'mixed method'. The crucial issue is the Laplace Transform inversion. A lot of methods are available in literature [1]. They can be roughly classified into two categories: the ones using complex values of the Laplace Transform and the ones using only uniquely real values. Here, rather than proposing a …