Derivative pricing as a transport problem: MPDATA solutions to Black–Scholes-type equations

Abstract

We discuss applications of the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) to numerically solve partial differential equations arising from stochastic models in quantitative finance. In particular, we develop a framework for solving Black–Scholes-type equations by transforming them into advection–diffusion problems. The equations are then numerically integrated backward in time using an iterative explicit finite-difference approach, in which the Fickian term is represented as an additional advective term. We discuss the correspondence between transport phenomena and financial models, uncovering the possibility of expressing the no-arbitrage principle as a conservation law. We show second-order accuracy in time and space of the embraced numerical scheme. This is done via a convergence analysis comparing MPDATA numerical solutions with classic Black–Scholes analytical formulæ for the valuation of European options. We demonstrate in addition a way of applying MPDATA to solve the free boundary problem (leading to a linear complementarity problem) for the valuation of American options. We finally comment on the potential of MPDATA methods with respect to more complex models typically used in quantitative finance.