Volume 5, Issue 6
Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem

Germán I. Ramírez-Espinoza1 ,  ∗ and Matthias Ehrhardt

10.4208/aamm.12-m1216

Adv. Appl. Math. Mech., 5 (2013), pp. 759-790.

Preview Full PDF BiBTex 2 154
  • Abstract

This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Peclet 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 method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a  put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.

  • History

Published online: 2013-05

  • AMS Subject Headings

65M10, 91B25

  • Cited by