This issue of The Journal of Computational Finance focuses on various novel numerical mathematical techniques. Two of the issue's papers deal with numerical simulation techniques for partial differential equations (PDEs) and the other two introduce numerical methods for option pricing stochastic differential equations (SDEs).In the first paper in the issue, "Quadratic finite element and preconditioning methods for options pricing in the SVCJ model", Ying-Ying Zhang, Hong-Kui Pang, Liming Feng and Xiao-Qing Jin present efficient preconditioners for a Krylov subspace method dealing with a stochastic volatility jump diffusion option pricing partial integrodifferential equation. A drastic reduction in CPU solution time is reported.
In the issue's second paper, "TR-BDF2 for fast stable American option pricing", Fabien Le Floc'h proposes employing the so-called trapezoidal rule with secondorder backward differentiation formula (TR-BDF2) for solving the Black-Scholes option PDE on a nonuniform grid. The method's convergence for American options, as well as the stability of the Greeks, is studied and compared with the currently popular numerical schemes. This is fundamental work that may be of interest to a broad audience. The Monte Carlo SDE simulation schemes in the two subsequent papers focus on systems of SDEs.
In our third paper, "Monte Carlo pricing in the Schöbel-Zhu model and its extensions", Alexander van Haastrecht, Roger Lord and Antoon Pelsser propose a simulation algorithm for the Schöbel-Zhu stochastic volatility model, and for its extension with stochastic interest rates. The scheme comes with a natural definition for a martingale correction, and the authors find that the new numerical scheme consistently outperforms the well-known Euler scheme.
The issue's fourth paper, "Robust and accurate Monte Carlo simulation of (cross-) Gammas for Bermudan swaptions in the LIBOR market model" by Ralf Korn and Qian Liang, deals with efficient simulation of interest rate products under the LIBOR market model. Two new efficient methods for the computation of specific Greeks are proposed. One of them is based on a combination of the finite difference method and pathwise Deltas in the context of an adjoint technique, while the other is based on a pure pathwise method and a robust unbiased simulation.
The aim of all the numerical papers in this issue is, as always, to combine numerical accuracy with computational efficiency. I hope you enjoy reading this issue of The Journal of Computational Finance.
Cornelis W. Oosterlee
CWI - Dutch Center for Mathematics and Computer Science, Amsterdam