Journal of Computational Finance

This issue of The Journal of Computational Finance has numerical partial differential equation discretization techniques as its central theme. Modern aspects like accurate time discretization for nonsmooth conditions, adaptivity and error estimates, dealing with high dimensionality, and dealing with partial integrodifferential equations are discussed in the four papers that make up the issue.

The first paper, "The damped Crank-Nicolson time-marching scheme for the adaptive solution of the Black-Scholes equation" by Christian Goll, Rolf Rannacher and Winnifried Wollner, deals with error estimators and mesh adaptation for a space-time finite element discretization of the basic Black-Scholes equation. An interesting modern numerical mathematical technique for a fundamental pricing equation in finance is explained.

"An efficient numerical partial differential equation approach for pricing foreign exchange interest rate hybrid derivatives" by Duy Minh Dang, Christina Christara, Kenneth Jackson and Asif Lakhany discusses long-dated foreign exchange interest rate hybrids under a three-factor multi-currency model. The resulting partial differential equation is high dimensional and is governed by exotic features, due to path dependency and barriers. Nonuniform spatial finite difference grids are combined with alternating direction implicit time discretization.

High dimensionality in a partial differential equation framework is also the theme of the paper by Christoph Reisinger and Rasmus Wissman: "Numerical valuation of derivatives in high-dimensional settings via partial differential equation expansions". A new numerical approach is presented, based on principal components analysis in combination with an expansion into solutions of low-dimensional partial differential equations. The resulting numerical solutions are carefully compared in terms of accuracy and run time to Monte Carlo methods.

Radha Krishn Coonjobeharry, Désiré Yannick Tangman and Muddun Bhuruth's paper, "A novel partial integrodifferential equation-based framework for pricing interest rate derivatives under jump-extended short-rate models", completes our lineup. The spatial differential terms as well as the integral are discretized by higher-order discrete schemes, and different time discretization schemes are evaluated. European and American options are priced under a jump-extended constant-elasticity-of-variance asset process. Fast run time and high accuracy are reported.

I wish you very enjoyable reading. In the meantime, we are in the process of switching to an automated electronic submission and manuscript handling system, making the submission of your future manuscripts for The Journal of Computational Finance easier.

Cornelis W. Oosterlee
CWI - Dutch Center for Mathematics and Computer Science, Amsterdam