Welcome to this first issue of The Journal of Computational Finance for 2018, in which we present four interesting research papers. There is one on a specific nonlinear partial differential equation, one on payoff smoothing of discontinuous payoffs for Monte Carlo methods, one on adjoint differentiation, and one on corridor variance swaps, making this issue quite wide-ranging in terms of content.
The first paper in the issue, “A new nonlinear partial differential equation in finance and a method of its solution” by Andrey Itkin, states that, in the context of optimization problems, such as optimal consumption, nonlinear partial differential equations (PDEs) may result. Two examples are discussed in detail. The properties of these equations are given, and Itkin proposes a numerical method that is both unconditionally stable and second order in space and time. Numerical experiments confirming this are presented.
In “Adjoint algorithmic differentiation tool support for typical numerical patterns in computational finance” by Uwe Naumann and Jacques du Toit, the flexibility and ease of algorithmic differentiation based on overloading is explained, keeping memory use within limits. This is a numerical technique that gives exact mathematical derivatives of a computer program. Adjoint algorithmic differentiation may compute gradients at a small computational cost. Applications in the PDE framework as well as in Monte Carlo simulation are presented.
Our third paper, “Monte Carlo payoff smoothing for pricing autocallable instruments”, is by Frank Koster and Achim Rehmet. The authors price instruments with discontinuous payoffs and nonsmooth trigger functions, and their method enhancement gives rise to stable Greeks via finite differences. Payoff smoothing is extended to the multivariate case by means of coordinate transformation and special handling of the dominant dimension. Numerical experiments confirm the theoretical findings.
“Efficient pricing and super-replication of corridor variance swaps and related products” by Christoph Burgard and Olaf Torné is this issue’s fourth and final paper. A corridor variance swap is a product in which realized variance is subjected to a weighting function that depends on the spot level. This paper focuses on hedging these products under market scenarios. Based on realistic assumptions, an explicit formula to replicate the weighted variance by vanilla instruments results. The hedge then involves a static component consisting of a strip of appropriately weighted European vanilla options, plus a dynamic part that is based on continuous rebalancing in the
I wish you very enjoyable reading of this diverse February 2018 issue of The Journal of Computational Finance.
Cornelis W. Oosterlee
CWI – Dutch Center for Mathematics and Computer Science, Amsterdam
In this paper, the author considers a special type of nonlinear PDE that arises by applying optimization to some financial problems.
Adjoint algorithmic differentiation tool support for typical numerical patterns in computational finance
This paper demonstrates the flexibility and ease in using C++ algorithmic differentiation (AD) tools based on overloading to numerical patterns (kernels) arising in computational finance.
This paper develops a Monte Carlo method to price instruments with discontinuous payoffs and non-smooth trigger functions, which allows a stable computation of Greeks via finite differences.
This paper proposes a method for overhedging weighted variance using only a finite number of maturities.