In this issue of The Journal of Computational Finance we encounter different aspects of numerical and computational techniques for some modern applications in finance. Two papers deal with regression and simulation within the Monte Carlo pricing context, one paper advocates the use of radial basis functions for calculating counterparty credit risk, and one paper elaborates on the use of partial differential equations for a real options application.
In their paper "Numerical algorithms for research and development stochastic control models", Chi Man Leung and Yue Kuen Kwok present a numerical scheme for a Hamilton Jacobi-Bellman formulation of a real options model regarding the optimal research and development (R&D) expenditure strategy to develop an innovative product. The R&D stochastic control partial differential equation model, which is explained in detail in the paper, is discretized and solved with an advanced nonlinear iteration scheme.
A radial basis functions framework is proposed by Matt Thompson in his paper "Counterparty credit risk pricing and measurement of swaption portfolios". The radial basis expansions in a partial differential equation framework can be incorporated into simulations for credit value adjustment and potential future exposure. Partial differential equation problems in as many as forty-eight spatial dimensions under lognormal and local volatility dynamics are solved.
The first of the two papers in the issue that concentrate on regression and simulation is by Lars Stentoft and is titled "Value function approximation or stopping time approximation: a comparison of two recent numerical methods for American option pricing using simulation and regression". Classical Monte Carlo regression-based methods - such as the Longstaff-Schwartz method and the methods by Carriere and by Tsitsiklis and van Roy - are analyzed and viewed from a different perspective. Based on the impact of cross-sectional regressions, the author tends to favor one of the classical approaches.
Our final paper also has least squares Monte Carlo simulation in its title: "Pricing American-style options by Monte Carlo simulation: alternatives to ordinary least squares" by Stathis Tompaidis and Chunyu Yang. In this paper ordinary least squares regression is compared with five alternative techniques. Four of the methods have been proposed in the literature, but the fifth - a new method proposed in the paper and called modified matching projection pursuit - is the recommended method.
I wish you enjoyable reading of this issue of The Journal of Computational Finance.
CWI - Dutch Center for Mathematics and Computer Science, Amsterdam
The authors consider the optimal strategy of research and development (R&D) expenditure adopted by a firm that engages in R&D to develop an innovative product to be launched in the market.
This paper introduces a technique for pricing and risk measurement of portfolios containing swaption contracts in the presence of counterparty credit risk, under general market model and volatility assumptions.
Value function approximation or stopping time approximation: a comparison of two recent numerical methods for American option pricing using simulation and regression
The authors investigate the performance of the ordinary least squares (OLS) regression method in Monte Carlo simulation algorithms for pricing American options.