In this first 2016 editorial for The Journal of Computational Finance, I am delighted to introduce two new members of our editorial board. Luis Ortiz Gracia, from the Centre de Recerca Matemàtica in Barcelona, Spain, is an expert on wavelets and their use in financial derivatives pricing as well as in risk management. At the same time, he works on so-called conditional Monte Carlo methods. This latter research topic is also one of the many competencies of the second new member of our editorial board: Duy-Minh Dang, from the University of Queensland, Australia. In addition to Monte Carlo methods, he is highly skilled in numerically solving linear and nonlinear partial differential equations. I wish Luis and Duy-Minh a warm welcome to our editorial board!
This issue is one containing diverse papers. We start with a paper by Johannes Vilsmeier that deals with "Updating the option implied probability of default methodology". Based on a stable objective function for the estimation of the risk-neutral density, a recent implied probability of default (iPoD) approach is improved upon. Unlike the original approach, this alternative procedure for the estimation of the iPoD, based on Lagrange multipliers, produces reliable results.
"Efficient solution of backward jump-diffusion partial integro-differential equations with splitting and matrix exponentials" is the title of Andrey Itkin's contribution. A unified approach for solving jump-diffusion partial integro-differential equations is proposed. The technique is based on second-order operator splitting, with a dimensional splitting for the diffusion operator appearing. The jump integral term is treated as a pseudo-differential operator. Appropriate first- and second-order discrete approximations to these operators are presented, resulting in an unconditionally stable time discretization and a preservation of the solution's positivity.
Automatic differentiation (AD) is the theme of the issue's third paper: "The efficient application of automatic differentiation for computing gradients in financial applications" by Wei Xu, Xi Chen and Thomas F. Coleman. In finance, the use of reversemode AD allows the computation of gradients in the same time required to evaluate an objective function itself. Memory requirement may, however, make reverse-mode AD expensive in some cases and slower than expected. The authors show that many functions in calibration and inverse problems exhibit a natural substitution structure. Significant speedup is achieved compared with common reverse-mode AD.
"B-spline techniques for volatility modeling" by Sylvain Corlay is this issue's final paper. The use of B-splines is advocated for volatility modeling within the calibration of stochastic local volatility models and for the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. Shape-constrained B-splines are used, for example, for the estimation of conditional expectations. A B-spline parameterization of the Radon-Nikodym derivative of the underlying risk-neutral probability density with respect to a roughly calibrated base model provides smooth arbitrage-free implied volatility surfaces.
All these topics are highly relevant in contemporary computational finance in
academia as well as in industrial practice. I wish you very enjoyable reading of this issue of The Journal of Computational Finance.
Cornelis W. Oosterlee
CWI - Dutch Center for Mathematics and Computer Science, Amsterdam
This paper updates the option implied probability of default (iPoD) approach recently suggested in the literature.
The efficient application of automatic differentiation for computing gradients in financial applications
Automatic differentiation is the theme of this paper. The authors show that many functions in calibration and inverse problems, exhibit a natural substitution structure. A significant speedup is achieved compared with common reverse-mode AD.
In this paper the use of B-splines is advocated for volatility modeling within the calibration of stochastic local volatility (SLV) models and for the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data.
Efficient solution of backward jump-diffusion partial integro-differential equations with splitting and matrix exponentials
A unified approach for solving jump-diffusion partial integro differential equations is proposed.