The papers in this issue of The Journal of Computational Finance cover a diverse range of applications and numerical techniques. Applications range from the valuation of financial derivatives to optimal investment, quadratic hedging and the valuation of electricity storage. The numerical techniques employed are also diverse, ranging from Monte Carlo methods to importance sampling, semi-Lagrange partial differential equation finite differences and partial integrodifferential equation discretization techniques.
In the issue's first paper, "Optimal investment: bounds and heuristics", Leonard "Chris" Rogers and Pavel Zaczkowski present a technique for finding upper bounds on the value of a portfolio in a (possibly high-dimensional) optimal investment problem. A dual method for finding lower bounds is also provided. Numerical evidence for the performance is given by looking at different optimal investment examples, from Merton's problem to an example with correlated assets and a no-short-selling constraint.
Our second paper is "Numerical methods for the quadratic hedging problem in Markov models with jumps" by Carmine De Franco, Peter Tankov and Xavier Warin. Algorithms are developed using the Hamilton-Jacobi-Bellman approach for parabolic partial integrodifferential equations related to the quadratic hedging strategy in incomplete markets. The finite-difference method is the numerical discretization of choice, and applications include electricity markets.
Electricity also plays a role in the third paper in the issue: "SLADI: a semi-Lagrangian alternating-direction implicit method for the numerical solution of advection-diffusion problems with application to electricity storage valuations" by Javier Hernández Ávalos, Paul V. Johnson and Peter W. Duck. In a finite-difference discretization, the semi-Lagrange approach for hyperbolic problems is combined with the alternating-direction implicit method for parabolic problems with diffusion. The method's stability, accuracy and numerical convergence are presented for the valuation of a four-dimensional storage option.
In the issue's fourth paper, "Importance sampling for jump processes and applications to finance" by Laetitia Badouraly Kassim, Jérôme Lelong and Imane Loumrhari, an adaptive importance sampling technique is generalized to jump processes by, first, a change of measure computation by Newton's method and, second, using the estimator of this measure change in an independent Monte Carlo method. The efficiency of the method is demonstrated through valuation of financial derivatives under several jump models.
In summary, this is a diverse issue, with papers detailing some high-quality state-of-the-art numerical techniques. 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
The authors present a technique for finding upper bounds on the value of a portfolio in a (possibly high-dimensional) optimal investment problem.
In this paper algorithms are developed using the Hamilton–Jacobi–Bellman approach for parabolic partial integrodifferential equations related to the quadratic hedging strategy in incomplete markets.
SLADI: a semi-Lagrangian alternating-direction implicit method for the numerical solution of advection–diffusion problems with application to electricity storage valuations
In this paper, an efficient and novel methodology for numerically solving advection–diffusion problems is presented.
Adaptive importance sampling techniques are widely known for the Gaussian setting of Brownian-driven diffusions. In this paper, the authors extend them to jump processes.