This issue of The Journal of Computational Finance contains four papers that are quite different in terms of their financial applications, and they stand out because of their remarkable mathematical techniques. We will briefly touch upon each of the interesting techniques proposed.
The first research paper in this issue is "Wiener chaos expansion and numerical solutions of the Heath-Jarrow-Morton interest rate model" by Evangelia A. Kalpinelli, Nikolaos E. Frangos and Athanasios N. Yannacopoulos from Athens, Greece. The solution of a stochastic differential equation is decomposed into a hierarchy of deterministic evolution equations. By doing this, the randomness can be separated from the deterministic part of a solution, and these two parts can therefore be computed separately. The statistical moments of the stochastic solution can be easily recovered from the deterministic propagator using simple formulas and avoiding time-consuming computations.
"Accelerated trinomial trees applied to American basket options and American options under the Bates model", by Conall O'Sullivan and Stephen O'Sullivan from Ireland, is our second paper. Based on the equivalence between a one-step binomial tree and a multistep trinomial tree, nonuniform time steps are included in a multistep trinomial tree. The resulting accelerated trinomial tree financial derivatives pricing method circumvents the restrictive time step condition inherent in standard trinomial trees and explicit finite-difference methods.
Akihiko Takahashi and Yukihiro Tsuzuki from Japan present "A new improvement scheme for approximation methods of probability density functions" as the issue's third paper. Accurate analytic approximations for probability density functions of asset prices based on insights from the Hilbert space projection theorem and implemented by means of Dykstra's cyclic projections algorithm are proposed. Application of the approximation to an asymptotic expansion option pricing method demonstrates the effectiveness under the stochastic alpha, beta, rho (SABR) model.
The fourth paper in the issue, by Nicolas Privault and JiadongYu from Singapore, is "Stratified approximations for the pricing of options on average". Stratified sampling is typically used as a variance reduction method in Monte Carlo simulations, but in the present setting it also improves numerical accuracy. A conditional expectation that is encountered can then be computed by means of a triple oscillating integral, but this does lack numerical stability in the small time case. The integral form is replaced by approximations that are based on the gamma and lognormal distributions. The stratified approximations based on the gamma and lognormal distributions for the pricing of options on an average, such as Asian options, result in robust and efficient solution techniques.
I wish you very enjoyable reading of this issue of The Journal of Computational Finance.
The authors propose stratified approximations of option prices using the gamma and lognormal distributions, with an application to bond pricing in the Dothan model.
This paper develops a new scheme for improving an approximation method of a probability density function.
Accelerated trinomial trees applied to American basket options and American options under the Bates model
This paper introduces accelerated trinomial trees, a novel efficient lattice method for the numerical pricing of derivative securities.
The authors propose an efficient, novel numerical scheme for solving the stochastic Heath–Jarrow–Morton interest rate model.