Technical Note—On Matrix Exponential Differentiation with Application to Weighted Sum Distributions

Abstract

In this note, we revisit the innovative transform approach introduced by Cai, Song, and Kou [(2015) A general framework for pricing Asian options under Markov processes. Oper. Res. 63(3):540–554] for accurately approximating the probability distribution of a weighted stochastic sum or time integral under general one-dimensional Markov processes. Since then, Song, Cai, and Kou [(2018) Computable error bounds of Laplace inversion for pricing Asian options. INFORMS J. Comput. 30(4):625–786] and Cui, Lee, and Liu [(2018) Single-transform formulas for pricing Asian options in a general approximation framework under Markov processes. Eur. J. Oper. Res. 266(3):1134–1139] have achieved an efficient reduction of the original double to a single-transform approach. We move one step further by approaching the problem from a new angle and, by dealing with the main obstacle relating to the differentiation of the exponential of a matrix, we bypass the transform inversion. We highlight the benefit from the new result by means of some numerical examples.

Keywords: Stochastic sum, probability distribution, matrix exponential and column vector differentiation, Pearson curve fit, pricing