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

Milan Kumar Das (), Henghsiu Tsai (), Ioannis Kyriakou () and Gianluca Fusai ()
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Milan Kumar Das: Institute of Statistical Science, Academia Sinica, Taiwan (ROC)
Henghsiu Tsai: Institute of Statistical Science, Academia Sinica, Taiwan (ROC)
Ioannis Kyriakou: Faculty of Actuarial Science & Insurance, Bayes Business School, City, University of London, London EC1Y 8TZ, United Kingdom
Gianluca Fusai: Dipartimento di Studi per l’Economia e l’Impresa, Università del Piemonte Orientale, 28100 Novara, Italy; Faculty of Finance, Bayes Business School, City, University of London, London EC1Y 8TZ, United Kingdom

Operations Research, 2022, vol. 70, issue 4, 1984-1995

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: Financial Engineering; stochastic sum; probability distribution; matrix exponential and column vector differentiation; Pearson curve fit; pricing (search for similar items in EconPapers)
Date: 2022
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