Limit theorems for prices of options written on semi-Markov processes

Scalas, Enrico and Toaldo, Bruno (2021) Limit theorems for prices of options written on semi-Markov processes. Theory of Probability and Mathematical Statistics, 105. pp. 3-33. ISSN 0094-9000

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Abstract

We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

Item Type: Article
Keywords: Semi-Markov processes, Fractional calculus, Option pricing
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
SWORD Depositor: Mx Elements Account
Depositing User: Mx Elements Account
Date Deposited: 06 Aug 2021 07:25
Last Modified: 14 Dec 2021 16:20
URI: http://sro.sussex.ac.uk/id/eprint/100967

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