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.

Limit theorems for prices of options written on semi-Markov processes / Scalas, E.; Toaldo, B.. - In: THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS. - ISSN 0094-9000. - 105:(2021), pp. 3-33. [10.1090/tpms/1153]

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

Scalas E.;Toaldo B.
2021

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.
2021
option pricing; financial mathematics; semi-Markov processes
01 Pubblicazione su rivista::01a Articolo in rivista
Limit theorems for prices of options written on semi-Markov processes / Scalas, E.; Toaldo, B.. - In: THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS. - ISSN 0094-9000. - 105:(2021), pp. 3-33. [10.1090/tpms/1153]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1667998
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