Open AccessLimit theorems for prices of options written on semi-Markov processes
Author(s) -
Enrico Scalas,
Bruno Toaldo
Publication year - 2021
Publication title -
theory of probability and mathematical statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.393
H-Index - 12
eISSN - 1547-7363
pISSN - 0094-9000
DOI - 10.1090/tpms/1153
Subject(s) - subordinator , mathematics , geometric brownian motion , martingale (probability theory) , laplace transform , markov process , brownian motion , martingale representation theorem , multiplicative function , mathematical analysis , mathematical economics , lévy process , diffusion process , statistics , knowledge management , innovation diffusion , computer science
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.