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

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.

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

“…In the specific case of the α-stable subordinator, one refers to such a new process as a fractional version of the parent one, due to the link between this time-change procedure and fractional calculus, as one can see from [11][12][13][14][15][16][17], to cite some of the several works on the topic. Together with a purely mathematical interest, let us stress that this procedure leads also to some interesting applications (see, for instance, [18] in economics, [19] in computing and [20] in computational neurosciences). In the context of queueing theory, this has been first done in [4], in the case of the M/M/1 queue, and then extended in [21,22] to the case of M/M/1 queues with catastrophes, [23] in the case of Erlang queues and [24] for the M/M/∞ queues.…”