Stochastic clock methods for trade duration in option pricing

Stochastic clock and in-business-time return models are useful in finance to recover Gaussianity of asset returns by means of re-normalisation with an increasing process (a time change) capturing the market activity (number of trades and their sizes) through time. When applied to a Brownian motion parent process, absolutely continuous time changes give raise to stochastic volatility return models, whereas if the stochastic clock is an independent Lévy subordinator, a new pure jump Lévy model is attained.

In this talk I introduce a third way of using stochastic clock techniques, specifically geared towards modelling stochasticity in the times of price revisions, i.e. trade duration, within a Lévy-based, continuous-time finance framework. By using as a time change the first hitting time process of a Lévy subordinator we turn a jump process into a continuous but locally constant one, thus obtaining a stochastic clock which can be interpreted as a calendar time where the price evolution is halted at random instants.

I illustrate some of the many applications such a technique may have, with a specific view to option pricing. I discuss three main topics: (i) modelling of regulatory and automated trading halts in equity markets; (ii); explicit identification of a market price of risk connected to liquidity factors; (iii) impact of dependence between duration and returns in the large maturity decay of the implied volatility skew. Explicit analytic pricing formulae can indeed be exhibited, making use of the fact that an analytic transform theory of appropriate inverse Lévy subordinators is fully developed.

I finally suggest that the statistical fitting of such models poses delicate questions. For example, for a principled option prices calibration, it is critical to understand which of the trade duration parameters can be derived from the market implied risk-neutral distributions, and which are instead inherent to the market microstructure, and must therefore appear in the minimisation problem as a constraining factor.