Sub-diffusion for Illiquid Markets

Abstract

The previous chapters have initiated our journey into the world of processes that are not martingales. We introduce in this chapter a new category of processes perfectly adapted for modeling illiquidity. In emerging or in small cap markets, the number of participants is often low, and thus transactions are sparse. The time series of stock prices in such conditions display characteristic periods in which they stay motionless. This phenomenon is also visible at high frequency: the sample paths of stock prices look like stepwise random processes rather than continuous ones. A similar behavior is observed in physical systems exhibiting sub-diffusion. The constant periods of financial processes correspond to the trapping events in which a heavy particle gets immobilized, see, e.g., Eliazar and Klafter (Physica D 187:30–50, 2004) or Metzler and Klafter (J Phys A Math General 37(31):R161, 2004). In statistical physics, this type of dynamic is modeled by a sub-diffusive Brownian motion. This process is obtained by observing a standard Brownian motion on a different scale of time. In this chapter, after introducing the detailed features of this stochastic clock, we show that the density of a sub-diffusion is described in terms of a fractional Fokker–Planck (FFP) equation. We evaluate European options in this setting. The chapter concludes by estimating the model using the particle MCMC algorithm presented in Chap. 3.