Stochastic Clock and Financial Markets

Brownian motion played a central role throughout the twentieth century in probability theory. The same statement is even truer in finance, with the introduction in 1900 by the French mathematician Louis Bachelier of an arithmetic Brownian motion (or a version of it) to represent stock price dynamics. This process was ‘pragmatically’ transformed by Samuelson in ([48, 49]; see also [50]) into a geometric Brownian motion ensuring the positivity of stock prices. More recently the elegant martingale property under an equivalent probability measure derived from the No Arbitrage assumption combined with Monroe’s theorem on the representation of semi-martingales have led to write asset prices as time-changed Brownian motion. Independently, Clark [14] had the original idea of writing cotton Future prices as subordinated processes, with Brownian motion as the driving process. Over the last few years, time changes have been used to account for different speeds in market activity in relation to news arrival as the stochastic clock goes faster during periods of intense trading. They have also allowed us to uncover new classes of processes in asset price modelling.