From measure changes to time changes in asset pricing

Abstract

The goal of the paper is to review the last 35 years of continuous-time finance by focusing on two major advances: (i) The powerful elegance of the martingale representation for primitive assets and attainable contingent claims in more and more general settings, thanks to the probabilistic tool of probability change and the economic flexibility in the choice of the numéraire relative to which prices are expressed. This numéraire evolved over time from the money market account to a zero-coupon bond or a stock price, lastly to strictly positive quantities involved in the Libor or swap market models and making the pricing of caps or swaptions quite efficient. (ii) The persistent central role of Brownian motion in finance across the 20th century: even when the underlying asset price is a very general semi-martingale, the no-arbitrage assumption and Monroe theorem [Monroe, I., 1978. Processes that can be embedded in Brownian motion. Annals of Probability 6, 42–56] allow us to write it as Brownian motion as long as we are willing to change the time. The appropriate stochastic clock can be shown empirically to be driven by the cumulative number of trades, hence by market activity. Consequently, starting with a general multidimensional stochastic process S defined on a probability space ( Ω, F , F t , P) and representing the prices of primitive securities, the no-arbitrage assumption allows, for any chosen numéraire, to obtain a martingale representation for S under a probability measure Q S equivalent to P. This route will be particularly beneficiary for the pricing of complex contingent claims. Alternatively, changing the clock, i.e., changing the filtration ( F t ), we can recover the Brownian motion and normality of returns. In all cases martingales appear as the central representation of asset prices, either through a measure change or through a time change.