Waiting for returns: using space–time duality to calibrate financial diffusions

Abstract

Historically, tests of the Geometric Brownian Motion (GBM) model for security prices—and for that matter any diffusion process—have been performed by selecting a fixed interval of time (one day, one week, one month) t and then using the increments in logarithmic price ln1⁄2P over the predetermined t. Under the classical specification of the GBM model, the logarithmic price increments ln1⁄2P should be statistically independent from each other and these increments ln1⁄2P should be Normally distributed with a mean and variance that is proportional to the time increment t. This approach has a long tradition in finance. Research done in the 1950s by Kendall (1953) and Osborne (1959) as well as the work by Fama (1970) all the way through to the contemporary work of Campbell et al. (1997) based on Lo and MacKinlay (1988) focuses on a particular time interval t. Thus, for example, Kendall (1953) looked at a time increment of t 1⁄4 one week on the New York Stock Exchange, and concluded that the logarithmic price increments ln1⁄2P have a statistically insignificant serial correlation in addition to being (approximately) normally distributed. In another study, Fama (1970) looked at the 30 Dow Jones Industrial stocks with a t 1⁄4 one day, and concluded that there is a statistically significant positive serial correlation in logarithmic price increments ln1⁄2P . Poterba and Summers (1988) found that for a t 1⁄4 three years, the logarithmic price increments ln1⁄2P exhibit a statistically significant negative serial correlation which translates into a long-term mean reversion in prices. Among the many recent studies that document violations of the GBM by looking at the time series properties of returns to various financial instruments are Bakshi et al. (2000), Bollerslev et al. (1992), Cont (2001), Cont and da Fonseca (2002), and Nelson (1991). Nevertheless, the broad unifying methodology of this large literature is to select a time interval and then investigate price increments vis a vis that time interval. Hence, it is quite common to hear that the GBM-Lognormal model is rejected for hourly data while it is accepted for monthly data but rejected again for yearly data or some combination thereof. In fact, this was the recent conclusion of Levy and Duchin (2004). In this paper we propose an alternative way of thinking about the appropriate distribution. We investigate the GBM model for fixed ln1⁄2P intervals as opposed to fixed t intervals. In other words, we start at the beginning of a time series and judiciously select a price increment ln1⁄2P 1⁄4 d (for example, 1%) and then measure the amount of time 1 it takes the security to move the pre-specified quantity. After the security has moved by ln1⁄2P 1⁄4 d, we measure the time 2 at which the security moves an additional ln1⁄2P 1⁄4 d and so on and so forth. The final result is a collection of time increments ð iþ1 iÞ for each pre-specified ln1⁄2P . We then compare (statistically) the empirical distribution of the ð iþ1 iÞ’s to the theoretical distribution they should obey under the GBM model. If, indeed, the price increments are normal, then the ð iþ1 iÞ’s—for each particular ln1⁄2P —should obey the Inverse Gaussian (or Wald) distribution as a result of the Space–Time duality that exists for Brownian motion. We select an entire spectrum of ln1⁄2P ’s (for example, from 1% all the way to 15%) and then extract the appropriate sample of i’s (for each ln1⁄2P ) so as to measure goodness of fit and estimate confidence intervals for the implied drift and diffusion coefficients. Our approach should not be confused with, and is very different from, the paradigm of spectral *Corresponding author. Email: milevsky@yorku.ca