Waiting for Returns: Using Space-Time Duality to Calibrate Financial Diffusions

Abstract

In this brief paper we propose an alternative methodology for testing and calibrating diffusion processes for financial time-series. The methodology focuses on the duality that exists between time and space for any given stochastic process. More specifically, we use the First Passage Time (FPT) which is the amount of time required by a stochastic process to travel a pre-specified distance. Thus, for example, we demonstrate that testing the hypothesis that (logarithmic) investment returns are independent and normally distributed is equivalent to testing the hypothesis that the FPT is Inverse Gaussian distributed. We apply this idea to calibrate geometric Brownian motion (GBM) parameters for the S&P 500 index over the period 1952 - 2004.

Keywords: Diffusion, duality, inverse guassian, GBM, First Passage Time

JEL Classification: G10, G13