Encyclopedia of Financial Models III

Change of Time Methods

ANATOLIY SWISHCHUK, PhD

Professor of Mathematics and Statistics, University of Calgary

Abstract: Change of time can be used in financial modeling to introduce stochastic volatility or solve many stochastic differential equations. The main idea of the change of time method is to change time from t to a nonnegative process T(t) with nondecreasing sample paths (e.g., subordinator). Many Lévy processes may be written as time-changed Brownian motion. Lévy processes can also be used as a time change for other Lévy processes (subordinators). Using change of time, we can get an option pricing formula for an asset following geometric Brownian motion (e.g., Black-Scholes formula) and obtain an explicit option pricing formula for an asset following the mean-reverting process (e.g., continous-time GARCH proccess).

In this entry, we provide an overview on change of time methods (CTM), and show how to solve many stochastic differential equations (SDEs) in finance (geometric Brownian motion [GBM], Ornstein-Uhlenbeck [OU], Vasiek, continuous-time GARCH, etc.) using the change of time method. As applications of CTM we present two different models: geometric Brownian motion (GBM) and mean-reverting models. The solutions of these two models are different. But the nice thing is that they can be solved by CTM like many other models mentioned in this entry. And moreover, we ...