Estimation of continuous-time models in finance

Abstract

INTRODUCTION Continuous-time stochastic processes arise in many applications in economics, but perhaps nowhere do they play as large a role as in finance. Following the pathbreaking work of Merton (1969, 1973) and Black and Scholes (1973), the use of continuous-time stochastic processes has become a common feature of many applications, especially asset pricing models. Even a casual comparison of the textbooks of the seventies (e.g., Fama and Miller (1972), Fama (1976)) with the current crop (e.g., Ingersoll (1987), Duffie (1988)) serves to demonstrate the remarkable speed with which the tools of stochastic process theory have been assimilated into mainstream finance. This survey will look at the specification and estimation of continuous-time stochastic processes. Although much of the discussion is relevant for other applications, I have chosen to write it from the perspective of someone interested in evaluating the empirical content of current continuous-time asset pricing models and in contributing to their future development. It is interesting to speculate on the reasons for the widespread adoption of continuous-time models in asset pricing. Although many come to mind, I would argue that they have been widely adopted not because of their empirical properties but in spite of them. The explosion and sophistication of theoretical research simply has not been matched by empirical work. Continuous-time asset pricing models typically involve restrictions linking the parameters of the price process to those of some underlying ‘forcing’ variables. In general equilibrium models, the forcing variables may be taste and technology. In option pricing models, they may be the term structure and/or the price of the underlying security. Tests of these models are invariably joint tests of ‘nuisance’ assumptions, including the specification of the forcing variable process.