Reconsidering the continuous time limit of the GARCH(1, 1) process

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Abstract

In this note we reconsider the continuous time limit of the GARCH(1, 1) process. Let Yk and σk2 denote, respectively, the cumulative returns and the volatility processes. We consider the continuous time approximation of the couple (Yk,σk2). We show that, by choosing different parameterizations, as a function of the discrete interval h, we can obtain either a degenerate or a non-degenerate diffusion limit. We then show that GARCH(1, 1) processes can be obtained as Euler approximations of degenerate diffusions, while any Euler approximation of a non-degenerate diffusion is a stochastic volatility process.

Introduction

In a seminal paper, Nelson (1990) analyzed the continuous time limit of ARCH models, as the discrete interval approaches zero. He showed that different classes of GARCH processes, e.g. GARCH(1, 1) and exponential ARCH, EARCH, after a proper reparameterization, as the time interval shrinks, converge in distribution to a two-dimensional non-degenerate diffusion; i.e. to a diffusion in R×R+ driven by two Brownian motions, whose covariance matrix is non-singular. More recently, Fornari and Mele (1996) analyze the continuous time limit for a class of nonlinear ARCH models proposed by Ding et al. (1993). Duan (1997) introduces a new general class of volatility models, called augmented GARCH, where the augmentation plays the role of a Box–Cox transformation of the conditional variance and analyzes their continuous time limit. Also Fornari and Mele, as well as Duan obtain as a continuous time limit a non-degenerate diffusion. Such a (correct) result is somewhat surprising. In fact, discrete time GARCH models are characterized by only one source of noise, nevertheless their continuous time counterpart is a diffusion process driven by two independent (or at least non perfectly correlated) Brownian motions. In this sense, the continuous time limit of GARCH processes is the same as the continuous time limit of stochastic volatility processes, that are instead characterized by two non-perfectly correlated sources of noise. On the other hand, Kallsen and Taqqu (1998) via a continuous time embedding tecnique, show that the continuous time counterpart of a GARCH process is a process driven by only one Brownian motion. Their methodology is somewhat different from the diffusion approximation technique followed in the papers cited above, in that they allow asset prices to move continuously, while they allow volatility to jump only at integer values of time.

The purpose of this note is to reconsider the continuous time approximation of the GARCH(1,1) process and to show that, depending on the specific continuous approximation we consider, as the time interval shrinks, either a non-degenerate or a degenerate diffusion limit may arise. Furthermore, we shall show that GARCH processes can be obtained as Euler approximations of degenerate diffusions, while any Euler approximation of a non-degenerate diffusion results in a stochastic volatility process.

Non-degenerate and degenerate diffusion limits have very different implications for option pricing. In fact, if volatility is not a tradeable asset, then the degeneracy of the diffusion preserves market completeness, thus allowing for unique preference independent prices for contingent claims (e.g. Hobson and Rogers, 1998). On the other hand, non-degenerate diffusions in general do not preserve market completeness, so that the pricing of options requires additional assumptions on, e.g. risk premia and/or investors preferences (see e.g. Hull and White, 1987; Melino and Turnbull, 1990). From an empirical point of view, market completeness and so the existence of an exact option pricing formula based only on the assumption of no arbitrage opportunities, is not a very big issue, given the rapidly growing literature on nonparametric option pricing (e.g. see the recent survey by Ghysels et al., 1998 and the nonparametric test for the relevance of ARCH effect in option pricing by Christoffersen and Hahn, 1997).

The rest of this note is organized as follows. Section 2 considers the continuous approximation of the couple (Yk,σk2) and show that, depending on which parameterization we choose, as a function of the time interval h, either a degenerate or a non-degenerate limit may arise. Section 3 analyzes the volatility models arising as Euler approximations to, respectively, non-degenerate and degenerate diffusions.

Section snippets

Continuous time limits for (Yk, σk2)

This section considers two different continuous time approximations of the GARCH(1, 1) process, one leading to a degenerate diffusion and another leading to a non-degenerate diffusion (Nelson's result).

Let YkYk−1, k=1,2,… be returns on a generic financial asset and so let Yk denote the cumulative returns. We begin by considering the following discrete time GARCH(1, 1) process, written as in Bollerslev et al. (1994),Yk−Yk−1k−1εk,σk201σk−122σk−12εk2with ω0,ω1,ω2>0 and ω1+ω2<1, εk∼iidN(0,1)

GARCH as diffusion approximations

In the previous section we started from a discrete time GARCH(1, 1) process and we derived two different diffusion approximation results, one leading to a degenerate diffusion and another one leading to a non-degenerate diffusion. We now move in the opposite direction, by considering an Euler approximation (it is well known that Euler approximations are not unique) of the two diffusion limits defined in , , degenerate case, and in , non-degenerate case. We shall see that Euler approximations of

Acknowledgements

I wish to thank an Associate Editor, two referees, as well as Frank Diebold, Rob Engle, Dean Foster, Christian Gourieroux, Jin Hahn, Grant Hillier, Allan Timmermann, and the seminar participants at Rice University-University of Houston, University of Pennsylvania, University of Southampton, CORE, LSE-Financial Market Group, and 1997 Winter Meeting of the Econometric Society for very useful comments and suggestions. I owe special thanks to Torben Andersen for having pointed out a mistake in an

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