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

In this note we reconsider the continuous time limit of the GARCH(1, 1) process. Let > k and p2 denote, respectively, the cumulative returns and the volatility processes. We consider the continuous time approximation of the couple (> k , p2 ). We show that, by choosing di!erent parameterizations, as a function of the discrete interval h, we can obtain either a degenerate or a non-degenerate di!usion limit. We then show that GARCH(1, 1) processes can be obtained as Euler approximations of degenerate di!usions, while any Euler approximation of a non-degenerate di!usion is a stochastic volatility process. ( 2000 Elsevier Science S.A. All rights reserved.

[1]  Daniel B. Nelson.  ARCH models as diffusion approximations , 1990 .

[2]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[3]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[4]  C. Granger,et al.  A long memory property of stock market returns and a new model , 1993 .

[5]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[6]  L. Arnold.  Stochastic Differential Equations: Theory and Applications , 1992 .

[7]  É. Renault,et al.  Nonparametric methods and option pricing , 1997 .

[8]  T. Nijman,et al.  Temporal Aggregation of GARCH Processes. , 1993 .

[9]  B. Werker,et al.  Closing the GARCH gap: Continuous time GARCH modeling , 1996 .

[10]  A. Melé,et al.  Recovering the Probability Density Function of Asset Prices using Garch as Diffusion Approximations , 2001 .

[11]  Murad S. Taqqu,et al.  Option Pricing in ARCH‐type Models , 1998 .

[12]  A. Melé,et al.  Weak convergence and distributional assumptions for a general class of nonliner arch models , 1997 .

[13]  É. Renault,et al.  Aggregations and Marginalization of Garch and Stochastic Volatility Models , 1998 .

[14]  L. Rogers,et al.  Complete Models with Stochastic Volatility , 1998 .

[15]  S. Turnbull,et al.  Pricing foreign currency options with stochastic volatility , 1990 .

[16]  Eduardo S. Schwartz,et al.  Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model , 1992 .

[17]  Jin-Chuan Duan,et al.  Augmented GARCH (p,q) process and its diffusion limit , 1997 .

[18]  Ruth J. Williams,et al.  Introduction to Stochastic Integration , 1994 .