Continuous Time Approximations to GARCH and Stochastic Volatility Models

We collect some continuous time GARCH models and report on how they approximate discrete time GARCH processes. Similarly, certain continuous time volatility models are viewed as approximations to discrete time volatility models. 1 Stochastic volatility models and discrete GARCH Both stochastic volatility models and GARCH processes are popular models for the description of financial time series. Recall that a discrete time stochastic volatility model (SV-model) is a process (Xn)n∈N0 together with a non-negative volatility process (σn)n∈N0 , such that Xn = σnεn, n ∈ N0, (1) where the noise sequence (εn)n∈N0 is a sequence of independent and identically distributed (i.i.d.) random variables, which is assumed to be independent of (σn)n∈N0 . Further information about these processes can be found e.g. in Shephard (2008) and Davis and Mikosch (2008). In contrast to stochastic volatility models, GARCH processes have the property that the volatility process is specified as a function of the past observations. The classical ARCH(1) process by Engle (1982) and the GARCH(1,1) process by Bollerslev (1986), for example, are processes (Xn)n∈N0 with a non-negative volatility process (σn)n∈N0 , such that Xn = σnεn, n ∈ N0, (2) Alexander M. Lindner Technische Universität Braunschweig, Institut für Mathematische Stochastik, Pockelsstraße 14, D-38106 Braunschweig, Germany e-mail: [email protected]