Issues of Aggregation Over Time of Conditional Heteroscedastic Volatility Models: What Kind of Diffusion Do We Recover?

Continuous-time models play a central role in the theory of finance whereas empirical finance makes use of discrete-time models. This article investigates the connection between the two classes of models, particularly between conditional heteroscedastic and diffusion processes. As was advocated earlier by Stroock and Varadhan (1979), under some sets of conditions ARCH-type models weakly (in distribution) converge to diffusion processes as the time interval shrinks to zero. We provide the required set of conditions that ensures such a convergence and focus on the kind of the diffusion limit recovered. In the general setting, the diffusion is bivariate and driven by two possibly correlated Brownian motions. We illustrate this result for particular GARCH(1,1) specifications, the augmented GARCH (1,1) and a non-linear specification CEV-ARCH. By imposing an alternate set of conditions regarding the speed of convergence of parameters, a degenerate case is obtained. In the latter, the diffusion limit is governed by a single Brownian motion characterizing the price process while the volatility process becomes deterministic. Finally, we propose a discrete-time heteroscedastic model which shares various properties with ARCH-type models and converges to the complete model with stochastic volatility (CMSV) introduced by Hobson and Rogers (1998) for which the price and the volatility processes are driven by the same Brownian motion. Our analysis bears directly on the market completeness and unicity of asset prices issues.