Moments structure of ℓ 1-stochastic volatility models

Abstract

We consider Taylor’s stochastic volatility model (SVM) when the innovations of the hidden log-volatility process have a Laplace distribution (l1 exponential density), rather than the standard Gaussian distribution (l2) usually employed. Recently many investigations have employed l1 metric to allow better modeling of the abrupt changes of regime observed in financial time series. However, the estimation of SVM is known to be difficult because it is a non-linear with an hidden markov process. Moreover, an additional difficulty yielded by the use of l1 metric is the not differentiability of the likelihood function. An alternative consists in using a generalized or efficient method-of-moments (GMM/EMM) estimation. For this purpose, we derive here the moments and autocovariance function of such l1-based stochastic volatility models.