Nonparametric stochastic volatility

Using recent advances in the nonparametric estimation of continuous-time processes under mild statistical assumptions as well as recent developments on nonparametric volatility estimation by virtue of market microstructure noise-contaminated high-frequency asset price data, we provide (i) a theory of spot variance estimation and (ii) functional methods for stochastic volatility modelling. Our methods allow for the joint evaluation of return and volatility dynamics with nonlinear drift and di¤usion functions, nonlinear leverage e¤ects, jumps in returns and volatility with possibly state-dependent jump intensities, as well as nonlinear riskreturn trade-o¤s. Our identi…cation approach and asymptotic results apply under weak recurrence assumptions and, hence, accommodate the persistence properties of variance in …nite samples. Functional estimation of a generalized (i.e., nonlinear) version of the square-root stochastic variance model with jumps in both volatility and returns for the SPt+1 = log(pt+1) log(pt) and consider the system: rt;t+dt = d log(pt) = ( 2 t )dt+ tdW r t + dJ r t ; (1) df( t ) = mf(:)( 2 t )dt+ f(:)( 2 t )dW t + dJ t ; (2) where fW r t ;W t g are possibly correlated Brownian motions, fJ t ; J t g are Poisson jump processes independent of each other and independent of fW r t ;W t g with intensities (:) and f(:)(:), and (:), mf(:)(:), and f(:)(:) are generic functions satisfying smoothness conditions laid out in the following sections. Our procedures have three main features. First, we …lter spot variance by localizing (in time) high-frequency estimates of integrated variance R sds. We then use spot variance to identify the parameters and functions driving variance dynamics (i.e., f(:)(:), mf(:)(:), f(:)(:) and, given parametric assumptions on the jump size distribution, the moments of the volatility jumps). Since the classical realized variance estimator (i.e., the sum of squared intra-daily returns over the day) may contain substantial contaminations due to market microstructure noise (as emphasized by Bandi and Russell, 2008, and Zhang at al., 2005, in recent work), we employ robust (to noise) integrated variance estimates. In other words, when possible, we allow for market microstructure noise and control for it.1 Second, di¤erently from much existing work on stochastic volatility modelling, we avoid imposing tight (possibly a¢ ne) parametric structures on f(:)(:), mf(:)(:); and f(:)(:). Speci…cally, we identify the relevant functions (through estimates of the system’s in…nitesimal moments) using nonparametric kernel methods for di¤usion and jump-di¤usion processes as proposed by Bandi and Nguyen (2003), Bandi and Phillips (2003), and Johannes (2004) in simpler frameworks, namely in the context of scalar models with observables. In order to lay out the main ideas in the context of a well-understood estimation problem, we use classical Nadaraya-Watson kernel estimates. However, as we illustrate below, extensions to alternative functional estimation methods are rather 1For recent surveys of nonparametric methods for integrated variance estimation using market microstructure noise-contaminated high-frequency asset price data, we refer the reader to the review papers by Bandi and Russell (2007), Barndor¤-Nielsen and Shephard (2007), and McAleer and Medeiros (2008).