Estimating the Leverage Parameter of Continuous-time Stochastic Volatility Models Using High Frequency S&P 500 and VIX

This paper proposes a new method for estimating continuous-time stochastic volatility (SV) models for the S&P 500 stock index process using intraday high-frequency observations of both the S&P 500 index and the Chicago Board of Exchange (CBOE) implied (or expected) volatility index (VIX). Intraday high-frequency observations data have become readily available for an increasing number of financial assets and their derivatives in recent years, but it is well known that attempts to estimate the parameters of popular continuous-time models can lead to nonsensical estimates due to severe intraday seasonality. A primary purpose of the paper is to estimate the leverage parameter, ρ, that is, the correlation between the two Brownian motions driving the diffusive components of the price process and its spot variance process, respectively. We show that, under the special case of Heston’s (1993) square-root SV model without measurement errors, the “realized leverage”, or the realized covariation of the price and VIX processes divided by the product of the realized volatilities of the two processes, converges to ρ in probability as the time intervals between observations shrink to zero, even if the length of the whole sample period is fixed. Finite sample simulation results show that the proposed estimator delivers accurate estimates of the leverage parameter, unlike existing methods.