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 directly apply popular continuous-time models to short intraday time intervals, and estimate the parameters using such data, can lead to nonsensical estimates due to severe intraday seasonality. A primary purpose of the paper is to provide a framework for using intraday high frequency data of both the index estimate, in particular, for improving the estimation accuracy of the leverage parameter, , that is, the correlation between the two Brownian motions driving the diffusive components of the price process and its spot variance process, respectively. As a special case, we focus on Heston’s (1993) square-root SV model, and propose the realized leverage estimator for , noting that, under this 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, is in-fill consistent for  . Finite sample simulation results show that the proposed estimator delivers more accurate estimates of the leverage parameter than do existing methods.Continuous time, high frequency data, stochastic volatility, S&P 500, implied volatility, VIX.

A Test for Dependence and Covariance Estimator of Market Microstructure Noise

There are many approaches for estimating an integrated variance and covariance in the presence of market microstructure noise. It is important to know a dependence of noise to construct the integrated variance and covariance estimators. We study a time dependence of bivariate noise processes in this paper. We propose a test statistic for the dependence of the noises and an autocovariance estimator of the noises and derive its asymptotic distribution. The asymptotic distribution of the autocovariance estimator provides us to another test statistic which is for significance of the autocovariances and for detection whether the noise exists or not. We obtain good performances of the test statistics and autocovariance estimator of the noises in a finite sample through Monte Carlo simulation. In empirical illustration, we confirm that the proposed statistics and estimators capture various dependence patterns of the market microstructure noises

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.Continuous time, high frequency data, stochastic volatility, S&P 500, implied volatility, VIX

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

Wald,LM and LR test statistics of linear hypothese in a strutural equation model

For the linear hypothesis in a strucural equation model, the properties of test statistics based on the two stage least squares estimator (2SLSE) have been examined since these test statistics are easily derived in the instrumental variable estimation framework. Savin (1976) has shown that inequalities exist among the test statistics for the linear hypothesis, but it is well known that there is no systematic inequality among these statistics based on 2SLSE for the linear hypothesis in a structural equation model. Morimune and Oya (1994) derived the constrained limited information maximum likelihood estimator (LIMLE) subject to general linear constraints on the coefficients of the structural equation, as well as Wald, LM and Lr Test statistics for the adequacy of the linear constraints. In this paper, we derive the inequalities among these three test statistics based on LIMLE and the local power functions based on Limle and 2SLSE to show that there is no test statistic which is uniformly most powerful, and the LR test statistic based on LIMLE is locally unbised and the other test statistics are not. Monte Carlo simulations are used to examine the actual sizes of these test statistics and some numerical examples of the power differences among these test statistics are given. It is found that the actual sizes of these test statistics are greater than the nominal sizes, the differences between the actual and nominal sizes of Wald test statistics are generally the greatest, those of LM test statistics are the smallest, and the power functions depend on the correlations between the endogenous explanatory variables and the error term of the structural equation, the asymptotic variance of estimator of coefficients of the structural equation and the number of restrictions imposed on the coefficients.

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 directly apply popular continuous-time models to short intraday time intervals, and estimate the parameters using such data, can lead to nonsensical estimates due to severe intraday seasonality. A primary purpose of the paper is to provide a fraework for using intraday high frequency data of both the index estimate, in particular, for improving the estimation accuracy of the leverage parameter, p, that is, the correlation between the two Brownian motions driving the diffusive components of the price process and its spot variance process, respectively. As a special case, we focus on Heston’s (1993) square-root SV model, and propose the realized leverage estimator for p, noting that, under this 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, is in-fill consistent for p. Finite sample simulation results show that the proposed estimator delivers more accurate estimates of the leverage parameter than do existing methods