The dynamics of stochastic volatility: evidence from underlying and options markets

Abstract

This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a time-varying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for shortdated options.JEL classification: G12, C11. Evidence from Underlying and Options MarketsThis paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and risk-neutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a time-varying correlation.Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for short-dated options.

Forecasting volatility

“…Jones (2003) also argues that the additional approximation error in (2) from introducing a leverage effect is likely to be small. 3 For a review of various volatility measures and their properties, see Andersen et al (2002). where s is the number of out-of-sample forecasts, denotes the forecast of the future average volatility at maturity t and a OE (-…”

Operator Methods for Continuous-Time Markov Processes

“…Even with a zero value of κ the pull measure µ/σ −σ /2 diverges to −∞ at the right boundary provided that γ is greater than one. Jones (2003) in fact estimates a value for γ that exceeds one. The pull measure also diverges at the left boundary to +∞.…”

“…Conley, Hansen, Luttmer, and Scheinkman (1997) study the constant volatility elasticity model but allowing for drift nonlinearity. Jones (2003) uses constant volatility elasticity models to extend Nelson (1990)'s model of the dynamic evolution of volatility.…”

“…Using parameter estimates from Jones (2003), we display the first five principal components for the volatility process in Figure 1. For the principal component extraction, we use the two weighting functions described previously.…”

Modeling Financial Security Returns Using Levy Processes

“…Favorable evidence based on time-series returns include Chacko and Viceira (2003), Ishida and Engle (2002), Javaheri (2005), and Jones (2003). Jones (2003) and Medvedev and Scaillet (2003) also find that the 3/2 specification prices option better. Bakshi, Ju, and Ou-Yang (2004) find that the statistical dynamics of V IX 2 is also closer to a 3/2 specification than to a square root specification.…”