This study extends the literature on the relation between trading activity and volatility by looking at a new asset class in the form of VIX futures, and by decomposing each side of the relation into two components. The results confirm several findings documented in prior studies: The number of...

In this paper we develop a general method for deriving closed form approximations of European option prices and equivalent implied volatilities in stochastic volatility models. Our method relies on perturbations of the model dynamics and we show how the expansion terms can be calculated using...

We explore the implications for asset prices and implied volatilities in an equilibrium model of commodity production. Production of the commodity can be carried out in one of two regimes. In the first regime the reserves are set in constant decline while in the second regime new additions to...

In this paper we examine the empirical performance of affine jump diffusion models with stochastic volatility in a time series study of crude oil prices. We compare four different models and estimate them using the Markov Chain Monte Carlo method. The support for a stochastic volatility model...

In this paper we derive closed form approximations of European option prices in different versions of the SABR model of Hagan et al. (2002). Our approach is based on perturbing the model dynamics and approximations of call prices are obtained from a second order Taylor expansion. The method is...