Volatility indices have been designed for many markets as gauges to measure investors' fear of market crash. The recent … market turmoil has produced historically high volatility levels, in some cases around four times higher than their previous … average levels. We take a look at the behavior of various volatility indices by including the recent market turmoil into the …

In this article, we show how to calibrate the widely-used SVI parameterization of the implied volatility smile in such … volatility surfaces with a simple closed-form representation. We demonstrate the high quality of typical SVI fits with a …

volatility on backward-looking term rates …

, Jarrow, and Morton (1992) framework for commodity futures prices that incorporates stochastic volatility and stochastic … interest rate and allows a correlation structure between the futures price process, the futures volatility process and the … interest rate process. The functional form of the futures price volatility is specified so that the model admits finite …

hedged. We numerically evaluate American call options under stochastic volatility, stochastic interest rates and jumps in … both the asset price and volatility. By employing the Method of Lines (Meyer (2015)), the option price, the early exercise … exercise the option, whilst including asset-volatility jumps elevates the free boundary and the option holder is less likely to …

volatility models. We develop closed-form expansions and sharp error bounds for VIX futures, options and implied volatilities. In … neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol-of-vol processes. Our results …

, these expansions are not quite suitable for volatility or variance densities as they inherently assign positive mass to the …