Stochastic Power Variations in Financial Returns

Abstract

For underlying asset motions calibrating skewness and kurtosis beyond the volatility it becomes possible to consider these entities as responding to their observations in past data. Models with stochastic skewness and kurtosis are constructed by allowing the second, third and fourth power variations to respond to their realized levels. The models are formulated in continuous time as OU equations reverting to a drift that is itself a TFLP, Tempered Fractional Lévy Process. Markovian discrete time approximations are simulated to incorporate stochasticity in all three entities, volatility, skewness and kurtosis. Estimation results are reported for time series and option data on SPY using simulated method of moments for stochasticity in volatility and skewness. Implications for a log normal volatility of volatility are presented along with the effects on periodic higher moment return term structures. For the physical process volatility is observed to get close to linear in time while skewness and kurtosis maintain high levels in perpetuity. Risk neutrally there is momentum in volatility and mean reversion in skewness. Risk neutral volatility and skewness are inversely related while kurtosis is positively related to volatility.

Keywords: Bilateral Gamma, Stochastic Exponential, Power Variation

JEL Classification: G10, G11, G12