Small-time asymptotics, moment explosion and the moderate deviations regime

Abstract

In ersten Teil "Small-Maturity Asymptotics for the At-the-Money Implied Volatility Slope in Lévy Models" betrachten wir die at-the-money Steigung der impliziten Volatilität, d.h. die Ableitung der impliziten Volatilität nach dem Ausübungspreis, wenn die Restlaufzeit der dazugehörigen Option gegen

In the first part "Small-Maturity Asymptotics for the At-the-Money Implied Volatility Slope in Lévy Models", we consider the at-the-money strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the behaviour of the slope for infinite activity exponential Lévy models including a Brownian component. As auxiliary results, we obtain asymptotic expansions of short maturity at-the-money digital call options, using Mellin transform asymptotics. In the second part "Option Pricing in the Moderate Deviations Regime", we consider call option prices close to expiry in diffusion models, in an asymptotic regime ("moderately out of the money") that interpolates between the well-studied cases of at-the-money and out-of-the-money regimes. First and higher order small-time moderate deviation estimates of call prices and implied volatilities are obtained. The third part "Moment Explosion in the Rough Heston Model" focuses on an extension of the well-known Heston model, where the paths of the volatility process are rougher than in the classic Heston model. We analyse the explosion time of the moment generating function for any parameter choice. In case of a finite explosion time, we give upper and lower bounds. Eventually, using these estimates, we show the finiteness of the critical moments in the rough Heston model. The last part "Large-Strike Asymptotics in the 3/2-Model" deals with the 3/2-model. At first, we consider the density function which can be expressed via Fourier transform as a contour integral in the complex plane. Choosing a transformed Hankel-type contour and using the explicitness of the moment generating function, we determine the asymptotic behaviour of the positive tail of the density function. Finally, we derive large-strike asymptotics for the implied volatility.