Calibrating the Italian Smile with Time-Varying Volatility and Heavy-Tailed Models

Abstract

In this paper we consider several time-varying volatility and/or heavy-tailed models to explain the dynamics of return time series and to fit the volatility smile for exchange-traded options where the underlying is the main ‘Borsa Italiana’ stock index. Given observed prices for the time period we investigate, we calibrate both continuous-time and discrete-time models. First, we estimate the models from a time-series perspective (i.e. under the historical probability measure) by investigating more than ten years of daily index price log-returns. Then, we explore the risk-neutral measure by fitting the values of the implied volatility for numerous strikes and maturities during the highly volatile period from April 1, 2007 (prior to the subprime mortgage crisis in the U.S.) to March 30, 2012. We assess the extent to which time-varying volatility and heavy-tailed distributions are needed to explain the behavior of the most important stock index of the Italian market.

Keywords: volatility smile, option pricing, non-Gaussian Ornstein-Uhlenbeck processes, Lévy processes, tempered stable processes and distributions, stochastic volatility models, time-changed Lévy processes, GARCH model, filtered historical simulation, particle filter

JEL Classification: C02, C46, C58, C61, C63