Short and Long Term Smile Effects: The Binomial Normal Mixture Diffusion Model

Abstract

This paper extends the normal mixture diffusion (NMD) local volatility model of Brigo and Mercurio (2000, 2001 a, b, 2002) so that it explains both short-term and long-term smile effects. Short-term smile effects are captured by a local volatility model where excess kurtosis in the price density decreases with maturity. This agrees with the 'stylised facts' of econometric analysis of ex-post returns of different frequencies and follows from the central limit theorem. We introduce the 'binomial' NMD model, so called because it is based on simple and intuitive assumptions that imply that the mixing law for the normal mixture log price density is binomial. This very parsimonious model can easily be calibrated to observed option prices, and it explains the short-term smile effect where leptokurtosis in the log price density decreases rapidly with time. However, smile effects in currency options often persist into fairly long maturities, and to capture at least some part of this it is necessary to introduce uncertainty. Longer-term smile effects that arise from uncertainty in the local volatility surface are modeled by a simple extension of the binomial NMD model. The results are illustrated by calibrating the model to a currency option smile surface.