Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects Carol Alexander * ISMA Centre, School of Business, University of Reading, UK Received 28 March 2003; accepted 27 October 2003 Available online 9 January 2004 Abstract This paper introduces a parameterization of the normal mixture diffusion (NMD) local vol- atility model that captures only a short-term smile effect, and then extends the model so that it also captures a long-term smile effect. We focus on the Ôbinomial’ NMD parameterization, so- called because it is based on simple and intuitive assumptions that imply the mixing law for the normal mixture log price density is binomial. With more than two possible states for volatility, the general parameterization is related to the multinomial mixing law. In this parsimonious class of complete market models, option pricing and hedging is straightforward since model prices and deltas are simple weighted averages of Black–Scholes prices and deltas. But they only capture a short-term smile effect, where leptokurtosis in the log price density decreases with term, in accordance with the Ôstylised facts’ of econometric analysis on ex-post returns of different frequencies and the central limit theorem. However, the last part of the paper shows that longer term smile effects that arise from uncertainty in the local volatility surface can be modeled by a natural extension of the binomial NMD parameterization. Results are illustrated by calibrating the model to several Euro–US dollar currency option smile surfaces. Ó 2003 Elsevier B.V. All rights reserved. JEL classification: G12; C16 Keywords: Local volatility; Stochastic volatility; Volatility uncertainty; Smile consistent models; Term structure of option prices; Normal variance mixtures * Tel.: +44-1189-316431. E-mail address: c.alexander@ismacentre.rdg.ac.uk (C. Alexander). 0378-4266/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2003.10.017 www.elsevier.com/locate/econbase Journal of Banking & Finance 28 (2004) 2957–2980