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

This paper extends the normal mixture diffusion (NMD) local volatility model of Brigo and Mercurio (2000, 2001a,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 follows from the central limit theorem and concords with the ‘stylised facts’ of econometric analysis of ex-post returns of different frequencies. We introduce a term structure for option prices in the NMD model by assuming there is a fixed probability of each volatility state occurring in every time interval Dt, and we show that with this assumption the mixing law for the price density is the multinomial density. This very parsimonious model can easily be calibrated to observed option prices. 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 stochastic volatility. The last part of this paper considers only two possible volatility states in each Dt with probabilities l and (1 - l). If l were fixed, the binomial mixing law model would only apply to short-term smile effects. But by making l stochastic, longer-term smile effects that arise from uncertainty in volatility are also captured by the model. The results are illustrated by calibrating the model with and without stochastic l, to a currency option smile surface