Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects

By Carol Alexander. 2004

62 citations recorded by the source corpus.

This paper introduces a parameterization of the normal mixture diffusion (NMD) local volatility 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 ‘binomial’ 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 ‘stylised 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.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/4c42de71f3d9836ce0d8fa2a1a4743ca22638824

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Citations in the corpus

Understanding volatility dynamics in the EU-ETS market (2009, background): “The advantage of this parametrization is its intuitive interpretation: the individual distributions in the mixture represent di¤erent regimes while the mixing law gives the probabilities of each regime (Alexander, 2004 and Alexander and Lazar, 2006).”
A simple approach for pricing equity options with Markov switching state variables (2006, background): “It complements the literature on trees with stochastic volatility (see e.g. Bollen 1998) as well as the normal mixture diffusion class of models (see e.g. Alexander (2004), Mercurio (2002), Brigo and Mercurio (2002) and Brigo et al. (2004)).”