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

Alexander, Carol (2004) Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects. Journal of Banking & Finance, 28 (12). pp. 2957-2980. ISSN 0378-4266

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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.

Item Type: Article
Keywords: Local volatility; Stochastic volatility; Volatility uncertainty; Smile consistent models; Term structure of option prices; Normal variance mixtures
Schools and Departments: School of Business, Management and Economics > Business and Management
Subjects: H Social Sciences > HG Finance
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Depositing User: Carol Alexander
Date Deposited: 17 Sep 2012 14:09
Last Modified: 17 Sep 2012 14:09
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