Short and Long Term Smile Effects: The Binomial Normal Mixture Diffusion Model
AbstractThis 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
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Bibliographic InfoPaper provided by Henley Business School, Reading University in its series ICMA Centre Discussion Papers in Finance with number icma-dp2003-06.
Length: 24 pages
Date of creation: Nov 2002
Date of revision: Mar 2003
Publication status: Published in Journal of Banking and Finance 2004, 28:12, 2957-2980
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Local volatility; stochastic volatility; smile consistent models; term structure of option prices; normal variance mixtures;
Find related papers by JEL classification:
- G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
- C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
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- Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
- Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
- Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
- Damiano Brigo & Fabio Mercurio & Giulio Sartorelli, 2003. "Alternative asset-price dynamics and volatility smile," Quantitative Finance, Taylor & Francis Journals, vol. 3(3), pages 173-183.
- Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
- Ritchey, Robert J, 1990. "Call Option Valuation for Discrete Normal Mixtures," Journal of Financial Research, Southern Finance Association & Southwestern Finance Association, vol. 13(4), pages 285-96, Winter.
- Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
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