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Alternative asset-price dynamics and volatility smile

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Author Info

  • Damiano Brigo
  • Fabio Mercurio
  • Giulio Sartorelli

Abstract

We review the general class of analytically tractable asset-price models that was introduced by Brigo and Mercurio (2001a Mathematical Finance—Bachelier Congr. 2000 (Springer Finance) ed H Geman, D B Madan, S R Pliska and A C F Vorst (Berlin: Springer) pp 151-74), where the considered asset can be an exchange rate, a stock index or even a forward Libor rate. The class is based on an explicit SDE under a given forward measure and includes models featuring (i) explicit asset-price dynamics, (ii) a virtually unlimited number of parameters and (iii) analytical formulae for European options. We also review the fundamental case where the asset-price density is given, at every time, by a mixture of log-normal densities with equal means. We then introduce two other cases: the first is still based on log-normal densities, but it allows for different means in the distributions; the second is based on processes of hyperbolic-sine type. Finally, we test the goodness of calibration to real market data of the considered models, choosing a particularly asymmetric volatility surface. As expected, the model based on hyperbolic-sine density mixtures achieves the lowest calibration error.

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File URL: http://www.tandfonline.com/doi/abs/10.1088/1469-7688/3/3/303
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Bibliographic Info

Article provided by Taylor & Francis Journals in its journal Quantitative Finance.

Volume (Year): 3 (2003)
Issue (Month): 3 ()
Pages: 173-183

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Handle: RePEc:taf:quantf:v:3:y:2003:i:3:p:173-183

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Cited by:
  1. Carol Alexander, 2002. "Short and Long Term Smile Effects: The Binomial Normal Mixture Diffusion Model," ICMA Centre Discussion Papers in Finance icma-dp2003-06, Henley Business School, Reading University, revised Mar 2003.
  2. Damiano Brigo, 2008. "The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation," Papers 0812.4052, arXiv.org.
  3. Schneider, Stefan & Schneider, Stefan, 2010. "Power Spot Price Models with negative Prices," MPRA Paper 29958, University Library of Munich, Germany.
  4. Marco Airoldi & Vito Antonelli & Bruno Bassetti & Andrea Martinelli & Marco Picariello, 2004. "Long Range Interaction Generating Fat-Tails in Finance," GE, Growth, Math methods 0404006, EconWPA, revised 27 Apr 2004.
  5. Dinghai Xu, 2009. "The Applications of Mixtures of Normal Distributions in Empirical Finance: A Selected Survey," Working Papers 0904, University of Waterloo, Department of Economics, revised Sep 2009.
  6. Damiano Brigo & Francesco Rapisarda & Abir Sridi, 2013. "The arbitrage-free Multivariate Mixture Dynamics Model: Consistent single-assets and index volatility smiles," Papers 1302.7010, arXiv.org.

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