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Alternative asset-price dynamics and volatility smile

Author

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  • 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.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:quantf:v:3:y:2003:i:3:p:173-183
    DOI: 10.1088/1469-7688/3/3/303
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    Citations

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    Cited by:

    1. Roman V. Ivanov, 2023. "On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model," Risks, MDPI, vol. 11(6), pages 1-23, June.
    2. Nappo, Giovanna & Marchetti, Fabio Massimo & Vagnani, Gianluca, 2023. "Traders’ heterogeneous beliefs about stock volatility and the implied volatility skew in financial options markets," Finance Research Letters, Elsevier, vol. 53(C).
    3. 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.
    4. Damiano Brigo & Camilla Pisani & Francesco Rapisarda, 2021. "The multivariate mixture dynamics model: shifted dynamics and correlation skew," Annals of Operations Research, Springer, vol. 299(1), pages 1411-1435, April.
    5. Alessandro Ramponi, 2011. "Mixture Dynamics and Regime Switching Diffusions with Application to Option Pricing," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 349-368, June.
    6. Iain J. Clark & Saeed Amen, 2017. "Implied Distributions from GBPUSD Risk-Reversals and Implication for Brexit Scenarios," Risks, MDPI, vol. 5(3), pages 1-17, July.
    7. 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.
    8. 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, revised Sep 2014.
    9. Schneider, Stefan & Schneider, Stefan, 2010. "Power Spot Price Models with negative Prices," MPRA Paper 29958, University Library of Munich, Germany.
    10. Marco Airoldi & Vito Antonelli & Bruno Bassetti & Andrea Martinelli & Marco Picariello, 2004. "Long Range Interaction Generating Fat-Tails in Finance," GE, Growth, Math methods 0404006, University Library of Munich, Germany, revised 27 Apr 2004.
    11. 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, University of Reading, revised Mar 2003.
    12. T. van der Zwaard & L. A. Grzelak & C. W. Oosterlee, 2024. "On the Hull-White model with volatility smile for Valuation Adjustments," Papers 2403.14841, arXiv.org.
    13. Ma, Chao & Ma, Qinghua & Yao, Haixiang & Hou, Tiancheng, 2018. "An accurate European option pricing model under Fractional Stable Process based on Feynman Path Integral," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 87-117.

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