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The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation

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  • Damiano Brigo

Abstract

In the present paper, given an evolving mixture of probability densities, we define a candidate diffusion process whose marginal law follows the same evolution. We derive as a particular case a stochastic differential equation (SDE) admitting a unique strong solution and whose density evolves as a mixture of Gaussian densities. We present an interesting result on the comparison between the instantaneous and the terminal correlation between the obtained process and its squared diffusion coefficient. As an application to mathematical finance, we construct diffusion processes whose marginal densities are mixtures of lognormal densities. We explain how such processes can be used to model the market smile phenomenon. We show that the lognormal mixture dynamics is the one-dimensional diffusion version of a suitable uncertain volatility model, and suitably reinterpret the earlier correlation result. We explore numerically the relationship between the future smile structures of both the diffusion and the uncertain volatility versions.

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  • 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.
    • Handle: RePEc:arx:papers:0812.4052
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    References listed on IDEAS

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    1. Mark Britten‐Jones & Anthony Neuberger, 2000. "Option Prices, Implied Price Processes, and Stochastic Volatility," Journal of Finance, American Finance Association, vol. 55(2), pages 839-866, April.
    2. Carol Alexander & Sujit Narayanan, 2001. "Option Pricing with Normal Mixture Returns: Modelling Excess Kurtosis and Uncertanity in Volatility," ICMA Centre Discussion Papers in Finance icma-dp2001-10, Henley Business School, University of Reading, revised Dec 2001.
    3. 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.
    4. 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 621-651, October.
    5. Nelson, Daniel B & Ramaswamy, Krishna, 1990. "Simple Binomial Processes as Diffusion Approximations in Financial Models," The Review of Financial Studies, Society for Financial Studies, vol. 3(3), pages 393-430.
    6. Damiano Brigo & Fabio Mercurio, 2008. "Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing," Papers 0812.4010, arXiv.org.
    7. Damiano Brigo & Fabio Mercurio, 2002. "Lognormal-Mixture Dynamics And Calibration To Market Volatility Smiles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(04), pages 427-446.
    8. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    9. Brigo, Damiano, 2000. "On SDEs with marginal laws evolving in finite-dimensional exponential families," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 127-134, August.
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    Cited by:

    1. Antoine Jacquier & Patrick Roome, 2015. "Black-Scholes in a CEV random environment," Papers 1503.08082, arXiv.org, revised Nov 2017.
    2. Lech A. Grzelak, 2022. "Randomization of Short-Rate Models, Analytic Pricing and Flexibility in Controlling Implied Volatilities," Papers 2211.05014, arXiv.org.
    3. Alexander, Carol, 2004. "Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects," Journal of Banking & Finance, Elsevier, vol. 28(12), pages 2957-2980, December.
    4. Carol Alexander & Andrew Scourse, 2004. "Bivariate normal mixture spread option valuation," Quantitative Finance, Taylor & Francis Journals, vol. 4(6), pages 637-648.
    5. 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.
    6. Kole, Erik & Koedijk, Kees & Verbeek, Marno, 2006. "Portfolio implications of systemic crises," Journal of Banking & Finance, Elsevier, vol. 30(8), pages 2347-2369, August.

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