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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


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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Paper provided by in its series Papers with number 0812.4052.

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Date of creation: Dec 2008
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Publication status: Published in Related publication in Brigo, D., Mercurio, F., and Sartorelli, G., Alternative Asset Price Dynamics and Volatility Smile, Quantitative Finance, Vol 3, N. 3. (2003) pp. 173-183
Handle: RePEc:arx:papers:0812.4052

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  1. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, University of Chicago Press, vol. 51(4), pages 621-51, October.
  2. Damiano Brigo & Fabio Mercurio & Giulio Sartorelli, 2003. "Alternative asset-price dynamics and volatility smile," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 3(3), pages 173-183.
  3. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  4. Damiano Brigo & Fabio Mercurio, 2008. "Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing," Papers 0812.4010,
  5. Mark Britten-Jones & Anthony Neuberger, 2000. "Option Prices, Implied Price Processes, and Stochastic Volatility," Journal of Finance, American Finance Association, American Finance Association, vol. 55(2), pages 839-866, 04.
  6. Carol Alexander & Sujit Narayanan, 2001. "Option Pricing with Normal Mixture Returns: Modelling Excess Kurtosis and Uncertanity in Volatility," ICMA Centre Discussion Papers in Finance, Henley Business School, Reading University icma-dp2001-10, Henley Business School, Reading University, revised Dec 2001.
  7. Nelson, Daniel B & Ramaswamy, Krishna, 1990. "Simple Binomial Processes as Diffusion Approximations in Financial Models," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 3(3), pages 393-430.
  8. Brigo, Damiano, 2000. "On SDEs with marginal laws evolving in finite-dimensional exponential families," Statistics & Probability Letters, Elsevier, Elsevier, vol. 49(2), pages 127-134, August.
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Cited by:
  1. Carol Alexander & Andrew Scourse, 2004. "Bivariate normal mixture spread option valuation," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 4(6), pages 637-648.
  2. Damiano Brigo & Francesco Rapisarda & Abir Sridi, 2013. "The arbitrage-free Multivariate Mixture Dynamics Model: Consistent single-assets and index volatility smiles," Papers 1302.7010,


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