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Valuation equations for stochastic volatility models

  • Erhan Bayraktar
  • Constantinos Kardaras
  • Hao Xing

We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of the state space. We allow for various types of model behavior: the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and the asset-price process may be a strict local martingale. Our main result is a necessary and sufficient condition on the uniqueness of classical solutions to the valuation equation: the value function is the unique nonnegative classical solution to the valuation equation among functions with at most linear growth if and only if the asset-price is a martingale.

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File URL: http://arxiv.org/pdf/1004.3299
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Paper provided by arXiv.org in its series Papers with number 1004.3299.

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Date of creation: Apr 2010
Date of revision: Dec 2011
Handle: RePEc:arx:papers:1004.3299
Contact details of provider: Web page: http://arxiv.org/

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  1. David Hobson, 2010. "Comparison results for stochastic volatility models via coupling," Finance and Stochastics, Springer, vol. 14(1), pages 129-152, January.
  2. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
  3. Erhan Bayraktar & Hao Xing, 2009. "On the uniqueness of classical solutions of Cauchy problems," Papers 0908.1086, arXiv.org, revised Sep 2009.
  4. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2009. "Strict Local Martingale Deflators and Pricing American Call-Type Options," Papers 0908.1082, arXiv.org, revised Dec 2009.
  5. Aleksandar Mijatovi\'c & Mikhail Urusov, 2011. "A note on a paper by Wong and Heyde," Papers 1105.3918, arXiv.org.
  6. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
  7. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
  8. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, January.
  9. 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.
  10. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
  11. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
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