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A model-free backward and forward nonlinear PDEs for implied volatility

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  • Peter Carr
  • Andrey Itkin
  • Sasha Stoikov

Abstract

We derive a backward and forward nonlinear PDEs that govern the implied volatility of a contingent claim whenever the latter is well-defined. This would include at least any contingent claim written on a positive stock price whose payoff at a possibly random time is convex. We also discuss suitable initial and boundary conditions for those PDEs. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach.

Suggested Citation

  • Peter Carr & Andrey Itkin & Sasha Stoikov, 2019. "A model-free backward and forward nonlinear PDEs for implied volatility," Papers 1907.07305, arXiv.org.
    • Handle: RePEc:arx:papers:1907.07305
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    References listed on IDEAS

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    3. Li, Bo & Lu, Benzhuo & Wang, Zhongming & McCammon, J. Andrew, 2010. "Solutions to a reduced Poisson–Nernst–Planck system and determination of reaction rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(7), pages 1329-1345.
    4. Andrey Itkin, 2017. "Modelling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps," Applied Mathematical Finance, Taylor & Francis Journals, vol. 24(6), pages 485-519, November.
    5. A Itkin, 2019. "Deep learning calibration of option pricing models: some pitfalls and solutions," Papers 1906.03507, arXiv.org.
    6. Jose Manuel Corcuera & Florence Guillaume & Peter Leoni & Wim Schoutens, 2009. "Implied Levy volatility," Quantitative Finance, Taylor & Francis Journals, vol. 9(4), pages 383-393.
    7. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
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