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Inverting the Markovian projection, with an application to local stochastic volatility models

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  • Daniel Lacker
  • Mykhaylo Shkolnikov
  • Jiacheng Zhang

Abstract

We study two-dimensional stochastic differential equations (SDEs) of McKean--Vlasov type in which the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. Such SDEs arise when one tries to invert the Markovian projection developed by Gy\"ongy (1986), typically to produce an It\^o process with the fixed-time marginal distributions of a given one-dimensional diffusion but richer dynamical features. We prove the strong existence of stationary solutions for these SDEs, as well as their strong uniqueness in an important special case. Variants of the SDEs discussed in this paper enjoy frequent application in the calibration of local stochastic volatility models in finance, despite the very limited theoretical understanding.

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  • Daniel Lacker & Mykhaylo Shkolnikov & Jiacheng Zhang, 2019. "Inverting the Markovian projection, with an application to local stochastic volatility models," Papers 1905.06213, arXiv.org.
    • Handle: RePEc:arx:papers:1905.06213
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    Cited by:

    1. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A Generative Adversarial Network Approach to Calibration of Local Stochastic Volatility Models," Risks, MDPI, vol. 8(4), pages 1-31, September.
    2. Acciaio, Beatrice & Guyon, Julien, 2020. "Short communication: inversion of convex ordering: local volatility does not maximise the price of VIX futures," LSE Research Online Documents on Economics 102984, London School of Economics and Political Science, LSE Library.
    3. Beatrice Acciaio & Julien Guyon, 2019. "Inversion of Convex Ordering: Local Volatility Does Not Maximize the Price of VIX Futures," Papers 1910.05750, arXiv.org.

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