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Markovian Stochastic Volatility With Stochastic Correlation €” Joint Calibration And Consistency Of Spx/Vix Short-Maturity Smiles

Author

Listed:
  • MARTIN FORDE

    (Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK)

  • BENJAMIN SMITH

    (Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK)

Abstract

In this paper, we show how to calibrate a general Markovian stochastic volatility model with stochastic correlation to the VIX implied volatility smile and the overall level, slope and curvature of the SPX smile in the T→0 limit. Explicit formulae are obtained for the asymptotic VIX smile for Heston and SABR-type models with mean reversion, and the Lewis CEV-p-model. We also discuss how the Bass martingale can be used to give an exact fit to a single VIX smile for T>0. In the second half of this paper, we derive a more involved integral equation for the correlation function Ï (y) to be perfectly consistent with the short-maturity SPX and VIX smiles at all strikes (or all strikes in an interval) as T→0, and discuss consistency conditions between the wings of the two asymptotic smiles and how to avoid |Ï (y)|>1 for the calibrated Ï (y) in practice.

Suggested Citation

  • Martin Forde & Benjamin Smith, 2023. "Markovian Stochastic Volatility With Stochastic Correlation €” Joint Calibration And Consistency Of Spx/Vix Short-Maturity Smiles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 26(02n03), pages 1-42, May.
    • Handle: RePEc:wsi:ijtafx:v:26:y:2023:i:02n03:n:s0219024923500073
    DOI: 10.1142/S0219024923500073
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