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The S&P 500 Index as a Sato Process Travelling at the Speed of the VIX

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  • Dilip Madan
  • Marc Yor

Abstract

The logarithm of the S&P 500 Index is modelled as a Sato process running at a speed proportional to the current level of the VIX. When the VIX is itself modelled as the exponential of a compound Poisson process with drift, we show that exact expressions are available for the prices of equity options, taken at an independent exponential maturity. The parameters for the compound Poisson process are calibrated from VIX options whereas the parameters for the Sato process driving the stock may be inferred from market option prices. Results confirm that both the S&P 500 index option surface and the parameters of the VIX time-changed Sato process have volatilities, skews and term volatility spreads that are responsive to the VIX level and the VIX option surface.

Suggested Citation

  • Dilip Madan & Marc Yor, 2011. "The S&P 500 Index as a Sato Process Travelling at the Speed of the VIX," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(3), pages 227-244.
    • Handle: RePEc:taf:apmtfi:v:18:y:2011:i:3:p:227-244
    DOI: 10.1080/1350486X.2010.486558
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    Cited by:

    1. Andrew Papanicolaou, 2021. "Extreme-Strike Comparisons and Structural Bounds for SPX and VIX Options," Papers 2101.00299, arXiv.org, revised Mar 2021.
    2. Alexander Schnurr, 2015. "An Ordinal Pattern Approach to Detect and to Model Leverage Effects and Dependence Structures Between Financial Time Series," Papers 1502.07321, arXiv.org.
    3. Andrea Barletta & Elisa Nicolato & Stefano Pagliarani, 2019. "The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 928-966, July.
    4. Andrew Papanicolaou & Ronnie Sircar, 2014. "A regime-switching Heston model for VIX and S&P 500 implied volatilities," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1811-1827, October.

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