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

Madan, Dilip B.; Yor, Marc

doi:10.2139/ssrn.1540816

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

Robert H. Smith School Research Paper No. RHS 06-117

22 Pages Posted: 28 Jan 2010 Last revised: 13 May 2010

See all articles by Dilip B. Madan

Dilip B. Madan

University of Maryland - Robert H. Smith School of Business

Marc Yor

Universite Paris

Date Written: April 4, 2010

Abstract

The logarithm of SPX is modeled as a Sato process running at a speed proportional to the current level of the VIX. When the logarithm of the VIX is an exponential compound Poisson process with drift one may obtain exact expressions for the prices of equity options taken at an independent exponential maturity. The parameters for the Lévy process are calibrated from VIX options while the parameters for the Sato process driving the stock may be calibrated from market option prices taken at an independent exponential maturity. Results confirm that both the option surface and the VIX time changed Sato process volatilities, skews and term volatility spreads are responsive to the VIX level and the VIX option surface.

Keywords: Exponential Compound Poisson, Asian Option, Laplace Transform

JEL Classification: G1, G12, G13

Suggested Citation: Suggested Citation

Madan, Dilip B. and Yor, Marc, The S&P 500 Index as a Sato Process Travelling at the Speed of the VIX (April 4, 2010). Robert H. Smith School Research Paper No. RHS 06-117, Available at SSRN: https://ssrn.com/abstract=1540816 or http://dx.doi.org/10.2139/ssrn.1540816