A regime-switching Heston model for VIX and S&P 500 implied volatilities

Abstract

Volatility products have become popular in the past 15 years as a hedge against market uncertainty. In particular, there is growing interest in options on the VIX volatility index. A number of recent empirical studies have examine whether there is significantly greater risk premium in VIX option prices compared with S&P 500 option prices. We address this issue by proposing and analysing a stochastic volatility model with regime switching. The basic Heston model cannot capture VIX-implied volatilities, as has been documented. We show that the incorporation of sharp regime shifts can bridge this shortcoming. We take advantage of asymptotic and Fourier methods to make the extension tractable, and we present a fit to data, both in times of crisis and relative calm, which shows the effectiveness of the regime switching.

Keywords:

• Applied mathematical finance
• Calibration of stochastic volatility
• VIX options
• Model calibration

JEL Classification:

• G12
• G13

Acknowledgements

The authors thank Lisa Goldberg for discussion on regime models, as well as Jim Gatheral and two anonymous referees for their helpful comments. Work by A.P. partially supported by NSF grant DMS-0739195, and R.S. partially supported by NSF grant DMS-1211906.

Notes

There is a VXTH white paper located at http://www.cboe.com/micro/VXTH/documents/VXTHWhitePaper.pdf.