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Joint pricing of VIX and SPX options with stochastic volatility and jump models

Thomas Kokholm (Department of Economics and Business, Aarhus University, Aarhus, Denmark)
Martin Stisen (Nordea - Nordea Markets, Copenhagen, Denmark)

Journal of Risk Finance

ISSN: 1526-5943

Article publication date: 19 January 2015




This paper studies the performance of commonly employed stochastic volatility and jump models in the consistent pricing of The CBOE Volatility Index (VIX) and The S&P 500 Index (SPX) options. With the existence of active markets for volatility derivatives and options on the underlying instrument, the need for models that are able to price these markets consistently has increased. Although pricing formulas for VIX and vanilla options are now available for commonly used models exhibiting stochastic volatility and/or jumps, it remains to be shown whether these are able to price both markets consistently. This paper fills this vacuum.


In particular, the Heston model, the Heston model with jumps in returns and the Heston model with simultaneous jumps in returns and variance (SVJJ) are jointly calibrated to market quotes on SPX and VIX options together with VIX futures.


The full flexibility of having jumps in both returns and volatility added to a stochastic volatility model is essential. Moreover, we find that the SVJJ model with the Feller condition imposed and calibrated jointly to SPX and VIX options fits both markets poorly. Relaxing the Feller condition in the calibration improves the performance considerably. Still, the fit is not satisfactory, and we conclude that one needs more flexibility in the model to jointly fit both option markets.


Compared to existing literature, we derive numerically simpler VIX option and futures pricing formulas in the case of the SVJ model. Moreover, the paper is the first to study the pricing performance of three widely used models to SPX options and VIX derivatives.



Presented at the 8th Bachelier Finance Society World Congress. The authors wish to thank two anonymous referees, Jan Baldeaux, Andy Guanghua Lian and Elisa Nicolato for comments.


Kokholm, T. and Stisen, M. (2015), "Joint pricing of VIX and SPX options with stochastic volatility and jump models", Journal of Risk Finance, Vol. 16 No. 1, pp. 27-48.



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