Abstracts

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 employed models exhibiting stochastic volatility and/or jumps, it remains to be shown whether these are able to price both markets consistently. This paper aims to fill 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 are studied. In all these models the characteristic function of log-returns is known in an analytically closed form, and options on the underlying index can be priced via Fourier transform techniques. Likewise, the three model specifications allow for tractable pricing of VIX options as demonstrated in [1]. Compared to the VIX option pricing formula in Lian and Zhu (2013), we derive a numerically simpler formula in the case of the Heston model with jumps in returns (but not variance).
We find that the full flexibility of having jumps in both returns and volatility added to a stochastic volatility model is essential. The jumps in returns allow for an improved fit to SPX options, while jumps in volatility are important to match the upward sloping implied volatility skew observed on VIX options. Moreover, we find that the SVJJ model with the Feller condition imposed and calibrated jointly to SPX and VIX options fails to fit both markets satisfactory with average relative pricing errors for the dates considered around 16-20%. Relaxing the Feller condition in the calibration improves the fit considerably and errors are down to around 8-11%. Still, the fit is not satisfactory for trading purposes and we conclude that one needs more flexibility in the model in order to jointly fit both options markets.

[1] Lian, G. and Zhu, S. (2013). Pricing VIX options with stochastic volatility and random jumps. Decisions in Economics and Finance, 36 (1): 71-88.