Consistent Modelling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model

Abstract

The paper demonstrates that a pure-diffusion 3/2 model is able to capture the observed upward-sloping implied volatility skew in VIX options. This observation contradicts a common perception in the literature that jumps are required for the consistent modelling of equity and VIX derivatives. The pure-diffusion model, however, struggles to reproduce the smile in the implied volatilities of short-term index options. The pronounced implied volatility smile produces artificially inflated fitted parameters, resulting in unrealistically high VIX option implied volatilities. To remedy these shortcomings, jumps are introduced. The resulting model is able to better fit short-term index option implied volatilities while producing more realistic VIX option implied volatilities, without a loss in tractability.

Trading and Investing in Volatility Products

“…Mean-reversion, stochastic volatility of volatility and volatility jumps are features considered to be important in these papers. For more recent studies in this area see, for instance, Kaeck and Alexander (2013), Drimus and Farkas (2013) and Baldeaux and Badran (2014) and references therein.…”

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

“…However, we see that all the models generate a positive skew in VIX option implied volatilities when the Feller condition is relaxed. This contrasts the conventional Gatheral (2008), Baldeaux and Badran (2014) and Bardgett et al (2013). Also, notice how the implied volatility curves of the SV and SVJ models have the same shape, as the SVJ model only adds flexibility to the SPX return distribution and not to the terminal distribution of the VIX.…”

“…Lian and Zhu (2013) extend this to the case of VIX options when the underlying index and its volatility are allowed to jump. The related paper by Baldeaux and Badran (2014) derives a semi-closed formula for VIX options in a 3/2 stochastic volatility model with jumps in the underlying index. Any continuous time model with stochastic volatility and/or jumps implies joint dynamics for the VIX index and its underlying index and will lead to prices for both VIX and index options.…”

“…With the existence of active markets for volatility derivatives and options on the underlying, the need for models that are able to price these markets consistently has increased. Although pricing formulas for VIX options/ futures and vanilla options are now available for models exhibiting stochastic volatility and jumps (Lian and Zhu, 2013;Sepp, 2008;Baldeaux and Badran, 2014), it remains to be shown whether these models are able to price the risks of both markets consistently. This paper aims to fill this vacuum.…”

Stochastic lattice models for valuation of volatility options

“…Later the Monte Carlo and PDE methods have been widely used to study the pricing of volatility options: Psychoyios and Skiadopoulos (2006) developed Monte Carlo method for pricing the volatility options and compared the hedging roles of the volatility options with the standard options; Sepp (2008) utilized the PDE approach to derive the pricing formulas for volatility options of MRSRP models with jump-diffusion process; Carr and Lee (2009) gave a survey to the studies of volatility options; Lin and Chang (2009) also used the PDE approach to study the VIX options under the ERAKER models; Psychoyios et al (2010) studied the European volatility options on spot VIX and forward VIX by PDE methods; Wang and Daigler (2011) gave an empirical evidence on the performance of VIX option pricing model; Chang (2010, 2012) investigated the VIX European options under the SVSCJ model; Baldeaux and Badran (2012) obtained the quasi-closed-form formula for the VIX options under the jump-diffusion process of 3/2 dimension using PDE approach; Graziano and Torricelli (2012) studied the target volatility options by a PDE method; and Zhu and Liang (2012) studied the analytical formulas for the VIX futures.…”