The Fine Structure of Variance: Consistent Pricing of VIX Derivatives

Instantaneous squared VIX and VIX derivatives

“…We model VIX as the weighted average of , , and . If plugging the definition of as a combination of and return jumps in , we obtain a similar formula for VIX as in Branger et al () and Bardgett et al (). However, by introducing in this way as a whole, we are able to focus on rather than its components, which means that there is no need to estimate these individual parameters.…”

“…For example, Branger et al () investigate a stochastic volatility model with volatility of volatility as another stochastic process and Bardgett et al () specify a stochastic volatility model with the intensity of volatility jumps given by another stochastic process. First, by considering the dynamics of SPX and modeling its diffusion variance as a stochastic volatility model, the VIX in Branger et al () and Bardgett et al () is a explicit function of the parameters of SPX dynamics including return jumps. We model VIX as the weighted average of , , and .…”

“…However, by introducing in this way as a whole, we are able to focus on rather than its components, which means that there is no need to estimate these individual parameters. Second, we can estimate the dynamics of by using VIX term structure data only while SPX and its options data are necessary to estimate parameters in Branger et al () and Bardgett et al (). Further, the SPX return parameters will enter into VIX derivatives pricing in Branger et al () and Bardgett et al () while there are no SPX return parameters in our VIX derivatives formula, as can be seen in a later section.…”

“…Second, we can estimate the dynamics of by using VIX term structure data only while SPX and its options data are necessary to estimate parameters in Branger et al () and Bardgett et al (). Further, the SPX return parameters will enter into VIX derivatives pricing in Branger et al () and Bardgett et al () while there are no SPX return parameters in our VIX derivatives formula, as can be seen in a later section. In other words, less parameter estimation and less latent information are required in our estimation.…”

Consistency between S&P500 and VIX derivatives: Insights from model‐free VIX futures pricing

“…The SPX options prices are calculated as in Duffie et al (2000), and the VIX derivatives are computed as in Branger, Kraftschik, and Völkert (2016). Since the model features stochastic volatility and jumps, it provides a valid market framework to study the model-free pricing formula for VIX futures.…”