Consistent Pricing and hedging with Two Volatility Surfaces

Using the joint characteristic function of equity price and state variables, we can price contingent claims on both equity and VIX consistently. Based on linear approximation of jump size, we show that one factor models implies all VIX future contract of different maturities are perfectly correlated in contrast to market observations. In the examples of multi-factor model, we demonstrate how to calculate the optimal hedging ratio for VIX future to hedge VIX option. We derived the unconditional correlation term structure of VIX future implied by the model based on the stationary distribution of state variables. We show multifactor models that are calibrated to the two voaltility surfaces will produce very different hedge ratios for VIX options.