Empirical model of VIX volatility and SPX variance

We introduce a toy model for VIX futures dynamics using the Shifted-Lognormal Model (SLM) that does an excellent job of fitting VIX option prices with only two parameters. Then using the results of Roger Lee, who studied SLM models in detail, we propose a way to extrapolate VIX volatility surface which by construction is arbitrage free. Then, we derive analytical formula for forward variance in this model and relate risks of VIX volatility surface with that of SPX volatility surface by matching forward variance swaps results from both markets. This allows one to relate risk parameters between VIX and SPX volatility surface, namely we derive a nonlinear equation that relates VIX skew and SPX skew. This can be used for cross market hedging, arbitrage and enhanced risk monitoring.