We study the SABR stochastic volatility model with the volatility-of-volatility parameter ν . We provide a method to expand the price C_SABR(S, K, ν, σ, τ ) of a European call in this model as a Taylor series in ν , C_SABR(S, K, ν, σ, τ ) = C_BS(S,K, σ, τ ) ν C₁ ν²C₂ . . . ν^kC_k O(ν<sup>k...

We show that the call-put implied volatility spread (IVS) outperforms many well-known predictors of the U.S. equity premium at return horizons up to six months over the period from 1996:1 to 2017:12. The predictive ability of the IVS is unrelated to the dividend yield and is useful in explaining...