SABR Models for Synthetic FX Rates

Abstract

Let Xij be the forward FX rate for currency I in terms of currency j. Suppose that the volatility smiles for Xik and Xjk modeled with the log normal (β= 1) SABR model. We show that the cross FX rate Xij = Xik/Xjk is then governed by the log normal SABR model, and find explicit formulas for its SABR parameters β, ρ, ν in terms of the SABR parameters for Xik and Xjk. These results are not exact, but they are accurate through O(ε2), the same accuracy as the original SABR implied volatility formulas. We also extend this analysis to self-consistent models for several currencies, where each currency pair can be driven by multiple factors.