Arbitrage-free smile construction on FX option markets using Garman-Kohlhagen deltas and implied volatilities

This paper addresses arbitrage-free FX smile construction from near-term implied volatility dynamics proposed by Carr (J Financ Econ, 120(1), 1–20, 2016). The approach is directly applicable to FX option market conventions. Prices of market benchmark contracts (risk reversals and butterflies) are identified as the roots of a cubic polynomial and ATM-volatility can be matched by construction. Implied volatilities are computed with respect to (non-premium adjusted) option deltas. The approach is compared to the Vanna Volga Approach, which does not guarantee arbitrage-free prices. An empirical application to a normal and a stress scenario demonstrates that arbitrage-free implied volatilities coincide with those from the Vanna Volga Approach when prices are interpolated between the Δ25-call and Δ25-put options. Differences are observed when implied volatilities are extrapolated to the wings. Empirically, these differences are particularly relevant in a stress scenario during the Coronavirus crises (2020).