Foreign exchange rate volatility smiles and smirks

We study the implied volatilities of three foreign exchange (FX) option markets: EUD/USD, GBP/USD, and AUD/USD. We find that they are distinct from each other. The implied volatilities of the EUD/USD market tend to be more U-shaped than those of other markets. Local volatility models such as the constant elasticity of variance (CEV) model and stochastic volatility models, such as the Heston model, may fail to capture this type of convexity. We choose a stochastic-local volatility model to obtain an implied volatility formula for the corresponding FX options. The formula is given by the CEV formula with additional terms reflecting the (pure) stochastic volatility nature of FX rates. Based on this result, we show that the stochastic-local volatility model is a suitable universal choice for the pricing of these FX options.