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.

中文翻译：

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汇率波动的微笑和傻笑

我们研究了三个外汇 (FX) 期权市场的隐含波动率：EUD/USD、GBP/USD 和 AUD/USD。我们发现它们彼此不同。EUD/USD 市场的隐含波动率往往比其他市场更呈 U 形。局部波动率模型，例如恒定方差弹性 (CEV) 模型和随机波动率模型，例如 Heston 模型，可能无法捕捉到这种类型的凸性。我们选择随机局部波动率模型来获得相应外汇期权的隐含波动率公式。该公式由 CEV 公式给出，附加项反映了外汇汇率的（纯）随机波动性。基于这一结果，我们表明随机局部波动率模型是这些 FX 期权定价的合适通用选择。