Five diffusion models are estimated using three different foreign exchange rates to find an appropriate model for each. Daily spot exchange rates expressed as the prices of 1 euro, 1 British pound and 100 Japanese yen in US dollars, respectively denoted by USD/EUR, USD/GBP, and USD/100JPY, are used. The maximum likelihood estimation method is implemented after deriving an approximate log-transition density function (log-TDF) of the diffusion processes because the true log-TDF is unknown. Of the five models, the most general model is the best fit for the USD/GBP, and USD/100JPY exchange rates, but it is not the case for the case of USD/EUR. Although we could not find any evidence of the mean-reverting property for the USD/EUR exchange rate, the USD/GBP, and USD/ 100JPY exchange rates show the mean-reversion behavior. Interestingly, the volatility function of the USD/EUR exchange rate is increasing in the exchange rate while the volatility functions of the USD/GBP and USD/100Yen exchange rates have a U-shape. Our results reveal that more care has to be taken when determining a diffusion model for the exchange rate. The results also imply that we may have to use a more general diffusion model than those proposed in the literature when developing economic theories for the behavior of the exchange rate and pricing foreign currency options or derivatives.

中文翻译：

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汇率连续时间扩散模型的最大似然估计

使用三种不同的外汇汇率估算了五个扩散模型，以便为每个模型找到合适的模型。使用以1欧元，1英镑和100日元的美元价格表示的每日即期汇率，分别用USD / EUR，USD / GBP和USD / 100JPY表示。由于真实对数TDF是未知的，因此在推导扩散过程的近似对数跃迁密度函数（log-TDF）之后，实施最大似然估计方法。在这五个模型中，最通用的模型最适合USD / GBP和USD / 100JPY汇率，但USD / EUR并非如此。尽管我们找不到美元/欧元汇率的均值回复性质的任何证据，但美元/英镑和美元/ 100日元汇率显示了均值回复行为。有趣的是 美元兑欧元汇率的波动率函数在汇率中正在增加，而美元兑英镑和美元/ 100日元汇率的波动率函数呈U形。我们的结果表明，在确定汇率扩散模型时必须格外小心。结果还暗示，在为汇率行为和定价外币期权或衍生工具的经济理论发展时，我们可能必须使用比文献中提出的模型更广泛的扩散模型。