While well-established literature argues that realized variances are close to a lognormal distribution, this study follows Benoit Mandelbrot by taking a fractal perspective. Using power laws to model realized foreign exchange rate variances, our findings indicate that power laws offer an alternative to the lognormal in terms of goodness-of-fit tests. Further, our analysis shows that estimated power law exponents for seven out of nine realized FX variances are α^<3, which indicates that the variance of realized variance is statistically undefined. We conclude that the foreign exchange rate market is far riskier than earlier believed. By implication, documented research in an enormous body of literature that draws conclusions from variance analyses stands on shaky grounds.

中文翻译：

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论外汇汇率的已实现风险：分形视角

虽然已有的文献认为已实现的方差接近对数正态分布，但本研究遵循了 Benoit Mandelbrot 的分形观点。使用幂律对已实现的外汇汇率差异进行建模，我们的研究结果表明，幂律在拟合优度检验方面提供了对数正态的替代方法。此外，我们的分析表明，九个已实现外汇方差中的七个的估计幂律指数为 α^<3，这表明已实现方差的方差在统计上未定义。我们的结论是，外汇市场的风险比之前认为的要高得多。言下之意是，从方差分析中得出结论的大量文献中记载的研究是站不住脚的。