In this study, we examine the fluctuation in the value of the Great Britain
Pound (GBP). We focus particularly on its relationship with the United States
Dollar (USD) and the Euro (EUR) currency pairs. Utilizing data from June 15,
2018, to June 15, 2023, we apply various mathematical models to assess their
effectiveness in predicting the 20-day variation in the pairs' daily returns.
Our analysis involves the implementation of Exponentially Weighted Moving
Average (EWMA), Generalized Autoregressive Conditional Heteroskedasticity
(GARCH) models, and Implied Volatility (IV) models. To evaluate their
performance, we compare the accuracy of their predictions using Root Mean
Square Error (RMSE) and Mean Absolute Error (MAE) metrics. We delve into the
intricacies of GARCH models, examining their statistical characteristics when
applied to the provided dataset. Our findings suggest the existence of
asymmetric returns in the EUR/GBP pair, while such evidence is inconclusive for
the GBP/USD pair. Additionally, we observe that GARCH-type models better fit
the data when assuming residuals follow a standard t-distribution rather than a
standard normal distribution. Furthermore, we investigate the efficacy of
different forecasting techniques within GARCH-type models. Comparing rolling
window forecasts to expanding window forecasts, we find no definitive
superiority in either approach across the tested scenarios. Our experiments
reveal that for the GBP/USD pair, the most accurate volatility forecasts stem
from the utilization of GARCH models employing a rolling window methodology.
Conversely, for the EUR/GBP pair, optimal forecasts are derived from GARCH
models and Ordinary Least Squares (OLS) models incorporating the annualized
implied volatility of the exchange rate as an independent variable.

通过应用数学模型和评估预测准确率，研究了英镑与美元和欧元货币对的波动性。发现 EUR/GBP 货币对存在不对称的回报，而 GBP/USD 货币对的证据不确定。使用 GARCH 模型时，假设残差遵循标准 t 分布而不是标准正态分布，效果更好。对于 GBP/USD 货币对，使用滚动窗口方法得到的 GARCH 模型产生最准确的波动率预测，而对于 EUR/GBP 货币对，最佳预测来自 GARCH 模型和 OLS 模型，其中 OLS 模型中的独立变量是年化的隐含波动率。