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.

翻译：本研究考察了英镑（GBP）价值的波动性，特别关注其与美元（USD）和欧元（EUR）货币对的关系。基于2018年6月15日至2023年6月15日的数据，我们应用多种数学模型评估其对货币对日收益率20天变动的预测效果。分析中采用指数加权移动平均（EWMA）、广义自回归条件异方差（GARCH）模型和隐含波动率（IV）模型。为评估模型性能，我们使用均方根误差（RMSE）和平均绝对误差（MAE）指标比较预测精度。研究深入探讨了GARCH模型的复杂性，检验了该模型在给定数据集上的统计特性。结果表明欧元/英镑（EUR/GBP）货币对存在非对称收益特征，而英镑/美元（GBP/USD）货币对的证据则不具备确定性。此外，我们观察到当残差服从标准t分布而非标准正态分布时，GARCH类模型对数据的拟合效果更优。进一步地，我们考察了GARCH类模型中不同预测技术的有效性。通过比较滚动窗口预测与扩展窗口预测，发现两种方法在测试场景中均未表现出绝对优势。实验表明，对GBP/USD货币对而言，最精确的波动率预测来自采用滚动窗口方法的GARCH模型；而对EUR/GBP货币对，最优预测则源自将汇率年化隐含波动率作为自变量的GARCH模型和普通最小二乘法（OLS）模型。