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題名：跳躍風險相關之匯率選擇權: 傅立葉轉換評價法、Martingale法與蒙地卡羅法之比較
作者：温晉祥
作者(外文)：Wen, Chin-Hsiang
校院名稱：國立政治大學
系所名稱：金融學系
指導教授：林士貴
學位類別：博士
出版日期：2020
主題關鍵詞：Amin and Jarrow model；外匯選擇權；相關跳躍風險；匯率；利率；跳躍風險；Amin and Jarrow model；currency option；correlated jump risks；exchange rate；interest rate；jump risks
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本論文觀察最近十多年來國際上幾個主要國家利率與匯率的走勢以及同一個期間內的跳躍情況，發現走勢有相關性存在，並且經常同時發生跳躍。為了此特殊性質，本研究建立一個考慮走勢與跳躍相關的模型來捕捉此特性，稱作考慮相關跳躍模型 (Amin and Jarrow model with correlated jump risks, AJ-CJ)。實證結果發現AJ-CJ比起幾何布朗運動 (Geometric Brownian motion, GBM)、Amin and Jarrow 模型 (Amin and Jarrow model, AJ)、考慮獨立跳躍模型 (Amin and Jarrow model with independent jump risks, AJ-IJ) 可以更加捕捉利率及匯率的特性。利用martingale法與傅立葉轉評價法推導出AJ-CJ下的匯率選擇權評價公式並且比較兩種方法與蒙地卡羅法之計算速度與準確度，發現三種方法的評價結果很接近，且傅立葉轉評價法計算速度比另外兩種方法快許多。實證發現，大多數的例子中，AJ-CJ改善了樣本內及樣本外定價誤差，也代表可以更精準地評價匯率選擇權。研究結果支持利率與匯率存在相關性及跳躍間也存在相關。
In this paper, we investigate the trends of interest rates and exchange rates in several major international countries in the past ten years and find that the trends are correlated and often jump at the same time. Given the characteristics of correlated jump risks in interest rates and exchange rates, we construct a new model called Amin and Jarrow model with correlated jump risks (AJ-CJ) to capture the movements. The empirical results in exchange rates and interest rates data with the log-likelihood value show that AJ-CJ can capture the interest rates and the exchange rates better than Geometric Brownian model (GBM), Amin and Jarrow model (AJ), and Amin and Jarrow model with independent jump risks (AJ-IJ). After finding the martingale condition, we derive the pricing formula for currency options under AJ-CJ with the traditional martingale method and generalized Fourier transform method. This study adds the Monte Carlo method to verify the evaluation results and compare calculating time. We found that the evaluation result of traditional martingale method and Fourier evaluation method is very close to the Monte Carlo method. The calculating time of Fourier evaluation method is much faster than traditional martingale method and the Monte Carlo method. In addition, the empirical performance of the option data finds that AJ-CJ improves the in-sample and out-of-sample pricing error performances in most cases. Therefore, we conclude that correlated jump risks between interest rates and exchange rates.
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