We use the Garman-Kohlhagen model to compute the hedge and price of a participating forward contract on the US dollar that is written by a Turkish Bank. The algorithm is computed using actual market data and a weekly updated hedge is computed. We note that despite a weekly update and many assumptions made on the volatility and the interest rates the model gives a very reasonable hedge.

Subject of the submitted thesis is the issue of currency options. The aim is the detailed analysis of currency options forcefully on dealing, characteristics, methods of pricing and their use for hedging strategies. The first part of the thesis presents an introduction into the option theory. The second part is about dealing, pricing and arbitrage relationships of currency options. In this part are two option pricing model extracted -- the binomial options pricing model for pricing currency options and the Garman-Kohlhagen model for pricing European currency options. In the third part is an example for a currency put option hedging strategy.

本研究利用GARCH動態過程的優點捕捉匯率報酬率之異質變異與波動度叢聚性質，並以GARCH動態過程為基礎，考慮跳躍風險服從Lévy過程，再利用特徵函數與快速傅立葉轉換方法推導出GARCH-Lévy動態過程下的歐式匯率選擇權解析解。以日圓兌換美元(JPY/USD)之歐式匯率選擇權為實證資料，比較基準GARCH選擇權評價模型與GARCH-Lévy選擇權評價模型對市場真實價格的配適效果與預測能力。實證結果顯示，考慮跳躍風險為無限活躍之Lévy過程，即GARCH-VG與GARCH-NIG匯率選擇權評價模型，不論是樣本內的評價誤差或是在樣本外的避險誤差皆勝於考慮跳躍風險為有限活躍Lévy過程的GARCH-MJ匯率選擇權評價模型。整體而言，本研究發現進行匯率選擇權之評價時，GARCH-NIG匯率選擇權評價模型有較小的樣本內及樣本外評價誤差。 / In this thesis, we make use of GARCH dynamic to capture volatility clustering and heteroskedasticity in exchange rate. We consider a jump risk which follows Lévy process based on GARCH model. Furthermore, we use characteristic function and fast fourier transform to derive the currency option pricing formula under GARCH-Lévy process. We collect the JPY/USD exchange rate data for our empirical analysis and then compare the goodness of fit and prediction performance between GARCH benchmark and GARCH-Lévy currency option pricing model. The empirical results show that either in-sample pricing error or out-of-sample hedging performance, the infinite-activity Lévy process, GARCH-VG and GARCH-NIG option pricing model is better than finite-activity Lévy process, GARCH-MJ option pricing model. Overall, we find using GARCH-NIG currency option pricing model can achieve the lower in-sample and out-of sample pricing error.

本篇論文以Regime Switching Stochastic Volatility（RSV）作為外匯選擇權市場的波動度模型，採用馬可夫鏈蒙地卡羅法（Markov Chain Monte Carlo）中的GibbS Sampling演算法估計RSV模型的參數，並預測外匯選擇權在RSV模型下的價格。 數值結果方面首先對GibbS Sampling參數估計的結果做討論，再對預測出的選擇權價格與Black and Scholes作比較，最後並提出笑狀波幅與隱含波動度平面的結果。 本研究所得到之結論： 1. RSV模型與MCMC模擬法的組合，具備產生笑狀波幅的能力，提供足夠證據顯示，RSV模型與MCMC演算法所計算出來的選擇權價格，確實反應且捕捉到了市場上選擇權價格所應具備的特色。 2. 本模型能有效解釋期限結構 （Term Stucture of Volatility）、笑狀波幅（Volatility Smile）的現象。 關鍵字：馬可夫鏈蒙地卡羅法、外匯選擇權、貝氏選擇權評價、MCMC、Regime switching Regine change、Gibbs Sampling、currency option、Markov Chain Montec Carlo