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.

本研究使用美國、德國、英國股票指數期貨之日內交易資料，從報酬率中分離出連續波動度與跳躍項，再以MLE法估計Hawkes相互激勵過程之參數，衡量跨市場跳躍傳染現象。擴展文獻中僅兩市場的分析至三市場模型，更能從整體的角度解釋市場間的關係及跳躍傳染途徑。實證結果顯示，美國能直接影響其他市場，而其他市場反過來不易干涉美國，呈現非對稱影響效果。歐洲兩國能互相傳染，英國對德國的影響較大，也更有能力影響美國，稱英國為歐洲的影響輸出國，德國為歐洲的影響輸入國。

本研究以Lévy過程為模型基礎，考慮Merton Jump及跳躍強度服從Hawkes Process的Merton Jump兩種跳躍風險，利用Particle Filter方法及EM演算法估計出模型參數並計算出對數概似值、AIC及BIC。以S&P500指數為實證資料，比較隨機波動度模型、Variance Gamma模型及兩種不同跳躍風險對市場真實價格的配適效果。實證結果顯示，隨機波動度模型其配適效果勝於Variance Gamma模型，且加入跳躍風險後可使模型配適效果提升，尤其在模型中加入跳躍強度服從Hawkes Process的Merton Jump，其配適效果更勝於Merton Jump。整體而言，本研究發現，以S&P500指數為實證資料時，SVHJ模型有較好的配適效果。 / This paper, based on the Lévy process, considers two kinds of jump risk, Merton Jump and the Merton Jump whose jump intensity follows Hawkes Process. We use Particle Filter method and EM Algorithm to estimate the model parameters and calculate the log-likelihood value, AIC and BIC. We collect the S&P500 index for our empirical analysis and then compare the goodness of fit between the stochastic volatility model, the Variance Gamma model and two different jump risks. The empirical results show that the stochastic volatility model is better than the Variance Gamma model, and it is better to consider the jump risk in the model, especially the Merton Jump whose jump intensity follows Hawkes Process. The goodness of fit is better than Merton Jump. Overall, we find SVHJ model has better goodness of fit when S&P500 index was used as the empirical data.

本論文目的在於探討「異質期望」(heterogeneous expectations)、「資訊密度」(information arrival intensity)、以及「跳躍風險」(jump risk) 這些因素對於期貨價格的影響，並且透由「跨期模型」(intertemporal models) 的建立，推導出具有封閉解形式的期貨價格理論公式。 誠如 Harrison and Kreps (1978) 所言：除非所有市場參與者的行為方式完全相同、而且他們都打算抱著股票直到永遠，否則「投機交易」(speculation transactions) 與「異質期望」就不可能自市場當中滅絕。有鑑於此，本論文在第二章中討論「異質期望」對於期貨價格的影響；同時為了反映交易者看法可能會隨時間演進而發生改變的可能性，「調整效果」(adjustment effects) 是本章另一個討論重點；第三、為了區別期貨契約與遠期契約的基本差異，「利率」這個隨機因子也被納入模型當中。由「部分均衡」(partial equilibrium) 觀點下具有封閉解形式的期貨價格公式來觀察，這三個重要因素以及彼此間存在著的複雜交互作用，可以協助解釋一些實證現象與重要變數之間的關係。 第三章主要是借用Clark (1973) 與Chang et al. (1988) 「資訊時間」(information time) 的概念，取代一般模型所使用的「日曆時間」(calendar time) 設定方法，並且額外納入「利率」與「便利所得」(convenience yield) 這兩個廣為一般期貨定價文獻所認定的重要隨機因素，推導出「部分均衡」觀點下的期貨價格封閉解。根據1998/7/21 至 2003/12/31 台灣期交所「台灣證券交易所總加權股價指數期貨」的實證結果來看，本章模型的定價績效不僅勝過「持有成本模型」(the cost of carry model)，也比同時考慮「利率」與「便利所得」兩個隨機因子的「日曆時間」模型要來的好。 第四章則是嘗試結合Hemler and Longstaff (1991) 的「無偏好模型」(preference-free model) 以及Merton (1976) 的「跳躍」(jumps) 設定，重新推導「一般均衡」(general equilibrium) 模型下、考慮「跳躍風險」(jump risk) 後的期貨價格封閉解。根據本章各種比較靜態與模擬分析的結果顯示，整個經濟體系或是「狀態變數」(state variables) 的安定程度，決定了市場變數間的關係；另一方面，這些關聯會因為「跳躍風險」規模的遞增 — 不管是肇因於「發生機率」(occurring probability) 或是「衝擊效果」(impulse effect) — 而變的更加不可預測。 / The dissertation contains three essays on intertemporal futures pricing models allowing for heterogeneous expectations, information-time based setting, and jump risk. As Harrison and Kreps (1978) have noted, unless traders are all identical and obliged to hold a stock forever, speculation would not extinguish in market, and heterogeneity in expectations yields whereby. The first essay develops intertemporal futures pricing formulas accounting for such reality, adjustment effect, and stochastic interest rate in a partial-equilibrium sense. The closed-form solutions show that the three factors complicated with each others can help to explain some existing empirics on relationships between futures prices and other important market variables such as indeterminate converging pattern. The second essay extends Chang et al. (1988) option pricing model to derive futures prices with information-time based processes. Stochastic interest rate and convenience yield are also taken into account to derive closed-form formulas. According to empirical results of transaction data of TAIEX index and its corresponding TFETX futures contract through 1998/7/21 to 2003/12/31, the analytic solution performs better than the cost of carry model and its calendar-time based counterpart, especially when information arrival intensity estimates become larger. The last essay combines Hemler and Longstaff’s (1991) preference-free model and Merton’s (1976) jump setting to measure effects from jump risk and a futures pricing formula is derived in its closed-form as well. According to miscellaneous comparative static and simulation results, the bounded degrees of state variables, or economy, affect co-varying extents among variables, while the increasing jump risk, including the size of occurring probability and its corresponding impulse effect, makes them un-tractable.

本研究利用美國芝加哥商品交易所針對 18 個城市發行之冷氣指數／暖氣指數衍生性商品與相對應之日均溫進行分析與評價。研究成果與貢獻如下：一、延伸 Alaton, Djehince, and Stillberg (2002) 模型，引入跳躍風險、隨機波動度、波動跳躍等因子，提出新模型以捕捉更多溫度指數之特徵。二、針對不同模型，分別利用最大概似法、期望最大演算法、粒子濾波演算法等進行參數估計。實證結果顯示新模型具有較好之配適能力。三、利用 Esscher 轉換將真實機率測度轉換至風險中立機率測度，並進一步利用　Feynman-Kac 方程式與傅立葉轉換求出溫度模型之機率分配。四、推導冷氣指數／暖氣指數期貨之半封閉評價公式，而冷氣指數／暖氣指數期貨之選擇權不存在封閉評價公式，則利用蒙地卡羅模擬進行評價。五、無論樣本內與樣本外之定價誤差，考慮隨機波動度型態之模型對於溫度衍生性商品皆具有較好之評價績效。六、實證指出溫度市場之市場風險價格為負，顯示投資人承受較高之溫度風險時會要求較高之風險溢酬。本研究可給予受溫度風險影響之產業，針對衍生性商品之評價與模型參數估計上提供較為精準、客觀與較有效率之工具。 / This study uses the daily average temperature index (DAT) and market price of the CDD/HDD derivatives for 18 cities from the CME group. There are some contributions in this study: (i) we extend the Alaton, Djehince, and Stillberg (2002)'s framework by introducing the jump risk, the stochastic volatility, and the jump in volatility. (ii) The model parameters are estimated by the MLE, the EM algorithm, and the PF algorithm. And, the complex model exists the better goodness-of-fit for the path of the temperature index. (iii) We employ the Esscher transform to change the probability measure and derive the probability density function of each model by the Feynman-Kac formula and the Fourier transform. (iv) The semi-closed form of the CDD/HDD futures pricing formula is derived, and we use the Monte-Carlo simulation to value the CDD/HDD futures options due to no closed-form solution. (v) Whatever in-sample and out-of-sample pricing performance, the type of the stochastic volatility performs the better fitting for the temperature derivatives. (vi) The market price of risk differs to zero significantly (most are negative), so the investors require the positive weather risk premium for the derivatives. The results in this study can provide the guide of fitting model and pricing derivatives to the weather-linked institutions in the future.