本論文考慮在跳躍擴散式的資產價格動態過程中，提出使價格跳躍項具有序列相關性的模型建構方法，並探討其在歐式選擇權訂價及選擇權避險投資組合上的應用。我們提出兩種建模方法，第一種方法是在相鄰的跳躍項中，引入均值回歸的特性以呈現跳躍項會隨著跳躍次數增加而逐漸趨向長期均值的負相關現象，所建立的模型為一個均值回歸式的跳躍擴散模型，其跳躍項間的負相關性可以用來描繪投資人面對市場未預期資訊可能產生的過度反應現象。在此模型下，我們推導其歐式選擇權之訂價公式並據此校正選擇權市場隱含波動度以反應此投資行為對選擇權價格可能產生之影響。在第二種建模方法中，我們進一步考慮資產價格處於多頭市場和空頭市場期間，投資人常採取動量策略使得資產價格跳躍項可能呈現正相關的情形。為使模型可以描繪此現象，我們假設前後跳躍項間具有自我相關的性質，以此建構出的跳躍擴散模型，能夠綜合考量動量策略下之價格跳躍項所呈現的正相關性，以及投資人過度反應資訊下進行反向策略而產生之價格跳躍項所呈現的負相關性。從在此模型的分析中，我們得到資產報酬分配的主要統計量和歐式選擇權價格以及避險參數的公式解。這些解析結果使我們得以分析跳躍相關性對避險投資組合所帶來的影響，以利投資人作為參考。

This dissertation considers two extensions of jump-diffusion models to incorporate the serially correlated jump sizes and discusses their applications in options pricing and hedging portfolios. In the first part of this study, we propose a jump-diffusion model where the bivariate jumps are serially correlated with a mean-reverting structure pulling the jump process toward its long-term mean. The negative serial correlation of the adjacent jumps could reflect the phenomenon of financial investors overreacting to unexpected information. Under this model, we derive the European option price in analytical form and provide the calibration examples for the implied volatility to investigate the effects of the investor’s behavior on the option market. In the second part of this study, we further consider serially correlated jump sizes with an autoregressive (AR) structure. It is intended to capture the asset price momentum during the bull/bear market making jump sizes positively correlated as seen in recent empirical evidences. The AR structure allows us to construct a general jump-diffusion model with both positive and correlation which reflect the combined effects from momentum and overreaction. The main statistics of asset return distribution, European option prices, and hedging parameters are derived analytically, with numerical examples visualizing the effects from the correlation parameter. Further examples are given to discuss its effects on hedging portfolios providing useful information for financial investors.

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