The jump phenomenons of many financial assets prices have been observed in many empirical papers. In this paper we extend the Heath-Jarrow-Morton model to include the jump component to derive the European-style pricing formula of the interest rate futures options. We use numerical method to simulate the options prices and analyze how each component of HJM model under jump-diffusion processes affects the interest rate futures options. Finally, we utilize LSM method which are presented by Longstaff and Schwartz to derive American options prices and compare it with European options.

股東價值函數意指公司未來預期股利的加總，其模型主要利用擴散過程來建構。因為公司可透過保留現金和發放股利兩大策略來調控公司價值－－過多現金保留將產生現金邊際效益降低；現金保留過少則會產生流動性風險甚至破產問題。論文主要探討公司在捨取現金保留時如果達大公司價值的極大化，並利用實際資料來檢視上市公司的價值是否有高低估之嫌。此外，亦在論文中提供以股東價值函數為評價方式的交易策略。 / We consider a problem of finding the optimal solution for maximizing the value of a firm in a stochastic environment. In the paper, we apply a methodology that determines a corporate value by using cash reserves as the main variables. We examine the model by employing two empirical approaches to compare the corporate value computed from the model to the market value. We find that there are slight differences between the stock value and shareholder value. Therefore, we may use the facts that we find to construct optimal trading strategies to exploit small price differences.

本文實證探討黃金期貨報酬率的特性並在標的黃金期貨價格遵循狀態轉換跳躍擴散過程時實現美式選擇權之評價。在這樣的動態過程下，跳躍事件被一個複合普瓦松過程與對數常態跳躍振幅所描述，以及狀態轉換到達強度是由一個其狀態代表經濟狀態的隱藏馬可夫鏈所捕捉。考量不同的跳躍風險假設，我們使用Merton測度與Esscher轉換推導出在一個不完全市場設定下的風險中立黃金期貨價格動態過程。為了達到所需的精確度，最小平方蒙地卡羅法被用來近似美式黃金期貨選擇權的價值。基於實際市場資料，我們提供實證與數值結果來說明這個動態模型的優點。 / This dissertation empirically investigates the characteristics of gold futures returns and achieves the valuation of American-style options when the underlying gold futures price follows a regime-switching jump-diffusion process. Under such dynamics, the jump events are described as a compound Poisson process with a log-normal jump amplitude, and the regime-switching arrival intensity is captured by a hidden Markov chain whose states represent the economic states. Considering the different jump risk assumptions, we use the Merton measure and Esscher transform to derive risk-neutral gold futures price dynamics under an incomplete market setting. To achieve a desired accuracy level, the least-squares Monte Carlo method is used to approximate the values of American gold futures options. Our empirical and numerical results based on actual market data are provided to illustrate the advantages of this dynamic model.

可轉換公司債是給予持有者於債券存續期間內行使轉換為股票之複合式證券，除了債券性質外，內嵌的股票選擇權便屬於美式選擇權。而在本文中，針對內含美式選擇權的公司債評價是使用最小平方蒙地卡羅的數值分析，主要原因在於可轉債本身的條款彈性高，加上可轉債可能涉及之標的資產為兩個以上或狀態變數也可能具有多個維度(dimension)。此外，針對可轉債發行公司本身的信用問題，本文則採用縮減式(reduced-form)模型來處理其違約風險問題。依據A. Takahashi, T. Kobayashi, and N. Nakagawa認為採用結構式(structured-form)的缺點為參數難以校準，並列出下面兩論點認為使用縮減式的優點在於： 1. 違約事件將可能造成股價跳躍(jump)現象。 2. 在Duffie and Singleton方法下，資產隨機過程不必設定jump term，仍可設定為擴散過程(diffusion process)。 至於在利率期間結構方面，雖然Brennan and Schwartz(1980)認為實務上，考量利率的隨機性除了降低評價的效率性之外，與利率設定為常數相比，其差異不大。但針對為何差異不大的原因，本文認為利率對於純粹債券之價值影響為負向關係，而對於股票買權則是正向關係，故使得最後可轉債的影響則不明顯。然而，在目前「可轉債資產交換」等可轉債相關衍生性商品相繼推陳出新之下，使得可轉債的純粹債券與選擇權的個別要素評價也是相當重要。所以本文在利率風險的建構上將使用BGM模型來描述利率的隨機過程。

Schwarz (1997)提出均數回復過程（Mean-Reverting Process, MR）捕捉能源價格的動態過程，而Lucia and Schwarz (2002)將此模型結合確定季節性函數，並推導出期貨價格封閉解。然而，能源價格常會因為未預期事件的發生而產生大幅度的變動，為了描述價格跳躍的現象，Clewlow and Strickland (2000)延伸Schwarz的模型提出均數回復跳躍擴散模型（Mean-reverting jump diffusion process, MRJD），此模型除了保留均數回復模型對能源價格會回復至長期水準的描述外，再加上跳躍項來描述價格的異常變動。而Cartea and Figueroa (2005)則同時考慮季節性和跳躍因子，並推導出期貨價格封閉解。另外，雖然台灣目前並非京都議定書所規範的國家，但環境保護是未來的趨勢，故在衡量能源產業財務風險時，除了考慮相關原料和產品，應考慮碳權交易之影響。為了探討財務風險管理在能源產業之應用，本文以煉油廠為例，將其表示成特定期貨部位的投資組合，並透過計算投資組合風險值來衡量煉油廠的財務風險。文中使用結合季節性的均數回復過程、均數回復跳躍擴散過程進行模型配適。實證結果顯示，均數回復跳躍擴散模型在回溯測試下表現最佳；另外，考慮碳權交易後會使得煉油廠的財務風險上升。 / Schwarz (1997) proposes the mean-reverting process (MR) to model energy spot price dynamics, and Lucia and Schwarz (2002) extend this model by including mean reversion and a deterministic seasonality. This model can capture the mean-reversion of energy price, but fail to account for the huge and non-negligible price movement in the market. Clewlow and Strickland (2000) extend Schwarz’s model to mean-reverting jump diffusion process (MRJD). Cartea and Figueroa (2005) present a model which captures the most importance characteristics of energy spot prices such as mean reversion, jumps and seasonality, and provide a closed-form solution for the forward. Although Taiwan is not the member of Kyoto Protocol, but Environmental Protection is a trend in the future. In order to measure the financial risk induced by energy industries, we should consider the effect of emission trading. In this paper, we discuss the implication of financial risk management in energy industries by analyzing the exposure of refinery which represented certain energy futures portfolios. We use MR and MRJD process with seasonality to model energy spot price dynamics, and calibrate the parameters to historical data. And, we consider the interaction of all of positions and calculate the Value-at-Risk of portfolios. The results show that among various approaches the MRJD presents more efficient results in back-testing, and emission trading poses additional risk factors which will increase the financial risk for refineries.

為了改進Black-Scholes模式的實證現象，許多其他的模型被建議有leptokurtic特性以及波動度聚集的現象。然而對於其他的模型分析的處理依然是一個問題。在本論文中，我們建議使用馬可夫跳躍擴散過程，不僅能整合leptokurtic與波動度微笑特性，而且能產生波動度聚集的與長記憶的現象。然後，我們應用Lucas的一般均衡架構計算選擇權價格，提供均衡下當跳躍的大小服從一些特別的分配時則選擇權價格的解析解。特別地，考慮當跳躍的大小服從兩個情況，破產與lognormal分配。當馬可夫跳躍擴散模型的馬可夫鏈有兩個狀態時，稱為轉換跳躍擴散模型，當跳躍的大小服從lognormal分配我們得到選擇權公式。使用轉換跳躍擴散模型選擇權公式，我們給定一些參數下研究公式的數值極限分析以及敏感度分析。 / To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address the leptokurtic feature of the asset return distribution, and the effects of volatility clustering phenomenon. However, analytical tractability remains a problem for most of the alternative models. In this dissertation, we propose a Markov jump diffusion model, that can not only incorporate both the leptokurtic feature and volatility smile, but also present the economic features of volatility clustering and long memory. Next, we apply Lucas's general equilibrium framework to evaluate option price, and to provide analytical solutions of the equilibrium price for European call options when the jump size follows some specific distributions. In particular, two cases are considered, the defaultable one and the lognormal distribution. When the underlying Markov chain of the Markov jump diffusion model has two states, the so-called switch jump diffusion model, we write an explicit analytic formula under the jump size has a lognormal distribution. Numerical approximations of the option prices as well as sensitivity analysis are also given.