本論文的研究目的，主要是希望利用新奇選擇權（exotic option）來降低保本型基金的高風險投資部分的權利金，提升參與率，藉此吸引投資人。因為近年來保本型基金面臨最大的問題就是『市場波動度變大，造成衍生性商品的價格上升，侵蝕了保本率。』(Lee,2001)，因為波動度和商品價格具有正比的關係。再加入浮動利率之考量之後，求出更精確的封閉解，以及本文所提『雙邊連動』，提升商品吸引力。 在精算科學界，Esscher transform是一種沿用已久的工具。Gerber and Shiu (1994)闡述在某些假設下評價衍生性證券時，Esscher transform是一種有效率的方法。本論文延伸『Esscher transform』方法來求出商品評價的公式解。 本論文的主要貢獻就是引用Esscher transform（Gerber and Shiu,1994架構傳統機率測度轉換並且求出上(下)出局、上(下)生效等保本型指數連動商品的封閉解，並且加入一個新的概念，『雙邊連動』，作為整篇論文的主要貢獻。基於上述原因，本論文研究成果可以分為下面幾項： 1.以『Esscher transform』為本論文的評價模型，加以說明驗證。 2.設計出雙邊保本的保本型指數連動商品，並且找出封閉解以及探討此種商品的可行性及市場性。 3.利用電腦模擬求算評價公式的避險參數。求出多元常態累積機率分配函數，以期能夠解出多資產連動商品的理論價格。並且整理出上下限型的機率密度整理表。 在程式應用的方面，本論文利用了『Mathematica』求取避險參數，因而不必再費時的計算就可以求出正確的避險參數，及利用計量軟體『R』來求算多元常態累積機率分配函數，使本論文的多因子分析不在只是理論。

Derivative pricing, and in particular the pricing of options, is an important area of current research in financial mathematics. Experts debate on the best method of pricing and the most appropriate model of a price process to use. In this thesis, a ``Switching Black-Scholes'' model of a price process is proposed. This model is based on the standard geometric Brownian motion (or Black-Scholes) model of a price process. However, the drift and volatility parameters are permitted to vary between a finite number of possible values at known times, according to the state of a hidden Markov chain. This type of model has been found to replicate the Black-Scholes implied volatility smiles observed in the market, and produce option prices which are closer to market values than those obtained from the traditional Black-Scholes formula. As the Markov chain incorporates a second source of uncertainty into the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no unique option pricing methodology exists. In this thesis, we apply the methods of mean-variance hedging, Esscher transforms and minimum entropy in order to price options on assets which evolve according to the Switching Black-Scholes model. C programs to compute these prices are given, and some particular numerical examples are examined. Finally, filtering techniques and reference probability methods are applied to find estimates of the model parameters and state of the hidden Markov chain. / Thesis (Ph.D.)--Applied Mathematics, 2003.

Derivative pricing, and in particular the pricing of options, is an important area of current research in financial mathematics. Experts debate on the best method of pricing and the most appropriate model of a price process to use. In this thesis, a ``Switching Black-Scholes'' model of a price process is proposed. This model is based on the standard geometric Brownian motion (or Black-Scholes) model of a price process. However, the drift and volatility parameters are permitted to vary between a finite number of possible values at known times, according to the state of a hidden Markov chain. This type of model has been found to replicate the Black-Scholes implied volatility smiles observed in the market, and produce option prices which are closer to market values than those obtained from the traditional Black-Scholes formula. As the Markov chain incorporates a second source of uncertainty into the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no unique option pricing methodology exists. In this thesis, we apply the methods of mean-variance hedging, Esscher transforms and minimum entropy in order to price options on assets which evolve according to the Switching Black-Scholes model. C programs to compute these prices are given, and some particular numerical examples are examined. Finally, filtering techniques and reference probability methods are applied to find estimates of the model parameters and state of the hidden Markov chain. / Thesis (Ph.D.)--Applied Mathematics, 2003.

Black and Scholes (1973)對於報酬率提出以B-S模型配適，但B-S模型無法有效解釋報酬率不對稱高狹峰、波動度微笑、波動度叢聚、長記憶性的性質。Merton (1976)認為不尋常的訊息來臨會影響股價不連續跳躍，因此發展B-S模型加入不連續跳躍風險項的跳躍擴散模型，該模型可同時描述報酬率不對稱高狹峰和波動度微笑兩性質。Charles, Fuh and Lin (2011)加以考慮市場狀態提出狀態轉換跳躍模型，除了保留跳躍擴散模型可描述報酬率不對稱高狹峰和波動度微笑，更可以敘述報酬率的波動度叢聚和長記憶性。本文進一步拓展狀態轉換跳躍模型，考慮不連續跳躍風險項的帄均數與市場狀態相關，提出狀態轉換跳躍相關模型。並以道瓊工業指數與S&P 500指數1999年至2010年股價指數資料，採用EM和SEM分別估計參數與估計參數共變異數矩陣。使用概似比檢定結果顯示狀態轉換跳躍相關模型比狀態轉換跳躍獨立模型更適合描述股價指數報酬率。並驗證狀態轉換跳躍相關模型也可同時描述報酬率不對稱高狹峰、波動度微笑、波動度叢聚、長記憶性。最後利用Esscher轉換法計算股價指數選擇權定價公式，以敏感度分析模型參數對於定價結果的影響，並且市場驗證顯示狀態轉換跳躍相關模型會有最小的定價誤差。 / Black and Scholes (1973) proposed B-S model to fit asset return, but B-S model can’t effectively explain some asset return properties, such as leptokurtic, volatility smile, volatility clustering and long memory. Merton (1976) develop jump diffusion model (JDM) that consider abnormal information of market will affect the stock price, and this model can explain leptokurtic and volatility smile of asset return at the same time. Charles, Fuh and Lin (2011) extended the JDM and proposed regime-switching jump independent model (RSJIM) that consider jump rate is related to market states. RSJIM not only retains JDM properties but describes volatility clustering and long memory. In this paper, we extend RSJIM to regime-switching jump dependent model (RSJDM) which consider jump size and jump rate are both related to market states. We use EM and SEM algorithm to estimate parameters and covariance matrix, and use LR test to compare RSJIM and RSJDM. By using 1999 to 2010 Dow-Jones industrial average index and S&P 500 index as empirical evidence, RSJDM can explain index return properties said before. Finally, we calculate index option price formulation by Esscher transformation and do sensitivity analysis and market validation which give the smallest error of option prices by RSJDM.

退休金是退休人未來生活的依靠，確保在退休後能得到適足的退休給付，政府在退休金上實施保證收益制度，此制度為最低保證利率與投資報酬率連結。本文探討退休金給付標準為確定提撥制，當退休金的投資報酬率是根據其連結之股價指數的表現來計算時，股價指數報酬率的模型假設為馬可夫調控跳躍過程模型，考慮市場狀態與布朗運動項、跳躍項的跳躍頻率相關，即為Elliot et al. (2007) 的模型特例。使用1999年至2012年的道瓊工業指數與S&P 500指數的股價指數對數報酬率作為研究資料，採用EM演算法估計參數及SEM演算法估計參數共變異數矩陣。透過概似比檢定說明馬可夫調控跳躍過程模型比狀態轉換模型、跳躍風險下狀態轉換模型更適合描述股價指數報酬率變動情形，也驗證馬可夫調控跳躍過程模型具有描述報酬率不對稱、高狹峰及波動叢聚的特性。最後，假設最低保證利率為固定下，利用Esscher轉換法計算不同模型下型I保證之確定提撥制退休金的評價公式，從公式中可看出受雇人提領的退休金價值可分為政府補助與個人帳戶擁有之退休金兩部分。以執行敏感度分析探討估計參數對於馬可夫調控跳躍過程模型評價公式的影響，而型II保證之確定提撥制退休金的價值則以蒙地卡羅法模擬並探討其敏感度分析結果。 / Pension plan make people a guarantee life in their retirement. In order to ensure the appropriate amount of pension plan, government guarantees associated with pension plan which ties minimum rate of return guarantees and underlying asset rate of return. In this paper, we discussed the pension plan with defined contribution (DC). When the return of asset is based on the stock indices, the return model was set on the assumption that markov-modulated jump diffusion model (MMJDM) could the Brownian motion term and jump rate be both related to market states. This model is the specific case of Elliot et al. (2007) offering. The sample observations is Dow-Jones industrial average and S&P 500 index from 1999 to 2012 by logarithm return of the stock indices. We estimated the parameters by the Expectation-Maximization (EM) algorithm and calculated the covariance matrix of the estimates by supplemented EM (SEM) algorithm. Through the likelihood ratio test (LRT), the data fitted the MMJDM better than other models. The empirical evidence indicated that the MMJDM could describe the asset return for asymmetric, leptokurtic, volatility clustering particularly. Finally, we derived different model's valuation formula for DC pension plan with type-I guarantee by Esscher transformation under rate of return guarantees is constant. From the formula, the value of the pension plan could divide into two segment: government supplement and employees deposit made pension to their personal bank account. And then, we done sensitivity analysis through the MMJDM valuation formula. We used Monte Carlo simulations to evaluate the valuation of DC pension plan with type-II guarantee and discussed it from sensitivity analysis.

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.

本文使用Esscher轉換法推導狀態轉換模型、跳躍獨立風險下狀狀態轉換模型及跳躍相關風險下狀態轉換模型的選擇權定價公式。藉由1999年至2011年道瓊工業指數真實市場資料使用EM演算法估計模型參數並使用概似比檢定得到跳躍相關風險下狀態轉換模型最適合描述報酬率資料。接著進行敏感度分析得知，高波動狀態的機率、報酬率的整體波動度及跳躍頻率三者與買權呈現正相關。最後由市場驗證可知，跳躍相關風險下狀態轉換模型在價平及價外的定價誤差皆是最小，在價平的定價誤差則略高於跳躍獨立風險下狀態轉換模型。 / In this paper, we derive regime switching model, regime switching model with independent jump and regime switching model with dependent jump by Esscher transformation. We use the data from 1999 to 2011 Dow-Jones industrial average index market price to estimate the parameter by EM algorithm. Then we use likelihood ratio test to obtain that regime switching model with dependent jump is the best model to depict return data. Moreover, we do sensitivity analysis and find the result that the probability of the higher volatility state , the overall volatility of rate of return , and the jump frequency are positively correlated with call option value. Finally, we enhance the empirical value of regime switching model with dependent jump by means of calculating the price error.

Chen, Chang, Wen and Lin (2013)提出馬可夫調控跳躍過程模型(MMJDM)描述股價指數報酬率，布朗運動項、跳躍項之頻率與市場狀態有關。然而，利率並非常數，本論文以狀態轉換模型配適零息債劵之動態過程，提出狀態轉換下的利率與具跳躍風險的股票報酬之二維模型(MMJDMSI)，並以1999年至2013年的道瓊工業指數與S&P 500指數和同期間之一年期美國國庫劵價格為實證資料，採用EM演算法取得參數估計值。經由概似比檢定結果顯示無論道瓊工業指數還是S&P 500指數，狀態轉換下利率與跳躍風險之股票報酬二維模型更適合描述報酬率。接著，利用Esscher轉換法推導出各模型下的股價指數之歐式買權定價公式，再對MMJDMSI模型進行敏感度分析以評估模型參數發生變動時對於定價公式的影響。最後，以實證資料對各模型進行模型校準及計算隱含波動度，結果顯示MMJDMSI在價內及價外時定價誤差為最小或次小，且此模型亦能呈現出波動度微笑曲線之現象。 / To model asset return, Chen, Chang, Wen and Lin (2013) proposed Markov-Modulated Jump Diffusion Model (MMJDM) assuming that the Brownian motion term and jump frequency are all related to market states. In fact, the interest rate is not constant, Regime-Switching Model is taken to fit the process of the zero-coupon bond price, and a bivariate model for interest rate and stock index return with regime-switching and dependent jump risks (MMJDMSI) is proposed. The empirical data are Dow Jones Industrial Average and S&P 500 Index from 1999 to 2013, together with US 1-Year Treasury Bond over the same period. Model parameters are estimated by the Expectation-Maximization (EM) algorithm. The likelihood ratio test (LRT) is performed to compare nested models, and MMJDMSI is better than the others. Then, European call option pricing formula under each model is derived via Esscher transformation, and sensitivity analysis is conducted to evaluate changes resulted from different parameter values under the MMJDMSI pricing formula. Finally, model calibrations are performed and implied volatilities are computed under each model empirically. In cases of in-the-money and out-the-money, MMJDMSI has either the smallest or the second smallest pricing error. Also, the implied volatilities from MMJDMSI display a volatility smile curve.