本論文研究如何由觀測的選擇權市場價格還原風險中立機率測度（等價平賭測度）。首先建構選擇權投資組合的套利模型，其中假設選擇權為單期，到期日時的狀態為離散點且個數有限，並且對應同一標的資產且不同履約價格。若市場不存在套利機會時，可使用拉格朗日乘數法則將選擇權套利模型導出拉格朗日乘子的可行性問題。將可行性問題作為限制式重新建構線性規劃模型以還原風險中立機率測度，並且利用此風險中立機率測度評價選擇權的公正價格。最後，我們以台指選擇權（TXO）為例，驗證此模型的評價能力。 / This thesis investigates how to recover the risk-neutral probability (equivalent martingale measure) from observed market prices of options. It starts with building an arbitrage model of options portfolio in which the options are assumed to be in one-period time, finite discrete-states, and corresponding to the same underlying asset with different strike prices. If there is no arbitrage opportunity in the market, we can use Lagrangian multiplier method to obtain a Lagrangian multiplier feasibility problem from the arbitrage model. We employ the feasibility problem as the constraints to construct a linear programming model to recover the risk-neutral probability, and utilize this risk-neutral probability to evaluate the fair price of options. Finally, we take TXO as an example to verify the pricing ability of this model.

本研究探討VIX 期貨價格所隱含的資訊對於S&P 500 指數波動度預測的解釋力。過去許多文獻主要運用線性預測模型探討歷史波動度、隱含波動度和風險中立偏態對於波動度預測的資訊內涵。然而過去研究顯示，波動度具有長期記憶與非線性的特性，因此本文主要研究非線性預測模型對於波動度預測的有效性。本篇論文特別著重在不同市場狀態下(高波動與低波動)的實現波動度及隱含波動度異質自我迴歸模型(HAR-RV-IV model)。因此，本研究以考慮馬可夫狀態轉化下的異質自我迴歸模型(MRS-HAR model)進行實證分析。 本研究主要目的有以下三點: (1) 以VIX期貨價格所隱含的資訊提升S&P 500波動度預測的準確性。(2) 結合風險中立偏態與VIX期貨的資訊內涵，進一步提升S&P 500 波動度預測的準確性。(3) 考慮狀態轉換後的波動度預測模型是否優於過去文獻的線性迴歸模型。 本研究實證結果發現: (1) 相對於過去的實現波動度及隱含波動度，VIX 期貨可以提供對於預測未來波動度的額外資訊。 (2) 與其他模型比較，加入風險中立偏態和VIX 期貨萃取出的隱含波動度之波動度預測模型，只顯著提高預測未來一天波動度的準確性。 (3) 考慮狀態轉換後的波動度預測模型優於線性迴歸模型。 / This paper explores whether the information implied from VIX futures prices has incremental explanatory power for future volatility in the S&P 500 index. Most of prior studies adopt linear forecasting models to investigate the usefulness of historical volatility, implied volatility and risk-neutral skewness for volatility forecasting. However, previous literatures find out the long-memory and nonlinear property in volatility. Therefore, this study focuses on the nonlinear forecasting models to examine the effectiveness for volatility forecasting. In particular, we concentrate on Heterogeneous Autoregressive model of Realized Volatility and Implied Volatility (HAR-RV-IV) under different market conditions (i.e., high and low volatility state). This study has three main goals: First, to investigate whether the information extracted from VIX futures prices could improve the accuracy for future volatility forecasting. Second, combining the information content of risk-neutral skewness and VIX futures to enhance the predictive power for future volatility forecasting. Last, to explore whether the nonlinear models are superior to the linear models. This study finds that VIX futures prices contain additional information for future volatility, relative to past realized volatilities and implied volatility. Out-of-sample analysis confirms that VIX futures improves significantly the accuracy for future volatility forecasting. However, the improvement in the accuracy of volatility forecasts is significant only at daily forecast horizon after incorporating the information of risk-neutral skewness and VIX futures prices into the volatility forecasting model. Last, the volatility forecasting models are superior after taking the regime-switching into account.

Brigo與Mercurio提出了三種新的資產價格過程，分別是位移CEV過程、位移對數常態過程與混合對數常態過程。在這三種過程中，資產價格的波動度不再是一個固定的常數，而是時間與資產價格的明確函數。而由這三種過程所推導出來的歐式選擇權評價公式，將會導致隱含波動度曲線呈現傾斜曲線或是微笑曲線，且提供了參數讓我們能夠配適市場的波動度結構。本文利用台指買權來實證Brigo與Mercurio所提出的三種歐式選擇權評價公式，我們發現校準結果以混合對數常態過程優於位移CEV過程，而位移CEV過程則稍優於位移對數常態過程。因此，在實務校準時，我們建議以混合對數常態過程為台指買權的評價模型，以達到較佳的校準結果。 / Brigo and Mercurio proposed three types of asset-price dynamics which are shifted-CEV process, shifted-lognormal process and mixture-of-lognormals process respectively. In these three processes, the volatility of the asset price is no more a constant but a deterministic function of time and asset price. The European option pricing formulas derived from these three processes lead respectively to skew and smile in the term structure of implied volatilities. Also, the pricing formula provides several parameters for fitting the market volatility term structure. The thesis applies Taiwan’s call option to verifying these three pricing formulas proposed by Brigo and Mercurio. We find that the calibration result of mixture-of-lognormals process is better than the result of shifted-CEV process and the calibration result of shifted-CEV process is a little better than the result of shifted-lognormal process. Therefore, we recommend applying the pricing formula derived from mixture-of-lognormals process to getting a better calibration.

台灣特殊的兩黨對立政治環境及幾乎每年都會有的固定選舉，使得政治的不確定性深深的影響著國內的投資環境及投資人心態。本研究便是要探討，2002/1/1~2006/1/16 研究期間台灣的投資人在選舉前後的投資行為，是否真如大家所預期的，會受到台灣選舉事件的影響。 本研究首先利用適當的機率密度函數模型及選擇權市場資訊來導出隱含的風險中立密度值。再利用這些風險中立密度值，求出各個選舉事件相對應的機率分配圖形，並透過其機率分配圖形及波動率指數等統計值於投票日前後的變化來觀察某一選舉事件前後投資者的反應。 研究結果發現：1. 選舉事件的發生確實會影響投資者的心理，且投資者會透過選擇權市場有效率的反應預期的未來股價指數分佈情況。2. 越大型、越具爭議且全國性的選舉結果，其選舉期間機率分配圖形及波動率指數具有較高的波動性。3. 一般而言，選舉過後市場不確定因素降低，將使投資者對於股市的預期較為一致和樂觀。而若這個選舉結果使投資者感到意外，因而增加了市場的不確定性，則選後機率分配圖形及波動率指數的改變反而會更為明顯。4. 在此研究下對數常態混合法比傳統的 Black-Scholes 方法產生較低的誤差值，因此就實證的分析上能提供更好的配適。 / This research examines the behavior of investors during election periods from January 1st 2002 to January 6th 2006 in Taiwan. The research includes a few steps. First, we adopted a proper probability density function composed of stock index options data to construct the implied distribution. Then, when changing the whole shape of the risk-neutral implied distribution, the volatility indexes, and the statistics of the implied distribution, we observed investors' response around a specific election event. According to the empirical results, we found that: 1. An election event would influence investors’ behavior, and investors tend to reflect their expectation of future stock index in the option market in an efficient way. 2. The result of a large-scale and more disputed nationwide election will cause a higher fluctuation in both the implied distribution and the volatility index. 3. In general, the factor resulting from investors’ uncertainty of the market is likely to reduce after the election, which makes investors’ relatively unanimous and optimistic expectation of the stock market. However, if this election result surprises investors, their uncertainty of the market will increase, and thus the changes of the implied distribution and the volatility index become quite obvious. 4. The in-sample performance of the lognormal mixtures method employed in the research is considerably better than that of the traditional Black-Scholes model by having a lower root mean squared error.

波動度對於金融市場影響甚多，同時為金融資產定價的重要參數以及市場穩 定度的衡量指標，尤其在金融危機發生時，波動度指數的驟升反映資產價格震盪。 本篇論文嘗試捕捉S&P500 指數實現波動度與VIX變動率未來之動態，並將VIX、 VIX 選擇權與VVIX 納入預測模型中，探討其資訊內涵。透過研究S&P500 指數 實現波動度，能夠預測S&P500 指數未來之波動度與報酬，除了能夠觀察市場變 動，亦能使未來選擇權定價更為準確；而藉由模型預測VIX，能夠藉由VIX 選 擇權或VIX 期貨，提供避險或投資之依據。文章採用2006 年至2011 年之S&P500 指數、VIX、VIX 選擇權與VVIX 資料。 在 S&P500 指數之實現波動度預測當中，本篇論文的模型改良自先前文獻， 結合實現波動度、隱含波動度與S&P500 指數選擇權之風險中立偏態，所構成之 異質自我回歸模型(HAR-RV-IV-SK model)。論文額外加入VIX 變動率以及VIX指數選擇權之風險中立偏態作為模型因子，預測未來S&P500 指數實現波動度。 研究結果表示，加入VIX 變動率作為S&P500 指數實現波動度預測模型變數後， 可增加S&P500 指數實現波動度預測模型之準確性。 在 VIX 變動率預測模型之中，論文採用動態轉換模型，作為高低波動度之 下，區分預測模型的方法。以VIX 過去的變動率、VIX 選擇權之風險中立動差 以及VIX 之波動度指數(VVIX)作為變數，預測未來VIX 變動率。結果顯示動態 轉換模型能夠提升VIX 預測模型的解釋能力，並且在動態轉換模型下，VVIX 與 VIX 選擇權之風險中立動差，對於VIX 預測具有相當之資訊隱涵於其中。 / This paper tries to capture the future dynamic of S&P 500 index realized volatility and VIX. We add the VIX change rate and the risk neutral skewness of VIX options into the Heterogeneous Autoregressive model of Realized Volatility, Implied Volatility and Skewness (HAR-RV-IV-SK) model to forecast the S&P 500 realized volatility. Also, this paper uses the regime switching model and joins the VIX, risk neutral moments of VIX options and VVIX variables to raise the explanatory ability in the VIX forecasting. The result shows that the VIX change rate has additional information on the S&P 500 realized volatility. By using the regime switching model, the VVIX and the risk neutral moments of VIX options variables have information contents in VIX forecasting. These models can be used for hedging or investment purposes.

本論文提出線性規劃的方法以還原隱藏於選擇權市場價格中的風險中立機率測度，並利用該機率測度計算選擇權的合理價格。模型中假設選擇權對應同一標的資產與到期日，資產價格於到期日的狀態為離散點且個數有限，當市場不具任何套利機會時，以極小化市場價格與合理價格之離差總和作為挑選風險中立機率測度的準則。最後，以臺指選擇權(TXO)的交易資料做為實證對象。實證中發現，加入平滑限制式與離差權重之線性規劃模型在評價歐式選擇權合理價格的效能最為優異。 / The thesis proposes a liner programming to recover the risk-neutral probability distribution of an underlying asset price from its associated market option prices, and we evaluate the fair prices of options via the resulting risk-neutral probability distribution. Assume that we face a series of European options with different exercise prices on the same maturity and underlying asset in this linear programming model. The criterion of choosing a risk-neutral probability distribution is minimizing the sum of total deviations subject to requiring that the fair prices of options are consistent with observed market option prices. Finally, we take the trading data of TXO as an empirical study. The empirical study indicates that the model with smooth constraints and weighted deviations has the best performance in pricing the rational price of European options.

我國金融市場積極發展多元化與國際化，將由台灣期貨交易所推出臺指選擇權，建立選擇權市場以完備金融市場。由於選擇權市場掛牌交易契約眾多與選擇權的交易特性，因此有賴造市者發揮提供流動性、風險移轉、價格效率性及價格資訊揭露等經濟功能。本研究以造市者之功能性逐項比較我國造市者制度與EUREX、SEHK、LIFFE等各大交易所造市者制度之差異，用以發現我國首度引進的造市者制度對於選擇權市場發展的影響。 本研究為瞭解選擇權造市者營運的知識與技術，整理造市者作業循環的流程，針對造市者業務進行分工，建立起造市風險分析架構。推導得出造市風險包含造市交易風險與部位管理風險，呈現出市場波性風險、報價及交易策略風險、造市交易之市場機制風險、市場行情變動風險、遇險交易策略風險、部位管理之市場機制風險及財務與作業風險等七大類的造市風險。 本研究亦提出對於我國選擇權市場未來發展方向的關切，以及建議重視造市者利基與市場運作之關係，並提出未來與造市者相關之研究建議。 / Taiwan's financial market has been developing constructively into a globalization and multiplicity market within these years. To create a more completed financial market, Taiwan Futures Exchange (TAIFEX) will issue "the Taiwan stock index option" recently. As the complex and various features of the options trading, the option market will be successful and efficient through the market maker's economic functions, which are adding liquidity, increasing price efficiency, transferring risk and proving price information. This survey tried to compare the different functions of market maker system with TAIFEX, EUREX, SEHK and LEFFE, and to disclose the influence of the new market maker system on Taiwan's option market. This exposure takes great interest in market maker's knowledge and technology. By summarizing and classifying the operation process cycle flow of market maker, this research further organizes an analytic structure in market maker's risks. Including the market making risks and the positions management risks, the discussion demonstrates the market change risk, pricing strategy risk, limitary quoting risk, market volatility risk, hedging risk, mechanism risk, and finance and operation risk. The consequent also furnishes the concerns about the future development of Taiwan option market, emphasizing the importance of the relationship between market makers and market' benefit, and the suggestions to the further research.

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.