本研究顯著地發展時間轉換Lévy過程在最適投資組合的運用性。在連續Lévy過程模型設定下，槓桿效果直接地產生跨期波動度不對稱避險需求，而波動度回饋效果則透過槓桿效果間接地發生影響。另外，關於無窮跳躍Lévy過程模型設定部分，槓桿效果仍扮演重要的影響角色，而波動度回饋效果僅在短期投資決策中發生作用。最後，在本研究所提出之一般化隨機波動度不對稱資產報酬動態模型下，得出在無窮跳躍的資產動態模型設定下，擴散項仍為重要的決定項。 / This study significantly extends the applicability of time-changed Lévy processes to the portfolio optimization. The leverage effect directly induces the intertemporal asymmetric volatility hedging demand, while the volatility feedback effect exerts a minor influence via the leverage effect under the pure-continuous time-changed Lévy process. Furthermore, the leverage effect still plays a major role while the volatility feedback effect just works over the short-term investment horizon under the infinite-jump Lévy process. Based on the proposed general stochastic asymmetric volatility asset return model, we conclude that the diffusion term is an essential determinant of financial modeling for index dynamics given infinite-activity jump structure.

由於一般投資人在買進或賣出選擇權時，並不會同時買進多個履約價的選擇權，故會造成選擇權隱含波動度的微笑曲線出現有不連續的現象。本文嘗試運用台指選擇權建構一個日內的隱含波動度微笑曲線交易策略，利用曲線配適的方法來捕捉瞬間時點下隱含波動度曲線發生不連續的現象，雖然最後出來的損益並不如預期但還是驗證了台指選擇權市場有多次這種不連續的機會且價格失衡的狀態會回歸正常。 / Option’s implied volatility smile curves discontinuous phenomenon exists when general investors buy or sell options, they won’t buy in every strike’s options. This paper attempts to use Taiwan Index Options (trading code: TXO) to construct a trading strategy based on the implied volatility. We use curve fitting method to capture volatility smile curve’s instant discontinuous. Although we find out that the strategy won’t make a profit, there were several times when TXO market’s implied volatility smile curves were discontinuous, and the market option price will eventually go back to the theoretical price.

一般研究外匯波動均以現貨的波動為主，但理論上衍生性金融商品由於成本低、市場限制較少，並且隱含波動度為「事前」波動度，隱含「預期」的意涵，因此衍生性金融商品的波動應該比現貨更能反映市場的資訊，市場資訊透過市場參與者的投資策略反映在市場，將會造成市場上的波動，且影響是不對稱和具有反轉現象的，所謂的「反轉」是指當價格變動幅度很大時，負向的價格變動比正向對波動度的影響要大，但當價格變動很小時，影響方向便會出現反轉 (reversal)，即小幅度的正向價格變動比負向價格變動對波動度的影響要大。 本研究以英磅、歐元、日圓及瑞士法郎四種外匯選擇權作為研究標的，探討外匯波動是否具有不對稱效果以及不對稱效果是否因價格變動幅度而有反轉現象，並且發展類似double-threshold GARCH模型的VS-VOLUME-GARCH模型，在控制交易量變數後，檢視不對稱及反轉的現象是否有所改變。實證結果發現市場訊息對英磅、歐元、日圓與瑞士法郎波動具有不對稱效果與反轉，但是方向與影響程度剛好與一般股市波動相反，即小幅度正向價格變動對波動度的影響較負向小，大幅度的正向價格變動對波動度的影響較負向大，其次，交易量的確可以用來解釋波動度不對稱及反轉但是僅能解釋部份原因，並且由實證結果可知交易量的確可以減輕波動度不對稱及反轉的程度，另外，實證結果也指出交易量只是造成不對稱及反轉效果的一個原因，除了交易量之外應該還存在其它重要因素。

金融資產報酬的波動一直都是財務市場熱衷研究的主題, 由於真正報酬的波動無法確知, 造成無法判斷何者為衡量報酬波動最佳的模型, 進而導致預測未來報酬的風險增加。因此, 本文利用靴帶抽樣法(Bootstrap)反覆抽樣的估計方式, 建立報酬與報酬波動的預測區間來衡量由估計模型參數產生的不確定性, 希望能藉此更瞭解資產報酬的變化以降低投資風險。鑒於目前衡量報酬波動的模型眾多, 文中將採用文獻上普遍最能掌握金融資產報酬波動現象的GARCH模型, 作為衡量報酬波動的方法, 再以靴帶抽樣方法估計其報酬與報酬波動的預測區間, 透過有限樣本的模擬將估計模型參數不確定性的靴帶抽樣方法與其他方法比較, 證明靴帶抽樣法最能適當的捕捉報酬波動真實的情況。最後, 由台灣上市股票市場中選取四支不同類股的各股以日報酬進行實證研究, 結果顯示各股的日報酬都具有波動變異的現象, 進一步估計樣本外不同範圍的波動預測區間, 發現利用估計模型參數不確定性的靴帶抽樣方法可以適當地涵蓋波動的變化。

這次的金融海嘯重擊結構型商品市場，結構型商品市場目前正處於過度時期，商品設計勢必朝向簡單化及透明化，故本文選擇市場上常見的利率及股權商品，期望從投資人角度出發，讓投資人對於商品有更進一步的了解。在利率商品中，本文挑選累計區間贖回連動債，該債券連結標的為十年期交換利率，其計息方式為觀察每天交換利率是否落入區間，故本文採用在即期測度下之BGM模型來模擬每天之遠期利率。波動度結構之估計是採用參數化波動度函數來模擬市場上波動度之駝峰現象；相關係數矩陣之校準則是採用遞迴投影。在股權商品中，本文選擇元大證券所發行之一年期機不可失商品，該商品為連結台灣電子產業中之四檔股票，其報酬型態對投資人來說看似優渥，然而在模擬過程中發現其實不然，讓本文了解到如果不對商品有深入之探討，往往也遭受到表面之矇騙。

自從槓桿型指數基金於各股票市場發行後，各國主管機關皆紛紛發出聲明表示，該商品並不適合長期持有；因此，該類型投資商品的公開說明書皆會註明不宜長期投資。然而，本研究實證結果發現，持有期間長短並非主要風險來源。雖然，如大家所知，槓桿型指數基金多是以「日」為單位追蹤指數，而導致複利效果 (Compounding Effect) 使基金長期報酬與槓桿倍數不同。 根據算出上述的報酬差異(Return Difference)可以發現不論是正向2倍或是反向1倍皆與台灣50報酬率的標準差有統計上顯著關係。反向1倍皆與台灣50報酬率的標準差有顯著負相關；反之，正向2倍與台灣50報酬率的標準差有顯著正向相關。然而，從已實現乘數(Realized Multiple)的分佈中可發現，不合理值並不隨投資期間越長而越多。意即儘管投資期間越長，並不一定會導致複利效果越大，而與目標槓桿倍數脫節。再者，隨著投資期間越長，波動度(volatility)的對於報酬差異的解釋力越強；因此，若想長期投資槓桿型指數資金，預測標的波動度的能力更顯為重要。 / When we browse the reports about the inverse and leveraged ETF, most of them emphasize that the LETF is not appropriate to long-term investors. However, in this research, we attempt to demonstrate the main factor of the performance of the leveraged and inverse ETF is not how long the LETF we hold, but the volatility of the underlying index or ETF. Observing the empirical test, no matter how long the investment horizon is, the coefficient of the variance of the Taiwan 50 is statistically significant both in the Taiwan 50 Bear -1X and the Taiwan 50 Bull. However, its effect on the Bear -1X is opposite to that on the Bull 2X. First, the relationship between the volatility and the return difference of Taiwan 50 and the Bear -1X is negative. In contrast, the relationship between the volatility and the return difference of Taiwan 50 and the Bull 2X is positive. However, in accordance with the distribution of the realized multiples, the frequency of either the Bear -1X or Bull 2X was not more and more when the holding period is longer. As a result, our research show the variance has a significant effect on both, no matter how long investors hold. If the volatility is moderate, the return difference may be close to zero; then the LETFs would be a convenient way to investors who desire to magnify the market return. Moreover, due to the increasing explanatory power of the volatility, we may make a further inference that whether the compounding effect is positive or negative depends on the volatility, especially within longer holding period. Therefore, without the great ability to forecast the variance, the LETFs are not recommended to the long-term investors.

由於隱含波動度具有前瞻性的特質，以往有許多學者探討隱含波動度與標的股價指數間的關聯性，但多利用線性模型。而本研究與其他文獻不同之處在於，本文利用馬可夫轉換模型分析隱含波動度VIX和VXN(VIX為S&P500指數的隱含波動度，而VXN為Nasdaq-100指數的隱含波動度)，馬可夫轉換模型為非線性模型，可捕捉不同區間轉換與不規則跳動，隱含波動度在特殊金融事件發生時會突然竄高，馬可夫轉換模型相對於一般線性模型更可捕捉此跳動，並將隱含波動度分為兩個區間。 　　經由多變量迴歸分析後，本研究也發現隱含波動度的變動以及技術指標的趨勢（偏離五天移動平均值）皆會影響標的股價指數的報酬，但隱含波動度變動對於股價指數報酬的影響高於技術指標，且不同區間存在不同現象。 / Implied volatility indices are forward-looking, and lots of researches discuss the relationship between the implied volatility and underlying stock market returns. Dif-ferent from other studies, we use Marcov switching model to examine the implied volatility indices: S&P 500 volatility index (VIX) and NASDAQ-100 volatility index (VXN), then we separately exploit the different regime behavior about the relationship between implied volatility change, technical indicators and stock market returns. As a result, S&P 500 index and NASDAQ-100 index respond in opposite direc-tions to positive and negative S&P 500 volatility index (VIX) and NASDAQ volatility index (VXN) changes, where technical indicators do not have that much influence on stock market returns. In addition, the impact of implied volatility change, technical indicators to stock market returns indeed depend on different regimes.

本篇論文的目的在驗證台股期貨報酬率與其波動度之間的相對應關係是由槓桿效果或是波動度回饋效果之因果關係所驅動，並且分別以日資料以及高頻率日內資料進行實證。實證結果發現在高頻率日內資料的應用下，能夠比日資料揭露出更詳細的波動度資訊，將報酬率與波動度間的對應關係描繪得更加明瞭。且在大多數資料期間內，同期下，台股期貨報酬率與其波動度之間會呈現負相關性，而負相關的程度會隨著報酬率遞延期數越長而逐漸遞減，因此可以發現報酬率與其波動度間呈現一個經由報酬率進而影響波動度的對應關係，與槓桿效果的因果關係雷同。最後，本文亦採用了常見的波動度預測模型，歷史模擬法、GARCH(1,1)模型、EGARCH(1,1)模型以及GJR-GARCH(1,1)模型，觀察這些波動度模型所預測出之波動度是否含有上述驗證的資訊意涵，並比較各波動度模型的預測能力，結果發現GJR-GARCH模型於樣本外期間所預測之波動度，其與報酬率之間不但具有槓桿效果的因果關係，且預測能力亦於四個波動度模型中表現最佳。

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.

本文考量波動度不對稱效果(Volatility Asymmetric Effect)與均數回歸(Mean Reverting)兩個特性，並考量台股市場特性，嘗試建立一個適合台灣市場的波動度交易策略。利用GARCH(1,1)波動度與VIX指標建構第一個交易訊號，並建立當日沖銷部位。以賺取日內行情為出發點，利用時間序列模型捕捉波動度的高估或低估且搭配純跨式(Pure Straddle)策略或根據Delta調整後的跨式(Adjusted Straddle)策略。第二個交易訊號則是利用市場敏感指標，觀察外資與自營商在交易部位與未平倉部位的變化，找出對於波動度的影響。建立由選擇權與期貨組成的Delta-Hedged部位，藉由觀察市場上主力籌碼的變化，動態調整部位契約，尋找波段之間的獲利機會。 實証部分以期交所公布的每日交易資料與VIX日資料，利用2007至2008兩年的歷史資料，估計參數與測試交易訊號。樣本外期間為2009年1月開始至3月結束共55個交易日。考量交易成本後，兩個不同型態的交易訊號，仍然能夠藉由本研究的策略，獲得正的報酬。本文認為台灣為一個淺碟市場，過度反應資訊的特性，讓波動度策略出現獲利的機會。藉由這個波動度交易系統的研究，除了讓資金豐沛的機構投資人使用外，也能夠讓一般投資大眾建立自己的波動度交易策略 關鍵字:波動度交易，選擇權交易策略，GARCH(1,1)，VIX，市場情緒指標 / Trying to apply a preliminary study of volatility trading strategies in Taiwan derivative market is the topic of this dissertation. Capturing the market movement or even the dynamic of underlying asset is a Pandora’s Box for academic researchers and industry participants. Mean-reverting and asymmetrical effects are the two special characteristics of volatility for us to design our trading system according to the previous empirical studies. In our study, we use different type of volatility signal to capture the trading opportunities. Use the new released information form TAIFEX including VIX and Position Structure of Institutional Traders to design our signal. We apply the idea to use pure option position and delta-hedged position as our trading tools in this volatility trading system and look for the opportunities between realized volatility and implied volatility. An over-reaction may rises the uncertainty and also lead the market volatility change coherently. We use history data from 2007 to 2008 test our trading signal and parameters. The out sample period is from 2009 January to 2009 March which has 55 trading days to simulate our strategies. In the end, we see a positive result in both trading signals which earns positive return after considering the trading cost. Key words: Volatility Trading, Market Sentiment Indices, Option Strategies, VIX, GARCH(1,1)