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Author: 李秀君
Hsiu-chun Lee
Thesis Title: 關於隨機模型的兩篇文章─離散時間佇列與波動度微笑曲線
Two Essays on Stochastic Modeling: Discrete-Time Queues and Volatility Smiles
Advisor: 繆維中
Daniel Wei-Chung Miao
Committee: 劉代洋
Day-Yang Liu
黃瑞卿
Rachel Juiching Huang
陳俊男
Chun-Nan Chen
陳宏
Hung Chen
周賢榮
Shyan-Rong Chou
黃彥聖
Yen-Sheng Huang
Degree: 博士
Doctor
Department: 管理學院 - 財務金融研究所
Graduate Institute of Finance
Thesis Publication Year: 2013
Graduation Academic Year: 101
Language: 英文
Pages: 98
Keywords (in Chinese): 離散時間佇列離散型自我迴歸過程自我相關函數常態─拉普拉斯複合分配厚尾峰態歐式選擇權隱含波動度
Keywords (in other languages): discrete-time queues, discrete autoregressive processes, autocorrelation function, normal-Laplace mixture distribution, tail-fatness, kurtosis, European options, implied volatility
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  • 本篇論文包含兩個主題,在第一個主題中,我們建立一個離散型二階自我迴歸過程(DAR(2)),相對於一階自我迴歸過程(DAR(1))我們的模型多增加一個參數的設定,就可以討論更多種類的自我相關序列,並且可以完整的捕捉到當衰退率很低或是有較久的記憶性之序列。我們藉由衡量佇列長度的一階動差以及二階動差,來判斷該佇列是否是一個變異數較小而穩定的佇列。在本文給定的假設之下,我們可以求出動差的封閉解以判斷佇列是否穩定,若是去除本文給定的假設,我們也可以找出近似解並且做蒙地卡羅模擬;本文的數值模擬結果,發現在DAR(2)的來源下,當系統繁忙、忽然有大量的封包來臨、或是序列高度自我相關的時候,在DAR(2)的過程下,佇列長度之變異數會比DAR(1)高出30%以上,顯示本文增加的該項參數之影響很重要而不可忽略。
    本篇論文的第二個主題使用常態─拉普拉斯複合分配,建構風險中立測度下有厚尾現象的報酬率分配模型,討論峰態係數對機率分配圖形之曲線形狀的影響,並且配適選擇權的隱含波動度微笑曲線。本文的模型中使用兩個參數,符合配適參數簡約理論,並且有以下兩個特色:(1) 峰態係數可以任意給定,沒有限制範圍,(2) 額外考慮一個參數用以調整分配之形狀,以期更符合市場上真實分配的行為。在本文的分配下建立歐式選擇權的封閉解,並且討論選擇權動態避險的行為,以及找出波動度微笑曲線。本文用迴歸分析說明增加此一參數的使用,可以更充分的解釋隱含波動度彎曲的厚尾行為。最後本文提供實證結果分析,我們的模型針對NASDAQ市場上交易的外匯選擇權,可以較完整地捕捉波動度微笑曲線的形狀。


    This thesis contains two essays on stochastic modeling applications. In the first essay, DAR(2) uses one more parameter in contrast with DAR(1) to provide a much richer pattern in the autocorrelation function and is able to capture slower decay rate and longer memory. To investigate how the additional traffic parameter in DAR(2) influences the queueing performance, this thesis provides an analysis of the discrete-time DAR(2)/D/1 queue. The performance measures concerned are the first two moments of queue size. Under a mild condition, these performance indices are derived in closed-form which allows for efficient computing. An approximate version is also developed to relax the condition and cover more general sources, and both versions serve as a simple tool set for performance evaluation. The numerical examples indicate that the effect from slower decay rate in autocorrelation is not negligible and using the extra parameter is necessary particularly when the queue is heavily loaded with correlated traffic.
    The second essay proposes to use a standardized version of the normal-Laplace mixture distribution for the modeling of tail-fatness in an asset return distribution and for the fitting of volatility smiles implied by option prices. Despite the fact that only two free parameters are used, the proposed mixture distribution allows arbitrarily high kurtosis and uses one shape parameter to adjust the density function within three standard deviations for any specified kurtosis. Mathematical analysis shows that the proposed mixture distribution offers two interesting properties about tail-fatness: (1) arbitrarily high kurtosis can be achieved, and (2) for a given kurtosis, there is an additional parameter affecting the shape of distribution function within three standard deviations. For an asset price model based on this distribution, the closed-form formulas for European option prices are derived, and subsequently the volatility smiles can be easily obtained. A regression analysis is conducted to examine the roles played by both tail-fatness parameters, showing that show that the kurtosis, which is commonly used as an index of tail-fatness, is unable to explain the smiles satisfactorily under the proposed model, because the additional shape parameter also significantly accounts for the deviations revealed in smiles. The effectiveness of the proposed parsimonious model is demonstrated in the practical examples where the model is fitted to the volatility smiles implied by the NASDAQ market traded foreign exchange options.

    Abstract i List of Figures vi List of Tables viii 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Review of implied probability distribution and implied volatility smile . . . 2 1.3 The Studies Conducted in this Thesis . . . . . . . . . . . . . . . . . . . . . 7 1.4 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Performance Analysis of DAR(2) Queues and the Marginal E ect of the Additional Tra c Parameter 10 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Slower Decay Rate in the Autocorrelation Function of DAR(2) . . . . . . . 13 2.3 Performance Analysis of the DAR(2)/D/1 Queue . . . . . . . . . . . . . . 19 2.3.1 The rst moment of queue size . . . . . . . . . . . . . . . . . . . . 21 2.3.2 The second moment of queue size . . . . . . . . . . . . . . . . . . . 24 2.3.3 The approximate formulas when the assumption (2.4) does not hold 27 2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.1 Numerical examples when (2.4) holds . . . . . . . . . . . . . . . . . 30 2.4.2 Numerical examples when (2.4) does not hold . . . . . . . . . . . . 32 2.4.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 Volatility Smiles Modeling under a Standardized Normal-Laplace Mix- ture Distribution 42 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 A Standardized Normal and Laplace Mixture Distribution . . . . . . . . . 48 3.3 Analytical Formula for Options Pricing and Hedging under the SNL Based Asset Price Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.1 Pricing formulas for European call and put options . . . . . . . . . 59 3.3.2 The in uences of tail fatness on option prices and hedging parameters 60 3.3.3 The in uences of and on volatility smiles . . . . . . . . . . . . . 63 3.4 Practical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4 Conclusions 83 4.1 Summaries of the Two Essays . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Potential Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 A Technical Details of DAR(2) 86 A.1 Derivations of the terms in 1 . . . . . . . . . . . . . . . . . . . . . . . . . 86 A.2 Derivations of the terms in 2 . . . . . . . . . . . . . . . . . . . . . . . . . 87 A.3 Derivations of the terms in 3 . . . . . . . . . . . . . . . . . . . . . . . . . 88 B Technical Details of SNL 90 B.1 Derivations of fi(a; b; c; ), gi(a; b; c; ), i = 1; 2 . . . . . . . . . . . . . . . . 90 B.2 The derivatives required for the calculation of hedging parameters . . . . . 92 Bibliography 94

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