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題名 在金融海嘯前中後波動度與跳躍風險在現貨市場與選擇權市場之研究
The implication of volatility and jump risks from spot and option markets before, during and after the recent financial crisis
作者 鍾長恕
Chung, Chang-Shu
貢獻者 林士貴
Lin, Shih-Kuei
鍾長恕
Chung, Chang-Shu
關鍵詞 隨機波動度
跳躍風險
風險溢酬
粒子濾波演算法
共同估計
Stochastic volatility
Jump risk
Risk premiums
Particle-Filtering algorithm
Joint estimation
日期 2018
上傳時間 2-Feb-2018 16:42:02 (UTC+8)
摘要 本文利用隨機波動度模型配合不同的跳躍動態配適S&P500 指數報酬率的變動過程，並試圖解決三個實證問題。第一個問題，平均而言，隨機波動度和報酬率跳躍分別佔了S&P500 指數總報酬率的變異多少比例？而那一個風險對總報酬率的變化影響程度較大？第二個問題，在現貨市場和選擇權市場上，無限跳躍模型的配適程度是否優於有限跳躍模型？第三個問題，投資者在什麼時候會要求較高的風險溢酬？波動度的風險溢酬和跳躍的風險溢酬在金融風暴的前、中、後期或是否會有顯著的變化？對於第一個問題，我們發現絕大部分的報酬率變異都是由隨機波動度所造成的，只有在金融危機爆發初期，跳躍風險造成的報酬率變異才會高於波動度的影響。針對第二個問題，我們採用粒子濾波演算法和期望值最大化演算法，配合動態共同估計，發現「具有雙指數跳躍搭配波動度相關跳躍的隨機波動度模型」和「具有常態逆高斯分佈跳躍過程的隨機波動率模型」對於S&P500 指數報酬率與選擇權有良好的配適能力。最後，對於第三個問題，我們透過風險溢酬時間序列觀察到金融危機爆發後，波動度和跳躍風險溢酬都有大幅增加的趨勢，也就是金融危機之後的平穩期，投資人更容易因為恐慌造成會要求更高的風險溢酬。
In this paper, we attempt to answer three questions: (i) On average, what does the proportion of the stochastic volatility and return jumps account for the total return variations in S&P500 index, respectively? In particular, which one has more influence than the other does on the total return variations? (ii) Is the fitting performance of infinite-activity jump models better than that of finite-activity jump models both in the spot and option markets? (iii) When will investors require significantly higher risk premiums? Specifically, were there significant changes in volatility risk premiums and in jump risk premiums before, during or after the financial crisis? For the first question, we find that most of the return variations are explained by the stochastic volatility. In fact, the return jump accounts for the higher percentage than the stochastic volatility at the beginning of financial crisis. To answer the second question, we adopt the expectation-maximization algorithm with the particle filtering algorithm and dynamic joint estimation to obtain the stochastic volatility model with double-exponential jumps and correlated jumps in volatility (SV-DEJ-VCJ) and the stochastic volatility model with normal inverse Gaussian jumps (SV-NIG) fit S&P500 index returns and options well in different criterions, respectively. Finally, for the third question, we observe that both the volatility and jump
risk premiums significantly increase after the financial crisis periods, that is, the panic in the post-crisis period causes more expected returns.
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描述 碩士
國立政治大學
金融學系
104352033
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0104352033
資料類型 thesis
dc.contributor.advisor 林士貴zh_TW
dc.contributor.advisor Lin, Shih-Kueien_US
dc.contributor.author (Authors) 鍾長恕zh_TW
dc.contributor.author (Authors) Chung, Chang-Shuen_US
dc.creator (作者) 鍾長恕zh_TW
dc.creator (作者) Chung, Chang-Shuen_US
dc.date (日期) 2018en_US
dc.date.accessioned 2-Feb-2018 16:42:02 (UTC+8)-
dc.date.available 2-Feb-2018 16:42:02 (UTC+8)-
dc.date.issued (上傳時間) 2-Feb-2018 16:42:02 (UTC+8)-
dc.identifier (Other Identifiers) G0104352033en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/115781-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 金融學系zh_TW
dc.description (描述) 104352033zh_TW
dc.description.abstract (摘要) 本文利用隨機波動度模型配合不同的跳躍動態配適S&P500 指數報酬率的變動過程，並試圖解決三個實證問題。第一個問題，平均而言，隨機波動度和報酬率跳躍分別佔了S&P500 指數總報酬率的變異多少比例？而那一個風險對總報酬率的變化影響程度較大？第二個問題，在現貨市場和選擇權市場上，無限跳躍模型的配適程度是否優於有限跳躍模型？第三個問題，投資者在什麼時候會要求較高的風險溢酬？波動度的風險溢酬和跳躍的風險溢酬在金融風暴的前、中、後期或是否會有顯著的變化？對於第一個問題，我們發現絕大部分的報酬率變異都是由隨機波動度所造成的，只有在金融危機爆發初期，跳躍風險造成的報酬率變異才會高於波動度的影響。針對第二個問題，我們採用粒子濾波演算法和期望值最大化演算法，配合動態共同估計，發現「具有雙指數跳躍搭配波動度相關跳躍的隨機波動度模型」和「具有常態逆高斯分佈跳躍過程的隨機波動率模型」對於S&P500 指數報酬率與選擇權有良好的配適能力。最後，對於第三個問題，我們透過風險溢酬時間序列觀察到金融危機爆發後，波動度和跳躍風險溢酬都有大幅增加的趨勢，也就是金融危機之後的平穩期，投資人更容易因為恐慌造成會要求更高的風險溢酬。zh_TW
dc.description.abstract (摘要) In this paper, we attempt to answer three questions: (i) On average, what does the proportion of the stochastic volatility and return jumps account for the total return variations in S&P500 index, respectively? In particular, which one has more influence than the other does on the total return variations? (ii) Is the fitting performance of infinite-activity jump models better than that of finite-activity jump models both in the spot and option markets? (iii) When will investors require significantly higher risk premiums? Specifically, were there significant changes in volatility risk premiums and in jump risk premiums before, during or after the financial crisis? For the first question, we find that most of the return variations are explained by the stochastic volatility. In fact, the return jump accounts for the higher percentage than the stochastic volatility at the beginning of financial crisis. To answer the second question, we adopt the expectation-maximization algorithm with the particle filtering algorithm and dynamic joint estimation to obtain the stochastic volatility model with double-exponential jumps and correlated jumps in volatility (SV-DEJ-VCJ) and the stochastic volatility model with normal inverse Gaussian jumps (SV-NIG) fit S&P500 index returns and options well in different criterions, respectively. Finally, for the third question, we observe that both the volatility and jump
risk premiums significantly increase after the financial crisis periods, that is, the panic in the post-crisis period causes more expected returns.		en_US
dc.description.tableofcontents 1 Introduction . . .1
2 Literature Review . . .7
2.1 The Background of Research Issue . . . . . . 7
2.2 The Stochastic Volatility and Jump Diffusion Processes . .. . . . 9
3 The Models . . . . 13
3.1 Stochastic Volatility Model . . . . . . . . . . . 13
3.2 The Characteristic Exponent of Levy Jump Processes . 15
3.3 Stochastic Volatility Model with Merton Jumps . . . 19
3.4 Stochastic Volatility Model with Independent Merton Jumps . . . . . . . 22
3.5 Stochastic Volatility Model with Correlated Merton Jumps . . . . . . . . 25
3.6 Stochastic Volatility Model with Double-Exponential Jumps . . . . . . . 28
3.7 Stochastic Volatility Model with Independent Double-Exponential Jumps 32
3.8 Stochastic Volatility Model with Correlated Double-Exponential Jumps . 36
3.9 Stochastic Volatility Model with Variance-Gamma Jumps . . . . . . . . . 40
3.10 Stochastic Volatility Model with Normal Inverse Gaussian Jumps . . . . 42
4 The Risk-Neutral Dynamics and Characteristic Functions 45
4.1 The Risk-Neutral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.1 Stochastic Volatility Model . . . . . . . . . . . . . . . . . . . . . . 46
4.1.2 Stochastic Volatility Model with Merton Jumps . . . . . . . . . . 46
4.1.3 Stochastic Volatility Model with Independent Merton Jumps . . . 47
4.1.4 Stochastic Volatility Model with Correlated Merton Jumps . . . . 47
4.1.5 Stochastic Volatility Model with Double-Exponential Jumps . . . 48
4.1.6 Stochastic Volatility Model with Independent Double-Exponential
Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.7 Stochastic Volatility Model with Correlated Double-Exponential
Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.8 Stochastic Volatility Model with Variance-Gamma Jumps . . . . . 51
4.1.9 Stochastic Volatility Model with Normal Inverse Gaussian Jumps 52
4.2 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.1 Stochastic Volatility Model . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 Stochastic Volatility Model with Merton Jumps . . . . . . . . . . 54
4.2.3 Stochastic Volatility Model with Independent Merton Jumps . . . 55
4.2.4 Stochastic Volatility Model with Correlated Merton Jumps . . . . 56
4.2.5 Stochastic Volatility Model with Double Exponential Jumps . . . 57
4.2.6 Stochastic Volatility Model with Independent Double Exponential
Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.7 Stochastic Volatility Model with Correlated Double Exponential
Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.8 Stochastic Volatility Model with Variance-Gamma Jumps . . . . . 60
4.2.9 Stochastic Volatility Model with Normal Inverse Gaussian Jumps 61
5 Numerical method 63
5.1 The Fourier Transform Methods for Derivatives Pricing . . . . . . . . . . 63
5.2 The Fourier Transform Methods of Out-of-the-Money (OTM) Option Pricing 65
5.3 European Option Pricing using the Fast Fourier Transform (FFT) . . . . 67
6 Estimation Method 69
6.1 Nonlinear Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 Particle Filtering Method . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.4 Smoothing using Backwards Simulation . . . . . . . . . . . . . . . . . . . 74
6.5 Parameter Estimation using EM Algorithm with Particle Filtering Method 75
6.6 Joint Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.7 Model Diagnostics and Comparisons . . . . . . . . . . . . . . . . . . . . 81
7 Empirical Analysis 85
7.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.2 Estimated Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.2.1 Model Parameters and Latent Volatility/Jump Variables . . . . . 87
7.2.2 Performances in Modeling the Spot Return . . . . . . . . . . . . . 91
7.2.3 Joint Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.4 Model Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4.1 In Sample Pricing Performance . . . . . . . . . . . . . . . . . . . 99
7.4.2 Out-of-Sample Pricing Performance . . . . . . . . . . . . . . . . . 103
8 Conclusion . . .108
Bibliography . . .109

Appendix A Change Measure: Stochastic Volatility Model 114
Appendix B Change Measure: Stochastic Volatility Model with Merton Jumps . . .117
Appendix C Change Measure: Stochastic Volatility Model with Indepen-dent Merton Jumps . . .121
Appendix D Change Measure: Stochastic Volatility Model with Corre-lated Merton Jumps . . .125
Appendix E Change Measure: Stochastic Volatility Model with Double-Exponential Jumps . . .129
Appendix F Change Measure: Stochastic Volatility Model with Indepen-dent Double-Exponential Jumps . . .133
Appendix G Change Measure: Stochastic Volatility Model with Corre-lated Double-Exponential Jumps . . .137
Appendix H Change Measure: Stochastic Volatility Model with Variance Gamma Process . . .143
Appendix I Change Measure: Stochastic Volatility Model with Normal Inverse Gaussian Process . . .146
Appendix J Characteristic Function: Stochastic Volatility . . .149
Appendix K Characteristic Function: Stochastic Volatility with Merton Jumps . . .151
Appendix L Characteristic Function: Stochastic Volatility with Indepen-dent Merton Jumps . . .153
Appendix M Characteristic Function: Stochastic Volatility with Correlated Merton Jumps . . .156
Appendix N Characteristic Function: Stochastic Volatility with Double-Exponential Jumps . . .159
Appendix O Characteristic Function: Stochastic Volatility with Indepen-dent Double-ExponentialJumps . . .161
Appendix P Characteristic Function: Stochastic Volatility with Correlated Double-Exponential Jumps . . .164
Appendix Q Characteristic Function: Stochastic Volatility with Variance Gamma Jumps . . .167
Appendix R Characteristic Function: Stochastic Volatility with Normal Inverse Gaussian Jumps . . .169		zh_TW
dc.format.extent 16662127 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0104352033en_US
dc.subject (關鍵詞) 隨機波動度zh_TW
dc.subject (關鍵詞) 跳躍風險zh_TW
dc.subject (關鍵詞) 風險溢酬zh_TW
dc.subject (關鍵詞) 粒子濾波演算法zh_TW
dc.subject (關鍵詞) 共同估計zh_TW
dc.subject (關鍵詞) Stochastic volatilityen_US
dc.subject (關鍵詞) Jump risken_US
dc.subject (關鍵詞) Risk premiumsen_US
dc.subject (關鍵詞) Particle-Filtering algorithmen_US
dc.subject (關鍵詞) Joint estimationen_US
dc.title (題名) 在金融海嘯前中後波動度與跳躍風險在現貨市場與選擇權市場之研究zh_TW
dc.title (題名) The implication of volatility and jump risks from spot and option markets before, during and after the recent financial crisisen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Andersen, T., L. Benzoni, and J. Lund. 2002, "An Empirical Investigation of
Continuous-Time Equity Return Models," The Journal of Finance, Vol. 57, 1239-
1284.
[2] Bates, D. S., 2001, "Jumps and Stochastic Volatility: Exchange Rate Processes
Implicit in Deutsche Mark Options," Review of Financial Studies, Vol. 9, 69-107.
[3] Bakshi, G., C. Cao, and Z. W. Chen. 1997, "Empirical Performance of Alternative
Option Pricing Models", Journal of Finance, Vol. 52, 2003-2049.
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