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 - 政大學術集成

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(1)國立政治大學金融學系碩士學位論文. 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. 政治大. 學. ‧ 國. The Implication 立 of Volatility and Jump Risks from Spot and Option Markets. ‧. Before, During and After the Recent. n. al. y. er. io. sit. Nat. Financial Crisis. Ch. engchi. i Un. v. 指導教授：林士貴博士研究生：鍾長恕撰. 中華民國一零七年一月.

(2) 誌謝在政大兩年半的時光如流星一般一閃即過，由衷感謝教導過我的老師、同學、實習的主管與同事在我碩士班就學期間的教導與照顧，雖然不管是在學業上還是實習工作上，都讓我的生活過得非常緊湊與忙碌，但我覺得這樣的生活過得非常充實，我很享受面對挑戰與困境時想辦法突破的時刻，這讓我在碩士生涯留下難得可貴美好的回憶。感謝指導老師林士貴博士在論文撰寫過程的指導與建議。在老師擔任系主任的期間還願意撥出時間帶領給我給我論文上修改意見以及幫我想未來發展方向，我覺得很幸運與榮幸能遇到這樣的指導老師。特別感謝陳亭甫學長在論文上的指引與不計成本不分晝夜的給我意見，常常半夜十一二點跟遠在台中擔任教授的學長討論論文，時常. 治政大方向，希望在未來可以將這篇論文投搞上期刊。立感謝國立政治大學金融所的同班同學：彥勳、仕紘、憲聰、威凱、吉祥、永澤、. 假日還要來政大跟我一起討論，讓我可以順利完成這篇論文以及論文後續發展延伸的. ‧ 國. 學. 宇澤、上豪。在碩士班期間一起讀書、一起比賽與一起爬山，讓我枯澀的碩士生活多了一些色彩與心靈上的放鬆；學長：明哲。給我許多論文的意見與文書編輯的技巧教. ‧. 學，讓我的論文更加精彩、豐富與美觀。還要感謝政大金融系羽的夥伴：澤琛、偉. y. Nat. 倫、承憲、秀樺、秀泓、雨晨。讓我在碩士期間可以繼續打我最喜歡的羽球。. sit. 最後，要把這份論文獻給我摯愛之家人，感謝您們長久以來的栽培、照顧與支持，. n. al. er. io. 讓我可以無後顧之憂地專注於學業上，在此將這份成就與喜悅與您們一起分享。. Ch. engchi. i Un. v. 鍾長恕謹誌於國立政治大學金融研究所中華民國一零七年一月. i.

(3) 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國立政治大學金融學系碩士學位論文指導教授：林士貴. 研究生：鍾長恕摘要. 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，. 政治大了 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立. 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. ‧ 國. 學. 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. sit. y. Nat. 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. er. io. 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. n. al. Ch. i Un. v. 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. engchi. 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。關鍵字：隨機波動度、跳躍風險、風險溢酬、粒子濾波演算法、共同估計. ii.

(4) The Implication of Volatility and Jump Risks from Spot and Option Markets Before, During and After the Recent Financial Crisis Department of Money and Banking, National Chengchi University Master Thesis Student: Chang-Shu Chung. Advisor: Shih-Kuei Lin. Abstract. 政治大. In this paper, we attempt to answer three questions: (i) On average, what does the. 立. proportion of the stochastic volatility and return jumps account for the total return vari-. ‧ 國. 學. ations 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. y. Nat. and option markets? (iii) When will investors require significantly higher risk premiums?. sit. Specifically, were there significant changes in volatility risk premiums and in jump risk. er. io. premiums before, during or after the financial crisis? For the first question, we find that. al. n. iv n C h ethan jump accounts for the higher percentage h i U volatility at the beginning of n gthec stochastic most of the return variations are explained by the stochastic volatility. In fact, the return. 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. Keywords: Stochastic Volatility, Jump Risk, Risk Premiums, Particle-Filtering Algorithm, Joint Estimation iii.

(5) Contents 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. 立. 3 The Models. 政治大. 13. ‧ 國. 學. Stochastic Volatility Model . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 3.2. The Characteristic Exponent of Lévy 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 . . . . . . . .. 3.6. Stochastic Volatility Model with Double-Exponential Jumps . . . . . . .. 3.7. Stochastic Volatility Model with Independent Double-Exponential Jumps. 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. ‧. 3.1. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. 4 The Risk-Neutral Dynamics and Characteristic Functions 4.1. 25 28 32. 45. 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. iv.

(6) 4.1.6. Stochastic Volatility Model with Independent Double-Exponential Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 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. 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. Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58. 4.2.6. 學. 4.2.7. 政治大 Stochastic Volatility Model with Independent Double Exponential 立. ‧ 國. 4.2. Stochastic Volatility Model with Correlated Double Exponential Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60. 4.2.9. Stochastic Volatility Model with Normal Inverse Gaussian Jumps. y. 61. sit. 63. er. io. al. iv n C h eofnOut-of-the-Money The Fourier Transform Methods g c h i U (OTM) Option Pricing n. 5.3. ‧. Stochastic Volatility Model with Variance-Gamma Jumps . . . . .. Nat. 5.2. 59. 4.2.8. 5 Numerical method 5.1. 49. The Fourier Transform Methods for Derivatives Pricing . . . . . . . . . .. European Option Pricing using the Fast Fourier Transform (FFT) . . . .. 6 Estimation Method. 63 65 67 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. v.

(7) 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. 立. Bibliography. 政治大. 108 109. ‧ 國. 學. Appendix A Change Measure: Stochastic Volatility Model. 114. Jumps. ‧. Appendix B Change Measure: Stochastic Volatility Model with Merton. Nat. y. 117. n. al. er. io. dent Merton Jumps. sit. Appendix C Change Measure: Stochastic Volatility Model with Indepen-. Ch. i Un. 121. v. Appendix D Change Measure: Stochastic Volatility Model with Correlated Merton Jumps. engchi. 125. Appendix E Change Measure: Stochastic Volatility Model with DoubleExponential Jumps. 129. Appendix F Change Measure: Stochastic Volatility Model with Independent Double-Exponential Jumps. 133. Appendix G Change Measure: Stochastic Volatility Model with Correlated Double-Exponential Jumps. 137. Appendix H Change Measure: Stochastic Volatility Model with Variance Gamma Process. 143 vi.

(8) 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 Independent Merton Jumps. 153. Appendix M Characteristic Function: Stochastic Volatility with Correlated Merton Jumps. 156. 治政大Volatility with DoubleAppendix N Characteristic Function: Stochastic 立 Exponential Jumps 159 ‧ 國. 學. Appendix O Characteristic Function: Stochastic Volatility with Indepen-. ‧. dent Double-ExponentialJumps. 161. sit. y. Nat. Appendix P Characteristic Function: Stochastic Volatility with Corre-. io. al. 164. er. lated Double-Exponential Jumps. v. n. Appendix Q Characteristic Function: Stochastic Volatility with Variance Gamma Jumps. Ch. engchi. i Un. 167. Appendix R Characteristic Function: Stochastic Volatility with Normal Inverse Gaussian Jumps. 169. vii.

(9) List of Tables 1. The Lévy Measures, Characteristic Functions, and References for Selected Pure Jump Lévy Processes. . . . . . . . . . . . . . . . . . . . . . . . . .. 171. 2. Summary of Measure Change . . . . . . . . . . . . . . . . . . . . . . . .. 172. 3. Descriptive Statistics of S&P500 Index and Return . . . . . . . . . . . .. 173. 4. S&P500 Call Option Data . . . . . . . . . . . . . . . . . . . . . . . . . .. 173. 5. 政治大 Estimated parameters立 for S&P500 index under the physical measure . . .. 174. Call Option Prices: DFT, FFT v.s. Monte Carlo with 95% CI . . . . . .. 176. 7. Call Option Pricing Error: DFT, FFT v.s. Monte Carlo . . . . . . . . . .. 179. 8. Weekly In-Sample Option Pricing Performance . . . . . . . . . . . . . . .. 182. 9. Weekly Out-of-Sample Option Pricing Performance . . . . . . . . . . . .. 183. 10. Weekly In-Sample Option Pricing Performance . . . . . . . . . . . . . . .. 11. Weekly Out-of-Sample Option Pricing Performance . . . . . . . . . . . .. 12. Mean of the Weekly In-Sample/Out-of-Sample Absolute/Relative Pricing. y. io. 187. n. er. Nat. al. 184. sit. ‧. ‧ 國. 學. 6. Ch. i Un. v. Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 190. 13. Estimated parameters for S&P500 Index and Option (Joint Estimation) .. 191. 14. Volatility and Jump Risk Premiums for Each Time Interval . . . . . . . .. 193. 15. Kolmogorov-Smirnov Test Statistics and p-Values for Each Model . . . .. 194. engchi. viii.

(10) List of Figures 1. Number of Option Contracts, Daily S&P500 Index, Return and At-theMoney Implied Black-Scholes Volatility. . . . . . . . . . . . . . . . . . . .. 195. 2. Monte Carlo v.s. Fast Fourier Transform . . . . . . . . . . . . . . . . . .. 196. 3. Filtered Latent Stochastic Volatility Variables of SV, SV-MJ, SV-MJ-VIJ,. SV-DEJ-VCJ, SV-VG, and SV-NIG . . . . . . . . . . . . . . . . . . . . .. 198. 學. 5. 197. ‧ 國. 4. 政治大 Filtered Latent Stochastic 立 Volatility Variables of SV-DEJ, SV-DEJ-VIJ,. and SV-MJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Filtered Latent Return Jump Variables of SV-MJ, SV-MJ-VIJ, and SV-. Nat. y. Filtered Latent Return Jump Variables of SV-DEJ, SV-DEJ-VIJ, and SVDEJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 200 201. 8. Filtered Latent Return Jump Number Variables of SV-MJ, SV-MJ-VIJ,. n. al. er. Filtered Latent Return Jump Variables of SV-VG and SV-NIG . . . . . .. io. 7. Ch. i Un. v. and SV-MJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. engchi. 205. Percentage of Latent Return Jump Variables of SV-MJ, SV-MJ-VIJ, and SV-MJ-VCJ Account for the Total Return Variance . . . . . . . . . . . .. 13. 204. Filtered Latent Stochastic Volatility Jump Number Variables of SV-MJVIJ, SV-MJ-VCJ, SV-DEJ-VIJ, and SV-DEJ-VCJ . . . . . . . . . . . . .. 12. 203. Filtered Latent Stochastic Volatility Jump Components Variables of SVMJ-VIJ, SV-MJ-VCJ, SV-DEJ-VIJ, and SV-DEJ-VCJ . . . . . . . . . .. 11. 202. Filtered Latent Return Jump Number Variables of SV-DEJ, SV-DEJ-VIJ, and SV-DEJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 199. sit. 6. ‧. MJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 206. Percentage of Latent Return Jump Variables of SV-DEJ, SV-DEJ-VIJ, and SV-DEJ-VCJ Account for the Total Return Variance . . . . . . . . .. ix. 207.

(11) 14. Percentage of Latent Return Jump Variables of SV-VG and SV-NIG Account for the Total Return Variance . . . . . . . . . . . . . . . . . . . . .. 208. 15. Kernel Density of the Residuals of Each Model . . . . . . . . . . . . . . .. 209. 16. QQ-Plots of the Residuals of Each Model . . . . . . . . . . . . . . . . . .. 210. 17. The Log-Likelihood Values in Joint Estimation for SV, SV-MJ, SV-MJVIJ, and SV-MJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. 211. The Log-Likelihood Values in Joint Estimation for SV, SV-DEJ, SV-DEJVIJ, and SV-DEJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 212. 19. The Log-Likelihood Values in Joint Estimation for SV-VG and SV-NIG .. 213. 20. The Time Series of Estimated Volatility Risk Premiums for SV, SV-MJ, SV-MJ-VIJ, and SV-MJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . .. 22. 政治大 DEJ-VIJ, and SV-DEJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . 立 The Time Series of Estimated Volatility Risk Premiums for SV-DEJ, SV-. 學. SV-NIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. ‧. 217. sit. y. Nat. The Time Series of Estimated Jump Risk Premiums for SV-DEJ, SV-DEJ-. io. er. VIJ, and SV-DEJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 218. The Time Series of Estimated Jump Risk Premiums for SV-VG and SV-NIG219. al. n. 25. 216. The Time Series of Estimated Jump Risk Premiums for SV-MJ, SV-MJVIJ, and SV-MJ-VCJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. 215. The Time Series of Estimated Volatility Risk Premiums for SV-VG and. ‧ 國. 21. 214. Ch. engchi. x. i Un. v.

(12) Chapter 1 Introduction In this paper, we extend from Kou, Yu, and Zhong (2016) whose research issues focus on. 政治大 We involve empirical evidences立 in the option market to analyze the volatility risk, jump how the jump risk affect the equity index returns before and during the financial crisis.. ‧ 國. 學. risk, and risk premiums before, during, and after the financial crisis. We ask the three questions and attempt to answer them: (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. Nat. sit. y. the total return variations? (ii) Is the fitting performance of infinite-activity jump models. al. er. io. better than that of finite-activity jump models both in the spot and option markets? (iii). n. When will investors require significantly higher risk premiums than before, during, or. Ch. i Un. v. after the 2008-2009 the financial crisis? Specifically, were there significant changes in. engchi. volatility risk premiums or in jump risk premiums, or both? For the first question, the index returns was significant changes during financial crisis, and these return variations may cause to operate the financial instrument improperly. Therefore, we attempt to understand the composition of return variations so that we can find the suitable option pricing model for pricing and hedging in the financial market. In this paper, we utilize the affine jump-diffusion models and Lévy jump models with stochastic volatility to fit the dynamic of index returns, and thus the return variations are consisted of two components: stochastic volatility risk and jump risk. Thus, in order to fit our models well, there is an important issue that how to set up the proportion of these two components in total return variations. Moreover, we also want to investigate which type of jump models has better fitting performance in the spot market. Further, a 1.

(13) appropriate model of return dynamics is essential for option pricing and risk management, because different model specifications lead to different option pricing results for equity options. Thus, if one model fit index returns well, will this model has superior forecasting ability in the option market? For the second question, Kou et al. (2016) documents the stochastic volatility model with double-exponential jump outperforms that with other jump-size specifications. We further extend their works to compare different jump-types model between infiniteactivity jump models and finite-activity jump models in capturing the joint dynamics of index returns and option prices, simultaneously. We attempt to involve broader jump models with more market information for a comprehensive comparison to know how the jump risk presents in the financial market with respect to the specific models and periods.. 政治大 rolling window approach in the spot and option markets, we can acquire extra parameters, 立 With dynamic joint estimation, using the particle filtering to jointly estimate with. that is, time varying risk premiums for volatility and jump process. In contrast to the. ‧ 國. 學. average estimated value of risk premiums for a certain period, it is not clear when investors will require more risk premiums, before, during or after the financial crisis. Thus, the. ‧. third question is interesting because we can extract the time series of the goodness of fit. sit. y. Nat. and risk premiums to obtain more implied market information in different time periods.. io. er. To answer the three questions, we focus on data sets of the S&P500 daily returns and weekly call option contracts from January 1, 2007 to August 31, 2017, which cov-. n. al. Ch. i Un. v. ers the 2008-2009 financial crisis and the European sovereign debt crisis. We compare. engchi. the empirical results of nested models as follows. The fundamental model is stochastic volatility model in Heston (1993), which can clearly capture volatility clustering during the financial crisis. Next, we adopt the finite-activity compound Poisson processes with normal jump sizes to the stochastic volatility model in order to capture the dramatic changes of index returns. However, there is a drawback of using the normal distribution to model jump sizes because its distribution can not have a monotone decreasing density on both side from 0 if the mean of jump size deviates from 0. This drawback may cause the poor fitting performance for small jumps documented in Kou (2002) and Kou et al. (2016). Thus, we consider the other two jump-types. First, the model with the time-changed Lévy processes (the variance gamma (VG) process and the normal inverse Gaussian (NIG). 2.

(14) process) with the stochastic volatility models which have the parameters that can control skewness and kurtosis more flexible than normal distribution. Further, by the definition of Lévy measure, we find that the time-changed Lévy processes can depict infinitely many small jumps. Li et al. (2008, 2011) show that the variance gamma jumps model fit small jumps in the equity index returns better than the affine jump-diffusion model in Eraker et al. (2003) with normal distributed jump sizes. Second, we consider the models based on the compound Poisson jump processes with double-exponential jump sizes which has the monotone structure and has better fitting performance for small jumps than normal distribution (see Kou et al. (2016)). Moreover, we adopt the extended model proposed in Duffie et al. (2000) and Eraker et al. (2003), that is, the index return jump process can be independent and correlated with volatility. 政治大 latent stochastic volatility to track the return variations more accuracy to enhance the 立 jumps. Jumps in volatility allow volatility to increase rapidly and it is helpful for the. fitting performance.. ‧ 國. 學. In summary, our answer to the first question is that most of return variations are dominated by the stochastic volatility. In fact, the return jump accounts for the higher. ‧. percentage than the stochastic volatility only at the beginning of financial crisis events.. sit. y. Nat. Take the stochastic volatility model with Merton jumps (SV-MJ) as example, the per-. io. er. centage of return jump accounts for the return variation up to 62.12 percent in September 15, 2008 and 91.10 percent in September 29, 2008 during the early stages of a financial. n. al. Ch. i Un. v. crisis. The average percentage of return jump account for the total return variation only. engchi. 0.91 percent in 2008. Based on the stochastic volatility model, we observe that the variations of index returns caused by significant jump are rare and the stochastic volatility can interpret not only small variations but also a part of jumps of return variations. On the other hand, we observe that most of small jumps of index returns can be examined by the stochastic volatility. Our empirical study emphasizes the low frequency character of jumps in index return. At the beginning of financial crisis, the dramatic index shocks are caused by both jump risk and volatility risk, and the jumps lead to more variations of return than volatility does. However, the return spillovers bring the increase of stochastic volatility so that it enhances the explanatory power of volatility to the variation of returns in the financial crisis.. 3.

(15) To answer the second question, we adopt the expectation-maximization algorithm with the particle filtering and dynamic joint estimation to indicate that the stochastic volatility with correlated double-exponential jumps model (SV-DEJ-VCJ) and the stochastic volatility with normal inverse Gaussian process jumps model (SV-NIG) perform well in the S&P500 index return and option markets. When we choose the log-likelihood function to be the criterion, the finite-activity jump model (SV-DEJ-VCJ) has the best fitting performance in both spot and option markets. If we use the Bayesian information criterion (BIC), which considers a penalty term for both the number of parameters and sample sizes in the model as the criterion for model selection, the infinite-activity jumps model (SV-NIG) will represent the best performance. For the further examination, the out-of-sample test reports that the SV-DEJ-VCJ. 政治大 Our findings indicate the benefit of specifications with the 立. model has the best forecasting ability since it has less absolute pricing error in weekly option prices on average.. double-exponential jump components and the correlated jumps in volatility, with the. ‧ 國. 學. more richly parameterized model performing relative better both for the in-sample fitness and for the out-of-sample prediction test.. ‧. For the third question, we observe interesting phenomenon to the risk premiums of. sit. y. Nat. both volatilities and jumps that significantly increase after the financial crisis periods.. io. er. Take the SV-DEJ-VCJ model as an example, we find the average annualized risk premiums of volatilities and jumps are 0.00729% and 0.0170% respectively during the financial. n. al. Ch. i Un. v. crisis. However, after the financial crisis, the average annualized risk premiums of volatil-. engchi. ities and jumps significantly rise to 0.046% and 0.110%, respectively. The evidence seem to illustrate the panic in the post-crisis period causes more expected returns. There are some contributions in this study: (i) We find that the return jumps account for a large proportion of the total return variations in S&P500 index returns during the early stages of a financial crisis. After these days, the stochastic volatility becomes the major impact factor to the total return variations. In fact, we also find that most of the small jumps in index returns which are explained by the stochastic volatility. Therefore, we only need to focus on capturing the large jumps of index returns when we consider the model with the stochastic volatility. (ii) In terms of model specification, we dynamically estimate a series of the stochastic volatility models with finite-activity jumps and infinite-activity jumps to obtain joint log-. 4.

(16) likelihood value for each day. Note that, when we estimate the model parameters in the option market, we employ the Esscher transform (1994) to change the probability measure and derive the option pricing formula with Fast Fourier Transform (FFT) scheme. As our empirical results, we find that the SV-DEJ-VCJ model have the best fitting performance in both the spot and option markets. The SV-DEJ-VCJ model has the superior performance owing to the double-exponential distribution which has the monotonicity and heavy-tail feature properties to capture small and large jumps well. Moreover, the correlated jumps in volatility make latent stochastic volatility rapidly track the return variations, that is, it has the better performance than the models without volatility jumps. With respect to the computational efficiency, the SV-NIG model has less parameters than the SV-DEJ-VCJ model though they have similar fitting performances. Therefore, the. 政治大 (iii) In terms of econometric methods, we use the sequential importance resampling 立. SV-NIG model has superior performance based on Bayesian information criterion.. algorithm, the particle filtering (PF) algorithm, to track the latent variables of the models,. ‧ 國. 學. such as jump sizes, jump times and stochastic volatility, and these filtered latent variables are beneficial for our analysis to obtain more implied market information. After we filter. ‧. the latent variables corresponding to each model, we adopt the expectation-maximization. sit. y. Nat. (EM) algorithm to maximize the log-likelihood value and obtain suitable parameters for. io. er. each model through an iterative process. Moreover, we use the rolling window approach to acquire the time series of model parameters and risk premiums of volatility and jump. al. n. risk.. Ch. engchi. i Un. v. (v) To calibrate more important information such as risk premiums containing in the option market. We use both S&P500 index returns and weekly call option prices from January 1, 2007 to August 31, 2017 for a long period empirical study. We observe that the risk premiums of volatility risk and jump risks are increasing after the crisis by dynamic jointly estimation, which shows that the panic in the post-crisis period causes more expected returns. The rest of the paper is organized as follows. Section 2 reviews the related literatures. Section 3 introduces the models for different distribution jump sizes. We address the risk neutralization procedure with Esscher transform in Section 4. Section 5 outline the numerical method, i.e., Fast Fourier Transform, for option pricing scheme. Section 6 introduce the particle filtering (PF) and the expectation-maximization (EM) algorithm. 5.

(17) utilized in our study and discuss model diagnostics to evaluate model performance. Section 7 presents the empirical results for S&P500 index and option market. Finally, Section 8 concludes this paper. The appendix provides all technical details.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 6. i Un. v.

(18) Chapter 2 Literature Review 2.1. The Background of Research 治 Issue. 立. 政. 大. After Black and Scholes (1973) submitted the Black-Scholes model, this model is widely. ‧ 國. 學. used as the fundamental option pricing model. In order to reduce model’s pricing error and hedge risk, several researchers added the different market factors and changed some. ‧. of the restrictive Black-Scholes model (BS) assumptions to test fitting performance for each market. Examples include (i) stochastic volatility (Hull & White (1987), Heston. y. Nat. sit. (1993) and Heston & Nandi (2000)); (ii) jump diffusion model (Merton (1976), Bates,. al. er. io. (1996), Bakshi, Cao, & Chen (1997) and Kou (2002)); (iii) correlated jumps in volatility. v. n. (Duffie, Pan, & Singleton (2000), Eraker (2003), Eraker, Johannes, & Polson (2004)); (iv). Ch. i Un. Lévy jump process (Madan & Seneta (1990), Madan, Carr, & Chang (1998), Barndorff-. engchi. Nielsen (1997), Carr, Geman, Madan, & Yor (2002), Li, Wells, and Yu (2008, 2011) and Ornthanalai (2014)). Thus, there is an important issue how to examine the fitting performance for each model. There are three main comparison methods. First, when we discuss the fitting performance of the time series of the equity index returns under the physical probability measure, we use the log-likelihood ratio test, Akaike information criterion (AIC), Bayesian information criterion (BIC), Deviance information criterion (DIC), Kolmogorov-Smirnov (KS) test, QQ-plot, kernel density, Markov Chain Monte Carlo, Kalman filtering algorithm, Particle filtering algorithm and so on. (Li, Wells, and Yu (2008), Johannes, Polson, and Stroud (2009), Daskalakis, Psychoyios and Markellos (2009), Christoffersen, Jacobs and Mimouni (2010), Nakajima and Ohashi (2012), Kaeck and Alexander (2013), Orn7.

(19) thanalai (2014), Diewald, Prokopczuk and Wese Simen (2015)). Second, we minimum the absolute pricing error between the implied volatility of the market option price and the theory option price under the risk-neutral measure. After calibrating the parameters, we can compute the pricing error to judge the performance for each model. (such as Bakshi, Cao, and Chen (1997), Chernov and Ghysels (2000), Broadie, Chernov, and Johannes (2007), Bakshi and Wu (2010), Li, Wells, and Yu (2011), and Hsu, Lin, and Chen (2014)). Third, we can also maximize the joint likelihood function which connect the physical probability measure with the risk-neutral measure. (such as Ornthanalai (2014)) That is, we apply the particle filtering algorithm to track the latent variables and then compute the log-liklihood value under the physical probability measure. On the other hand, we. 政治大 price and model’s price to obtain the implied informations from the option market, such 立 minimum the sum of the squared differences between the implied volatility of market’s. as the volatility and jump risk premiums. Therefore, we can obtain the joint estimation. ‧ 國. 學. value for each model.. For the empirical results, since the data sets of the commodity and financial index. ‧. markets are large and complex, it is not easy to transform and deal with them. Therefore,. sit. y. Nat. the existing literature only compares certain special cases of affine jump-diffusion model. io. er. and Lévy jump models in short periods or the fitting performance of equity index returns for each market. For example, Eraker, Johannes, & Polson (2003) find an affine jump-. n. al. Ch. i Un. v. diffusion model with stochastic volatility and correlated Merton jumps in returns and. engchi. volatility fit S&P500 index returns data from 1980 to 1999 well. Li, Wells, and Yu (2008) show that infinite-activity Lévy jumps are essential for modeling the S&P500 index returns data from 1980 to 2000. However, neither of these papers covers the period of 2008 financial crisis and 2011 European sovereign debt crisis. Kou, Yu, and Zhong (2016) therefore find that a simple affine jump-diffusion model with both stochastic volatility and double-exponential jump sizes in returns fits both the S&P500 and the NASDAQ-100 daily returns data from 1980 to 2013 well. But these papers mainly focus on the fitting performance of the index returns. Existing studies of the affine jump-diffusion and Lévy jump models using option prices, such as Huang and Wu (2004) use the S&P500 index options from April 6, 1999 to May 31, 2000. Li, Wells, and Yu (2011) examine the performance of several popular Lévy jump models and. 8.

(20) affine jump-diffusion models in capturing the joint dynamics of stock and option prices with daily spot and option prices of the S&P500 index from January, 1993 to December, 1993. But they do not compare the performance of Lévy jump models with that of the double-exponential jump in returns and volatility model in both the spot and option markets. Thus, the purpose of our paper is to fill this gap and conduct a comprehensive empirical study on the relative merits of option pricing models. To this goal, we follow the closed form option pricing model that admits the stochastic volatility model (SV) seen in Heston(1993), the stochastic volatility model with normal jumps (SV-MJ) seen in Bakshi (1997), double-exponential jumps (SV-DEJ) seen in Kou (2002), and correlated jump in volatility and normal jumps (SV-MJ-VCJ) seen in Eraker (2003). Moreover,. 政治大 with Variance-Gamma jumps (SV-VG), Normal Inverse Guassian jumps (SV-NIG), and 立. we develop the closed form option pricing model, such as the stochastic volatility model. correlated jump in volatility and double-exponential jumps (SV-DEJ-VCJ) in Kou, Yu,. ‧ 國. 學. and Zhong (2016) by Esscher Transforms (Gerber, Shiu, and Elias 1994).. ‧. The Stochastic Volatility and Jump Diffusion Pro-. io. sit. y. Nat. cesses. er. 2.2. al. n. iv n C are many hedging tools which use thehvolatility index U e n g c h i as underlying asset, such as VIX Volatility is an important role in the option pricing and also in hedging strategy. There. Futures, VIX Option and Variance futures. These derivatives indicate that option price. has been largely affected by the volatility. However, Black & Sholes (1973) model assume that the volatility is an constant that is too restrictive and therefore it cannot match the market requirements, such as the volatility smile and the volatility clustering. In order to find the right distribution assumption to close the market price, the researchers developed the series of the related volatility researches and applications. After Hull and White (1987) firstly developed the fundamental concern of the stochastic volatility whose variation dynamic follows the geometric Brownian motion, the series of the related stochastic volatility researches are generated by many financial researchers. Among these researches, Heston (1993) submitted the most popular model that the variation of index return follows the square-root process (used in Cox, Ingersoll, and Ross 9.

(21) (1985)). This model have several advantages: (i) It can avoid the negative variations. (ii) The variation of index return have the mean-reverting property. (iii) The correlated between volatility shocks and underlying stock returns serves to control to the level skewness and the volatility variation coefficient serves to the level of kurtosis. These benefits can make it fit index returns well. Moreover, Heston (1993) use the Fourier transform to obtain the closed form of European call option price by the CIR model. However, since the stochastic volatility model belong to continuous-time model, it cannot capture the heavy-tailed feature, i.e. large return jumps, in short-term period and thus prices short-term options properly is limited. Therefore, Merton (1976) firstly added the compound Poisson normal jump diffusion model to the Black & Sholes (1973) model. Bates (1996) based on the the stochastic volatility model and Merton (1973). 政治大 applied this model to American options and used the deutsche mark foreign currency 立 jump to develop the stochastic volatility with normal jump model (SV-MJ). Bates (1996). options to examine the fitting performance. Bakshi et al. (1997) compare the pricing. ‧ 國. 學. error and hedging error of the Black-Sholes, stochastic volatility, stochastic volatility stochastic interest rate and stochastic volatility with normal jump models. They find. ‧. that stochastic volatility with normal jump model outperform in the option pricing error.. sit. y. Nat. However, for hedging purposes, the SV model achieve the best hedging results among. io. er. all the models. There are several advantages of the compound Poisson jump diffusion model. First, it can capture the property of implied volatility smile. Second, it can. n. al. Ch. i Un. v. describe the heavy-tail feature of index return. That is, when the jump amplitude be-. engchi. come more larger, the return distribution which is constructed by the model will also become more heavier (Matsuda (2004), Gatheral (2006), Hull (2014)). Third, the compound Poisson jump diffusion model have analytical property, and we can solve the partial integro-differential equation (PIDE) to get the closed form formula which can make pricing and hedging more effective than monte carlo method. (such as Hilliard and Reis (1999), Koekebakker and Lien (2004), Cartea and Figueroa (2005), Chevallier (2013), Wilmot and Mason (2013), Schmitz et al. (2014), Mayer et al. (2015), Xiao et al. (2015)). On the other hand, Kou (2002) applied double-exponential distribution and equilibrium theory to acquire the option pricing model and used the Japanese LIBOR Caplets in May 1998 to illustrate that the model cam produce implied volatility smile. Since the den-. 10.

(22) sity of double-exponential distribution for jump sizes is monotonically which is decreasing for both negative jumps and positive jumps, it better fit for small jumps than Merton jumps (seen in Kou et al. (2016)). Further, asymmetric heavy-tail (e.g. leptokurtosis) feature of double-exponential distribution also helps fit both positive and negative large jumps. Therefore, many literatures start to use double-exponential distribution to model the return distribution. Ramezani and Zeng (2002) indicated that double-exponential jump diffusion model can fit better than normal jump model when the index return have the jump events around the time peiod. Moreover, Kou and Wang (2004) demonstrate that a double-exponential jump diffusion model can lead to an analytic approximation for finite-horizon American options and analytic solutions for popular path-dependent options.. 政治大 (2016) consider the stochastic volatility model incorporating jumps in returns 立. Duffie, Pan, and Singleton (2000), Eraker (2004), Eraker et al. (2003), and Kou et al.. and in volatility. The empirical result of Eraker (2003) show that both of these jump. ‧ 國. 學. components are important for both S&P500 and Nasdaq100 index from January 2, 1980, to December 31, 1999 and September 24, 1985, to December 31, 1999. Models with. ‧. only diffusive stochastic volatility and jumps in returns are misspecified, because they. sit. y. Nat. do not have a component driving the conditional volatility of returns, which is rapidly. io. er. moving. However, Kou et al. (2016) show that a simple affine jump-diffusion model with the stochastic volatility and double-exponential jumps fits both the S&P500 and. n. al. Ch. i Un. v. the NASDAQ-100 daily returns from 1980 to 2013 well. That is, the SV-DEJ-VCJ model. engchi. may not outperform the SV-DEJ model, since by construction, jumps in volatility occur at the same time as jumps in returns, a constraint that impedes the double-exponential distribution in picking up small jumps. Madan and Seneta (1990), Madan, Carr, and Chang (1998) and Huang and Wu (2004) submitted the Variance Gamma (VG) process which is obtained by evaluated Brownian motion (with constant drift and volatility) at a random time change given by a gamma process. That is, VG process allow non-normal increment as compared to normal increments of Brownian motion, and thus the jump component of variance gamma process is much more flexible than a compound Poisson process. Moreover, Li, Wells, and Yu (2008, 2011) develop efficient MCMC methods for estimating parameters and latent volatility/jump variables of the Lévy jump model using stock and option prices.. 11.

(23) Li, Wells, and Yu (2011) show that the VG model of Madan, Carr, and Chang (1998) with stochastic volatility has the best performance among of all the models they consider in capturing both the physical and risk-neutral dynamics of the S&P500 index. Barndoeff-Nielsen (1997, 1998) submitted normal inverse Gaussian (NIG) process which is obtained by evaluated Brownian motion (with constant drift and volatility) at a random time change given by a inverse Gaussian process. That is, NIG process allow non-normal increment as compared to normal increments of Brownian motion, and thus the jump component of NIG process is much more flexible than a compound Poisson process.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 12. i Un. v.

(24) Chapter 3 The Models 3.1. Stochastic Volatility Model 治. 政. 立. 大. This model is shown in Heston (1993) who assume that the variation of index return. ‧ 國. 學. follows the square-root process (used by Cox, Ingersoll, and Ross (1985)). This model have several characteristics for the index return: (i) It can avoid the negative variations.. ‧. (ii) The variation of index return have the mean-reverting property. (iii) The correlated between volatility and underlying spot returns serves to control to the skewness. (v) The. y. Nat. sit. volatility variation coefficient serves to the kurtosis. This model capture an important. al. er. io. features of equity index return dynamics, namely stochastic volatility, and also provide. v. n. analytical tractability for option pricing in Section 4.1.1 and model estimation in Section 6.. Consider a probability space. Ch . engchi. i Un. Ω, F, P, {Ft }Tt=0 , where Ω is the sample set of all. outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability measure. Let {Ft }Tt=0 be the filtration generated by the pair of the Wiener processes. P P dWy,t , dWv,t at time t, 0 ≤ t ≤ T . According to above definition, the index returns, yt , follows the stochastic volatility (SV) model, and the stochastic volatility vt follows the mean-reverting square-root process. Then its dynamic process under physical probability measure P is: √ 1 P dyt = µ − vt dt + vt dWy,t , 2 √ P dvt = κ (θ − vt ) dt + σv vt dWv,t ,. (3.1). where µ is the drift term which is a constant. The stochastic volatility vt follows a mean13.

(25) reverting square root process with long term level θ > 0, the speed of adjustment is P κ > 0, and the variation coefficient of stochastic volatility is σv > 0. Moreover, dWy,t P are a pair of correlated Wiener processes with correlation coefficient ρdt, i.e., and dWv,t P P P P ) = ρdt. Therefore, by the , dWv,t ∼ N (0, dt), and Corr(dWy,t ∼ 1 N (0, dt), dWv,t dWy,t. Cholesky decomposition, the correlated Wiener processes can be decomposed into two p P P P P P independent Wiener processes, i.e., dWy,t = dB1,t and dWv,t = ρdB1,t + 1 − ρ2 dB2,t . Thus, the equation (3.1) can be rewritten as following √ 1 P dyt = µ − vt dt + vt dB1,t , 2 p √ P P + 1 − ρ2 dB2,t , dvt = κ (θ − vt ) dt + σv vt dB1,t We transform (3.1) into discrete-time version: p 1 yt = yt−1 + µ − vt ∆ + vt−1 ∆εPy,t , 2 p vt = vt−1 + κ (θ − vt−1 ) ∆ + σv vt−1 ∆εPv,t ,. 立. (3.2). 政治大. (3.3). ‧ 國. 學. where ∆ is the time interval, µ is the drift term. The stochastic volatility vt follows a. ‧. mean-reverting square root process with long term level θ > 0, the speed of adjustment. y. Nat. κ > 0, and the variation coefficient of stochastic volatility σv > 0. Moreover, εPy,t and. sit. εPv,t are diffusion noises with standard normal distribution with correlation coefficient ρ,. n. al. er. io. i.e., εPy,t ∼ N (0, 1), εPv,t ∼ N (0, 1), and Corr(εPy,t , εPv,t ) = ρ. Therefore, by the Cholesky. i Un. v. decomposition, the correlated Wiener processes can be decomposed into two independent p Wiener processes, i.e., εPy,t = εP1,t and εPv,t = ρεP1,t + 1 − ρ2 εP2,t . Thus, under P measure,. Ch. engchi. the equation (3.3) can be rewritten as following p 1 yt = yt−1 + µ − vt ∆ + vt−1 ∆εP1,t , 2 p p P P 2 vt = vt−1 + κ (θ − vt−1 ) ∆ + σv vt−1 ∆ ρε1,t + 1 − ρ ε2,t ,. (3.4). For this dynamics process, we have observations {yt }Tt=0 , latent volatility variables {vt }Tt=0 and model parameters space Θ = {µ, κ, θ, σv , ρ}. There is a drawback under the SV model. Since the SV model belong to continuous-time model, it cannot capture the heavy-tailed feature, i.e. large return jumps, in short-term and thus it cannot explain the smirkiness exhibited in the cross-sectional option data (Bakshi et al., (1997), Bates 1 The notation ∼ N (µ, σ 2 ) means that the distribution is normal with parameter µ, σ 2 and the density √ 2 2 1/ 2πσ 2 e−((x−µ) /2σ ) , where x ∈ R. 14.

(26) (2000)). For modifying this drawback, we introduce the stochastic volatility with Lévy jumps model in the next section.. 3.2. The Characteristic Exponent of L´ evy Jump Processes. In order to solve the problem of the SV model, we introduce the Lévy process to measure the jump risk. In this paper, we consider two types of pure Lévy jump: finite-activity and infinite activity. The forms of return jump are finite activity if there are finitely many jumps in a fixed interval. That is, Lévy measure is less than infinite, i.e., v (R) < ∞. On the other hand, the forms of jump are infinite-activity if there are infinitely many. 政治大. jumps in a fixed interval. That is, Lévy measure is equal to infinite, i.e., v (R) = ∞, R where v is the Lévy measure on R, that satisfies v ({0}) = 0 and R (1 ∧ |x|2 ) < ∞.. 立. ‧ 國. 學. The Lévy measure describes the expected number of jumps of a certain height in a time integral of length one. The Lévy measure has no mass at the origin, while singularities. ‧. (i.e. infinitely many jumps) can occur around the origin (i.e. small jumps). Moreover, the mass away from the origin is bounded (i.e. only a finite number of big jumps can. y. Nat. sit. occur). In this paper, we consider the Compound Poisson process with Merton jump and. er. io. Double-Exponential jump which are finite-activities and Variance Gamma process and. al. n. iv n C The first Lévy jump type is thehincrement of a U e n g c h i Compound Poisson process with. Normal Inverse Gaussian process which are infinite-activities. (1).. Merton jumps (MJ) which assumes that each jump size is independent drawn from the normal distribution with mean γ and variance δ 2 . Hence the distribution of the jump size has density, ". (x − γ)2 fM J (x) = √ exp − 2δ 2 2πδ 2 1. #. Moreover, the number of jumps nt+1 arriving between periods t and t + 1 follow a Poisson process with the intensity λ and v (x) = λ · fM J (x) is a Lévy measure. Since the number of jumps that occur within an interval is finite, the Merton jump belong to finite-activity type. Thus the characteristic exponent of the a Compound Poisson process with Merton jumps is . . 1 ψ (u) = λ · exp γiu − u2 δ 2 2 15. . −1.

(27) (2). The second Lévy jump type is the increment of a Compound Poisson process with correlated Merton jumps (MJ-VCJ) in Duffie, Pan, and Singleton (2000) and Eraker, Johannes, and Polson (2003), i.e. jump in return and in volatility, which assumes that each jump size is independent drawn from the conditional normal distribution with mean γ+ρJ V and variance δ 2 where V is a jump size of volatility follows exponential distribution with mean µv and ρJ is correlated coefficient between return jump and volatility jump. Hence the distribution of the return jump size has density " # 1 (x − (γ + ρJ V ))2 fM J−V CJ (x) = √ exp − 2δ 2 2πδ 2 Moreover, the number of jumps nt+1 arriving between periods t and t + 1 follow a Poisson process with the intensity λ and v (x) = λ · fM J (x) is a Lévy measure. Since the number. 政治大 finite-activity type. Thus the characteristic exponent of the a Compound Poisson process 立 of jumps that occur within an interval is finite, the correlated Merton jump belong to. ψ (u) = λ ·. 學. ‧ 國. with correlated Merton jumps is. ! exp γiu − 12 u2 δ 2 −1 1 − ρJ µv. ‧. (3). The third Lévy jump type is the increment of a Compound Poisson process with. Nat. sit. y. Double-Exponential jumps (DEJ) in Kou (2002) which assumes that each jump size is. io. η − . Hence the distribution of the jump size has density,. n. al. Ch. er. independent drawn from the double exponential distribution with parameters p, η + , and. i Un. v. 1 −x 1 x fDE (x) = p + e η+ 1(x≥0) + (1 − p) − e η− 1(x<0) η η. engchi. Moreover, the number of jumps nt+1 arriving between periods t and t + 1 follow a Poisson process with the intensity λ and v (x) = λ · fDE (x) is a Lévy measure. Since the number of jumps that occur within an interval is finite, the Double-Exponential jump belong to finite-activity type. Thus the characteristic exponent of the a Compound Poisson process with Double-Exponential jumps is ψ (u) = λ ·. p 1−p + −1 1 − η + iu 1 + η − iu. (4). The fourth Lévy jump type is the increment of a Compound Poisson process with the correlated Double-Exponential jumps (DEJ-VCJ) in Kou, Yu, and Zhong (2016) which assumes that each jump size is independent drawn from an asymmetric conditional double exponential distribution with the density fCDE (x) with parameters p, η + , η − , and µv . 16.

(28) Hence the distribution of the jump size has density,. fCDE (x) =.                       . h x x i − + − − µ xρ 1 1 η v J p η+ −µ − e + q e e η+ 1{x≥0} + − η +µv ρJ v ρJ h x i 1 q − if ρJ > 0, e η− 1{x<0} , h η +µv ρJ x i − 1 p η+ −µ e η+ 1{x≥0} + v ρJ h i x − x − µ xρ 1 1 η − − e µv ρJ v J p η+ −µ + q 1{x<0} , if ρJ < 0, e e η − +µv ρJ v ρJ h i h i x x − − p η1+ e η+ 1{x≥0} + q η1− e η− 1{x<0} , if ρJ = 0,. where µv > 0, 0 < η + < 1, 0 < η − , 0 ≤ p, q ≤ 1, p + q = 1, p and q are probabilities of upwards and downward jumps, respectively. Moreover, the number of jumps nt+1 arriving between periods t and t + 1 follow a Poisson process with the intensity λ and v (x) = λ. fCDE (x) is a Lévy measure Since the number of jumps that occur within an. 政治大 Thus the characteristic exponent 立of the a Compound Poisson process with the correlated interval is finite, the correlated Double-Exponential jump belong to finite-activity type.. . − 1, if ρJ > 0 or ρJ < 0, if ρJ = 0,. ‧. ‧ 國. q 1+η − iu. 學. Double-Exponential jumps is  1  · 1−ηp+ iu + 1−µv ρJ ψ (u) = λ · p  + 1+ηq− iu − 1, 1−η + iu. sit. y. Nat. Besides the well-known Merton jump and Double-Exponential jump, we consider two infinite-activity Lévy jump: the Variance Gamma and Normal Inverse Gaussian process.. io. n. al. er. (5). First, we consider the Variance Gamma (VG) process introduced by Madan and. i Un. v. Seneta (1990) and Madan, Carr, and Chang (1998). The Variance Gamma process is. Ch. engchi. obtained by evaluating an arithmetic Brownian processes at an stochastic time intervals Tt with drift γ and variance σJ by an independent gamma process. That is, X (Tt ; γ, σJ ) = √ γTt +σJ Tt εPJ,t , where εt is a standard normal distribution which is independent of Tt , and conditional on Tt which follows gamma distribution with 1 shape and ν > 0 scale., i.e., Tt (t; 1, ν) ∼ 2 Gamma (1, ν). The process therefore provides two dimensions of control on the distribution. We observe below that control is attained over the skew via gamma and over kurtosis ν. Hence the distribution of the jump size has density, ! t Z ∞ g ν −1 exp − νg (X − γg)2 1 √ · fX(t) (x) = exp − t 2σ 2 g σ 2πg v ν Γ νt 0 2. The notation ∼ Gamma (α, β) means that the distribution is gamma distribution with parameters. α and β and the density(1/ (β α Γ (α))) xα−1 e−x/β , Γ is the gamma function.. 17.

(29) The characteristic function for the process is 1 2 2 1 ψ (u) = − ln 1 − iuγν + σ u ν ν 2 The process can also be expressed as difference of two independent increasing gamma processes X (t; γ, σJ , 1, ν) = Tp (t; µp , νp ) − Tn (t; µn , νn ) with. r 1 2σ 2 θ µp = γ2 + + , 2 ν 2 !2 r 1 2σ 2 θ νp = + γ2 + ν, 2 ν 2. 1 µn = 2 νn =. r γ2 + 1 2. r. 2σ 2 θ − , ν 2. 2σ 2 θ − γ2 + ν 2. !2 ν,. When viewed as the difference of two gamma processes, we may write the Lévy measure. 政治大. for Variance Gamma process X (t; γ, σJ , 1, ν) employing as  µp  µ2p · exp − νp x , for x ≥ 0, νp x ν (x) =  µ2n exp(− µνnn |x|) · , for x < 0, νn |x|. 立. ‧ 國. 學. We observe from Lévy measure of VG process inherits the property of an infinite arrival. ‧. rate of price jumps, from the gamma process. Thus, the VG jump belong to infinite-. sit. y. Nat. activity type.. (6). Second, we consider the Normal Inverse Gaussian process introduced by Barndorff-. io. n. al. er. Nielsen (1997, 1998). The Normal Inverse Gaussian process is obtained by evaluating. i Un. v. an arithmetic Brownian process at an stochastic time intervals Tt with drift βδ 2 and. Ch. engchi. variance δ by an independent Inverse Gaussian process with parameter δ, α and β. That √ is, X (Tt ; α, β, δ) = βδ 2 Tt + δ Tt εPJ,t , where εPJ,t is a standard normal distribution which is independent of Tt , and conditional on Tt which follows Inverse Gaussian distribution, i.e., p 2 2 Tt ∼ IG 1, δ α − β , where the probability density of Inverse Gaussian distribution is 2 3 p 1 δ − 2 2 fIG (x, δ, α, β) = (2π) δ exp δ α2 − β 2 x 2 exp − +x α −β 2 x p 3 where x > 0, and mean EIG = δ/ α2 − β 2 and variance V arIG = δ/ (α2 − β 2 ) 2 , and − 21. the probability density of Normal Inverse Gaussian distribution is q 2 2 K1 α (x − µ) + δ p αδ q fN IG (x, δ, α, β) = exp δ α2 − β 2 + β (x − µ) π (x − µ)2 + δ 2 18.

(30) where x > 0 and K1 is the modified Bessel function of third order and index 1, i.e., Z 1 1 ∞ −1 exp − u t + t dt K1 (u) = 2 0 2 Furthermore, α, β, µ and δ are parameters, satisfying 0 ≤ |β| ≤ α, µinR and δ > 0. The characteristic function for the NIG process is p q 2 2 2 2 ψ (u) = δ α − β − α − (β + iu) The Lévy measure for NIG process employing as X (Tt ; α, β, δ) employing as ν (x) =. αδ eβ|x| · K1 (α|x|) · π |x|. We observe from Lévy measure of NIG process inherits the property of an infinite arrival. 政治大. rate of return jumps, from the Inverse Gaussian process. Thus, the NIG jump belong to. 立. infinite-activity type.. ‧ 國. 學. 3.3. Stochastic Volatility Model with Merton Jumps. ‧. To solve the problem of the SV model, Bates (1996, 2000), Bakshi, Cao, and Chen (1997),. Nat. sit. y. and Pan (2002) based on the the SV model and Merton (1973) to develop an affine jump-. al. er. io. diffusion model with the stochastic volatility and Gaussian jumps. This model captures. n. two important features of equity index return dynamics, namely stochastic volatility and. Ch. i Un. v. return jumps, and also provide analytical tractability for option pricing in Section 4.1.2. engchi. and model estimation in Section 6. Jump in index returns can generate large movements such as the crash of 2008. That is, return jumps can capture the heavy-tailed feature, i.e. large return jumps, in short-term and also explain the smirkiness feature exhibited in the cross-sectional option data. Consider a probability space. . Ω, F, P, {Ft }Tt=0 , where Ω is the sample set of all. outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability measure. Let {Ft }Tt=0 be the filtration generated by the pair of the Wiener processes. P P dWy,t , dWv,t at time t, 0 ≤ t ≤ T . According to above definition, the index returns, yt , follows the stochastic volatility with Merton jumps (SV-MJ) model, and the stochastic volatility vt follows the mean-reverting square-root process. Then its dynamic process. 19.

(31) under physical probability measure P is: √ 1 P P P dyt = µ − vt − MJy dt + vt dWy,t + dJy,t , 2 √ P dvt = κ (θ − vt ) dt + σv vt dWv,t ,. (3.5). where µ is the drift term which is a constant, the compound Poisson process of return P P P jump dJy,t = ξy,t dNy,t , the jump size ξy,t is a random variable, dNy,t is used to depict the. number of the return jump which follows a Poisson process with the intensity λy which P ∼ 3 P oisson (λy dt), and MJPy is the compensator of the return is a constant, i.e., dNy,t P jump and the average jump amplitude, i.e., MJPy = E P dJy,t . The stochastic volatility. vt follows a mean-reverting square root process with long term level θ > 0, the speed of adjustment is κ > 0, and the variation coefficient of stochastic volatility is σv > 0. More-. 政治大 ∼ N (0, dt), and Corr(dW. P P are a pair of correlated Wiener processes with correlation coefficient and dWv,t over, dWy,t. 立. P P ∼ N (0, dt), dWv,t ρdt, i.e., dWy,t. P P y,t , dWv,t ). = ρdt. Therefore,. ‧ 國. 學. by the Cholesky decomposition, the correlated Wiener processes can be decomposed into p P P P P P two independent Wiener processes, i.e., dWy,t = dB1,t and dWv,t = ρdB1,t + 1 − ρ2 dB2,t .. ‧. Thus, the equation (3.5) can be rewritten as following √ 1 P P P dyt = µ − vt − MJy dt + vt dB1,t + dJy,t , 2 p √ P P 2 dvt = κ (θ − vt ) dt + σv vt dB1,t + 1 − ρ dB2,t ,. er. io. sit. y. Nat. (3.6). al. n. iv n C 2 deviation δ, i.e., ξy,t has a symmetric normal h e ndistribution, g c h i U i.e., ξy,t ∼ N (γ, δ ). Therefore,. P We suppose that the jump sizes dJy,t are normal distributed with mean γ and standard. the compensator of the return jump,. P 1 2 P (1) − 1 = λy MJPy = ψ (−i) = E P edJy,t − 1 = mdJy,t eγ+ 2 δ − 1 P P (h) is the moment generating function of dJy,t . In other word, where mdJy,t.  P = d dJy,t = ξy,t dNy,t. NP. y,t X.  j  ξy,t. j=1 iid. j 0 where return jump sizes ξy,t ∼ N (γ, σ 2 ), for j = 0, 1, ..., ∞, ξy,t = 0. In addition, a pair P P of the Wiener processes dWy,t , dWv,t is independent of ξy,t . We transform (3.5) into 3. eλ. The notation ∼ P oisson (λ) means that the distribution is Poisson with parameter λ and the density λk /k! , where k ≥ 0 and k! = k × (k − 1) × (k − 2) ×, ..., ×2 × 1 is the factorial of k.. 20.

(32) discrete-time version: p 1 P P yt = yt−1 + µ − vt − MJy ∆ + vt−1 ∆εPy,t + Jy,t , 2 p vt = vt−1 + κ (θ − vt−1 ) ∆ + σv vt−1 ∆εPv,t ,. (3.7). where ∆ is the time interval, µ is the drift term, the compound Poisson process of return P P P jump Jy,t = ξy,t Ny,t , the jump size ξy,t is a random variable, Ny,t is used to depict the. number of the return jump which follows a Poisson process with the intensity λy which is P a constant, i.e., dNy,t ∼ P oisson (λy ∆), and MJPy is the compensator of the return jump P and the average jump amplitude, i.e., MJPy = E P Jy,t . The stochastic volatility vt follows. a mean-reverting square root process with long term level θ > 0, the speed of adjustment κ > 0, and the variation coefficient of stochastic volatility σv > 0. Moreover, εPy,t and. 政治大 ∼ N (0, 1), and Corr(ε , ε ) = ρ. Therefore, by the Cholesky ∼ N (0, 1), i.e., 立 decomposition, the correlated Wiener processes can be decomposed into two independent εPv,t are diffusion noises with standard normal distribution with correlation coefficient ρ, P v,t. P y,t. εPv,t. Wiener processes, i.e., εPy,t = εP1,t and εPv,t = ρεP1,t +. 學. ‧ 國. εPy,t. p 1 − ρ2 εP2,t . Thus, under P measure,. ‧. the equation (3.7) can be rewritten as following p 1 P P yt = yt−1 + µ − vt − MJy ∆ + vt−1 ∆εP1,t + Jy,t , 2 p p P P 2 vt = vt−1 + κ (θ − vt−1 ) ∆ + σv vt−1 ∆ ρε1,t + 1 − ρ ε2,t ,. io. sit. y. Nat. (3.8). n. al. er. P where the jump sizes Jy,t are normal distributed, i.e., ξy,t has a symmetric normal distri-. i Un. v. bution, i.e., ξy,t ∼ N (γ, δ 2 ). Therefore, the compensator of the return jump, P γ+ 21 δ 2 MJy = ψ (−i) = λy e −1. Ch. engchi. In other word, NP. P = dJy,t = ξy,t Ny,t. y,t X. j ξy,t. j=1 iid. j 0 where return jump sizes ξy,t ∼ N (γ, σ 2 ), for j = 0, 1, ..., ∞, ξy,t = 0. In addition, a pair of the Wiener processes εP2,t , εP2,t is independent of ξy,t . For this dynamics process, we P T have observations {yt }Tt=0 , latent volatility variables {vt }Tt=0 , jump times {Ny,t }t=0 , jump. sizes {ξy,t }Tt=0 and model parameters space Θ = {µ, κ, θ, σv , ρ, γ, δ, λy }.. There are some drawbacks under the SV-MJ model: (i) The impact of a return jump is transient, that is, a jump in return today will not cause impact on the future 21.

(33) distribution of returns. In fact, the shocks of the index returns not only cause the large jump movement in index return, but also bring out the volatility increasing sharply and rapidly. (ii) Under the SV and SV-MJ models, the diffusive stochastic volatility can only increase gradually with a sequence of small normally distributed increments, that is, it cannot measure the rapid shock in the volatility. (iii) When we use normal distribution with negative mean to model jump sizes, such a distribution does not have a monotone decreasing density for both positive and negative jumps so that it may lead to a poor fitting for small jumps (see Kou et al. (2016)). To solve the first and second problems, we introduce the independent and correlated jump in volatility model in the Section 3.4 and 3.5. On the other hand, in order to solve the third question, we introduce the double-exponential and Lévy jump models in Section 3.9 and 3.10.. Stochastic Volatility Model with Independent Mer立. 學. ‧ 國. 3.4. 政治大. ton Jumps. ‧. The shocks of the index returns not only cause the large jump movement in index return, but also bring out the volatility increasing sharply and rapidly. The diffusive stochastic. y. Nat. sit. volatility can only increase gradually with a sequence of small normally distributed in-. er. io. crements, that is, it cannot measure the rapid shock in the volatility. Duffie et al. (2000). al. n. iv n C Thus, jump revises this drawback and capture h einnvolatility gchi U. and Eraker et al. (2003) consider the stochastic volatility model with independent jumps in returns and in volatility.. the rapid movements of jumps.. Consider a probability space. . Ω, F, P, {Ft }Tt=0 , where Ω is the sample set of all. outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability measure. Let {Ft }Tt=0 be the filtration generated by the pair of the Wiener processes. P. P P dWy,t , dWv,t and the pair of compound Poisson processes dJy,t , dJv,t at time t, 0 ≤ t ≤ T . According to above definition, the index returns, yt , follows the stochastic volatility with independent jumps in return and in volatility (SV-MJ-VIJ) model, and the stochastic volatility vt follows the mean-reverting square-root process. Then its dynamic process. 22.

(34) under physical probability measure (P) is: √ 1 P P P , + dJy,t dyt = µ − vt − MJy dt + vt dWy,t 2 √ P dvt = κ (θ − vt ) dt + σv vt dWv,t + dJv,t + dJv,t ,. (3.9). P P where µ is the drift term, the compound Poisson process of return jump dJy,t = ξy,t dNy,t , P is used to depict the number of the return the jump size ξy,t is a random variable, dNy,t. jump which follows a Poisson process with the intensity λy which is a constant, i.e., P dNy,t ∼ P oisson (λy dt), and MJPy is the compensator of the return jump and the average P jump amplitude, i.e., MJPy = E P dJy,t . The stochastic volatility vt follows a mean-. reverting square root process with long term level θ > 0, the speed of adjustment is κ > 0, and the variation coefficient of stochastic volatility is σv > 0. The compound. 政治大  . Poisson process of volatility jump,. NP. P dJv,t = ξv,t dNv,t = d. v,t X. j  ξv,t. j=1. 學. ‧ 國. 立. where the volatility jump size ξv,t is a random variable which follows the exponential. ‧. P distribution with mean µv , i.e. ξv,t ∼ exp (µv ), dNv,t is used to depict the number of. sit. y. Nat. the volatility jump which follows a Poisson process with the intensity λv which is a. io. er. P P P are a pair of correlated and dWv,t ∼ P oisson (λv dt). Moreover, dWy,t constant, i.e., dNv,t P P ∼ N (0, dt), Wiener processes with correlation coefficient ρdt, i.e., dWy,t ∼ N (0, dt), dWv,t. n. al. Ch. i Un. v. P P and Corr(dWy,t , dWv,t ) = ρdt. Therefore, by the Cholesky decomposition, the correlated. engchi. P Wiener processes can be decomposed into two independent Wiener processes, i.e., dWy,t = p P P P P and dWv,t = ρdB1,t + 1 − ρ2 dB2,t dB1,t . Thus, the equation (3.9) can be rewritten as. following . . √ P P dt + vt dB1,t + dJy,t , p √ P P dvt = κ (θ − vt ) dt + σv vt ρdB1,t + 1 − ρ2 dB2,t + dJv,t ,. dyt =. 1 µ − vt − MJPy 2. (3.10). P We suppose that the jump sizes dJy,t are normal distributed with mean γ and standard. deviation δ, i.e., ξy,t has a symmetric normal distribution, i.e., ξy,t ∼ N (γ, δ 2 ). Therefore, the compensator of the return jump, MJPy. = ψ (−i) = E. P. . e. P dJy,t. γ+ 21 δ 2 P (1) − 1 = λy − 1 = mdJy,t e −1. 23.

(35) P P (h) is the moment generating function of dJ where mdJy,t y,t . In other word,. . P Ny,t. . P = d dJy,t = ξy,t dNy,t. X. j  ξy,t. j=1 iid. j 0 = 0. In addition, a pair where return jump sizes ξy,t ∼ N (γ, σ 2 ), for j = 0, 1, ..., ∞, ξy,t P P is independent of ξy,t and ξv,t . We transform (3.9) , dWv,t of the Wiener processes dWy,t. into discrete-time version: p 1 P P yt = yt−1 + µ − vt − MJy ∆ + vt−1 ∆εPy,t + Jy,t , 2 p vt = vt−1 + κ (θ − vt−1 ) ∆ + σv vt−1 ∆εPv,t + Jv,t ,. (3.11). where ∆ is the time interval, µ is the drift term, the compound Poisson process of return. 政治大 number of the return jump which 立 follows a Poisson process with the intensity λ. P P P jump Jy,t = ξy,t Ny,t , the jump size ξy,t is a random variable, Ny,t is used to depict the y. which. ‧ 國. 學. P is a constant, i.e., dNy,t ∼ P oisson (λy ∆), and MJPy is the compensator of the return P . The stochastic volatility jump and the average jump amplitude, i.e., MJPy = E P Jy,t. ‧. vt follows a mean-reverting square root process with long term level θ > 0, the speed of adjustment κ > 0, and the variation coefficient of stochastic volatility σv > 0. The. Nat. NP. P Jv,t = ξv,t Nv,t =. n. al. Ch. v,t X. j ξv,t. j=1. engchi. er. io. sit. y. compound Poisson process of volatility jump,. i Un. v. where the volatility jump size ξv,t is a random variable which follows the exponential P distribution with mean µv , i.e. ξv,t ∼ exp (µv ), dNv,t is used to depict the number of the. volatility jump which follows a Poisson process with the intensity λv which is a constant, P i.e., dNv,t ∼ P oisson (λv dt). Moreover, εPy,t and εPv,t are diffusion noises with standard. normal distribution with correlation coefficient ρ, i.e., εPy,t ∼ N (0, 1), εPv,t ∼ N (0, 1), and Corr(εPy,t , εPv,t ) = ρ. Therefore, by the Cholesky decomposition, the correlated Wiener processes can be decomposed into two independent Wiener processes, i.e., εPy,t = εP1,t and p εPv,t = ρεP1,t + 1 − ρ2 εP2,t . Thus, under P measure, the equation (3.11) can be rewritten as following p 1 P P yt = yt−1 + µ − vt − MJy ∆ + vt−1 ∆εP1,t + Jy,t , 2 p p P P 2 vt = vt−1 + κ (θ − vt−1 ) ∆ + σv vt−1 ∆ ρε1,t + 1 − ρ ε2,t + Jv,t , 24. (3.12).