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| 研究生： |
鍾長恕 Chung, Chang-Shu |
|---|---|
| 論文名稱： |
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 |
| 指導教授： |
林士貴
Lin, Shih-Kuei |
| 學位類別： |
碩士
Master |
| 系所名稱： |
商學院 - 金融學系 Department of Money and Banking |
| 論文出版年： | 2018 |
| 畢業學年度： | 106 |
| 語文別： | 中文 |
| 論文頁數： | 219 |
| 中文關鍵詞： | 隨機波動度 、跳躍風險 、風險溢酬 、粒子濾波演算法 、共同估計 |
| 外文關鍵詞： | Stochastic volatility, Jump risk, Risk premiums, Particle-Filtering algorithm, Joint estimation |
| 相關次數： | 點閱：33 下載：11 |
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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 signicant changes in volatility risk premiums and in jump risk premiums before, during or after the nancial 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.
1 Introduction 1
2 Literature Review 7
2.1 The Background of Research Issue 7
2.2 The Stochastic Volatility and Jump Diusion 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 Jumps60
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
[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.
[4] Barndor-Nielsen, O.E. 1997, "Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling", Journal of Statistics, Vol. 24, 1-13.
[5] Barndor-Nielsen, O.E. 1998, "Processes of Normal Inverse Gaussian Type", Finance And Stochastics, Vol. 2, 41-68.
[6] Black, F., Scholes, M., 1973, "The pricing of options and corporate liabilities", Journal of Political Economy, Vol. 81, 637-654.
[7] Bakshi, G., Cao, C., Chen, Z. W., 1997, "Empirical performance of alternative option pricing models", Journal of Finance, Vol. 52, 2003-2049.
[8] Broadie, M., Chernov, M., Johannes, M., 2007, "Model specication and risk premia: Evidence from futures options", Journal of Finance, Vol. 62, 1453-1490.
[9] Bakshi, G., Wu, L., 2010, "The behavior of risk and market prices of risk over the Nasdaq bubble period", Management Science, Vol. 56, 2251-2264.
[10] Cox, J.C., J.E. Ingersoll, and S.A. Ross., 1985, "A Theory of the Term Structure of Interest Rates", Econometrica, Vol. 53, 385-408.
[11] Carr, P., and D. Madan., 1998, "Option Valuation Using the Fast Fourier Transform", Journal of Computational Finance, Vol. 2, 61{73.
[12] Carr, P., Geman, H., Madan, D., Yor, M., 2002, "The fine structure of asset returns: an empirical investigation", Journal of Business, Vol. 75, 305-332.
[13] Christoersen, P., Jacobs, K., Mimouni, K., 2010, "Volatility dynamics for the S&P500: Evidence from realized volatility, daily returns, and option prices", Re-view of Financial Studies, Vol. 23, 3141-3189.
[14] Chernov, M., Ghysels, E., 2000, "A study towards a unied approach to the joint estimation of objective and risk-neutral measures for the purpose of option valuation", Journal of Financial Economics, Vol. 56, 407-458.
[15] Cartea, A., Figueroa, M. G., 2005, "Pricing in electricity markets: A mean reverting jump diusion model with seasonality", Applied Mathematical Finance, Vol. 12, 313-335.
[16] Chevallier, J., Ielpo, F., 2013, "Twenty years of jumps in commodity markets", International Review of Applied Economics, Vol. 28, 64-82.
[17] Diebold, F., Mariano, R., 1995, "Comparing Predictive Accuracy", Journal of Busi-ness & Economic Statistics, Vol. 13, 134-144.
[18] Due, D., Pan, J., Singleton, K., 2000, "Transform analysis and asset pricing for ane jump-diusions", Econometrica, Vol. 68, 1343-1376.
[19] Daskalakis, G., Psychoyios, D., Markellos, R. N., 2009, "Modeling CO2 emission allowance prices and derivatives: Evidence from the European trading scheme", Journal of Banking & Finance, Vol. 33, 1230-1241.
[20] Diewald, L., Prokopczuk, M., Wese Simen, C., 2015, "Time-variations in commodity price jumps", Journal of Empirical Finance, Vol. 31, 72-84.
[21] Eraker B, Johannes M, Polson N, 2003, "The impact of jumps in volatility and returns", The Journal of Finance, Vol. 58, 1269-1300.
[22] Eraker, B., 2004, "Do stock prices and volatility jump? Reconciling evidence from spot and option prices", Journal of Finance, Vol. 59, 1367-1404.
[23] EGodsill, S. J., A. Doucet , M. West, 2004, "Monte Carlo Smoothing for Nonlinear Time Series", Journal of the American Statistical Association, Vol. 99, 156-168.
[24] Gerber, H. U., Shiu, Elias S. W, 1994, "Option pricing by Esscher transforms", Transactions of the Society of Actuaries, Vol. 46, 99{191.
[25] Hu, F., Zidek, J.V, 2002, "The weighted likelihood", Canadian Journal of Statistics, Vol. 30, 347{371.
[26] Harvey, D., Leybourne, S., and Newbold, P, 1997, "Testing the equality of prediction mean squared errors", International Journal of Forecasting, Vol. 13, 281-291.
[27] Heston, S. L., Nandi, S., 2000, "A closed-form GARCH option valuation model", Review of Financial Studies, Vol. 13, 585-625.
[28] Heston, S. L, 1993, "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options". Review of Financial Studies, Vol. 6, 327-343.
[29] Hull, J., White, A., 1987, "The pricing of options on assets with stochastic volatilities", Journal of Finance, Vol. 42, 281-300.
[30] Hull, J, C., 2014, "Options, futures, and other derivatives", Prentice Hall, 9th edition.
[31] Hsu, C. C., Lin, S. K., Chen, T. F., 2014, "Pricing and hedging European energy derivatives: A case study of WTI crude oil options", Asia-Pacic Journal of Finan-cial Studies, Vol. 43, 317-355.
[32] Huang, J. Z., Wu, Z., 2004, "Specication analysis of option pricing models based on time-changed Levy processes", Journal of Finance, Vol. 59, 1405-1439.
[33] Hilliard, J. E., Reis, J. A., 1999, "Jump processes in commodity futures prices and options pricing", American Journal of Agricultural Economics, Vol. 81, 273-286.
[34] Johannes, M. S., Polson, N. G., Stroud, J. R., 2009, "Optimal ltering of jump diusions: Extracting latent states from asset prices", Review of Financial Studies, Vol. 22, 2759-2799.
[35] Kou S, 2002, "A Jump-Diusion Model for Option Pricing". Management Science, Vol. 48, 1086-1101.
[36] Kou S, Yu, Zhong, 2016, "Jumps in Equity Index Returns Before and During the Recent Financial Crisis: A Bayesian Analysis", Management Science, Vol. 4, 988-1010.
[37] Kaeck A., Alexander, C., 2013, "Stochastic volatility jump-diusions for European equity index dynamics", European Financial Management, Vol. 19, 470-496.
[38] Koekebakker, S., Lien, G., 2004, "Volatility and price jumps in agricultural futures prices-evidence from wheat options", American Journal of Agricultural Economics, Vol. 86, 1018-1031.
[39] Li H, Wells M, Yu C, 2008, "A Bayesian Analysis of Return Dynamics with Levy Jumps", The Review of Financial Studies, Vol. 21, 2345-2378.
[40] Li H, Wells M, Yu C, 2011, "MCMC estimation of Levy jump models using stock and option prices", Mathematical Finance, Vol. 21, 383-422.
[41] Merton, R. C, 1976, "Option Pricing with Underlying Stock Returns are Discontinuous", Journal of Financial Economics, Vol. 3, 124-144.
[42] Madan, D., P. Carr, and E. Chang, 1998, "The Variance Gamma Process and Option Pricing", European Finance Review, Vol. 2, 79{105.
[43] Madan, D.B., Seneta, E, 1990, "The Variance Gamma (V.G.) Model for Share Market Returns", The Journal of Business, Vol. 63, 511-24.
[44] Matsuda, K., 2004, "Introduction to Merton jump diusion model", Department of Economics. The Graduate Center, The City University of New York.
[45] Mayer, K., Schmid, T., Weber, F., 2015, "Modeling electricity spot prices: Combining mean reversion, spikes, and stochastic volatility", European Journal of Finance, Vol. 21, 292-315.
[46] Pan, J., 2002, "The jump-risk premia implicit in options: evidence from an integrated time-series study", Journal of Financial Economics, Vol. 63, 3-50.
[47] Ramezani, C. A., Zeng, Y., 2002, "Maximum likelihood estimation of asymmetric jump-diusion process: Application to security prices", Working paper.
[48] Nakajima, K., Ohashi, K., 2012, "A cointegrated commodity pricing model", Journal of Futures Markets, Vol. 32, 995-1033.
[49] Ornthanalai, C., 2014, "Levy Jump Risk: Evidence from Options and Returns", Journal of Financial Economics, Vol. 112, 69-90.
[50] Stein, C., 1956, "Inadmissibility of the Usual Estimator for the Mean of a Multivariate Normal Distribution", Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 197-206.
[51] Schmitz, A., Wang, Z., Kimn, J. H., 2014, "A jump diusion model for agricultural commodities with Bayesian analysis", Journal of Futures Market, Vol. 34, 235-260.
[52] Wilmot, N. A., Mason, C. F., 2013, "Jump processes in the market for crude oil", The Energy Journal, Vol. 34, 33-48.
[53] Xiao, Y., Colwell, D. B., Bhar, R., 2015, "Risk premium in electricity prices: Evidence from the PJM market", Journal of Futures Markets, Vol. 35, 776-793.
