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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 2-二月-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\nrisk premiums significantly increase after the financial crisis periods, that is, the panic in the post-crisis period causes more expected returns.參考文獻 [1] Andersen, T., L. Benzoni, and J. Lund. 2002, "An Empirical Investigation of\nContinuous-Time Equity Return Models," The Journal of Finance, Vol. 57, 1239-\n1284.\n[2] Bates, D. S., 2001, "Jumps and Stochastic Volatility: Exchange Rate Processes\nImplicit in Deutsche Mark Options," Review of Financial Studies, Vol. 9, 69-107.\n[3] Bakshi, G., C. Cao, and Z. W. Chen. 1997, "Empirical Performance of Alternative\nOption Pricing Models", Journal of Finance, Vol. 52, 2003-2049.\n[4] Barndorff-Nielsen, O.E. 1997, "Normal Inverse Gaussian Distributions and Stochastic\nVolatility Modelling", Journal of Statistics, Vol. 24, 1-13.\n[5] Barndorff-Nielsen, O.E. 1998, "Processes of Normal Inverse Gaussian Type", Finance\nAnd Stochastics, Vol. 2, 41-68.\n[6] Black, F., Scholes, M., 1973, "The pricing of options and corporate liabilities", Journal of Political Economy, Vol. 81, 637-654.\n[7] Bakshi, G., Cao, C., Chen, Z. W., 1997, "Empirical performance of alternative option\npricing models", Journal of Finance, Vol. 52, 2003-2049.\n[8] Broadie, M., Chernov, M., Johannes, M., 2007, "Model specification and risk premia:\nEvidence from futures options", Journal of Finance, Vol. 62, 1453-1490.\n[9] Bakshi, G., Wu, L., 2010, "The behavior of risk and market prices of risk over the\nNasdaq bubble period", Management Science, Vol. 56, 2251-2264.\n[10] Cox, J.C., J.E. Ingersoll, and S.A. Ross., 1985, "A Theory of the Term Structure of\nInterest Rates", Econometrica, Vol. 53, 385-408.\n[11] Carr, P., and D. Madan., 1998, "Option Valuation Using the Fast Fourier Transform",\nJournal of Computational Finance, Vol. 2, 61{73.\n[12] Carr, P., Geman, H., Madan, D., Yor, M., 2002, "The fine structure of asset returns:\nan empirical investigation", Journal of Business, Vol. 75, 305-332.\n[13] Christoffersen, P., Jacobs, K., Mimouni, K., 2010, "Volatility dynamics for the\nS&P500: Evidence from realized volatility, daily returns, and option prices", Re-\nview of Financial Studies, Vol. 23, 3141-3189.\n[14] Chernov, M., Ghysels, E., 2000, "A study towards a uniifed approach to the joint estimation\nof objective and risk-neutral measures for the purpose of option valuation",\nJournal of Financial Economics, Vol. 56, 407-458.\n[15] Cartea, A., Figueroa, M. G., 2005, "Pricing in electricity markets: A mean reverting jump diffusion model with seasonality", Applied Mathematical Finance, Vol. 12, 313-\n335.\n[16] Chevallier, J., Ielpo, F., 2013, "Twenty years of jumps in commodity markets",\nInternational Review of Applied Economics, Vol. 28, 64-82.\n[17] Diebold, F., Mariano, R., 1995, "Comparing Predictive Accuracy", Journal of Busi-\nness & Economic Statistics, Vol. 13, 134-144.\n[18] Duffie, D., Pan, J., Singleton, K., 2000, "Transform analysis and asset pricing for\naffine jump-diffusions", Econometrica, Vol. 68, 1343-1376.\n[19] Daskalakis, G., Psychoyios, D., Markellos, R. N., 2009, "Modeling CO2 emission\nallowance prices and derivatives: Evidence from the European trading scheme",\nJournal of Banking & Finance, Vol. 33, 1230-1241.\n[20] Diewald, L., Prokopczuk, M., Wese Simen, C., 2015, "Time-variations in commodity\nprice jumps", Journal of Empirical Finance, Vol. 31, 72-84.\n[21] Eraker B, Johannes M, Polson N, 2003, "The impact of jumps in volatility and\nreturns", The Journal of Finance, Vol. 58, 1269-1300.\n[22] Eraker, B., 2004, "Do stock prices and volatility jump? Reconciling evidence from\nspot and option prices", Journal of Finance, Vol. 59, 1367-1404.\n[23] EGodsill, S. J., A. Doucet , M. West, 2004, "Monte Carlo Smoothing for Nonlinear\nTime Series", Journal of the American Statistical Association, Vol. 99, 156-168.\n[24] Gerber, H. U., Shiu, Elias S. W, 1994, "Option pricing by Esscher transforms",\nTransactions of the Society of Actuaries, Vol. 46, 99{191.\n[25] Hu, F., Zidek, J.V, 2002, "The weighted likelihood", Canadian Journal of Statistics,\nVol. 30, 347{371.\n[26] Harvey, D., Leybourne, S., and Newbold, P, 1997, "Testing the equality of prediction\nmean squared errors", International Journal of Forecasting, Vol. 13, 281-291.\n[27] Heston, S. L., Nandi, S., 2000, "A closed-form GARCH option valuation model",\nReview of Financial Studies, Vol. 13, 585-625.\n[28] Heston, S. L, 1993, "A Closed-Form Solution for Options with Stochastic Volatility\nwith Applications to Bond and Currency Options". Review of Financial Studies, Vol.\n6, 327-343.\n[29] Hull, J., White, A., 1987, "The pricing of options on assets with stochastic volatilities",\nJournal of Finance, Vol. 42, 281-300.\n[30] Hull, J, C., 2014, "Options, futures, and other derivatives", Prentice Hall, 9th edition.\n[31] Hsu, C. C., Lin, S. K., Chen, T. F., 2014, "Pricing and hedging European energy\nderivatives: A case study of WTI crude oil options", Asia-Pacific Journal of Finan-\ncial Studies, Vol. 43, 317-355.\n[32] Huang, J. Z., Wu, Z., 2004, "Specification analysis of option pricing models based\non time-changed Levy processes", Journal of Finance, Vol. 59, 1405-1439.\n[33] Hilliard, J. E., Reis, J. A., 1999, "Jump processes in commodity futures prices and\noptions pricing", American Journal of Agricultural Economics, Vol. 81, 273-286.\n[34] Johannes, M. S., Polson, N. G., Stroud, J. R., 2009, "Optimal filtering of jump\ndiffusions: Extracting latent states from asset prices", Review of Financial Studies,\nVol. 22, 2759-2799.\n[35] Kou S, 2002, "A Jump-Diffusion Model for Option Pricing". Management Science,\nVol. 48, 1086-1101.\n[36] Kou S, Yu, Zhong, 2016, "Jumps in Equity Index Returns Before and During the\nRecent Financial Crisis: A Bayesian Analysis", Management Science, Vol. 4, 988-\n1010.\n[37] Kaeck A., Alexander, C., 2013, "Stochastic volatility jump-diffusions for European\nequity index dynamics", European Financial Management, Vol. 19, 470-496.\n[38] Koekebakker, S., Lien, G., 2004, "Volatility and price jumps in agricultural futures\nprices-evidence from wheat options", American Journal of Agricultural Economics,\nVol. 86, 1018-1031.\n[39] Li H, Wells M, Yu C, 2008, "A Bayesian Analysis of Return Dynamics with Levy\nJumps", The Review of Financial Studies, Vol. 21, 2345-2378.\n[40] Li H, Wells M, Yu C, 2011, "MCMC estimation of Levy jump models using stock\nand option prices", Mathematical Finance, Vol. 21, 383-422.\n[41] Merton, R. C, 1976, "Option Pricing with Underlying Stock Returns are Discontinuous",\nJournal of Financial Economics, Vol. 3, 124-144.\n[42] Madan, D., P. Carr, and E. Chang, 1998, "The Variance Gamma Process and Option\nPricing", European Finance Review, Vol. 2, 79{105.\n[43] Madan, D.B., Seneta, E, 1990, "The Variance Gamma (V.G.) Model for Share Market\nReturns", The Journal of Business, Vol. 63, 511-24.\n[44] Matsuda, K., 2004, "Introduction to Merton jump diffusion model", Department of\nEconomics. The Graduate Center, The City University of New York.\n[45] Mayer, K., Schmid, T., Weber, F., 2015, "Modeling electricity spot prices: Combining\nmean reversion, spikes, and stochastic volatility", European Journal of Finance,\nVol. 21, 292-315.\n[46] Pan, J., 2002, "The jump-risk premia implicit in options: evidence from an integrated\ntime-series study", Journal of Financial Economics, Vol. 63, 3-50.\n[47] Ramezani, C. A., Zeng, Y., 2002, "Maximum likelihood estimation of asymmetric\njump-diffusion process: Application to security prices", Working paper.\n[48] Nakajima, K., Ohashi, K., 2012, "A cointegrated commodity pricing model", Journal\nof Futures Markets, Vol. 32, 995-1033.\n[49] Ornthanalai, C., 2014, "Levy Jump Risk: Evidence from Options and Returns",\nJournal of Financial Economics, Vol. 112, 69-90.\n[50] Stein, C., 1956, "Inadmissibility of the Usual Estimator for the Mean of a Multivariate\nNormal Distribution", Proceedings of the Third Berkeley Symposium on\nMathematical Statistics and Probability, Vol. 1, 197-206.\n[51] Schmitz, A., Wang, Z., Kimn, J. H., 2014, "A jump diffusion model for agricultural\ncommodities with Bayesian analysis", Journal of Futures Market, Vol. 34, 235-260.\n[52] Wilmot, N. A., Mason, C. F., 2013, "Jump processes in the market for crude oil",\nThe Energy Journal, Vol. 34, 33-48.\n[53] Xiao, Y., Colwell, D. B., Bhar, R., 2015, "Risk premium in electricity prices: Evidence\nfrom the PJM market", Journal of Futures Markets, Vol. 35, 776-793. 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-Kuei en_US dc.contributor.author (作者) 鍾長恕 zh_TW dc.contributor.author (作者) Chung, Chang-Shu en_US dc.creator (作者) 鍾長恕 zh_TW dc.creator (作者) Chung, Chang-Shu en_US dc.date (日期) 2018 en_US dc.date.accessioned 2-二月-2018 16:42:02 (UTC+8) - dc.date.available 2-二月-2018 16:42:02 (UTC+8) - dc.date.issued (上傳時間) 2-二月-2018 16:42:02 (UTC+8) - dc.identifier (其他 識別碼) G0104352033 en_US dc.identifier.uri (URI) https://ah.lib.nccu.edu.tw/item?item_id=133183 - 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) 104352033 zh_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\nrisk 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\n2 Literature Review . . .7\n2.1 The Background of Research Issue . . . . . . 7\n2.2 The Stochastic Volatility and Jump Diffusion Processes . .. . . . 9\n3 The Models . . . . 13\n3.1 Stochastic Volatility Model . . . . . . . . . . . 13\n3.2 The Characteristic Exponent of Levy Jump Processes . 15\n3.3 Stochastic Volatility Model with Merton Jumps . . . 19\n3.4 Stochastic Volatility Model with Independent Merton Jumps . . . . . . . 22\n3.5 Stochastic Volatility Model with Correlated Merton Jumps . . . . . . . . 25\n3.6 Stochastic Volatility Model with Double-Exponential Jumps . . . . . . . 28\n3.7 Stochastic Volatility Model with Independent Double-Exponential Jumps 32\n3.8 Stochastic Volatility Model with Correlated Double-Exponential Jumps . 36\n3.9 Stochastic Volatility Model with Variance-Gamma Jumps . . . . . . . . . 40\n3.10 Stochastic Volatility Model with Normal Inverse Gaussian Jumps . . . . 42\n4 The Risk-Neutral Dynamics and Characteristic Functions 45\n4.1 The Risk-Neutral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 45\n4.1.1 Stochastic Volatility Model . . . . . . . . . . . . . . . . . . . . . . 46\n4.1.2 Stochastic Volatility Model with Merton Jumps . . . . . . . . . . 46\n4.1.3 Stochastic Volatility Model with Independent Merton Jumps . . . 47\n4.1.4 Stochastic Volatility Model with Correlated Merton Jumps . . . . 47\n4.1.5 Stochastic Volatility Model with Double-Exponential Jumps . . . 48\n4.1.6 Stochastic Volatility Model with Independent Double-Exponential\nJumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n4.1.7 Stochastic Volatility Model with Correlated Double-Exponential\nJumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50\n4.1.8 Stochastic Volatility Model with Variance-Gamma Jumps . . . . . 51\n4.1.9 Stochastic Volatility Model with Normal Inverse Gaussian Jumps 52\n4.2 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 53\n4.2.1 Stochastic Volatility Model . . . . . . . . . . . . . . . . . . . . . . 53\n4.2.2 Stochastic Volatility Model with Merton Jumps . . . . . . . . . . 54\n4.2.3 Stochastic Volatility Model with Independent Merton Jumps . . . 55\n4.2.4 Stochastic Volatility Model with Correlated Merton Jumps . . . . 56\n4.2.5 Stochastic Volatility Model with Double Exponential Jumps . . . 57\n4.2.6 Stochastic Volatility Model with Independent Double Exponential\nJumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58\n4.2.7 Stochastic Volatility Model with Correlated Double Exponential\nJumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59\n4.2.8 Stochastic Volatility Model with Variance-Gamma Jumps . . . . . 60\n4.2.9 Stochastic Volatility Model with Normal Inverse Gaussian Jumps 61\n5 Numerical method 63\n5.1 The Fourier Transform Methods for Derivatives Pricing . . . . . . . . . . 63\n5.2 The Fourier Transform Methods of Out-of-the-Money (OTM) Option Pricing 65\n5.3 European Option Pricing using the Fast Fourier Transform (FFT) . . . . 67\n6 Estimation Method 69\n6.1 Nonlinear Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . 69\n6.2 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71\n6.3 Particle Filtering Method . . . . . . . . . . . . . . . . . . . . . . . . . . 71\n6.4 Smoothing using Backwards Simulation . . . . . . . . . . . . . . . . . . . 74\n6.5 Parameter Estimation using EM Algorithm with Particle Filtering Method 75\n6.6 Joint Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80\n6.7 Model Diagnostics and Comparisons . . . . . . . . . . . . . . . . . . . . 81\n7 Empirical Analysis 85\n7.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85\n7.2 Estimated Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86\n7.2.1 Model Parameters and Latent Volatility/Jump Variables . . . . . 87\n7.2.2 Performances in Modeling the Spot Return . . . . . . . . . . . . . 91\n7.2.3 Joint Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 93\n7.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97\n7.4 Model Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98\n7.4.1 In Sample Pricing Performance . . . . . . . . . . . . . . . . . . . 99\n7.4.2 Out-of-Sample Pricing Performance . . . . . . . . . . . . . . . . . 103\n8 Conclusion . . .108\nBibliography . . .109\n\nAppendix A Change Measure: Stochastic Volatility Model 114\nAppendix B Change Measure: Stochastic Volatility Model with Merton Jumps . . .117\nAppendix C Change Measure: Stochastic Volatility Model with Indepen-dent Merton Jumps . . .121\nAppendix D Change Measure: Stochastic Volatility Model with Corre-lated Merton Jumps . . .125\nAppendix E Change Measure: Stochastic Volatility Model with Double-Exponential Jumps . . .129\nAppendix F Change Measure: Stochastic Volatility Model with Indepen-dent Double-Exponential Jumps . . .133\nAppendix G Change Measure: Stochastic Volatility Model with Corre-lated Double-Exponential Jumps . . .137\nAppendix H Change Measure: Stochastic Volatility Model with Variance Gamma Process . . .143\nAppendix I Change Measure: Stochastic Volatility Model with Normal Inverse Gaussian Process . . .146\nAppendix J Characteristic Function: Stochastic Volatility . . .149\nAppendix K Characteristic Function: Stochastic Volatility with Merton Jumps . . .151\nAppendix L Characteristic Function: Stochastic Volatility with Indepen-dent Merton Jumps . . .153\nAppendix M Characteristic Function: Stochastic Volatility with Correlated Merton Jumps . . .156\nAppendix N Characteristic Function: Stochastic Volatility with Double-Exponential Jumps . . .159\nAppendix O Characteristic Function: Stochastic Volatility with Indepen-dent Double-ExponentialJumps . . .161\nAppendix P Characteristic Function: Stochastic Volatility with Correlated Double-Exponential Jumps . . .164\nAppendix Q Characteristic Function: Stochastic Volatility with Variance Gamma Jumps . . .167\nAppendix 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/#G0104352033 en_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 volatility en_US dc.subject (關鍵詞) Jump risk en_US dc.subject (關鍵詞) Risk premiums en_US dc.subject (關鍵詞) Particle-Filtering algorithm en_US dc.subject (關鍵詞) Joint estimation en_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 crisis en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) [1] Andersen, T., L. Benzoni, and J. Lund. 2002, "An Empirical Investigation of\nContinuous-Time Equity Return Models," The Journal of Finance, Vol. 57, 1239-\n1284.\n[2] Bates, D. S., 2001, "Jumps and Stochastic Volatility: Exchange Rate Processes\nImplicit in Deutsche Mark Options," Review of Financial Studies, Vol. 9, 69-107.\n[3] Bakshi, G., C. Cao, and Z. W. Chen. 1997, "Empirical Performance of Alternative\nOption Pricing Models", Journal of Finance, Vol. 52, 2003-2049.\n[4] Barndorff-Nielsen, O.E. 1997, "Normal Inverse Gaussian Distributions and Stochastic\nVolatility Modelling", Journal of Statistics, Vol. 24, 1-13.\n[5] Barndorff-Nielsen, O.E. 1998, "Processes of Normal Inverse Gaussian Type", Finance\nAnd Stochastics, Vol. 2, 41-68.\n[6] Black, F., Scholes, M., 1973, "The pricing of options and corporate liabilities", Journal of Political Economy, Vol. 81, 637-654.\n[7] Bakshi, G., Cao, C., Chen, Z. 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