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題名 跳躍風險與隨機波動度下溫度衍生性商品之評價
Pricing Temperature Derivatives under Jump Risks and Stochastic Volatility
作者 莊明哲
Chuang, Ming Che
貢獻者 林士貴
Lin, Shih Kuei
莊明哲
Chuang, Ming Che
關鍵詞 日均溫
冷氣指數／暖氣指數衍生性商品
風險中立評價法
隨機波動度
跳躍風險
粒子濾波演算法
期望最大演算法
daily average temperature index
CDD/HDD derivatives
risk-neutral pricing method
stochastic volatility
jump risk
particle filter algorithm
expectation-maximization algorithm
日期 2015
上傳時間 17-Aug-2015 15:02:39 (UTC+8)
摘要 本研究利用美國芝加哥商品交易所針對 18 個城市發行之冷氣指數／暖氣指數衍生性商品與相對應之日均溫進行分析與評價。研究成果與貢獻如下：一、延伸 Alaton, Djehince, and Stillberg (2002) 模型，引入跳躍風險、隨機波動度、波動跳躍等因子，提出新模型以捕捉更多溫度指數之特徵。二、針對不同模型，分別利用最大概似法、期望最大演算法、粒子濾波演算法等進行參數估計。實證結果顯示新模型具有較好之配適能力。三、利用 Esscher 轉換將真實機率測度轉換至風險中立機率測度，並進一步利用　Feynman-Kac 方程式與傅立葉轉換求出溫度模型之機率分配。四、推導冷氣指數／暖氣指數期貨之半封閉評價公式，而冷氣指數／暖氣指數期貨之選擇權不存在封閉評價公式，則利用蒙地卡羅模擬進行評價。五、無論樣本內與樣本外之定價誤差，考慮隨機波動度型態之模型對於溫度衍生性商品皆具有較好之評價績效。六、實證指出溫度市場之市場風險價格為負，顯示投資人承受較高之溫度風險時會要求較高之風險溢酬。本研究可給予受溫度風險影響之產業，針對衍生性商品之評價與模型參數估計上提供較為精準、客觀與較有效率之工具。
This study uses the daily average temperature index (DAT) and market price of the CDD/HDD derivatives for 18 cities from the CME group. There are some contributions in this study: (i) we extend the Alaton, Djehince, and Stillberg (2002)`s framework by introducing the jump risk, the stochastic volatility, and the jump in volatility. (ii) The model parameters are estimated by the MLE, the EM algorithm, and the PF algorithm. And, the complex model exists the better goodness-of-fit for the path of the temperature index. (iii) We employ the Esscher transform to change the probability measure and derive the probability density function of each model by the Feynman-Kac formula and the Fourier transform. (iv) The semi-closed form of the CDD/HDD futures pricing formula is derived, and we use the Monte-Carlo simulation to value the CDD/HDD futures options due to no closed-form solution. (v) Whatever in-sample and out-of-sample pricing performance, the type of the stochastic volatility performs the better fitting for the temperature derivatives. (vi) The market price of risk differs to zero significantly (most are negative), so the investors require the positive weather risk premium for the derivatives. The results in this study can provide the guide of fitting model and pricing derivatives to the weather-linked institutions in the future.
參考文獻 [1] Alaton, P., B. Djehince, and D. Stillberg, 2002, "On Modelling and Pricing Weather Derivatives," Applied Mathematical Finance, Vol. 9, 1-20.
[2] Andricopoulos, A. D., M. Widdicks, P. W. Duck, and D. P. Newton, 2003, "Universal Option Valuation Using Quadrature Methods," Journal of Financial Economics, Vol. 67, 447-471.
[3] Andricopoulos, A. D., M. Widdicks, D. P. Newton, and P. W. Duck, 2007, "Extending Quadrature Methods to Value Multi-Asset and Complex Path Dependent Options," Journal of Financial Economics, Vol. 83, 471-499.
[4] Bates, D. S., 2001, "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Vol. 9, 69-107.
[5] Barndorff-Nielsen, O. E. and N. Shephard, 2001, "Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses in Financial Economics," Journal of the Royal Statistical Society. Series B, Vol. 63, 167-241.
[6] Bellini, F., 2005, "The Weather Derivatives Market: Modelling and Pricing Temperature," Ph.D. thesis, University of Lugano.
[7] Benth, F. E. and J. Saltyte-Benth, 2005, "Stochastic Modelling of Temperature Variations with a View Towards Weather Derivatives," Applied Mathematical Finance, Vol. 12, 53-85.
[8] Benth, F. E. and J. Saltyte-Benth, 2007, "The Volatility of Temperature and Pricing of Weather Derivatives," Quantitative Finance, Vol. 7, 553-561.
[9] Benth, F. E. and J. Saltyte-Benth, 2011, "Weather Derivatives and Stochastic Modelling of Temperature," International Journal of Stochastic Analysis, Vol. 2011, 1-21.
[10] Bibby, B. M., and M. Sorensen, 1995, "Martingale Estimation Functions for Discretely Observed Diffusion Processes," Bernoulli, Vol. 1, 17-39.
[11] Black, F. and M. Scholes, 1973, "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, Vol. 81, 637-654.
[12] Bollerslev, T, 1986, "Generalized Autoregressive Conditional Heteroscedasticity," Journal of Econometrics, Vol. 31, 307-327.
[13] Brody, D. C., J. Syroka, and M. Zervos, 2002, "Dynamical Pricing of Weather Derivatives," Quantitative Finance, Vol. 2, 189-198.
[14] Campbell, S. D. and F. X. Diebold, 2005, "Weather Forecasting for Weather Derivatives," Journal of the American Statistical Association, Vol. 100, 6-16.
[15] Cao, M. and J. Wei, 1999, "Pricing Weather Derivatives: An Equilibrium Approach," Working Paper, Rotman Graduate School of Management, The University of Toronto.
[16] Cao, M. and J. Wei, 2000, "Pricing the Weather," Risk Weather Risk Special Report, Energy Power Risk Manage, 67-70.
[17] Cao, M. and J. Wei, 2004, "Weather Derivatives Valuation and Market Price of Weather Risk," Journal of Futures Markers, Vol. 24, 1065-1089.
[18] Carpenter, J., P. Clifford, and P. Fearnhead, 1999, "Improved Particle Filter for Nonlinear Problems," Radar, Sonar and Navigation, IEE Proceedings, Vol. 46, 2-7.
[19] Chen, D., H. J. Harkonen, and D. P. Newton, 2014, "Advancing the Universality of Quadrature Methods to any Underlying Process for Option Pricing," Journal of Financial Economics, Vol. 114, 600-612.
[20] Christoffersen, P., K. Jacobs, and K. Mimouni, 2010, "Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns, and Option Prices," Review of Financial Studies, Vol. 23, 3141-3189.
[21] 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.
[22] Davis, M., 2001, "Pricing Weather Derivatives by Marginal Value," Quantitative Finance, Vol. 1, 1-4.
[23] Diebold, F. X., 2001, "Modeling the Persistence of Conditional Variances: A Comment," Econometric Reviews, Vol. 5, 51-56.
[24] Doucet, A., S. Godsill, and C. Andrieu, 2000, "On Sequential Monte Carlo Sampling Methods for Bayesian Filtering," Statistics and Computing, Vol. 10, 197-208.
[25] Duan, J. C., P. Ritchken, and Z. Sun, 2004, "Jump Starting GARCH: Pricing and Hedging Options with Jumps in Returns and Volatilities," Working Paper, University of Toronto and Case Western Reserve University.
[26] Duan, J. C., P. Ritchken, and Z. Sun, 2006, "Approximating GARCH-Jump Models, Jump-Diffusion Processes, and Option Pricing," Mathematical Finance, Vol. 16, 21-52.
[27] Duffie, D., J. Pan, and K. Singleton, 2000, "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Vol. 68, 1343-1376.
[28] Engle, R. F, 1990, "Stock Volatility and the Crash of `87: Discussion," Review of Financial Studies, Vol. 3, 103-106.
[29] Engle, R. F. and T. Bollerslev, 1986, "Modelling the Persistence of Conditional Variances," Econometric Reviews, Vol. 5, 1-50.
[30] Feller, W, 1951, "Two Singular Diffusion Problems," Annals of Mathematics, Vol. 54, 173-182.
[31] Geweke, J., 1986, "Modeling the Persistence of Conditional Variances: A Comment," Econometric Review, Vol. 5, 57-61.
[32] Glosten, L., R. Jagannathan, and D. Runkle, 1993, "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, Vol. 48, 1779-1801.
[33] Godsill, S. J., A. Doucet, and M. West, 2004, "Monte Carlo Smoothing for Nonlinear Time Series," Journal of the American Statistical Association, Vol. 99, 156-168.
[34] Gordon, N. J., D. J. Salmond, and A. F. M. Smith, 1993, "Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation," Radar and Signal Processing, IEE Proceedings F, Vol. 140, 107-113.
[35] 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.
[36] Heston, S. L. and S. Nandi, 2000, "The Closed-Form GARCH Option Valuation Model," Review of Financial Studies, Vol. 13, 585-625.
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[38] Hull, J. and A, White, 1990, "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies, Vol. 3, 573-592.
[39] Lau, K. M. and H.Weng, 1995, "Climate Signal Detection UsingWavelet Transform: How to Make a Time Series Sing," Bulletin of the American Meteorological Society, Vol. 76, 2391-2402.
[40] Liu, J. S., R. Chen, and W. H. Wong, 1998, "Rejection Control For Sequential Importance Sampling," Journal of the American Statistical Association, Vol. 93, 1022-1031.
[41] Lucas, R. E., 1978, "Asset Prices in an Exchange Economy," Econometrica, Vol. 46, 1429-1445.
[42] Morlet, J., 1983, "Sampling Theory and Wave Propagation," NATO ASI Series, Vol. 1, Springer, 233-261.
[43] Nelson, D., 1990, "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Vol. 59, 347-370.
[44] Nelson, D. B. and S. P. Foster, 1994, "Asymptotic Filtering Theory for Univariate Arch Models," Econometrica, Vol. 62, 1-41.
[45] Pillaya, E. and J.G. O`Hara, 2011, "FFT Based Option Pricing under A Mean Reverting Process with Stochastic Volatility and Jumps," Journal of Computational and Applied Mathematics, Vol. 235, 3378-3384.
[46] Pitt, M. K. and N. Shephard, 1999, "Filtering via Simulation: Auxiliary Particle Filters," Journal of the American Statistical Association, Vol. 94, 590-599.
[47] Schwert, G. W, 1990, "Stock Volatility and the Crash of `87," Review of Financial Studies, Vol. 3, 77-102.
[48] Sentana, E., 1995, "Quadratic ARCH Models," Review of Economic Studies, Vol. 62, 639-661.
[49] Vasicek, O. A., 1977, "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics, Vol. 5, 177-188.
[50] Wong, H. Y. and Y. W. Lo, 2009, "Option Pricing with Mean Reversion and Stochastic Volatility," European Journal of Operational Research, Vol. 197, 179-187.
[51] Zakoian, J., 1994, "Threshold Heteroskedastic Models," Journal of Economic Dynamics and Control, Vol. 18, 931-955.
[52] Zapranis, A. and A. Alexandridis, 2008, "Modelling the Temperature Time Dependent Speed of Mean Reversion in the Context of Weather Derivatives Pricing," Applied Mathematical Finance, Vol. 15, 355-368.
[53] Zapranis, A. and A. Alexandridis, 2009a, "Modeling and Forecasting CAT and HDD Indices For Weather Derivative Pricing," In: EANN 2009, London, Springer, 210-222.
[54] Zapranis, A. and A. Alexandridis, 2009b, "Weather Derivatives Pricing: Modeling the Seasonal Residual Variance of an Ornstein-Uhlenbeck Temperature Process with Neural Networks," Neurocomputing, Vol. 73, 37-48.
描述 博士
國立政治大學
金融研究所
100352503
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0100352503
資料類型 thesis
dc.contributor.advisor 林士貴zh_TW
dc.contributor.advisor Lin, Shih Kueien_US
dc.contributor.author (Authors) 莊明哲zh_TW
dc.contributor.author (Authors) Chuang, Ming Cheen_US
dc.creator (作者) 莊明哲zh_TW
dc.creator (作者) Chuang, Ming Cheen_US
dc.date (日期) 2015en_US
dc.date.accessioned 17-Aug-2015 15:02:39 (UTC+8)-
dc.date.available 17-Aug-2015 15:02:39 (UTC+8)-
dc.date.issued (上傳時間) 17-Aug-2015 15:02:39 (UTC+8)-
dc.identifier (Other Identifiers) G0100352503en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/77626-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 金融研究所zh_TW
dc.description (描述) 100352503zh_TW
dc.description.abstract (摘要) 本研究利用美國芝加哥商品交易所針對 18 個城市發行之冷氣指數／暖氣指數衍生性商品與相對應之日均溫進行分析與評價。研究成果與貢獻如下：一、延伸 Alaton, Djehince, and Stillberg (2002) 模型，引入跳躍風險、隨機波動度、波動跳躍等因子，提出新模型以捕捉更多溫度指數之特徵。二、針對不同模型，分別利用最大概似法、期望最大演算法、粒子濾波演算法等進行參數估計。實證結果顯示新模型具有較好之配適能力。三、利用 Esscher 轉換將真實機率測度轉換至風險中立機率測度，並進一步利用　Feynman-Kac 方程式與傅立葉轉換求出溫度模型之機率分配。四、推導冷氣指數／暖氣指數期貨之半封閉評價公式，而冷氣指數／暖氣指數期貨之選擇權不存在封閉評價公式，則利用蒙地卡羅模擬進行評價。五、無論樣本內與樣本外之定價誤差，考慮隨機波動度型態之模型對於溫度衍生性商品皆具有較好之評價績效。六、實證指出溫度市場之市場風險價格為負，顯示投資人承受較高之溫度風險時會要求較高之風險溢酬。本研究可給予受溫度風險影響之產業，針對衍生性商品之評價與模型參數估計上提供較為精準、客觀與較有效率之工具。zh_TW
dc.description.abstract (摘要) This study uses the daily average temperature index (DAT) and market price of the CDD/HDD derivatives for 18 cities from the CME group. There are some contributions in this study: (i) we extend the Alaton, Djehince, and Stillberg (2002)`s framework by introducing the jump risk, the stochastic volatility, and the jump in volatility. (ii) The model parameters are estimated by the MLE, the EM algorithm, and the PF algorithm. And, the complex model exists the better goodness-of-fit for the path of the temperature index. (iii) We employ the Esscher transform to change the probability measure and derive the probability density function of each model by the Feynman-Kac formula and the Fourier transform. (iv) The semi-closed form of the CDD/HDD futures pricing formula is derived, and we use the Monte-Carlo simulation to value the CDD/HDD futures options due to no closed-form solution. (v) Whatever in-sample and out-of-sample pricing performance, the type of the stochastic volatility performs the better fitting for the temperature derivatives. (vi) The market price of risk differs to zero significantly (most are negative), so the investors require the positive weather risk premium for the derivatives. The results in this study can provide the guide of fitting model and pricing derivatives to the weather-linked institutions in the future.en_US
dc.description.tableofcontents 1 Introduction 1
2 Literature Review 7
2.1 Temperature Model 7
2.2 CDD/HDD Derivatives Pricing Model 10
3 The Models 12
3.1 Seasonal Mean-Reversion Model (S-MR) 12
3.2 Seasonal Mean-Reversion Model with Seasonal Volatility (S-MR-S) 13
3.3 Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Volatility (S-MR-JD-S) 15
3.4 Seasonal Mean-Reversion Model with Seasonal Stochastic Volatility (SMR-S-SV) 16
3.5 Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Stochastic Volatility (S-MR-JD-S-SV) 18
3.6 Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Stochastic Volatility and Jump Risk (S-MR-JD-S-SVJ) 19
4 Temperature Derivatives Pricing Formula 21
4.1 Underlying Asset 22
4.2 Temperature Derivatives Markets 23
4.2.1 Chicago Mercantile Exchange 23
4.2.2 Over-the-Counter 25
4.3 Pricing Formula 25
4.3.1 Equivalent Probability Measure 26
4.3.2 Expectation of HDD/CDD Index 29
4.3.3 CDD/HDD Futures Pricing Formula 35
4.3.4 CDD/HDD Futures Options Pricing Formula 35
4.3.5 CDD/HDD Index Options Pricing Formula 37
5 Estimation Method 39
5.1 Maximum Likelihood Estimation (MLE) 40
5.2 Expectation-Maximization (EM) Algorithm 41
5.3 Particle Filter (PF) Algorithm 42
5.3.1 Monte Carlo Filter 43
5.3.2 Resampling 43
5.3.3 Smoothing 44
5.3.4 EM Algorithm 44
6 Empirical Analysis 46
6.1 Data 46
6.1.1 Temperature Index 46
6.1.2 CDD/HDD Futures and Futures Options 48
6.2 Estimated Parameters 49
6.3 Model Performance 50
6.3.1 Market Price of Risk and In-Sample Pricing Performance 51
6.3.2 Out-of-Sample Pricing Performance 53
6.3.3 Comparison of Performances cross Regions 54
7 Conclusions 55
Bibliography 57
Appendix A Change Measure: Deterministic Volatility Model 62
Appendix B Change Measure: Stochastic Volatility Model 64
Appendix C Characteristic Function: Deterministic Volatility Model 67
Appendix D Characteristic Function: Stochastic Volatility Model 69		zh_TW
dc.format.extent 5047827 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0100352503en_US
dc.subject (關鍵詞) 日均溫zh_TW
dc.subject (關鍵詞) 冷氣指數／暖氣指數衍生性商品zh_TW
dc.subject (關鍵詞) 風險中立評價法zh_TW
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) 粒子濾波演算法zh_TW
dc.subject (關鍵詞) 期望最大演算法zh_TW
dc.subject (關鍵詞) daily average temperature indexen_US
dc.subject (關鍵詞) CDD/HDD derivativesen_US
dc.subject (關鍵詞) risk-neutral pricing methoden_US
dc.subject (關鍵詞) stochastic volatilityen_US
dc.subject (關鍵詞) jump risken_US
dc.subject (關鍵詞) particle filter algorithmen_US
dc.subject (關鍵詞) expectation-maximization algorithmen_US
dc.title (題名) 跳躍風險與隨機波動度下溫度衍生性商品之評價zh_TW
dc.title (題名) Pricing Temperature Derivatives under Jump Risks and Stochastic Volatilityen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] Alaton, P., B. Djehince, and D. Stillberg, 2002, "On Modelling and Pricing Weather Derivatives," Applied Mathematical Finance, Vol. 9, 1-20.
[2] Andricopoulos, A. D., M. Widdicks, P. W. Duck, and D. P. Newton, 2003, "Universal Option Valuation Using Quadrature Methods," Journal of Financial Economics, Vol. 67, 447-471.
[3] Andricopoulos, A. D., M. Widdicks, D. P. Newton, and P. W. Duck, 2007, "Extending Quadrature Methods to Value Multi-Asset and Complex Path Dependent Options," Journal of Financial Economics, Vol. 83, 471-499.
[4] Bates, D. S., 2001, "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Vol. 9, 69-107.
[5] Barndorff-Nielsen, O. E. and N. Shephard, 2001, "Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses in Financial Economics," Journal of the Royal Statistical Society. Series B, Vol. 63, 167-241.
[6] Bellini, F., 2005, "The Weather Derivatives Market: Modelling and Pricing Temperature," Ph.D. thesis, University of Lugano.
[7] Benth, F. E. and J. Saltyte-Benth, 2005, "Stochastic Modelling of Temperature Variations with a View Towards Weather Derivatives," Applied Mathematical Finance, Vol. 12, 53-85.
[8] Benth, F. E. and J. Saltyte-Benth, 2007, "The Volatility of Temperature and Pricing of Weather Derivatives," Quantitative Finance, Vol. 7, 553-561.
[9] Benth, F. E. and J. Saltyte-Benth, 2011, "Weather Derivatives and Stochastic Modelling of Temperature," International Journal of Stochastic Analysis, Vol. 2011, 1-21.
[10] Bibby, B. M., and M. Sorensen, 1995, "Martingale Estimation Functions for Discretely Observed Diffusion Processes," Bernoulli, Vol. 1, 17-39.
[11] Black, F. and M. Scholes, 1973, "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, Vol. 81, 637-654.
[12] Bollerslev, T, 1986, "Generalized Autoregressive Conditional Heteroscedasticity," Journal of Econometrics, Vol. 31, 307-327.
[13] Brody, D. C., J. Syroka, and M. Zervos, 2002, "Dynamical Pricing of Weather Derivatives," Quantitative Finance, Vol. 2, 189-198.
[14] Campbell, S. D. and F. X. Diebold, 2005, "Weather Forecasting for Weather Derivatives," Journal of the American Statistical Association, Vol. 100, 6-16.
[15] Cao, M. and J. Wei, 1999, "Pricing Weather Derivatives: An Equilibrium Approach," Working Paper, Rotman Graduate School of Management, The University of Toronto.
[16] Cao, M. and J. Wei, 2000, "Pricing the Weather," Risk Weather Risk Special Report, Energy Power Risk Manage, 67-70.
[17] Cao, M. and J. Wei, 2004, "Weather Derivatives Valuation and Market Price of Weather Risk," Journal of Futures Markers, Vol. 24, 1065-1089.
[18] Carpenter, J., P. Clifford, and P. Fearnhead, 1999, "Improved Particle Filter for Nonlinear Problems," Radar, Sonar and Navigation, IEE Proceedings, Vol. 46, 2-7.
[19] Chen, D., H. J. Harkonen, and D. P. Newton, 2014, "Advancing the Universality of Quadrature Methods to any Underlying Process for Option Pricing," Journal of Financial Economics, Vol. 114, 600-612.
[20] Christoffersen, P., K. Jacobs, and K. Mimouni, 2010, "Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns, and Option Prices," Review of Financial Studies, Vol. 23, 3141-3189.
[21] 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.
[22] Davis, M., 2001, "Pricing Weather Derivatives by Marginal Value," Quantitative Finance, Vol. 1, 1-4.
[23] Diebold, F. X., 2001, "Modeling the Persistence of Conditional Variances: A Comment," Econometric Reviews, Vol. 5, 51-56.
[24] Doucet, A., S. Godsill, and C. Andrieu, 2000, "On Sequential Monte Carlo Sampling Methods for Bayesian Filtering," Statistics and Computing, Vol. 10, 197-208.
[25] Duan, J. C., P. Ritchken, and Z. Sun, 2004, "Jump Starting GARCH: Pricing and Hedging Options with Jumps in Returns and Volatilities," Working Paper, University of Toronto and Case Western Reserve University.
[26] Duan, J. C., P. Ritchken, and Z. Sun, 2006, "Approximating GARCH-Jump Models, Jump-Diffusion Processes, and Option Pricing," Mathematical Finance, Vol. 16, 21-52.
[27] Duffie, D., J. Pan, and K. Singleton, 2000, "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Vol. 68, 1343-1376.
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