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題名 狀態轉換跳躍相關模型下選擇權定價:股價指數選擇權之實證
Option pricing under regime-switching jump model with dependent jump sizes: evidence from stock index option作者 李家慶
Lee, Jia-Ching貢獻者 劉惠美<br>林士貴
Liu, Hui Mei<br>Lin, Shih Kuei
李家慶
Lee, Jia-Ching關鍵詞 股價指數選擇權
狀態轉換跳躍相關模型
波動聚集
波動度微笑
EM演算法
Esscher轉換法
index option
regime-switching jump dependent model
volatility clustering
volatility smile
EM algorithm
Esscher transformation日期 2010 上傳時間 5-Sep-2013 15:12:19 (UTC+8) 摘要 Black and Scholes (1973)對於報酬率提出以B-S模型配適，但B-S模型無法有效解釋報酬率不對稱高狹峰、波動度微笑、波動度叢聚、長記憶性的性質。Merton (1976)認為不尋常的訊息來臨會影響股價不連續跳躍，因此發展B-S模型加入不連續跳躍風險項的跳躍擴散模型，該模型可同時描述報酬率不對稱高狹峰和波動度微笑兩性質。Charles, Fuh and Lin (2011)加以考慮市場狀態提出狀態轉換跳躍模型，除了保留跳躍擴散模型可描述報酬率不對稱高狹峰和波動度微笑，更可以敘述報酬率的波動度叢聚和長記憶性。本文進一步拓展狀態轉換跳躍模型，考慮不連續跳躍風險項的帄均數與市場狀態相關，提出狀態轉換跳躍相關模型。並以道瓊工業指數與S&P 500指數1999年至2010年股價指數資料，採用EM和SEM分別估計參數與估計參數共變異數矩陣。使用概似比檢定結果顯示狀態轉換跳躍相關模型比狀態轉換跳躍獨立模型更適合描述股價指數報酬率。並驗證狀態轉換跳躍相關模型也可同時描述報酬率不對稱高狹峰、波動度微笑、波動度叢聚、長記憶性。最後利用Esscher轉換法計算股價指數選擇權定價公式，以敏感度分析模型參數對於定價結果的影響，並且市場驗證顯示狀態轉換跳躍相關模型會有最小的定價誤差。
Black and Scholes (1973) proposed B-S model to fit asset return, but B-S model can’t effectively explain some asset return properties, such as leptokurtic, volatility smile, volatility clustering and long memory. Merton (1976) develop jump diffusion model (JDM) that consider abnormal information of market will affect the stock price, and this model can explain leptokurtic and volatility smile of asset return at the same time. Charles, Fuh and Lin (2011) extended the JDM and proposed regime-switching jump independent model (RSJIM) that consider jump rate is related to market states. RSJIM not only retains JDM properties but describes volatility clustering and long memory. In this paper, we extend RSJIM to regime-switching jump dependent model (RSJDM) which consider jump size and jump rate are both related to market states. We use EM and SEM algorithm to estimate parameters and covariance matrix, and use LR test to compare RSJIM and RSJDM. By using 1999 to 2010 Dow-Jones industrial average index and S&P 500 index as empirical evidence, RSJDM can explain index return properties said before. Finally, we calculate index option price formulation by Esscher transformation and do sensitivity analysis and market validation which give the smallest error of option prices by RSJDM.參考文獻 中文文獻[1] 吳聲杰，(2009)。狀態轉換跳躍模型下Supplemented Expectation Maximization 演算法與Gibbs Sampling演算法之參數估計之變異數估計，國立高雄大學統計學研究所碩士論文。[2] 林晉煜，(2010)。狀態轉換跳躍模型下權益指數年金之評價公式：股價指數之實證，國立高雄大學統計學研究所碩士論文。[3] 康怡禎，(2008)。跳躍幅度與跳躍頻率相依下馬可夫跳躍擴散模型在財務金融之實證分析，國立東華大學應用數學系碩士論文。英文文獻[1] Bailey, W., and Stulz, R., (1989). The pricing of stock index options in a general equilibrium model. Journal of Financial and Quantitative Analysis 24, 01, 1-12.[2] Bakshi, G., Cao, C., and Chen, Z., (1997). Empirical performance of alternative option pricing models. The Journal of Finance 52, 5, 2003-2049.[3] Ball, C., and Torous, W., (1983). A simplified jump process for common stock returns. Journal of Financial and Quantitative Analysis 18, 01, 53-65.[4] Ballotta, L., (2005). A Lévy process-based framework for the fair valuation of participating life insurance contracts. Insurance: Mathematics and Economics 37, 2, 173-196.[5] Barndorff-Nielsen, O., and Shephard, N., (2004). Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2, 1, 1.[6] Bates, D., (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies 9, 1, 69.[7] Beckers, S., (1981). A note on estimating the parameters of the diffusion-jump model of stock returns. Journal of Financial and Quantitative Analysis 16, 01, 127-140.[8] Black, F., and Scholes, M., (1973). The pricing of options and corporate liabilities. The Journal of Political Economy 81, 3, 637-654.[9] Charles, C., Fuh, C., D., and Lin, S., K., (2011). A Tale of Two Regimes: Theory and Empirical Evidence for a Markov-Switch Jump Diffusion Model and Derivative Pricing Implications. Working paper.[10] Cont, R., (2007). Volatility clustering in financial markets: empirical facts and agent-based models. Long Memory in Economics, 289-310.[11] Cont, R., and Tankov, P., (2004). Financial Modelling with Jump Processes. Chapman & Hall.[12] Corrado, C., and Su, T., (1997). Implied volatility skews and stock index skewness and kurtosis implied by S&P 500 index option prices. The Journal of Derivatives 4, 4, 8-19.[13] Cox, J., Ingersoll Jr, J., and Ross, S., (1985). A theory of the term structure of interest rates. Econometrica: Journal of the Econometric Society, 385-407.[14] Day, T., and Lewis, C., (1988). The behavior of the volatility implicit in the prices of stock index options* 1. Journal of Financial Economics 22, 1, 103-122.[15] Ding, Z., Granger, C., and Engle, R., (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 1, 83-106.[16] Elliott, R., Chan, L., and Siu, T., (2005). Option pricing and Esscher transform under regime switching. Annals of Finance 1, 4, 423-432.[17] Elliott, R., and Madan, D., (1998). A discrete time equivalent martingale measure. Mathematical Finance 8, 2, 127-152.[18] Eraker, B., (2004). Do stock prices and volatility jump? Reconciling evidence from spot and option prices. The Journal of Finance 59, 3, 1367-1404.[19] Eraker, B., Johannes, M., and Polson, N., (2003). The impact of jumps in volatility and returns. The Journal of Finance 58, 3, 1269-1300.[20] Gerber, H., and Shiu, E., (1994). Option pricing by Esscher transforms. Transactions of the Society of Actuaries 46, 99, 140.[21] Glasserman, P., and Kou, S., (2003). The term structure of simple forward rates with jump risk. Mathematical finance 13, 3, 383-410.[22] Heston, S., (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6, 2, 327.[23] Heston, S., and Nandi, S., (2000). A closed-form GARCH option valuation model. Review of Financial Studies 13, 3, 585.[24] Hull, J., and White, A., (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance 42, 2, 281-300.[25] Jarrow, R., and Rosenfeld, E., (1984). Jump risks and the intertemporal capital asset pricing model. The Journal of Business 57, 3, 337-351.[26] Jiang, G., and Oomen, R., (2008). Testing for jumps when asset prices are observed with noise-a. Journal of Econometrics 144, 2, 352-370.[27] Kallsen, J., and Shiryaev, A., (2002). The cumulant process and Esscher`s change of measure. Finance and Stochastics 6, 4, 397-428.[28] Kou, S., (2002). A jump-diffusion model for option pricing. Management Science, 1086-1101.[29] Lange, K. A, (1995). gradient algorithm locally equivalent to the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological) 57, 2, 425-437.[30] Last, G., and Brandt, A., (1995). Marked Point Processes on the Real Line: The Dynamic Approach. Springer.[31] Lau, J., and Siu, T., (2008). On option pricing under a completely random measure via a generalized Esscher transform. Insurance: Mathematics and Economics 43, 1, 99-107.[32] Liew, C., and Siu, T., (2010). A hidden Markov regime-switching model for option valuation. Insurance: Mathematics and Economics.[33] Mandelbrot, B., (1963). The variation of certain speculative prices. The Journal of Business 36, 4, 394-419.[34] Merton, R., (1976). Option pricing when underlying stock returns are discontinuous* 1. Journal of Financial Economics 3, 1-2, 125-144.[35] Mixon, S., (2007). The implied volatility term structure of stock index options. Journal of Empirical Finance 14, 3, 333-354.[36] Press, S., (1967). A compound events model for security prices. Journal of Business, 317-335.[37] Scott, L., (1997). Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: Applications of Fourier inversion methods. Mathematical Finance 7, 4, 413-426.[38] Stein, E., and Stein, J., (1991). Stock price distributions with stochastic volatility: an analytic approach. Review of Financial Studies 4, 4, 727.[39] Tauchen, G., and Zhou, H., (2006). Realized jumps on financial markets and predicting credit spreads. Journal of Econometrics.[40] Ward, A., et al. (1992). Numerical inversion of probability generating functions. Operations Research Letters 12, 4, 245-251. 描述 碩士
國立政治大學
統計研究所
98354006
99資料來源 http://thesis.lib.nccu.edu.tw/record/#G0098354006 資料類型 thesis dc.contributor.advisor 劉惠美<br>林士貴 zh_TW dc.contributor.advisor Liu, Hui Mei<br>Lin, Shih Kuei en_US dc.contributor.author (Authors) 李家慶 zh_TW dc.contributor.author (Authors) Lee, Jia-Ching en_US dc.creator (作者) 李家慶 zh_TW dc.creator (作者) Lee, Jia-Ching en_US dc.date (日期) 2010 en_US dc.date.accessioned 5-Sep-2013 15:12:19 (UTC+8) - dc.date.available 5-Sep-2013 15:12:19 (UTC+8) - dc.date.issued (上傳時間) 5-Sep-2013 15:12:19 (UTC+8) - dc.identifier (Other Identifiers) G0098354006 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/60439 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 統計研究所 zh_TW dc.description (描述) 98354006 zh_TW dc.description (描述) 99 zh_TW dc.description.abstract (摘要) Black and Scholes (1973)對於報酬率提出以B-S模型配適，但B-S模型無法有效解釋報酬率不對稱高狹峰、波動度微笑、波動度叢聚、長記憶性的性質。Merton (1976)認為不尋常的訊息來臨會影響股價不連續跳躍，因此發展B-S模型加入不連續跳躍風險項的跳躍擴散模型，該模型可同時描述報酬率不對稱高狹峰和波動度微笑兩性質。Charles, Fuh and Lin (2011)加以考慮市場狀態提出狀態轉換跳躍模型，除了保留跳躍擴散模型可描述報酬率不對稱高狹峰和波動度微笑，更可以敘述報酬率的波動度叢聚和長記憶性。本文進一步拓展狀態轉換跳躍模型，考慮不連續跳躍風險項的帄均數與市場狀態相關，提出狀態轉換跳躍相關模型。並以道瓊工業指數與S&P 500指數1999年至2010年股價指數資料，採用EM和SEM分別估計參數與估計參數共變異數矩陣。使用概似比檢定結果顯示狀態轉換跳躍相關模型比狀態轉換跳躍獨立模型更適合描述股價指數報酬率。並驗證狀態轉換跳躍相關模型也可同時描述報酬率不對稱高狹峰、波動度微笑、波動度叢聚、長記憶性。最後利用Esscher轉換法計算股價指數選擇權定價公式，以敏感度分析模型參數對於定價結果的影響，並且市場驗證顯示狀態轉換跳躍相關模型會有最小的定價誤差。 zh_TW dc.description.abstract (摘要) Black and Scholes (1973) proposed B-S model to fit asset return, but B-S model can’t effectively explain some asset return properties, such as leptokurtic, volatility smile, volatility clustering and long memory. Merton (1976) develop jump diffusion model (JDM) that consider abnormal information of market will affect the stock price, and this model can explain leptokurtic and volatility smile of asset return at the same time. Charles, Fuh and Lin (2011) extended the JDM and proposed regime-switching jump independent model (RSJIM) that consider jump rate is related to market states. RSJIM not only retains JDM properties but describes volatility clustering and long memory. In this paper, we extend RSJIM to regime-switching jump dependent model (RSJDM) which consider jump size and jump rate are both related to market states. We use EM and SEM algorithm to estimate parameters and covariance matrix, and use LR test to compare RSJIM and RSJDM. By using 1999 to 2010 Dow-Jones industrial average index and S&P 500 index as empirical evidence, RSJDM can explain index return properties said before. Finally, we calculate index option price formulation by Esscher transformation and do sensitivity analysis and market validation which give the smallest error of option prices by RSJDM. en_US dc.description.tableofcontents 1. 介 紹 …………………………………………………………………….12. 文獻回顧…………………………………………………………………………52.1 股價指數選擇權…………………………………………………………..52.2 跳躍擴散模型…………………………………………………………….62.3 隨機波動度模型…………………………………………………………112.4 Esscher 轉換…………………………………………………………….133. 股價指數模型…………………………………………………………………153.1 跳躍擴散模型……………………………………………………………153.2 狀態轉換跳躍獨立模型…………………………………………………163.3 狀態轉換跳躍相關模型…………………………………………………183.4 馬可夫調控普瓦松過程…………………………………………………203.5 狀態轉換跳躍相關模型之估計與檢定…………………………………214. 股價指數選擇權定價…………………………………………………………254.1 Esscher 轉換……………………………………………………………254.1.1 跳躍擴散模型下Esscher 轉換…………………………………264.1.2 狀態轉換跳躍獨立模型下Esscher 轉換………………………274.1.3 狀態轉換跳躍相關模型下Esscher 轉換………………………304.2股價指數選擇權定價……………………………………………………334.2.1 跳躍擴散模型下股價指數選擇權定價…………………………334.2.2 狀態轉換跳躍獨立模型下股價指數選擇權定價………………344.2.3 狀態轉換跳躍相關模型下股價指數選擇權定價………………355. 實證分析、敏感度分析與市場驗證………………………………………365.1 實證分析…………………………………………………………………365.1.1 估計與檢定 ………………………………………………………365.1.2 模型經濟分析……………………………………………………395.1.3 峰態與偏態………………………………………………………405.1.4 波動叢聚…………………………………………………………415.1.5 波動度微笑………………………………………………………415.2 敏感度分析………………………………………………………………425.3 市場驗證…………………………………………………………………446. 結論……………………………………………………………………………45參考文獻……………………………………………………………………………47中文文獻………………………………………………………………………47英文文獻………………………………………………………………………47附錄…………………………………………………………………………………51附錄A：跳躍擴散模型歐式買權定價公式………………………………51附錄B：狀態轉換跳躍獨立模型歐式買權定價公式………………………56附錄C：狀態轉換跳躍相關模型歐式買權定價公式……………………62附錄D：狀態轉換跳躍相關模型自我相關函數公式………………………68表目錄表1.1：1999年至2010年道瓊工業指數報酬率敘述統計表.................................2表5.1：道瓊工業指數與S&P 500模型參數估計與檢定…………………69表5.2：跳躍擴散模型動差公式………………………………………………70表5.3：狀態轉換跳躍獨立動差公式……………………………………71表5.4：狀態轉換跳躍相關模型動差公式…………………………………72表5.5：道瓊工業指數與S&P 500模型動差估計……………………………73表5.6：轉移機率敏感度分析…..……………………………………………74表5.7：布朗運動項標準差與跳躍幅度標準差敏感度分析……………74表5.8：跳躍幅度平均數敏感度分析………………………………………74表5.9：跳躍頻率敏感度分析………………………………………………74表5.10：道瓊工業指數歐式買權樣本外定價誤差…………………….75表5.11：S&P 500指數歐式買權樣本外定價誤差……………………………75圖目錄圖1.1：芝加哥選擇權交易所各年度指數選擇權交易量…………………1圖1.2：道瓊工業指數報酬率動態圖………………………………3圖5.1：道瓊工業指數股價、報酬率、狀態及跳躍機率動態圖...........76圖5.2：S&P 500指數股價、報酬率、狀態及跳躍機率動態圖...........77圖5.3：道瓊工業指數報酬率動態、資料與RSJDM自我相關函數.........78圖5.4：S&P 500指數報酬率動態、資料與RSJDM自我相關函數.........78圖5.5：道瓊工業指數隱含波動度微笑………....................79圖5.6：S&P 500指數隱含波動度微笑………………..……………..……79 zh_TW dc.format.extent 1537277 bytes - dc.format.mimetype application/pdf - dc.language.iso en_US - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0098354006 en_US dc.subject (關鍵詞) 股價指數選擇權 zh_TW dc.subject (關鍵詞) 狀態轉換跳躍相關模型 zh_TW dc.subject (關鍵詞) 波動聚集 zh_TW dc.subject (關鍵詞) 波動度微笑 zh_TW dc.subject (關鍵詞) EM演算法 zh_TW dc.subject (關鍵詞) Esscher轉換法 zh_TW dc.subject (關鍵詞) index option en_US dc.subject (關鍵詞) regime-switching jump dependent model en_US dc.subject (關鍵詞) volatility clustering en_US dc.subject (關鍵詞) volatility smile en_US dc.subject (關鍵詞) EM algorithm en_US dc.subject (關鍵詞) Esscher transformation en_US dc.title (題名) 狀態轉換跳躍相關模型下選擇權定價:股價指數選擇權之實證 zh_TW dc.title (題名) Option pricing under regime-switching jump model with dependent jump sizes: evidence from stock index option en_US dc.type (資料類型) thesis en dc.relation.reference (參考文獻) 中文文獻[1] 吳聲杰，(2009)。狀態轉換跳躍模型下Supplemented Expectation Maximization 演算法與Gibbs Sampling演算法之參數估計之變異數估計，國立高雄大學統計學研究所碩士論文。[2] 林晉煜，(2010)。狀態轉換跳躍模型下權益指數年金之評價公式：股價指數之實證，國立高雄大學統計學研究所碩士論文。[3] 康怡禎，(2008)。跳躍幅度與跳躍頻率相依下馬可夫跳躍擴散模型在財務金融之實證分析，國立東華大學應用數學系碩士論文。英文文獻[1] Bailey, W., and Stulz, R., (1989). The pricing of stock index options in a general equilibrium model. Journal of Financial and Quantitative Analysis 24, 01, 1-12.[2] Bakshi, G., Cao, C., and Chen, Z., (1997). Empirical performance of alternative option pricing models. The Journal of Finance 52, 5, 2003-2049.[3] Ball, C., and Torous, W., (1983). A simplified jump process for common stock returns. Journal of Financial and Quantitative Analysis 18, 01, 53-65.[4] Ballotta, L., (2005). A Lévy process-based framework for the fair valuation of participating life insurance contracts. Insurance: Mathematics and Economics 37, 2, 173-196.[5] Barndorff-Nielsen, O., and Shephard, N., (2004). Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2, 1, 1.[6] Bates, D., (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies 9, 1, 69.[7] Beckers, S., (1981). A note on estimating the parameters of the diffusion-jump model of stock returns. Journal of Financial and Quantitative Analysis 16, 01, 127-140.[8] Black, F., and Scholes, M., (1973). The pricing of options and corporate liabilities. The Journal of Political Economy 81, 3, 637-654.[9] Charles, C., Fuh, C., D., and Lin, S., K., (2011). A Tale of Two Regimes: Theory and Empirical Evidence for a Markov-Switch Jump Diffusion Model and Derivative Pricing Implications. Working paper.[10] Cont, R., (2007). Volatility clustering in financial markets: empirical facts and agent-based models. Long Memory in Economics, 289-310.[11] Cont, R., and Tankov, P., (2004). Financial Modelling with Jump Processes. Chapman & Hall.[12] Corrado, C., and Su, T., (1997). Implied volatility skews and stock index skewness and kurtosis implied by S&P 500 index option prices. The Journal of Derivatives 4, 4, 8-19.[13] Cox, J., Ingersoll Jr, J., and Ross, S., (1985). A theory of the term structure of interest rates. Econometrica: Journal of the Econometric Society, 385-407.[14] Day, T., and Lewis, C., (1988). The behavior of the volatility implicit in the prices of stock index options* 1. Journal of Financial Economics 22, 1, 103-122.[15] Ding, Z., Granger, C., and Engle, R., (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 1, 83-106.[16] Elliott, R., Chan, L., and Siu, T., (2005). Option pricing and Esscher transform under regime switching. Annals of Finance 1, 4, 423-432.[17] Elliott, R., and Madan, D., (1998). A discrete time equivalent martingale measure. Mathematical Finance 8, 2, 127-152.[18] Eraker, B., (2004). Do stock prices and volatility jump? Reconciling evidence from spot and option prices. The Journal of Finance 59, 3, 1367-1404.[19] Eraker, B., Johannes, M., and Polson, N., (2003). The impact of jumps in volatility and returns. 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