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題名 狀態轉換下利率與跳躍風險股票報酬之歐式選擇權評價與實證分析
Option Pricing and Empirical Analysis for Interest Rate and Stock Index Return with Regime-Switching Model and Dependent Jump Risks作者 巫柏成
Wu, Po Cheng貢獻者 陳麗霞<br>林士貴
Chen, Li Shya<br>Lin, Shih Kuei
巫柏成
Wu, Po Cheng關鍵詞 狀態轉換下利率與跳躍相關風險之股票報酬二維模型
EM演算法
Esscher轉換法
歐式買權定價公式
敏感度分析
模型校準
波動度微笑曲線
MMJDMSI model
EM algorithm
Esscher Transformation
European call option pricing formula
sensitivity analysis; model
model calibration
volatility smile curve日期 2015 上傳時間 1-Sep-2015 16:09:16 (UTC+8) 摘要 Chen, Chang, Wen and Lin (2013)提出馬可夫調控跳躍過程模型(MMJDM)描述股價指數報酬率，布朗運動項、跳躍項之頻率與市場狀態有關。然而，利率並非常數，本論文以狀態轉換模型配適零息債劵之動態過程，提出狀態轉換下的利率與具跳躍風險的股票報酬之二維模型(MMJDMSI)，並以1999年至2013年的道瓊工業指數與S&P 500指數和同期間之一年期美國國庫劵價格為實證資料，採用EM演算法取得參數估計值。經由概似比檢定結果顯示無論道瓊工業指數還是S&P 500指數，狀態轉換下利率與跳躍風險之股票報酬二維模型更適合描述報酬率。接著，利用Esscher轉換法推導出各模型下的股價指數之歐式買權定價公式，再對MMJDMSI模型進行敏感度分析以評估模型參數發生變動時對於定價公式的影響。最後，以實證資料對各模型進行模型校準及計算隱含波動度，結果顯示MMJDMSI在價內及價外時定價誤差為最小或次小，且此模型亦能呈現出波動度微笑曲線之現象。
To model asset return, Chen, Chang, Wen and Lin (2013) proposed Markov-Modulated Jump Diffusion Model (MMJDM) assuming that the Brownian motion term and jump frequency are all related to market states. In fact, the interest rate is not constant, Regime-Switching Model is taken to fit the process of the zero-coupon bond price, and a bivariate model for interest rate and stock index return with regime-switching and dependent jump risks (MMJDMSI) is proposed. The empirical data are Dow Jones Industrial Average and S&P 500 Index from 1999 to 2013, together with US 1-Year Treasury Bond over the same period. Model parameters are estimated by the Expectation-Maximization (EM) algorithm. The likelihood ratio test (LRT) is performed to compare nested models, and MMJDMSI is better than the others. Then, European call option pricing formula under each model is derived via Esscher transformation, and sensitivity analysis is conducted to evaluate changes resulted from different parameter values under the MMJDMSI pricing formula. Finally, model calibrations are performed and implied volatilities are computed under each model empirically. In cases of in-the-money and out-the-money, MMJDMSI has either the smallest or the second smallest pricing error. Also, the implied volatilities from MMJDMSI display a volatility smile curve.參考文獻 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, 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.Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Eeconomy 81, 3, 637-654.5.Bo, L., Wang, Y., and Yang, X. (2010). Markov-modulated jump-diffusion for currency option pricing. Insurance: Mathematics and Economics 46, 461-469.6.Charles, C., Fuh, C.D., and Lin, S.K. (2013). A tale of two regimes: Theory and empirical evidence for a markov-modulated jump diffusion model of equity returns and derivative pricing implications. Working paper.7.Chen, L.S., Chang, Y.H., Wen, C.H., and Lin, S.K. (2016). Valuation of a defined contribution pension plan: evidence from stock indices under markov-modulated jump diffusion model. Journal of the Chinese Statistical Association 53, 2, 79-106.8.Duan, J.C., Popova, I., and Ritchken, P. (2002). Option pricing under regime switching. Quantitative Finance 2, 116-132.9.Elliott, R.J., Chan, L., and Siu, T. (2005). Option pricing and Esscher transform under regime switching. Annals of Finance 1, 4, 423-432.10.Elliott, R.J., Siu, T.K., and Chan, L. (2007). Pricing options under a generalized markov-modulated jump-diffusion model. Stochastic Analysis and Application 25, 4, 821-843.11.Gerber, H. and Shiu, E. (1994). Option pricing by Esscher transforms. Transactions of the Society of Actuaries 46, 99, 140.12.Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357-384.13.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, 327.14.Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance 42, 2, 281-300.15.Jarrow, R. and Rosenfeld, E. (1984).Jump risks and the intertemporal capital asset pricing model. The Journal of Business 57, 3, 337-351.16.Lange, K. A. (1995). Gradient algorithm locally equivalent to the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological) 57, 2, 425-437.17.Lin, S.K. Shyu, S.D. and Wang, S.Y. (2013). Option pricing under stock market cycles with jump risks: evidence from Dow Jones industrial average index and S&P 500 index. Working paper. 18.Lin, S. K., Liu, H., and Lee, J.C. (2013). Option pricing under regime-switching jump model with dependent jump sizes: evidence from stock index options. Working paper.19.Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous Journal of Financial Economics 3, 1-2, 125-144.20.Mixon, S. (2007). The implied volatility term structure of stock index options. Journal of Empirical Finance 14, 3, 333-354.21.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, 413-426.22.Stein, E. and Stein, J. (1991).Stock price distributions with stochastic volatility: an analytic approach. Review of Financial Studies 4, 4, 727. 描述 碩士
國立政治大學
統計研究所
102354017資料來源 http://thesis.lib.nccu.edu.tw/record/#G0102354017 資料類型 thesis dc.contributor.advisor 陳麗霞<br>林士貴 zh_TW dc.contributor.advisor Chen, Li Shya<br>Lin, Shih Kuei en_US dc.contributor.author (Authors) 巫柏成 zh_TW dc.contributor.author (Authors) Wu, Po Cheng en_US dc.creator (作者) 巫柏成 zh_TW dc.creator (作者) Wu, Po Cheng en_US dc.date (日期) 2015 en_US dc.date.accessioned 1-Sep-2015 16:09:16 (UTC+8) - dc.date.available 1-Sep-2015 16:09:16 (UTC+8) - dc.date.issued (上傳時間) 1-Sep-2015 16:09:16 (UTC+8) - dc.identifier (Other Identifiers) G0102354017 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/78053 - 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) 102354017 zh_TW dc.description.abstract (摘要) Chen, Chang, Wen and Lin (2013)提出馬可夫調控跳躍過程模型(MMJDM)描述股價指數報酬率，布朗運動項、跳躍項之頻率與市場狀態有關。然而，利率並非常數，本論文以狀態轉換模型配適零息債劵之動態過程，提出狀態轉換下的利率與具跳躍風險的股票報酬之二維模型(MMJDMSI)，並以1999年至2013年的道瓊工業指數與S&P 500指數和同期間之一年期美國國庫劵價格為實證資料，採用EM演算法取得參數估計值。經由概似比檢定結果顯示無論道瓊工業指數還是S&P 500指數，狀態轉換下利率與跳躍風險之股票報酬二維模型更適合描述報酬率。接著，利用Esscher轉換法推導出各模型下的股價指數之歐式買權定價公式，再對MMJDMSI模型進行敏感度分析以評估模型參數發生變動時對於定價公式的影響。最後，以實證資料對各模型進行模型校準及計算隱含波動度，結果顯示MMJDMSI在價內及價外時定價誤差為最小或次小，且此模型亦能呈現出波動度微笑曲線之現象。 zh_TW dc.description.abstract (摘要) To model asset return, Chen, Chang, Wen and Lin (2013) proposed Markov-Modulated Jump Diffusion Model (MMJDM) assuming that the Brownian motion term and jump frequency are all related to market states. In fact, the interest rate is not constant, Regime-Switching Model is taken to fit the process of the zero-coupon bond price, and a bivariate model for interest rate and stock index return with regime-switching and dependent jump risks (MMJDMSI) is proposed. The empirical data are Dow Jones Industrial Average and S&P 500 Index from 1999 to 2013, together with US 1-Year Treasury Bond over the same period. Model parameters are estimated by the Expectation-Maximization (EM) algorithm. The likelihood ratio test (LRT) is performed to compare nested models, and MMJDMSI is better than the others. Then, European call option pricing formula under each model is derived via Esscher transformation, and sensitivity analysis is conducted to evaluate changes resulted from different parameter values under the MMJDMSI pricing formula. Finally, model calibrations are performed and implied volatilities are computed under each model empirically. In cases of in-the-money and out-the-money, MMJDMSI has either the smallest or the second smallest pricing error. Also, the implied volatilities from MMJDMSI display a volatility smile curve. en_US dc.description.tableofcontents 第一章 緒論 1第二章 文獻回顧 62.1股價指數選擇權 62.2股價報酬率模型 72.3其他模型 9第三章 模型與估計檢定 113.1 模型 113.1.1狀態轉換下利率與股價二維模型(RSMSI) 113.2.2狀態轉換下利率與跳躍風險之股價二維模型(RSMJSI) 123.2.3狀態轉換下利率與跳躍相關風險之股票報酬二維模型(MMJDMSI)143.2狀態轉換下利率與跳躍相關風險之股票報酬二維模型之估計與檢定15第四章 股價指數選擇權評價 184.1 Esscher轉換 184.1.1狀態轉換下利率與股價二維模型Esscher轉換 184.1.2狀態轉換下利率與跳躍風險之股價二維模型Esscher轉換204.1.3狀態轉換下利率與跳躍相關風險之股票報酬二維模型Esscher轉換234.2股價指數選擇權評價 264.2.1狀態轉換下利率與股價二維模型之選擇權定價公式 264.2.2狀態轉換下利率與跳躍風險之股價二維模型之選擇權定價公式 274.2.3狀態轉換下利率與跳躍相關風險之股票報酬二維模型之選擇權定價公式28第五章 實證分析 295.1 實證分析 295.1.1模型參數估計與檢定 295.1.2狀態與跳躍動態分析 365.2 敏感度分析 385.3模型校準 425.4隱含波動度 48第六章 結論 50參考文獻 53附錄 56附錄A EM演算法估計模型參數之過程 56附錄B狀態轉換下利率與股價二維模型之選擇權定價公式 58附錄C狀態轉換下利率與跳躍風險之股價二維模型之選擇權定價公式 64附錄D狀態轉換下利率與跳躍相關風險之股票報酬二維模型之選擇權定 71表目錄表 1 1999年至2013年道瓊工業指數報酬率之統計 2表 2 1999年至2013年一年期美國國庫劵價格報酬率之統計 2表 3 道瓊工業指數日報酬及一年期國庫劵價格日報酬在四種模型中之參數估計與檢定結果 30表 4 S&P500指數日報酬及一年期國庫劵價格日報酬在四種模型中之參數 估計與檢定結果 31表 5 股價與債劵價格敏感度分析 39表 6 狀態轉移機率敏感度分析 39表 7 股價布朗運動項標準差敏感度分析 40表 8 零息債劵價格布朗運動項標準差敏感度分析 40表 9 布朗運動項相關係數敏感度分析 40表10 跳躍幅度平均數與標準差敏感度分析 41表11 跳躍頻率敏感度分析 41表12 道瓊工業指數買權樣本內參數估計 46表13 道瓊工業指數買權樣本外定價誤差 46表14 S&P500指數買權樣本內參數估計 47表15 S&P500指數買權樣本外定價誤差 47圖目錄圖1 芝加哥選擇權交易所各年度指數選擇權交易量 1圖2 道瓊工業指數與一年期美國國庫劵指數動態圖 2圖3 道瓊工業指數報酬率與一年期美國國庫劵指數報酬率動態圖 4圖4 道瓊工業指數、股價報酬率、國庫劵價格、債劵報酬率、狀態機率及跳躍機率動態圖 36圖5 S&P500指數、股價報酬率、國庫劵價格、債劵報酬率、狀態機率及跳躍機率動態圖 37圖6 道瓊工業指數選擇權隱含波動度微笑曲線 49圖7 S&P500指數選擇權隱含波動度微笑曲線 49 zh_TW dc.format.extent 1358003 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0102354017 en_US dc.subject (關鍵詞) 狀態轉換下利率與跳躍相關風險之股票報酬二維模型 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) 歐式買權定價公式 zh_TW dc.subject (關鍵詞) 敏感度分析 zh_TW dc.subject (關鍵詞) 模型校準 zh_TW dc.subject (關鍵詞) 波動度微笑曲線 zh_TW dc.subject (關鍵詞) MMJDMSI model en_US dc.subject (關鍵詞) EM algorithm en_US dc.subject (關鍵詞) Esscher Transformation en_US dc.subject (關鍵詞) European call option pricing formula en_US dc.subject (關鍵詞) sensitivity analysis; model en_US dc.subject (關鍵詞) model calibration en_US dc.subject (關鍵詞) volatility smile curve en_US dc.title (題名) 狀態轉換下利率與跳躍風險股票報酬之歐式選擇權評價與實證分析 zh_TW dc.title (題名) Option Pricing and Empirical Analysis for Interest Rate and Stock Index Return with Regime-Switching Model and Dependent Jump Risks en_US dc.type (資料類型) thesis en dc.relation.reference (參考文獻) 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, 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.Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Eeconomy 81, 3, 637-654.5.Bo, L., Wang, Y., and Yang, X. (2010). Markov-modulated jump-diffusion for currency option pricing. Insurance: Mathematics and Economics 46, 461-469.6.Charles, C., Fuh, C.D., and Lin, S.K. (2013). A tale of two regimes: Theory and empirical evidence for a markov-modulated jump diffusion model of equity returns and derivative pricing implications. Working paper.7.Chen, L.S., Chang, Y.H., Wen, C.H., and Lin, S.K. (2016). Valuation of a defined contribution pension plan: evidence from stock indices under markov-modulated jump diffusion model. Journal of the Chinese Statistical Association 53, 2, 79-106.8.Duan, J.C., Popova, I., and Ritchken, P. (2002). Option pricing under regime switching. Quantitative Finance 2, 116-132.9.Elliott, R.J., Chan, L., and Siu, T. (2005). Option pricing and Esscher transform under regime switching. Annals of Finance 1, 4, 423-432.10.Elliott, R.J., Siu, T.K., and Chan, L. (2007). Pricing options under a generalized markov-modulated jump-diffusion model. Stochastic Analysis and Application 25, 4, 821-843.11.Gerber, H. and Shiu, E. (1994). Option pricing by Esscher transforms. Transactions of the Society of Actuaries 46, 99, 140.12.Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357-384.13.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, 327.14.Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance 42, 2, 281-300.15.Jarrow, R. and Rosenfeld, E. (1984).Jump risks and the intertemporal capital asset pricing model. The Journal of Business 57, 3, 337-351.16.Lange, K. A. (1995). Gradient algorithm locally equivalent to the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological) 57, 2, 425-437.17.Lin, S.K. Shyu, S.D. and Wang, S.Y. (2013). Option pricing under stock market cycles with jump risks: evidence from Dow Jones industrial average index and S&P 500 index. Working paper. 18.Lin, S. K., Liu, H., and Lee, J.C. (2013). Option pricing under regime-switching jump model with dependent jump sizes: evidence from stock index options. Working paper.19.Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous Journal of Financial Economics 3, 1-2, 125-144.20.Mixon, S. (2007). The implied volatility term structure of stock index options. Journal of Empirical Finance 14, 3, 333-354.21.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, 413-426.22.Stein, E. and Stein, J. (1991).Stock price distributions with stochastic volatility: an analytic approach. Review of Financial Studies 4, 4, 727. zh_TW
