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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 Kueien_US
dc.contributor.author (Authors) 巫柏成zh_TW
dc.contributor.author (Authors) Wu, Po Chengen_US
dc.creator (作者) 巫柏成zh_TW
dc.creator (作者) Wu, Po Chengen_US
dc.date (日期) 2015en_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) G0102354017en_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) 102354017zh_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 6
2.1股價指數選擇權 6
2.2股價報酬率模型 7
2.3其他模型 9
第三章 模型與估計檢定 11
3.1 模型 11
3.1.1狀態轉換下利率與股價二維模型(RSMSI) 11
3.2.2狀態轉換下利率與跳躍風險之股價二維模型(RSMJSI) 12
3.2.3狀態轉換下利率與跳躍相關風險之股票報酬二維模型(MMJDMSI)14
3.2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 18
4.1 Esscher轉換 18
4.1.1狀態轉換下利率與股價二維模型Esscher轉換 18
4.1.2狀態轉換下利率與跳躍風險之股價二維模型Esscher轉換20
4.1.3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23
4.2股價指數選擇權評價 26
4.2.1狀態轉換下利率與股價二維模型之選擇權定價公式 26
4.2.2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 27
4.2.3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 29
5.1 實證分析 29
5.1.1模型參數估計與檢定 29
5.1.2狀態與跳躍動態分析 36
5.2 敏感度分析 38
5.3模型校準 42
5.4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/#G0102354017en_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 modelen_US
dc.subject (關鍵詞) EM algorithmen_US
dc.subject (關鍵詞) Esscher Transformationen_US
dc.subject (關鍵詞) European call option pricing formulaen_US
dc.subject (關鍵詞) sensitivity analysis; modelen_US
dc.subject (關鍵詞) model calibrationen_US
dc.subject (關鍵詞) volatility smile curveen_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 Risksen_US
dc.type (資料類型) thesisen
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