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題名 跳躍風險相關之匯率選擇權: 傅立葉轉換評價法、Martingale法與蒙地卡羅法之比較 作者 温晉祥
Wen, Chin-Hsiang貢獻者 林士貴
Lin, Shih-Kuei
温晉祥
Wen, Chin-Hsiang關鍵詞 Amin and Jarrow model
外匯選擇權
相關跳躍風險
匯率
利率
跳躍風險
Amin and Jarrow model
currency option
correlated jump risks
exchange rate
interest rate
jump risks日期 2020 上傳時間 5-Feb-2020 17:30:51 (UTC+8) 摘要 本論文觀察最近十多年來國際上幾個主要國家利率與匯率的走勢以及同一個期間內的跳躍情況，發現走勢有相關性存在，並且經常同時發生跳躍。為了此特殊性質，本研究建立一個考慮走勢與跳躍相關的模型來捕捉此特性，稱作考慮相關跳躍模型 (Amin and Jarrow model with correlated jump risks, AJ-CJ)。實證結果發現AJ-CJ比起幾何布朗運動 (Geometric Brownian motion, GBM)、Amin and Jarrow 模型 (Amin and Jarrow model, AJ)、考慮獨立跳躍模型 (Amin and Jarrow model with independent jump risks, AJ-IJ) 可以更加捕捉利率及匯率的特性。利用martingale法與傅立葉轉評價法推導出AJ-CJ下的匯率選擇權評價公式並且比較兩種方法與蒙地卡羅法之計算速度與準確度，發現三種方法的評價結果很接近，且傅立葉轉評價法計算速度比另外兩種方法快許多。實證發現，大多數的例子中，AJ-CJ改善了樣本內及樣本外定價誤差，也代表可以更精準地評價匯率選擇權。研究結果支持利率與匯率存在相關性及跳躍間也存在相關。
In this paper, we investigate the trends of interest rates and exchange rates in several major international countries in the past ten years and find that the trends are correlated and often jump at the same time. Given the characteristics of correlated jump risks in interest rates and exchange rates, we construct a new model called Amin and Jarrow model with correlated jump risks (AJ-CJ) to capture the movements. The empirical results in exchange rates and interest rates data with the log-likelihood value show that AJ-CJ can capture the interest rates and the exchange rates better than Geometric Brownian model (GBM), Amin and Jarrow model (AJ), and Amin and Jarrow model with independent jump risks (AJ-IJ). After finding the martingale condition, we derive the pricing formula for currency options under AJ-CJ with the traditional martingale method and generalized Fourier transform method. This study adds the Monte Carlo method to verify the evaluation results and compare calculating time. We found that the evaluation result of traditional martingale method and Fourier evaluation method is very close to the Monte Carlo method. The calculating time of Fourier evaluation method is much faster than traditional martingale method and the Monte Carlo method. In addition, the empirical performance of the option data finds that AJ-CJ improves the in-sample and out-of-sample pricing error performances in most cases. Therefore, we conclude that correlated jump risks between interest rates and exchange rates.參考文獻 1. Amin, K. and R. A. Jarrow, 1991, “Pricing foreign currency options under stochastic interest rates,” Journal of International Money and Finance, Vol. 10, 310-329.2. Bailey, W and R.M. Stulz, 1989, “The pricing of stock index options in a general equilibrium Model,” Journal of Financial and Quantitative Analysis, Vol. 24, 1-12.3. Bates, D., 1991, “The crash of 87: Was it expected? The evidence from options markets,” Journal of Finance, Vol. 46, 1009-1044.4. Bates, D., 1996a, “Dollar jump fears, 1984-1992: Distributional abnormalities implicit in currency futures options,” Journal of International Money and Finance, Vol. 15, 65-93.5. Bates, D., 1996b, “Jumps and stochastic volatility: Exchange rate process implicit in Deutsche Mark options,” Review of Financial Studies, Vol. 9, 69-107.6. Bakshi, G., C. Cao, and Z. Chen, 1997, “Empirical performance of alternative option pricing models,” Journal of Finance, 52, 2003–2049.7. Bollen, N. P. B., S. F. Gary, and R. E. Whaley, 2000,” Regime switching in foreign exchange rates Evidence from currency option prices,” Journal of Econometrics , Vol. 94,239-276.8. Brigo, D. and F., Mercurio, 2006, Interest rate models—Theory and practice. Springer Verlag, Berlin.9. Ornthanalai, G., 2014, “Lévy jump risk: Evidence from options and returns,” Journal of Financial Economics, Vol. 112, 69-90.10. Bo. L, Y. Wang, and X. Yang, 2010, “Markov-modulated jump-diffusions for currency option pricing,” Insurance: Mathematics and Economics, Vol. 46, 461-469.11. Chan, W. H., 2004, “Conditional correlated jump dynamics in foreign exchange,” Economics Letters, Vol. 83, 23-28.12. Chang, M. A., D. C. Cho, and L. Park, 2007, “The pricing of foreign currency options under jump-diffusion processes,” The Journal of Futures Markets, Vol. 27, No. 7, 669-695 .13. Doffou, A., and J. E. Hilliard, 2001, “Pricing currency options under stochastic interest rates and jump-diffusion processes,” The Journal of Financial Research, Vol. 24, No. 4, 565-585.14. Feiger, G., and B. Jacquillat, 1979, “Currency option bonds, puts and calls on spot exchange and the hedging of contingent foreign earnings,” Journal of Finance, Vol. 34, 1129-1139.15. Grabbe, O., 1983, “The pricing of call and put options on foreign exchange,” Journal of International Money and Finance, Vol. 2, 239-253.16. Garman, M. B. and S. W. Kohlhagen, 1983. “Foreign currency option values,” Journal of International Money and Finance, Vol. 2, 231-237.17. Gerber, H. U. and E. S. W. Shiu, 1994, “Option pricing by esscher transforms,” Transactions of the Society of Actuaries, Vol. 46, 99-140.18. Guo, J. H. and M. W. Hung, 2007. “Pricing American options on foreign currency with stochastic volatility, jumps, and stochastic interest rates,” The Journal of Futures Markets, Vol. 27, No. 9, 867-891.19. Harrison, J.M. and S.R. Pliska, 1981 “Martingales and Stochastic Integrals in the Theory of Continuous Trading,” Stochastic Processes and Their Applications, Vol. 11, 215-26020. Hull, J. and A. White, 1990. “Pricing interest rate derivative securities,” Review of Financial Studies, Vol. 29, 347-368.21. Heston, S. and S. Nandi, 2000. A closed-form GARCH option pricing model. Review of Financial Studies, Vol.13, 585–626.22. Jorion, P., 1988, “On jump processes in the foreign exchange and stock markets,” The Review of Financial Studies, Vol. 1, No.4, 427-445.23. Jarrow, R., and Y. Yildirim, 2003, “Pricing treasury inflation protected securities and related derivatives using an HJM model,” Journal of Financial and Quantitative Analysis, Vol. 38, 337-357.24. Li, X. P., Y. Feng, C. F. Wu, and W. D. Xu, 2013, “Response of the term structure of forward exchange rate to jump in the interest rate,” Economic Modelling, Vol. 30, 863–87425. Lin. C. H., S. K., Lin, and A. C., Wu, 2015, “Foreign exchange option pricing in the currency cycle with jump risks,” Review of Quantitative Finance and Accounting, Vol. 44, 755-789.26. Merton, R. C., 1976, “Option pricing when underlying stock returns are discontinuous,” Journal of Financial Economics, Vol. 63, 3-50.27. Musiela, M. and M. Rutkowski, 1998, “Martingale Methods in Financial Modelling,” Journal of the American Statistical Association, Springer-Verlag, Berlin.28. Shokrollahi F, A. Kılıçman, M. Magdziarz , 2016, “Pricing European options and currency options by time changed mixed fractional Brownian motion with transaction costs,” International Journal of Financial Engineering, Vol.3, No.1, 1650003.29. Xiao, W. L., W. G. Zang, X. L. Zang, and Y. L. Wang, 2010, “Pricing currency options in a fractional Brownian motion with jumps,” Economic Modelling, 935-942 描述 博士
國立政治大學
金融學系
100352501資料來源 http://thesis.lib.nccu.edu.tw/record/#G0100352501 資料類型 thesis dc.contributor.advisor 林士貴 zh_TW dc.contributor.advisor Lin, Shih-Kuei en_US dc.contributor.author (Authors) 温晉祥 zh_TW dc.contributor.author (Authors) Wen, Chin-Hsiang en_US dc.creator (作者) 温晉祥 zh_TW dc.creator (作者) Wen, Chin-Hsiang en_US dc.date (日期) 2020 en_US dc.date.accessioned 5-Feb-2020 17:30:51 (UTC+8) - dc.date.available 5-Feb-2020 17:30:51 (UTC+8) - dc.date.issued (上傳時間) 5-Feb-2020 17:30:51 (UTC+8) - dc.identifier (Other Identifiers) G0100352501 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/128564 - dc.description (描述) 博士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 金融學系 zh_TW dc.description (描述) 100352501 zh_TW dc.description.abstract (摘要) 本論文觀察最近十多年來國際上幾個主要國家利率與匯率的走勢以及同一個期間內的跳躍情況，發現走勢有相關性存在，並且經常同時發生跳躍。為了此特殊性質，本研究建立一個考慮走勢與跳躍相關的模型來捕捉此特性，稱作考慮相關跳躍模型 (Amin and Jarrow model with correlated jump risks, AJ-CJ)。實證結果發現AJ-CJ比起幾何布朗運動 (Geometric Brownian motion, GBM)、Amin and Jarrow 模型 (Amin and Jarrow model, AJ)、考慮獨立跳躍模型 (Amin and Jarrow model with independent jump risks, AJ-IJ) 可以更加捕捉利率及匯率的特性。利用martingale法與傅立葉轉評價法推導出AJ-CJ下的匯率選擇權評價公式並且比較兩種方法與蒙地卡羅法之計算速度與準確度，發現三種方法的評價結果很接近，且傅立葉轉評價法計算速度比另外兩種方法快許多。實證發現，大多數的例子中，AJ-CJ改善了樣本內及樣本外定價誤差，也代表可以更精準地評價匯率選擇權。研究結果支持利率與匯率存在相關性及跳躍間也存在相關。 zh_TW dc.description.abstract (摘要) In this paper, we investigate the trends of interest rates and exchange rates in several major international countries in the past ten years and find that the trends are correlated and often jump at the same time. Given the characteristics of correlated jump risks in interest rates and exchange rates, we construct a new model called Amin and Jarrow model with correlated jump risks (AJ-CJ) to capture the movements. The empirical results in exchange rates and interest rates data with the log-likelihood value show that AJ-CJ can capture the interest rates and the exchange rates better than Geometric Brownian model (GBM), Amin and Jarrow model (AJ), and Amin and Jarrow model with independent jump risks (AJ-IJ). After finding the martingale condition, we derive the pricing formula for currency options under AJ-CJ with the traditional martingale method and generalized Fourier transform method. This study adds the Monte Carlo method to verify the evaluation results and compare calculating time. We found that the evaluation result of traditional martingale method and Fourier evaluation method is very close to the Monte Carlo method. The calculating time of Fourier evaluation method is much faster than traditional martingale method and the Monte Carlo method. In addition, the empirical performance of the option data finds that AJ-CJ improves the in-sample and out-of-sample pricing error performances in most cases. Therefore, we conclude that correlated jump risks between interest rates and exchange rates. en_US dc.description.tableofcontents 第一章 緒論 6第一節 研究背景 6第二節 研究動機 6第二章 文獻回顧與研究架構 9第一節 文獻回顧 9第二節 研究架構 11第三章 研究方法與模型假設 11第一節 考慮相關跳躍之動態過程 11第二節 EM演算法 13第三節 概似比檢定 14第四節 測度轉換 14第五節 匯率選擇權評價公式-Martingale法 16第六節 匯率選擇權評價公式-傅立葉轉換法 17第四章 政府債券與匯率市場實證分析 17第一節 LIBORs 18第二節 匯率 19第三節 相關性 19第四節 參數估計 19第五節 模型選擇 21第五章 匯率選擇權實證分析 22第一節 匯率選擇權市場價格 22第二節 樣本內定價誤差 22第三節 樣本外定價誤差 23第四節 模型計算時間比較 24第六章 結論 24參考文獻 25附錄A: 測度轉換 27附錄B：平賭條件 31附錄C：Martingale法及匯率選擇權評價公式 32附錄D：傅立葉轉換法及匯率選擇權評價公式 34 zh_TW dc.format.extent 1841173 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0100352501 en_US dc.subject (關鍵詞) Amin and Jarrow model zh_TW dc.subject (關鍵詞) 外匯選擇權 zh_TW dc.subject (關鍵詞) 相關跳躍風險 zh_TW dc.subject (關鍵詞) 匯率 zh_TW dc.subject (關鍵詞) 利率 zh_TW dc.subject (關鍵詞) 跳躍風險 zh_TW dc.subject (關鍵詞) Amin and Jarrow model en_US dc.subject (關鍵詞) currency option en_US dc.subject (關鍵詞) correlated jump risks en_US dc.subject (關鍵詞) exchange rate en_US dc.subject (關鍵詞) interest rate en_US dc.subject (關鍵詞) jump risks en_US dc.title (題名) 跳躍風險相關之匯率選擇權: 傅立葉轉換評價法、Martingale法與蒙地卡羅法之比較 zh_TW dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) 1. Amin, K. and R. A. Jarrow, 1991, “Pricing foreign currency options under stochastic interest rates,” Journal of International Money and Finance, Vol. 10, 310-329.2. Bailey, W and R.M. Stulz, 1989, “The pricing of stock index options in a general equilibrium Model,” Journal of Financial and Quantitative Analysis, Vol. 24, 1-12.3. Bates, D., 1991, “The crash of 87: Was it expected? The evidence from options markets,” Journal of Finance, Vol. 46, 1009-1044.4. Bates, D., 1996a, “Dollar jump fears, 1984-1992: Distributional abnormalities implicit in currency futures options,” Journal of International Money and Finance, Vol. 15, 65-93.5. Bates, D., 1996b, “Jumps and stochastic volatility: Exchange rate process implicit in Deutsche Mark options,” Review of Financial Studies, Vol. 9, 69-107.6. Bakshi, G., C. Cao, and Z. Chen, 1997, “Empirical performance of alternative option pricing models,” Journal of Finance, 52, 2003–2049.7. Bollen, N. P. B., S. F. Gary, and R. E. Whaley, 2000,” Regime switching in foreign exchange rates Evidence from currency option prices,” Journal of Econometrics , Vol. 94,239-276.8. Brigo, D. and F., Mercurio, 2006, Interest rate models—Theory and practice. Springer Verlag, Berlin.9. Ornthanalai, G., 2014, “Lévy jump risk: Evidence from options and returns,” Journal of Financial Economics, Vol. 112, 69-90.10. Bo. L, Y. Wang, and X. Yang, 2010, “Markov-modulated jump-diffusions for currency option pricing,” Insurance: Mathematics and Economics, Vol. 46, 461-469.11. Chan, W. H., 2004, “Conditional correlated jump dynamics in foreign exchange,” Economics Letters, Vol. 83, 23-28.12. Chang, M. A., D. C. Cho, and L. Park, 2007, “The pricing of foreign currency options under jump-diffusion processes,” The Journal of Futures Markets, Vol. 27, No. 7, 669-695 .13. Doffou, A., and J. E. Hilliard, 2001, “Pricing currency options under stochastic interest rates and jump-diffusion processes,” The Journal of Financial Research, Vol. 24, No. 4, 565-585.14. Feiger, G., and B. Jacquillat, 1979, “Currency option bonds, puts and calls on spot exchange and the hedging of contingent foreign earnings,” Journal of Finance, Vol. 34, 1129-1139.15. Grabbe, O., 1983, “The pricing of call and put options on foreign exchange,” Journal of International Money and Finance, Vol. 2, 239-253.16. Garman, M. B. and S. W. Kohlhagen, 1983. “Foreign currency option values,” Journal of International Money and Finance, Vol. 2, 231-237.17. Gerber, H. U. and E. S. W. Shiu, 1994, “Option pricing by esscher transforms,” Transactions of the Society of Actuaries, Vol. 46, 99-140.18. Guo, J. H. and M. W. Hung, 2007. “Pricing American options on foreign currency with stochastic volatility, jumps, and stochastic interest rates,” The Journal of Futures Markets, Vol. 27, No. 9, 867-891.19. Harrison, J.M. and S.R. Pliska, 1981 “Martingales and Stochastic Integrals in the Theory of Continuous Trading,” Stochastic Processes and Their Applications, Vol. 11, 215-26020. Hull, J. and A. White, 1990. “Pricing interest rate derivative securities,” Review of Financial Studies, Vol. 29, 347-368.21. Heston, S. and S. Nandi, 2000. A closed-form GARCH option pricing model. Review of Financial Studies, Vol.13, 585–626.22. Jorion, P., 1988, “On jump processes in the foreign exchange and stock markets,” The Review of Financial Studies, Vol. 1, No.4, 427-445.23. Jarrow, R., and Y. Yildirim, 2003, “Pricing treasury inflation protected securities and related derivatives using an HJM model,” Journal of Financial and Quantitative Analysis, Vol. 38, 337-357.24. Li, X. P., Y. Feng, C. F. Wu, and W. D. Xu, 2013, “Response of the term structure of forward exchange rate to jump in the interest rate,” Economic Modelling, Vol. 30, 863–87425. Lin. C. H., S. K., Lin, and A. C., Wu, 2015, “Foreign exchange option pricing in the currency cycle with jump risks,” Review of Quantitative Finance and Accounting, Vol. 44, 755-789.26. Merton, R. C., 1976, “Option pricing when underlying stock returns are discontinuous,” Journal of Financial Economics, Vol. 63, 3-50.27. Musiela, M. and M. Rutkowski, 1998, “Martingale Methods in Financial Modelling,” Journal of the American Statistical Association, Springer-Verlag, Berlin.28. Shokrollahi F, A. Kılıçman, M. Magdziarz , 2016, “Pricing European options and currency options by time changed mixed fractional Brownian motion with transaction costs,” International Journal of Financial Engineering, Vol.3, No.1, 1650003.29. Xiao, W. L., W. G. Zang, X. L. Zang, and Y. L. Wang, 2010, “Pricing currency options in a fractional Brownian motion with jumps,” Economic Modelling, 935-942 zh_TW dc.identifier.doi (DOI) 10.6814/NCCU202000074 en_US
