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題名 Lévy過程下Stochastic Volatility與Variance Gamma之模型估計與實證分析
Estimation and Empirical Analysis of Stochastic Volatility Model and Variance Gamma Model under Lévy Processes
作者 黃國展
Huang, Kuo Chan
貢獻者 林士貴<br>翁久幸
Lin, Shih Kuei<br>Weng, Chiu Hsing
黃國展
Huang, Kuo Chan
關鍵詞 隨機波動度模型
波動度聚集
Lévy過程
跳躍風險
粒子濾波器
Stochastic volatility model
Volatility clustering
Lévy-process
Jump risk
Particle filter
日期 2017
上傳時間 31-Jul-2017 10:57:21 (UTC+8)
摘要 本研究以Lévy過程為模型基礎，考慮Merton Jump及跳躍強度服從Hawkes Process的Merton Jump兩種跳躍風險，利用Particle Filter方法及EM演算法估計出模型參數並計算出對數概似值、AIC及BIC。以S&P500指數為實證資料，比較隨機波動度模型、Variance Gamma模型及兩種不同跳躍風險對市場真實價格的配適效果。實證結果顯示，隨機波動度模型其配適效果勝於Variance Gamma模型，且加入跳躍風險後可使模型配適效果提升，尤其在模型中加入跳躍強度服從Hawkes Process的Merton Jump，其配適效果更勝於Merton Jump。整體而言，本研究發現，以S&P500指數為實證資料時，SVHJ模型有較好的配適效果。
This paper, based on the Lévy process, considers two kinds of jump risk, Merton Jump and the Merton Jump whose jump intensity follows Hawkes Process. We use Particle Filter method and EM Algorithm to estimate the model parameters and calculate the log-likelihood value, AIC and BIC. We collect the S&P500 index for our empirical analysis and then compare the goodness of fit between the stochastic volatility model, the Variance Gamma model and two different jump risks. The empirical results show that the stochastic volatility model is better than the Variance Gamma model, and it is better to consider the jump risk in the model, especially the Merton Jump whose jump intensity follows Hawkes Process. The goodness of fit is better than Merton Jump. Overall, we find SVHJ model has better goodness of fit when S&P500 index was used as the empirical data.
參考文獻 [1] Bakshi, G., Cao, C., & Chen, Z. W., 1997. Empirical performance of alternative option pricing models. Journal of Finance, 52: 2003-2049.
[2] Bates, D. S., 1996. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies, 9: 69-107.
[3] Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31: 307-327.
[4] Christoffersen, P., Jacobs, K., & Mimouni, K., 2010. Volatility dynamics for the S&P500: Evidence from realized volatility, daily returns, and option prices. Review of Financial Studies, 23: 3141-3189.
[5] Eraker, B., 2004. Do stock prices and volatility jump? Reconciling evidence from spot and option prices. Journal of Finance, 59, 1367-1404.
[6] Heston, S. L., 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6: 327-343.
[7] Hull, J., White, A., 1987. The pricing of options on assets with stochastic volatilities. Journal of Finance, 42(2): 281-300.
[8] Madan, D. B., Milne, F., 1991. Option Pricing With V.G. Martingale Components. Mathematical Finance, 1(4): 39–55.
[9] Madan, D. B., Seneta, E., 1990. The variance gamma (V.G.) model for share market returns. Journal of Business, 63: 511-524.
[10] Madan, D. B., Carr, P. P.,Chang, E.C., 1998. The variance gamma (V.G.) model for share market returns. European Finance Review ,2: 79–105.
[11] Merton, R. C., 1976. Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 63: 3-50.
[12] Ornthanalai, C., 2014. Lévy jump risk: Evidence from options and returns. Journal of Financial Economics, 112: 69-90.
[13] Pitt, M., Shephard, N., 1999. Filtering via simulation based on auxiliaryparticle filters. J. Am. Stat. Assoc. 94: 590-599.
[14] Pitt, M., 2002. Smooth particle filters for likelihood evaluation and maximization.Unpublished working paper. University of Warwick.
描述 碩士
國立政治大學
統計學系
104354023
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0104354023
資料類型 thesis
dc.contributor.advisor 林士貴<br>翁久幸zh_TW
dc.contributor.advisor Lin, Shih Kuei<br>Weng, Chiu Hsingen_US
dc.contributor.author (Authors) 黃國展zh_TW
dc.contributor.author (Authors) Huang, Kuo Chanen_US
dc.creator (作者) 黃國展zh_TW
dc.creator (作者) Huang, Kuo Chanen_US
dc.date (日期) 2017en_US
dc.date.accessioned 31-Jul-2017 10:57:21 (UTC+8)-
dc.date.available 31-Jul-2017 10:57:21 (UTC+8)-
dc.date.issued (上傳時間) 31-Jul-2017 10:57:21 (UTC+8)-
dc.identifier (Other Identifiers) G0104354023en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/111446-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 104354023zh_TW
dc.description.abstract (摘要) 本研究以Lévy過程為模型基礎，考慮Merton Jump及跳躍強度服從Hawkes Process的Merton Jump兩種跳躍風險，利用Particle Filter方法及EM演算法估計出模型參數並計算出對數概似值、AIC及BIC。以S&P500指數為實證資料，比較隨機波動度模型、Variance Gamma模型及兩種不同跳躍風險對市場真實價格的配適效果。實證結果顯示，隨機波動度模型其配適效果勝於Variance Gamma模型，且加入跳躍風險後可使模型配適效果提升，尤其在模型中加入跳躍強度服從Hawkes Process的Merton Jump，其配適效果更勝於Merton Jump。整體而言，本研究發現，以S&P500指數為實證資料時，SVHJ模型有較好的配適效果。zh_TW
dc.description.abstract (摘要) This paper, based on the Lévy process, considers two kinds of jump risk, Merton Jump and the Merton Jump whose jump intensity follows Hawkes Process. We use Particle Filter method and EM Algorithm to estimate the model parameters and calculate the log-likelihood value, AIC and BIC. We collect the S&P500 index for our empirical analysis and then compare the goodness of fit between the stochastic volatility model, the Variance Gamma model and two different jump risks. The empirical results show that the stochastic volatility model is better than the Variance Gamma model, and it is better to consider the jump risk in the model, especially the Merton Jump whose jump intensity follows Hawkes Process. The goodness of fit is better than Merton Jump. Overall, we find SVHJ model has better goodness of fit when S&P500 index was used as the empirical data.en_US
dc.description.tableofcontents 第一章 緒論 1
1.1 研究動機 1
1.2 研究目的 2
第二章 文獻回顧 3
2.1 Lévy Process 3
2.2 隨機波動度模型 3
2.3 Particles Filter 5
第三章 研究方法 6
3.1 Lévy Process 6
3.2 報酬率模型 9
3.3 Particle Filter 13
3.4 Expectation-Maximization Algorithm 16
第四章 實證分析 18
4.1 模擬分析 19
4.2 實證結果 19
第五章 結論 22
參考文獻 23

附錄
附錄 A：概似函數推導 25		zh_TW
dc.format.extent 1979226 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0104354023en_US
dc.subject (關鍵詞) 隨機波動度模型zh_TW
dc.subject (關鍵詞) 波動度聚集zh_TW
dc.subject (關鍵詞) Lévy過程zh_TW
dc.subject (關鍵詞) 跳躍風險zh_TW
dc.subject (關鍵詞) 粒子濾波器zh_TW
dc.subject (關鍵詞) Stochastic volatility modelen_US
dc.subject (關鍵詞) Volatility clusteringen_US
dc.subject (關鍵詞) Lévy-processen_US
dc.subject (關鍵詞) Jump risken_US
dc.subject (關鍵詞) Particle filteren_US
dc.title (題名) Lévy過程下Stochastic Volatility與Variance Gamma之模型估計與實證分析zh_TW
dc.title (題名) Estimation and Empirical Analysis of Stochastic Volatility Model and Variance Gamma Model under Lévy Processesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Bakshi, G., Cao, C., & Chen, Z. W., 1997. Empirical performance of alternative option pricing models. Journal of Finance, 52: 2003-2049.
[2] Bates, D. S., 1996. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies, 9: 69-107.
[3] Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31: 307-327.
[4] Christoffersen, P., Jacobs, K., & Mimouni, K., 2010. Volatility dynamics for the S&P500: Evidence from realized volatility, daily returns, and option prices. Review of Financial Studies, 23: 3141-3189.
[5] Eraker, B., 2004. Do stock prices and volatility jump? Reconciling evidence from spot and option prices. Journal of Finance, 59, 1367-1404.
[6] Heston, S. L., 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6: 327-343.
[7] Hull, J., White, A., 1987. The pricing of options on assets with stochastic volatilities. Journal of Finance, 42(2): 281-300.
[8] Madan, D. B., Milne, F., 1991. Option Pricing With V.G. Martingale Components. Mathematical Finance, 1(4): 39–55.
[9] Madan, D. B., Seneta, E., 1990. The variance gamma (V.G.) model for share market returns. Journal of Business, 63: 511-524.
[10] Madan, D. B., Carr, P. P.,Chang, E.C., 1998. The variance gamma (V.G.) model for share market returns. European Finance Review ,2: 79–105.
[11] Merton, R. C., 1976. Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 63: 3-50.
[12] Ornthanalai, C., 2014. Lévy jump risk: Evidence from options and returns. Journal of Financial Economics, 112: 69-90.
[13] Pitt, M., Shephard, N., 1999. Filtering via simulation based on auxiliaryparticle filters. J. Am. Stat. Assoc. 94: 590-599.
[14] Pitt, M., 2002. Smooth particle filters for likelihood evaluation and maximization.Unpublished working paper. University of Warwick.		zh_TW