基于GMM方法跳聚集的短期利率模型参数估计
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摘要
构建具有自我激励机制跳的短期利率模型,应用随机跳的强度来描述自我激励机制跳的过程,即当短期利率发生跳时,同时跳的强度也相应地发生跳,从而刻画跳的聚集现象.文章将以美国国债收益率作为研究目标,通过广义矩估计方法(GMM)给出了模型的参数估计和统计推断.借鉴随机微分算子Taylor展开方法,从理论上给出了精确的矩函数,并通过辅助粒子滤波器(APF)给出随机跳的强度估计.实证结果揭示了文章所构建的模型不仅能够很好地刻画极端事件对于短期利率的冲击,而且也很好地描述跳的聚集现象.此外,实证结果也表明了跳的强度可作为市场压力测试的一个重要指标.
This paper proposes a short-term interest model with the self-exciting jump process described by stochastic jump intensity. In the model, a jump of shortterm interests increases the intensity of jumps during the same period, which captures the jump cluster. After researching into U.S. T-Bill rate, this paper will also make parametric estimations and statistical inference through employing the Generalized Moment Method(GMM). The moment function will be theoretically provided under the framework of the stochastic Taylor expansion of the differential operator.Meanwhile, the filtered values of the stochastic jump intensity will be estimated by applying the auxiliary particle filter algorithm(APF). The empirical results show that the model established not only sufficiently describes the impact of the extreme event on short-term interest, but also characterizes the jump cluster phenomena. In addition, the filtered jump intensity is an important indicator of financial market stress measurement.
This paper proposes a short-term interest model with the self-exciting jump process described by stochastic jump intensity. In the model, a jump of shortterm interests increases the intensity of jumps during the same period, which captures the jump cluster. After researching into U.S. T-Bill rate, this paper will also make parametric estimations and statistical inference through employing the Generalized Moment Method(GMM). The moment function will be theoretically provided under the framework of the stochastic Taylor expansion of the differential operator.Meanwhile, the filtered values of the stochastic jump intensity will be estimated by applying the auxiliary particle filter algorithm(APF). The empirical results show that the model established not only sufficiently describes the impact of the extreme event on short-term interest, but also characterizes the jump cluster phenomena. In addition, the filtered jump intensity is an important indicator of financial market stress measurement.
引文
[1]邓国和.双跳跃仿射扩散模型的美式看跌期权定价.系统科学与数学,2017, 37(7):1646-1663.(Deng G H. Valuation on American put option in an affine diffusion model with double jumps.Journal of Systems Science and Mathematical Sciences, 2017, 37(7):1646-1663.)
[2]邢文婷,吴胜利.天然气价格跳跃下的期货定价动态模型及实证研究,系统科学与数学,2017, 37(9):1988-1998.(Xing W T, Wu S L. Dynamic model of futures pricing under jump behavior of natural gas price and its empirical research. Journal of Systems Science and Mathematical Sciences, 2017, 37(9):1988-1998.)
[3]赵涤非,唐勇.基于高频数据视角的上证综指跳跃原因分析.系统科学与数学,2015, 35(1):85-98.(Zhao D F, Tang Y. Analysis on the reason of jumps in Shanghai Composite Index:Based on the viewpoints of high frequency data. Journal of Systems Science and Mathematical Sciences,2015, 35(1):85-98.)
[4] Eraker B, Johannes M, Polson N G. The impact of jumps in returns and volatility. Journal of Finance, 2003, 58(3):1269-1300.
[5] Duffie D, Pan J,Singleton K. Transform analysis and asset pricing for affine jump-diffusions.Econometrica, 2000, 68(6):1343-1376.
[6] A(i|¨)t-Sahalia Y. Disentangling diffusion from jumps. Journal of Financial Economics, 2004, 74(3):487-528
[7] A(i|¨)t-Sahalia Y, Julio C D, Laeven R J A. Modeling financial contagion using mutually exciting jump processes. The Journal of Financial Economics, 2015, 117(3):585-606.
[8] Johannes M. The statistical and economic role of jumps in continuous-time interest rate models.Journal of Finance, 2004, 59(7):227-260.
[9] Hawkes A G. Spectra of some self-exiting and mutually exciting point processes. Biometrika,1971, 58(1):83-90.
[10] Hawkes A G. Hawkes processes and their applications to finance:A review. Quantitative Finance,2017, 18(1):193-198.
[11] Zhu L. Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps. Journal of Applied Probability, 2014, 51(3):699-712.
[12] Hainaut D. A model for interest rates with clustering effects. Quantitative Finance, 2016, 16(8):1469-7696.
[13]吴奔,张波.Hawkes过程分支比估计——一种简单的非参数方法.统计研究,2015, 32(3):92-99.(Wu B, Zhang B. Branching ratio estimation for the Hawkes process—An improved nonparametric methods. Statisitcal Research, 2015, 32(3):92-99.)
[14] Lord R, Koekkoek R, Dijk D V. A comparison of biased simulation schemes for stochastic volatility models. Quantitative Finance, 2010, 10(2):177-194.
[15] Paseka A, Koulies T, Thavaneswaran. Inference for interest rate models using Milstein's approximation. Journal of Mathematical Finance, 2013, 3(1):110-118.
[16] Filipovi D, Eberhard M, Schneiderc P. Density approximations for multivariate affine jumpdiffusion processes. Journal of Econometrics, 2013, 176(2):93-111.
[17] Joslin S, Singleton K J, Zhu H. A new perspective on Gaussian dynamic term structure models.Review of Financial Studies, 2011, 24(3):926-970.
[18] Cont R, Tankov P. Financial modelling with Jump Processes. New York:Champan&Hall Press,2004.
[19] Stanton R. A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance, 1997, 52(5):1973-2002.
[20] Fonseca J, Zaatour R. Hawkes Process:Fast calibration, application to trade clustering, and diffusive limit. The Journal of Futures Martkets, 2014, 34(6):548-579.
[21] Das S R. The surprise element:Jumps in interest rates. Journal of Econometrics, 2002, 106(1):27-65.
[22] Hansen L P. Large sample properties of generalized method of moments estimators. Econometrica, 1984, 50(4):1029-1054.
[23] Hayashi F. Econometrics. New Jersey:Princeton University Press, 2000.
[24] Chan K C, Karolyi A, Longstaff F A, et al. An empirical comparison of alternative models of the short-term interest rate. The Journal of Fiannce, 1992, 47(3):1209-1227.
[25] Johannes M S, Polson N G, Stroud J R. Optimal filtering of jump diffusions:Extracting latent states from asset prices. The Review of Financial Studies, 2009, 22(7):2759-2799.
[26] Pitt M K, Malik S, Doucet A. Simulated likelihood inference for stochastic volatility models using continuous particle filtering. Annals of the Institute of Statistical Mathematics, 2014, 66(3):527-552.
[27J Kim J, Stoffer D S. Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the EM algorithm. Journal of Time Series Analysis, 2008, 29(5):811-833.
[28] Christoffersen P, Jacobs K, Mimouni K. Volatility dynamics for the S&P500:Evidence from realized volatility, daily returns, and option prices. Review of Financial Studies, 2010, 23(8):3141-3189.
1.在实证部分,当使用美国国债一个月到期每天交易收益率数据时,在2015年9-10月出现负的利率.如2015年10月1日收益率为-0.02%. Hainaut~([12])也说明了欧元隔夜指数平均值在2015年也出现了负的利率.
2.注意在Hainaut~([12])文章,他考虑的跳的幅度是随机的,而且是双指数跳的问题,同时他也引入一个阈值来描述跳的发生和不发生的问题,从而应用两步估计方法给出了精确的极大似然估计.本文仅考虑跳幅度是常数,而跳的N_t是满足经典的Poisson过程的模型来描述跳的聚集现象,而且直接通过样本数据给出参数估计和模型推断.
3. Zhu~([11])研究跳的幅度是一个常数的单因子短期利率模型,相应跳的强度具有仿射性结构性质,从而得出短期利率模型具有自我激励机制跳的性质,而跳是依赖于短期利率r作为内生的变量.本文引入随机跳强度不依赖于短期利率r,而是作为外生变量来驱动短期利率跳的过程.
4.由于我们主要目的是考虑跳及自我激励机制跳对短期利率模型的影响,因此不去考虑漂移项对跳的敏感性问题.事实上,根据Das~([21])的研究结果发现,引入跳的因子会使得漂移项线性化.
5.注意所得矩函数是有偏误的,但是根据定理3和4,这种偏误随着离散的区间减小而减少,而且该偏误也非常的小,如在实证部分用的δ=1/(262),相应的平方阶误差为1.46×10~(-5),是非常小的一个数.
6.似然率LR不是真正的似然率,然而根据Hayashi~([23])的研究结果,对于有效的GMM算法,LR的值可以当做似然率来测试一些嵌入的模型有效性.
7.一方面,由于作者无法获得我国完全的短期债券数据,因此使用了美国国债的数据.另一方面,根据Das~([21])的论述,可以使用美国债券收益数据作为无风险的短期利率.
8. A?t-Sahalia等~([7])说明了不像波动率可以通过恐慌指数来近似,而且在市场也有一些针对波动率的衍生品,因此波动率可以通过这些衍生品直接估计,然而对跳的因素在市场上没有任何的衍生品对冲,因此无法通过市场上的产品来近似替代.
[2]邢文婷,吴胜利.天然气价格跳跃下的期货定价动态模型及实证研究,系统科学与数学,2017, 37(9):1988-1998.(Xing W T, Wu S L. Dynamic model of futures pricing under jump behavior of natural gas price and its empirical research. Journal of Systems Science and Mathematical Sciences, 2017, 37(9):1988-1998.)
[3]赵涤非,唐勇.基于高频数据视角的上证综指跳跃原因分析.系统科学与数学,2015, 35(1):85-98.(Zhao D F, Tang Y. Analysis on the reason of jumps in Shanghai Composite Index:Based on the viewpoints of high frequency data. Journal of Systems Science and Mathematical Sciences,2015, 35(1):85-98.)
[4] Eraker B, Johannes M, Polson N G. The impact of jumps in returns and volatility. Journal of Finance, 2003, 58(3):1269-1300.
[5] Duffie D, Pan J,Singleton K. Transform analysis and asset pricing for affine jump-diffusions.Econometrica, 2000, 68(6):1343-1376.
[6] A(i|¨)t-Sahalia Y. Disentangling diffusion from jumps. Journal of Financial Economics, 2004, 74(3):487-528
[7] A(i|¨)t-Sahalia Y, Julio C D, Laeven R J A. Modeling financial contagion using mutually exciting jump processes. The Journal of Financial Economics, 2015, 117(3):585-606.
[8] Johannes M. The statistical and economic role of jumps in continuous-time interest rate models.Journal of Finance, 2004, 59(7):227-260.
[9] Hawkes A G. Spectra of some self-exiting and mutually exciting point processes. Biometrika,1971, 58(1):83-90.
[10] Hawkes A G. Hawkes processes and their applications to finance:A review. Quantitative Finance,2017, 18(1):193-198.
[11] Zhu L. Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps. Journal of Applied Probability, 2014, 51(3):699-712.
[12] Hainaut D. A model for interest rates with clustering effects. Quantitative Finance, 2016, 16(8):1469-7696.
[13]吴奔,张波.Hawkes过程分支比估计——一种简单的非参数方法.统计研究,2015, 32(3):92-99.(Wu B, Zhang B. Branching ratio estimation for the Hawkes process—An improved nonparametric methods. Statisitcal Research, 2015, 32(3):92-99.)
[14] Lord R, Koekkoek R, Dijk D V. A comparison of biased simulation schemes for stochastic volatility models. Quantitative Finance, 2010, 10(2):177-194.
[15] Paseka A, Koulies T, Thavaneswaran. Inference for interest rate models using Milstein's approximation. Journal of Mathematical Finance, 2013, 3(1):110-118.
[16] Filipovi D, Eberhard M, Schneiderc P. Density approximations for multivariate affine jumpdiffusion processes. Journal of Econometrics, 2013, 176(2):93-111.
[17] Joslin S, Singleton K J, Zhu H. A new perspective on Gaussian dynamic term structure models.Review of Financial Studies, 2011, 24(3):926-970.
[18] Cont R, Tankov P. Financial modelling with Jump Processes. New York:Champan&Hall Press,2004.
[19] Stanton R. A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance, 1997, 52(5):1973-2002.
[20] Fonseca J, Zaatour R. Hawkes Process:Fast calibration, application to trade clustering, and diffusive limit. The Journal of Futures Martkets, 2014, 34(6):548-579.
[21] Das S R. The surprise element:Jumps in interest rates. Journal of Econometrics, 2002, 106(1):27-65.
[22] Hansen L P. Large sample properties of generalized method of moments estimators. Econometrica, 1984, 50(4):1029-1054.
[23] Hayashi F. Econometrics. New Jersey:Princeton University Press, 2000.
[24] Chan K C, Karolyi A, Longstaff F A, et al. An empirical comparison of alternative models of the short-term interest rate. The Journal of Fiannce, 1992, 47(3):1209-1227.
[25] Johannes M S, Polson N G, Stroud J R. Optimal filtering of jump diffusions:Extracting latent states from asset prices. The Review of Financial Studies, 2009, 22(7):2759-2799.
[26] Pitt M K, Malik S, Doucet A. Simulated likelihood inference for stochastic volatility models using continuous particle filtering. Annals of the Institute of Statistical Mathematics, 2014, 66(3):527-552.
[27J Kim J, Stoffer D S. Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the EM algorithm. Journal of Time Series Analysis, 2008, 29(5):811-833.
[28] Christoffersen P, Jacobs K, Mimouni K. Volatility dynamics for the S&P500:Evidence from realized volatility, daily returns, and option prices. Review of Financial Studies, 2010, 23(8):3141-3189.
1.在实证部分,当使用美国国债一个月到期每天交易收益率数据时,在2015年9-10月出现负的利率.如2015年10月1日收益率为-0.02%. Hainaut~([12])也说明了欧元隔夜指数平均值在2015年也出现了负的利率.
2.注意在Hainaut~([12])文章,他考虑的跳的幅度是随机的,而且是双指数跳的问题,同时他也引入一个阈值来描述跳的发生和不发生的问题,从而应用两步估计方法给出了精确的极大似然估计.本文仅考虑跳幅度是常数,而跳的N_t是满足经典的Poisson过程的模型来描述跳的聚集现象,而且直接通过样本数据给出参数估计和模型推断.
3. Zhu~([11])研究跳的幅度是一个常数的单因子短期利率模型,相应跳的强度具有仿射性结构性质,从而得出短期利率模型具有自我激励机制跳的性质,而跳是依赖于短期利率r作为内生的变量.本文引入随机跳强度不依赖于短期利率r,而是作为外生变量来驱动短期利率跳的过程.
4.由于我们主要目的是考虑跳及自我激励机制跳对短期利率模型的影响,因此不去考虑漂移项对跳的敏感性问题.事实上,根据Das~([21])的研究结果发现,引入跳的因子会使得漂移项线性化.
5.注意所得矩函数是有偏误的,但是根据定理3和4,这种偏误随着离散的区间减小而减少,而且该偏误也非常的小,如在实证部分用的δ=1/(262),相应的平方阶误差为1.46×10~(-5),是非常小的一个数.
6.似然率LR不是真正的似然率,然而根据Hayashi~([23])的研究结果,对于有效的GMM算法,LR的值可以当做似然率来测试一些嵌入的模型有效性.
7.一方面,由于作者无法获得我国完全的短期债券数据,因此使用了美国国债的数据.另一方面,根据Das~([21])的论述,可以使用美国债券收益数据作为无风险的短期利率.
8. A?t-Sahalia等~([7])说明了不像波动率可以通过恐慌指数来近似,而且在市场也有一些针对波动率的衍生品,因此波动率可以通过这些衍生品直接估计,然而对跳的因素在市场上没有任何的衍生品对冲,因此无法通过市场上的产品来近似替代.