谢谢两位, 请问warecucff, 这么说即期利率还是个三因素的线形模型, 关键是如何用kalman filter 获得这三因素对吧? 即使有三因素了,那么affine model的系数,rho0 rho' 是从哪获得呢? (有三因素,系数才能得到r(t))

那么bloomberg上的1-yr,3-yr,10yr...之类的是即期利率么? 如果是的话那不是能直接得到了么?

when you pricing a bond[zero coupon bond for example and also for conventionality], what do you take as the discount factor? The short rate, and now we assume there are three latent factors [not always, as some studies find two factors could explain 99% variation for certain countries]explaining linearly the dynamics of short rate, we could price the bond prices by substituting the equations of factors and short rates into the bond pricing equation. Therefore you have the analytical solutions for the bond prices. These analytical solutions contains the roh0 and rho you mentioned above, and it is just for illustrating these analytical solutions, when it comes to implementing the model, you solve backwards from bond prices.

A good model usually has few structural parameters that enable varieties, those restrictions pn model only help to generate reasonable results but not to complicate it. You will get to know better when you are trying to replicate the empirical.

I don't know what they offer from bloomberg, but conventionaly you should find something called yield rate of zero coupon bond for different maturities, and these yield rates are mostly interpolated from other type of coupon bonds. Check Piazzesi(2003) for data selections.

Finally, Campbell, J.Y., A.W. Lo and A.C. MacKinlay (1997) The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press

Good luck.

我想确认一下你的意思:

比如说1年zero-coupon bond 票面价值100元,5% rate, 那么bond price就是 100/(1+5%), 你说的 "solve backwards from bond prices" 是指已知bond price,算出r(t)吧,至于三因素x(t), 我一开始以为用PCA从1-yr,2-yr...10yr(打比方)中算出,但他用kalman filter(我没学过,也是从相同数据中能算出么? 我会再看看), 有r(t),x(t),做regression 就能得出系数rho了吧.

我的问题是, 我只要求term premium和bond yield rate的关系, term premium is "Forward Rate - instantaneous interest rate". 你说的对,bloomberg上是不同maturity date 的 yield rate, 每天都在变. 这是不是就是r(t)了呢? 如果是的话, 那就不需要求三因素和建立回归模型了吧, 因为这个instantaneous interest rate 都是已知给我们了, 可能他是interpolate出来的(先不管). 至于forward rate, 给定这些rate之后, 可以用我上面说的方法求出在无套利情况下的rate.

其实我也没用过bloomberg,只是别人给我看了一下他有这么一些数据. 我现在defination有些没搞清, 比如在现实生活中哪些是已知,哪些是要求的. 还有bond yield rate 和 instanteous interest rate 和 r(t)是不是一样的等等.

不好意思 麻烦你了! 谢谢!

更正一下,

n-year zero coupon term premium 应该是 yield on an n-year zero coupon bond - average of expected future short rate. (what's this?)

Their 2004 paper[Term structure estimation with survey data on interest rate forecasts, www.federalreserve.gov/PUBS/feds/2005/200548/200548pap.pdf] describes the model in detail. You can see the estimation procedures from this paper, which could help you to understand the implementation.

Back to your questions, the instantaneous interest rate only exists in theory, in reality we call it short rate and it is approximated. How do we estimate the short rate? We assume the short rate is a affine function of three factors, and use the yield rates[zero coupon bond yield rate]to infere. Since the yield rates are affine in these three factors,we call it affine model. How do we get these factors? Using Kalman filter and Maximum Likelihood estimation.The by product of inverting these factors you also have the factor loadings, which are the coefficients in the factors dynamic equations. Using these you calculate the expected average future short rates. Since we believe there is a term premium associate with longer term bonds, which means that people demand compensation for holding longer term bonds over shorter ones. Therefore the term premium is calculated as (the yield rate on an n-year zero coupon bond - expected average of expected future short rate).

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那篇文章实在太难懂了,我看了一些别的文章,上传了一篇中文的,意思差不多的,能不能麻烦你看一下?

r(t),就是理论上的即时利率,是三因素的affine mode, r(t)=rho0+rho1*X, X是三个因素,

我现在比如有1-yr,2-yr,3-yr,...5-yr五组数据Y, Y=C0+C1*X1+C2*X2+C3*X3,也是affine model, 用kalman filter MLE,直接代入Y就能估计出参数C, 然后乘逆矩阵就能得到X了? 另外这里的C应该是个有时间变量的参数,代入不同t就能得到不同的Y是吧?

但我还是不明白如何得到rho...

另外,如果不在乎理论r(t)的话,不是直接把1-yr bond减掉1-yr 利率就能得到term premium了么?

多谢耐心指教!