Dear Carol,
This is my first time encountering basis spline by studying your book. But i'm determined to get to the bottom for all you've written in the yield curve section.
My questions are:
a. the knot points themselves are actually the discount factors, NOT time as in the example of using natual cubic spline to fit the yield curve in Vol. 1. Is this correct?
b. each segment of the yield curve between the knot points is built using a linear combination of some basis functions. Per my understanding, the basis functions within a linear combination for each segment do not have the same order. So, if the knot points are really few, then within a linear combination it can consist of basis functions of order 1, order 2, order 3 and up to order 5, lets say. But if the knot points are many, then it could consist of basis function of only order 1 and order 2, for example. Did I understand correctly here?
Thank you for your help!
risk taker
basis spline for yield curve fitting
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 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: basis spline for yield curve fitting
Hi
The x_i have the same interpretation (time) but the yields y of the example in Vol I are replaced by discount factors \delta
Basis functions are defined iteratively, starting from 1st order basis functions which are just horizontal lines at value 1, one 1st order basis function for each interval between knot points. Then the 2nd order basis function for the ith interval between knot points is a linear combination of 1st order basis functions for the ith and i+1th interval. And so on, using (III.i.64) . Here is a step by step explanation with nice pictures of the way we build the Bspline basis functions:
http://www.cs.mtu.edu/~shene/COURSES/cs ... basis.html
One you have the basis functions, then you use these in place of cubic polynomials. Higher order basis functions are more bendy, just like higher order polynomials. If you have only two know points you can fit them with a line, i.e. first order basis functions, but if you have lots of know points the basis functions need to be more bendy.
Hope this helps, Carol
The x_i have the same interpretation (time) but the yields y of the example in Vol I are replaced by discount factors \delta
Basis functions are defined iteratively, starting from 1st order basis functions which are just horizontal lines at value 1, one 1st order basis function for each interval between knot points. Then the 2nd order basis function for the ith interval between knot points is a linear combination of 1st order basis functions for the ith and i+1th interval. And so on, using (III.i.64) . Here is a step by step explanation with nice pictures of the way we build the Bspline basis functions:
http://www.cs.mtu.edu/~shene/COURSES/cs ... basis.html
One you have the basis functions, then you use these in place of cubic polynomials. Higher order basis functions are more bendy, just like higher order polynomials. If you have only two know points you can fit them with a line, i.e. first order basis functions, but if you have lots of know points the basis functions need to be more bendy.
Hope this helps, Carol

 Posts: 41
 Joined: Wed Aug 10, 2011 12:52 pm
Re: basis spline for yield curve fitting
Hi Carol,
Given a certain number of knot points, lets say there are 10 knot points. The maximum order of basis functions in this case is 9. But how do we decide which order of basis function to use?
i looked at the link you posted. In the third picture, when the order of basis is 2, or the degree of basis is 1, there are overlapping functions. But in yield curve fitting, we dont want overlapping functions. So how do we know which order of basis functions to use that avoids overlapping i.e.each curve segment joins up at the knot points which form a smooth curve?
The knot points themselves are NOT the discount factors, but are time??? But as you define the basis function of order 1 for ith interval, when delta falls within the ith interval, then the value of the basis is 1. So it doent make sense to me that discount factor falls within a time interval, which is a bit inconsistent. It would be consistent if discount factor falls within a discount factor interval or if time falls within a time interval. Where did i get it wrong?
When I read up on other sources on introducing BSplines, control points are introduced and they are in the BSpline function. But how can we know what's the appropriate number of control points and what are there actual values?? My guess is that the appropriate number of control points and their actual values are parameters of BSpline function, which are calibrated in the optimization algo. Am i right?
Sorry for these quesitons. As its my first time encountering BSpline, there are quite a few nitty gritty details I am just not sure and clear about. Appreciate hugely if you could shed some light.
risk taker
Given a certain number of knot points, lets say there are 10 knot points. The maximum order of basis functions in this case is 9. But how do we decide which order of basis function to use?
i looked at the link you posted. In the third picture, when the order of basis is 2, or the degree of basis is 1, there are overlapping functions. But in yield curve fitting, we dont want overlapping functions. So how do we know which order of basis functions to use that avoids overlapping i.e.each curve segment joins up at the knot points which form a smooth curve?
The knot points themselves are NOT the discount factors, but are time??? But as you define the basis function of order 1 for ith interval, when delta falls within the ith interval, then the value of the basis is 1. So it doent make sense to me that discount factor falls within a time interval, which is a bit inconsistent. It would be consistent if discount factor falls within a discount factor interval or if time falls within a time interval. Where did i get it wrong?
When I read up on other sources on introducing BSplines, control points are introduced and they are in the BSpline function. But how can we know what's the appropriate number of control points and what are there actual values?? My guess is that the appropriate number of control points and their actual values are parameters of BSpline function, which are calibrated in the optimization algo. Am i right?
Sorry for these quesitons. As its my first time encountering BSpline, there are quite a few nitty gritty details I am just not sure and clear about. Appreciate hugely if you could shed some light.
risk taker

 Posts: 41
 Joined: Wed Aug 10, 2011 12:52 pm
Re: basis spline for yield curve fitting
Hi Carol, I have added another question at the end of the previous post. Cheers!

 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: basis spline for yield curve fitting
Hi Risk Taker,
Sorry for delayed reply, I've been camping!
In general, the problem with any fitting methodology is that the shape can get distorted if the functions are too flexible. For this reason I have favored Hermite splines in my research for the last 10 years. Plus, I don't get involved in nittygritty implementation details myself (I drive the research with the big picture and my students do the programming and implementation) so I am not able to answer your furter questions. In general though, I would use a low order for the basis functions, but high enough for errors to be small. In the end it will be a question of trial and error plus good judgement.
Best wishes, Carol
Sorry for delayed reply, I've been camping!
In general, the problem with any fitting methodology is that the shape can get distorted if the functions are too flexible. For this reason I have favored Hermite splines in my research for the last 10 years. Plus, I don't get involved in nittygritty implementation details myself (I drive the research with the big picture and my students do the programming and implementation) so I am not able to answer your furter questions. In general though, I would use a low order for the basis functions, but high enough for errors to be small. In the end it will be a question of trial and error plus good judgement.
Best wishes, Carol
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