Calibrating one-factor CIR to caps price with Newton-Raphson

A

Alice_bxl

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3/20/13
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Hello to everyone !

For my Master's thesis I've made a code that price a particular interest rates derivative using MC with the CIR model.

The code is fine, I now need to calibrate the model.

I'm not hiding the fact that I really have difficulties to understand the process I should code for this part: the algorithm is easy to understand of course but I don't know the formulas I should use.

Here is how I see the process:

f(x,..,..,..,..,etc.) = NPV_CAP_CIR - NPV_CAP_BlackSholes

where x is the variable I want to calibrate (the pull-back variable for instance)
δf '(x,..,..,..,..,etc.)/ δx = ............................
Click to expand...

NPV_CAP_CIR is the pricing formula of bond option with the CIR model
NPV_CAP_BlackSholes is the pricing formula of cap with Black

Is that correct ?

Also, is the pricing formula of bond option with CIR model is given by :

http://isc.temple.edu/economics/wkpapers/Valuing Bonds/C-I-R.pdf

P13: C(r,t,T,s,K) ?

I didn't have quantitative oriented classes and I must confess that I'm a bit lost.
I'm statisfied about the MC and pricing code because I've never coded before but a non-calibated CIR is useless.


PS1: for the moment these are my questions, I'll certainly need to ask other questions later ;)
PS2: sorry about my English (I'm french !)

Thanks in advance

Alice
 
Hello Daniel and thanks for your help !

Everything is know except x ? In the CIR model, there are three parameters:

a : the speed of adjustment
b : the long run rate
volatility

I thought that I could use newton to calibrate a (the speed of adjustment). It would be my x.
But I really don't know if it makes sense (what about the other parameters ? )

x is a vol parameters ? x would be the a (speed of adjustment)


Here is my NPV_CAP_CIR (a is phi in the formula)
http://imageshack.us/f/850/circap.png/

It's a closed solution that gives cap price.

I've read again about the calibration. Here is how I now see the problem:

f(x,..,..,..,..,etc.) = (NPV_CAP_number1_CIR (with the cap maturity and strike of the chosen cap) - MarketPriceOfTheCapNumber1)+(NPV_CAP_number2_CIR (with the cap maturity and strike of the chosen cap) - MarketPriceOfTheCapNumber2) =0

where x is the variable I want to calibrate (the pull-back variable for instance)
δf '(x,..,..,..,..,etc.)/ δx = ............................
Click to expand...

Because the CIR model must be calibrated with a basket of cap (and not just a cap)

Thanks (as you can see, I'm kind of lost)

//
Thanks !
You teach me the calibration and I teach you French then ? ;)
 
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