Calibrating a historical rate structure . Based on the history of the European curve nominal rates of maturity from 1 to 30 years over the period August 1998 to December 2013 , it is " fitter " price zero coupon ( ZC ) with the CIR model 1-factor CIR model and the 2 factors.
1 ) Method of least squares on the closed form of zero -coupon price
Choose two of the following three methods.
- Nlin proc
- Optmodel proc
- Numerical approach ( finite tuple of parameters)
2 ) Calibration per time step ( month after month )
For a given month it is to find the model parameters minimizing the least squares on the yield curve with 30 points ZC price of 1-30 years.
3 ) Analysis of the values taken by each parameter
Calculating central moments of order 1 and 2 of each parameter. Draw each parameter as a function of time (history) .
4) To achieve a single tuple of parameters
From the descriptive statistics of the previous point , set the parameter whose standard deviation is smaller. Repeat the study point 2-3 . Analyze the results. Set the second parameter whose standard deviation is smaller. Repeat the study to obtain a single tuple of parameters.
5) Residue Analysis
Test of normality ( at least two different ) residue ( model price and historical price).
6 ) Simulation and Backtesting
Considering the diffusion process of the "r" instantaneous rate , simulate 1000 trajectories mesh monthly from January 2009. Deduct each time rates 1 year , 10 years and 30 years. We would like 95% of the simulated prices contain each time the prices recorded in the past. Whether or not this assertion.
- At first the entire study will cover only the CIR model one factor ;
- The spread of instantaneous and the closed price formula ZC rate is on page 3 of the pdf;
- In the first minimization by the least squares method , restart several times the same model to see if it is converging . If this is not the case, you should consider a series of random draws (100 should be enough ) input parameters . (macro language required. sas available Code).