calculating term structure of interest rates, suppose that we want to predict interest rate term structure for the coming 12 months. Let initial interest rate be 1% and assume the annual volatility of interest rate be sigma=2.5%, long run rate be theta= 8% which is mean reverting level of interest rate that is interest rate reverts to long run mean level, strength of reversion that is how fast the interest rate drifts back to mean reverting value on average which is k=.6, and time step is monthly can be daily also dt=1/12 For the first month, the random uniform value is 0.71 such that the random standard normal is 0.5534 and dw = 0.5534*SQRT(1/12) = 0.160. dr = k*(theta - r)*dt + sigma*sqrt(r)*dw = 0.60*(8.0% - 1.0%)*1/12 + 2.5%*SQRT(1.0%)*0.160 = + 0.350% due to mean-reverting drift + 0.040% due to random shock = +0.3900% which is the change in interest rate from previous month as predicted by the model. such that r(1/12) = 1.0% + 0.390% =1.390% which is the interest rate predicted as per the model in this way continue in this way to end up with the term structure of interest rate for coming 12 months.
I'm hoping someone can help me with a conceptual question about CIR model: dr = k(theta - r)dt + sigma*sqrt(r)*dw
This seems inconsistent since as r -->0, volatility will be decreasing since you will only have one-sided volatility as you approach 0.. this makes sense, however....
When looking at the good old convexity charts plotting price against yield (rate), this seems to imply exactly the opposite... that only a slight change in price will deliver greater variations in the yield (rate), and thus (I) would infer greater interest rate volatility. Can someone help?