In the past I have calibrated simple short rate models to the term structure by using maximum likelihood to get the parameters of the Vasicek/CIR sde, and then use the ZCB formula and the current yield curve to calibrate the market price of risk.
I am interested in doing the same for the Black-Karansinksi model. However, to my limited knowledge, it has no closed form ZCB prices. I imagine this means that the market price of risk needs to be otpimized numerically.
My very poor attempt at a recipe is -:
- Project a bunch of realizations of the sde using a grid of values for the market price of risk.
- Discount back using the realized short rate for all the time points for which we have bond prices and take an average to give the estimated expectation.
- Implement some sort of grid search for the best parameter that minimizes norm between the estimated and the actual values?
How should this be done properly?