# Black–Karasinski - Market Price of Risk

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?

• Just out of curiosity, is there a reason why you need to get the market price of risk? For pricing purposes, market price of risk shouldn't matter at all. Empirically, calibrating the market price of risk is very difficult and imprecise. You can get a few local maxima using MLE, but they could have completely different economic implications. Jun 19 '14 at 18:41
• For pricing purposes in general I think this is true. However, in some markets that have a poorly developed fixed income market, prices of swaptions/caps etc are not easily available and all you have is the short rate. This is problematic for insurance companies that have written embedded derivatives in their products - often unknowingly - and are now forced to value them market consistently. A second reason in my experience is that is useful for risk management purposes to have a model calibrated under P and Q via the market price of risk - even though this is difficult as you say. Jun 22 '14 at 7:10
• Have you tried a Trinomial Tree? As [Brigo & Mercurio illustrate][1], a numerical procedure to evaluate the Black and Karasinki model has been presented in Hull and White, "Branching Out", Risk magazine, 1994. Hull and White make use of a trinomial tree, which may indeed more computationally efficient than Monte Carlo, especially if you have to iterate over the parameter space for calibration purposes. Let me know if you find the original paper by Hull-White or Brigo-Mercurio. [1]: amazon.com/Interest-Rate-Models-Practice-Inflation/dp/… Jun 22 '14 at 20:01