Calibrating Black–Karasinski Market Price of Risk with a Trinomial Tree
Summary
The document asks how to calibrate the market price of risk in the Black–Karasinski short-rate model when zero-coupon bond prices lack a closed-form expression. The proposed Monte Carlo approach simulates short-rate paths, discounts cash flows along them, averages the results, and searches over candidate risk-premium values to fit observed bond prices.
The response points to a trinomial tree as a numerical alternative, citing a brief micro-test that found it fast enough. It says Hull and White described a tree procedure and suggests it may be more efficient than Monte Carlo when repeatedly evaluating parameters during calibration. The document does not provide the tree construction, calibration equations, or a comparison of accuracy and runtime, and it does not report detailed test results. Its guidance is therefore a pointer toward numerical pricing rather than a complete calibration recipe.
Key ideas
- Black–Karasinski bond pricing can be handled numerically when a closed-form zero-coupon bond formula is unavailable.
- A proposed Monte Carlo calibration estimates discounted bond values across simulated short-rate paths for candidate market prices of risk.
- A trinomial tree is presented as a potentially efficient alternative when parameter calibration requires repeated pricing.
- The document offers a literature pointer and a brief speed observation but no detailed implementation or validation.
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Full text
# Black–Karasinski - Market Price of Risk # 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? ## Answer by pincopallino (score 1) https://quant.stackexchange.com/a/12787 I made a micro-test and the trinomial tree seems to be fast enough. (From my original comment) As described in Brigo & Mercurio illustrate, 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.
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