Insight to the Black Karasinski Calibration problem

Hi, my name is Johannes Fürst and I am one of the financial engineers in the UnRisk developer team. Together with my colleagues, I have been working on a wide range of computational finance projects. My primary duties are the implementation and improvement of numerical methods and algorithms used for the pricing of instruments and model calibration in the UnRisk software package.

In my first blog post, I would like to give a little insight to the calibration of the Black Karasinski model, which assumes that the short interest rate process follows the stochastic differential equation
Since the distribution of the short rate is assumed to be lognormal, the interest rate never falls below zero. Before complex instruments can be priced, the model parameters have to be calibrated to the current term structure of interest rates and to the market prices of caps and swaptions.

To be able to fit the current term structure of interest, ϑ is chosen to be time dependent, whereas the reversion speed parameter η and the volatility parameter σ - used for the calibration to option data - are chosen to be constant.

Since (even for simple instruments) there is no analytical formula available in this model, numerical methods have to be applied (even in the calibration process). Using the Ito formula and no arbitrage arguments, the pricing equation - which is a parabolic partial differential equation  - can be derived:

The prices of bonds, caps and swaptions are obtained by solving the upper PDE backwards in time, taking into consideration the instrument specific interface conditions and using the instrument specific payoff as terminal condition at the product maturity date T. For the numerical solution of the PDE, we use a finite difference sheme (taking into account the direction of the flow using upwind techniques) on a two dimensional grid with respect to r and t.

More details of the calibration process and results for the Black Karasinski model will be presented in the blog next monday.