Perturbation Methods in Credit Derivatives

CHAPTER 5Black–Karasinski Short‐Rate Model

We consider next the short‐rate model of Black and Karasinski [1991]. We shall start, as in the Hull–White case, with an auxilary process defined by (4.1). Under the Black–Karasinski model, this auxiliary process is related to the instantaneous short rate by

(5.1)

with an (‐integrable) function to be determined by calibration to fit the instantaneous forward curve and given by (4.10) above. We shall again seek to obtain an exact Green's function solution. On this occasion, a closed form solution turns out not to be possible; instead, the result obtained takes the form of a perturbation expansion in powers of the short rate. As we shall see, only the first one or two correction terms are ever likely to be needed, so the solution enjoys most ...