Analytic Option Prices for the Black-Karasinski Short Rate Model

We consider a one-parameter family of short rate models which encompasses both Hull-White (normal) and Black-Karasinski (lognormal) models. We deduce a general form for the relevant Green's function as an asymptotic series, assuming only that the deviations of the short rate from the forward curve are on average small in absolute terms, and show how this solution can be parametrised in such a way as to fit the model to a term structure of zero coupon bond prices. We use the derived Green's function to calculate conditional bond prices and pricing formulae for caps and floors to second order accuracy. The results are seen to take a form which is straightforward to compute using quadrature and even the first order expressions achieve highly favourable comparison with benchmark Monte Carlo computations for a wide range of market conditions with both long and short cap/floor maturities