Analytic Option Prices for the Black-Karasinski Short Rate Model

15 Pages Posted: 14 Oct 2018 Last revised: 13 Apr 2020

See all articles by Blanka Horvath

Blanka Horvath

ETH Zürich - Department of Mathematics

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Colin Turfus

Deutsche Bank

Date Written: July 22, 2018


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.

Keywords: Perturbation Methods, Asymptotic Expansion, Hull-White, Black-Karasinski, Short Rate Model, Caplet Pricing, Green’s Function

Suggested Citation

Horvath, Blanka and Jacquier, Antoine and Turfus, Colin, Analytic Option Prices for the Black-Karasinski Short Rate Model (July 22, 2018). Available at SSRN: or

Blanka Horvath

ETH Zürich - Department of Mathematics ( email )

R¨amistrasse 101
Raemistr. 101
Z¨urich, 8092

Antoine Jacquier

Imperial College London ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom


The Alan Turing Institute ( email )

British Library, 96 Euston Road
London, NW12DB
United Kingdom

Colin Turfus (Contact Author)

Deutsche Bank ( email )

Winchester House
1 Great Winchester Street
London, EC2N 2DB
United Kingdom

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