An Effective Approximation for Zero-Coupon Bonds and Arrow-Debreu Prices in the Black-Karasinski Model

Abstract

We present an accurate and easy-to-compute approximation of zero-coupon bonds and Arrow-Debreu (AD) prices for the Black-Karasinski model of interest rates or default intensities. Through this procedure, dubbed exponent expansion, AD prices are obtained as a power series in time to maturity. This provides remarkably accurate results - for time horizons up to several years - even when truncated to the first few terms. For larger time horizons the exponent expansion can be combined with a fast numerical convolution to obtain extremely accurate results.

Keywords: Stochastic processes, Black-Karasinski, derivative pricing, power series expansions