Exact Arrow-Debreu Pricing for the Black-Karasinski Short Rate Model

Abstract

We consider the Black-Karasinski short rate model and provide a systematic derivation of an Arrow-Debreu pricing formula for European-style options using operator formalism combined with exponential expansion formulae. Our approach gives rise to an analytic expression involving an infinite series in powers of the interest rate (not of its lognormal volatility). We propose that this can be used to provide results to a chosen level of accuracy by truncating the power series at a suitable point; and further that the use of terms up to second or third order should in practice suffice.

Keywords: Arrow-Debreu, Perturbation Methods, Black-Karasinski, Short Rate Model, Pricing Kernel; Analytic Solution