Perturbation Expansion for Arrow-Debreu Pricing with Hull-White Interest Rates and Black-Karasinski Credit Intensity

Abstract

We consider a rates-credit hybrid model with the rates governed by a Hull-White short-rate model and the credit intensities by a Black-Karasinski short-rate model. We provide a systematic derivation of a pricing kernel (also known as an Arrow-Debreu formula) for European-style options and/or protection payments, using operator formalism combined with exponential expansion formulae. Our approach gives rise to an analytic expression involving an infinite series in powers of the credit spread (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 give explicit expressions for all terms up to second order, which level we suggest should in practice suffice. We apply our first-order result to calculate the impact of rates-credit correlation on the pricing of credit default swaps (CDS), extinguishing interest rate swaps, survival-contingent capped Libor flows and contingent CDS with interest rate swap underlying. Very simple analytic expressions are obtained in all cases. Highly favourable comparison is found between even the first order approximations and Monte Carlo simulations.