Abstract:
We propose a novel application of the approach employed successfully by Hagan et al. (2005)* in deriving perturbation expansion solutions for option prices in the SABR stochastic volatility model to capture the price impact of stochastic rates and/or credit default intensities on a wide class of derivatives. Our starting point is the beta blend short rate model which interpolates between the Hull-White normal model (beta = 1) and the Black-Karasinski lognormal model (beta = 0). The latter is used as the credit model. We allow for correlation among interest rates, credit default intensities and other lognormal spot underlyings (FX, equity, inflation). We also allow that any spot underlying may jump at default of the named debt issuer. We proceed by expressing the PDE governing the price of securities with contingent payoffs at maturity and/or default in perturbed linear operator format, under the assumption that deviations of rates and/or credit intensities from their forward values are small. We derive by standard techniques a Green’s function solution as a perturbation expansion. We illustrate the approach by application to the pricing of a) caps and floors under the Black-Karasinski model, b) contingent CDS, and c) quanto CDS with an imposed loss cap. We demonstrate that even with just first order terms the resultant expansions are highly accurate.