Risky Caplet Pricing with Backward-Looking Rates

We extend the Hull-White short rate model to include the integrated short rate as a separate independent variable and incorporate credit default risk, governed by a Black-Karasinski model, into cash flows. We derive an analytic representation of the associated pricing kernel and apply it to the pricing of risky compounded interest rate payments, including with caps and floors. We illustrate our results numerically applying them to the pricing of extinguishing fix-float swaps.