A Semi-Explicit Approach to Canary Swaptions in HJM One-Factor Model

Abstract

Leveraging the explicit formula for European swaptions and coupon-bond options in HJM one-factor model, we develop a semi-explicit formula for 2-Bermudan options (also called Canary options). We first extend the European swaption formula to future times. So equipped, we are able to reduce the valuation of a 2-Bermudan swaption to a single numerical integration at the first expiry date. In that integration the most complex part of the embedded European swaptions valuation has been simplified to perform it only once and not for every point. In a special but very common in practice case, we also provide a semi-explicit formula. Those results lead to a significantly more precise implementation of swaption valuation. The improvements extend even more favorable to sensitivity calculations.

Keywords: Bermudan option, swaption, bond option, HJM model, Hull-White model, one-factor model, explicit formula, numerical integration

JEL Classification: G13, E43