Analytic Pricing of Bermudan Swaptions with Few Exercise Dates

Abstract

In this paper, we consider pricing a Bermudan swaption with a small number of exercise dates. We begin with the case of two exercise dates. In this limit, we show that the Bermudan price decomposes into the sum of short-dated European swaptions, setting an upper bound, minus a correction term. This correction is expressed as an integral involving a forward volatility agreement type payoff with start at the first exercise date, and it can be evaluated in closed form. The magnitude of the correction is smaller when variance is front loaded and larger when it is back-loaded. We extend to three-exercise Bermudans via backward induction under rolling forward measures. A key feature is boundary linearity enabling further analytic steps. The exercise boundary of options splits into a strike-dependent term and a variance term; together they determine optimal exercise. The linear term is negative, supressing the exponentials in subsequent steps and aiding analytic calculations. This boundary linearity extends to multiple exercise dates and yields pricing formulas with the same decomposition, showing how optionality accumulates across exercise dates. We conclude that the Bermudan can be reconstructed by adding, at each exercise date, the initial short swaption with an increasingly higher strike and subtracting the integrated payoffs of all forward-starting receiver swaptions starting at that date. The corresponding double and higher-order integrals decrease rapidly and, in the presence of only a few exercise dates, can be safely neglected without materially impacting the valuation. The general case is discussed at the end.