quant research // summarizing the paperLOADING
Emiliano Papa · 2026-06-22
The paper derives closed-form (analytic) pricing formulas for Bermudan swaptions that have only a few exercise dates. It shows the Bermudan price can be broken into a sum of simpler short-dated European swaptions (an upper bound) minus a correction term that can be computed exactly, and that the correction depends on whether volatility is front-loaded or back-loaded. It extends this decomposition from two to three and more exercise dates using backward induction and a 'boundary linearity' property, arguing higher-order correction integrals shrink fast and can be neglected when exercise dates are few.
Why it matters: For rates desks and quants dealing with interest-rate derivatives, an analytic approximation could speed up valuation of Bermudan swaptions versus slower lattice or Monte Carlo methods, and offers intuition about how optionality builds across exercise dates. The insight that the correction depends on the term structure of variance (front- vs back-loaded) may aid pricing intuition and risk understanding.
⚠ This is a specialized derivatives-pricing result relevant mainly to rates quants, relies on model assumptions and the approximation only holds for a small number of exercise dates.
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.
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AI summary generated from the paper’s public abstract via arXiv; it may miss nuance — read the source before relying on it. Thank you to arXiv for its open-access interoperability; StockTools is not affiliated with arXiv, and all rights remain with the authors. Educational only, not financial advice.