Interest Rate Swaps and Their Derivatives: A Practitioner's Guide

CHAPTER 9

Bermudan-Style Options

A prominent class of derivatives is the class of American-style exercise options, in which the owner of the option has flexibility in choosing the time to exercise. A true American-style option allows the owner to exercise the option over any time during the exercise window. In interest-rate derivatives, true American-style options are rare, and the flexible-exercise feature is limited to a finite number of dates, typically reset/coupon dates (quarterly, semiannual) of the underlying instrument (bond, swap, . . . ), for example, a 30-year callable bond, callable at par after 10 years, and semiannually thereafter. This class of interest-rate options is known as Bermudan-style options.

The general pricing framework for Bermudan-style options is still risk-neutral valuation with t-prices provided as

(0 ≤ t ≤ T) C(t, ω) =ω)]

but augmented to include the owner′s option to choose the exercise time. The fixed exercise date T = t_e is replaced by a random exercise time chosen by the owner, T(ω). The exercise-time decision can be any rule/strategy chosen by the owner, and can depend on the current and history of the asset evolution, but not on the future, that is, it should be non-anticipative. In probability theory, the technical term for a nonanticipative strategy is a stopping time.

Given such a nonanticipative exercise strategy T(·), we evaluate the ...