Interest-Rate Option Pricing Revisited

December 1, 1996

When valuing derivative securities, if no arbitrage opportunities exist, then the value of the derivative must equal the value of a portfolio of fundamental securities that replicates the payoffs of the contract being valued. The purpose of this note is to point out the care that must be exercised when choosing fundamental securities with which to replicate the cash flows of more complex securities. Often, the choice of fundamental, or replicating, securities is driven by the existence of a known solution for their value.
An example of this approach is Turnbull and Milne (1991), who use the contingent claims framework to derive closed-form solutions for options written on a variety of interest-rate contingent securities. They value each interest-rate contingent security relative to the exogenously given term structure of interest rates. Using this approach, they are able to derive closed-form solutions not only for simple contracts, such as options on discount Treasury bonds, but also for more complex contracts, such as options on interest rate futures contracts.

As shown by Boyle and Turnbull (1989) or Turnbull and Milne (1991) a European put option on a T-bill can be used to replicate the payoff on an interest rate cap. They use this relationship because there are known solutions for the value of a European option on a T-bill. In replicating the interest rate cap, the appropriate number of put options on a T-bill must be purchased. In addition, the put options must have a strike price that is related to the strike rate for the interest rate cap. An exact solution for the number and characteristics of the put options is demonstrated by Hull and White (1990). For many interest rate caps it is common to determine the cash flow at the reset date but to delay payment until the end of the reset period just prior to the next reset date. This type of cap, called a deferred cap, is considered by Hull and White. This paper presents their results as a starting point and then compares the results to nondeferred caps.

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