Background
One can value a bond by discounting each of its cash flows at its own zero-coupon ("spot") rate. This procedure if equivalent to discounting the cash flows at a sequence of one-period forward rates. When a bond has one of more embedded options, however, its cash flow is uncertain. If a callable bond is called by the issuer, for example, its cash flow will be truncated.
To value such a bond, one must consider the volatility of interest rates, as their volatility will affect the possibility of the call option being exercised. One can do so by constructing a binomial interest rate tree that models the random evolution of future interest rates. The volatility-dependent one-period forward rates produced by this tree can be used to discount the cash flows of any bond in order to arrive at a bond value.
Given the values of bonds with and without an embedded option, one can obtain the value of the embedded option itself. The procedure can be used to value multiple or interrelated embedded options, as well as stand-alone risk control instruments such as swaps, swaptions, caps and floors.
Introduction
In the good old days, bond valuation was relatively simple. Not only did interest rates exhibit little day-to-day volatility, but in the long run they inevitably drifted up, rather than down. Thus the ubiquitous call option on long-term corporate bonds hardly ever required the attention of the financial manager. Those days are gone. Today, investors face volatile interest rates, a historically steep yield curve, and complex bond structures with one or more embedded options. The framework used to value bonds in a relatively stable interest rate environment is inappropriate for valuing bonds today. This article sets forth a general model that can be used to value any bond in any interest rate environment.
thnx.
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