Callable U.S. Treasury bonds: optimal calls, anomalies, and implied volatilities

Previous studies on interest rate derivatives have been limited by the relatively short history of most traded derivative securities. The prices for callable U.S. Treasury securities, available for the period 1926–95, provide the sole source of evidence concerning the implied volatility of interest rates over this extended period. Using the prices of callable, as well as non-callable, Treasury instruments, this paper estimates implied interest rate volatilities for the past seventy years. Our technique for estimating implied volatilities enables us to address two important issues concerning callable bonds: the apparent presence of negative embedded option values and the optimal policy for calling these, and similarly structured, deferred-exercise embedded option bonds. ; In examining the issue of negative option value callable bonds, our technique enables us to extend significantly both the sample period and sample breadth beyond those covered by other investigators of this phenomenon and to resolve the inconsistencies in their results. We show that the frequency of mispriced bonds is time-varying and that there also exist irrationally underpriced bonds. Critically, both anomalies are shown to be related to volatility-insensitive, away-from-the-money bonds. ; In contrast to the naive call decision rules suggested by previous authors, we develop the option-theoretic optimal call policy for deferred-exercised "Bermuda"-style options for which prior notification of intent to call is required. We do this by introducing the concept of "threshold volatility" to measure the point at which the time value of the embedded call option has been eroded to zero. By using this concept, we address the valuation of such bonds and document the frequent optimality of the Treasury's past call decisions for U.S. government obligations.