In this article, the authors propose an alternative approach for pricing bond options and swaptions under the one-factor Hull–White model. Their proposal differs from the existing models used to evaluate these type of instruments when the evolution of the term structure of interest rates is modeled by short-rate models. Assuming that the forward price of a coupon bond is a martingale under the forward risk-neutral measure, the proposed model involves deriving the volatility of the coupon bond as a function of its stochastic duration. Once the volatility function is defined, the price of options on coupon bonds can be derived by simply applying the standard no-arbitrage pricing theory in the context of the one-factor Hull–White model. Given the equivalence between the price of a coupon bond and the price of the corresponding swaptions, this approach can be adopted to calibrate the parameters of the one-factor Hull–White model using swaptions quoted in the market. The new solution proposed for the calibrati...

Monetary policy based on “quantitative easing” has been followed actively in the United States and Europe in recent years. It has led to short-term interest rates that have become negative in a number of countries. Most of the standard theoretical interest rate models have the restriction that rates are not allowed to go negative within the model, which makes it impossible to calibrate them to market prices in the current environment. This problem is particularly acute in the market for caps, floors, and swaptions, but it is more pervasive because these markets are often used to calibrate models for other securities. Practitioners either refuse to quote short maturity swaptions when the short rate is negative, or they use one of several alternative approaches for valuation: retaining the Black model but only using positive forward rates; modeling rate shocks as normal rather than lognormal, which allows rates to become arbitrarily negative in the model; or adding a shift to a lognormal density to push it ...