Calibrating Short Interest Rate Models in Negative Rate Environments

Abstract

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 into the region of positive rates. In this article, the authors document the problem of quoting swaptions when rates are negative and compare the three alternatives for calibrating standard interest rate models empirically.