Pricing Interest Rate Contingent Claims in Markets with Uncertain Volatility

Abstract

We consider a financial market where the volatility of the interest rate is not known exactly, but rather it is assumed to lie within two a-priori known bounds. These bounds represent the extreme values of the volatility implied by traded options. In this market, the interest rate process which allows no arbitrage and fits exactly the initial term structure of the forward interest rates, is not determined uniquely: for each volatility path in a band between the minimal and maximal volatility, there exists a different interest rate process.The asking and the bidding prices in our model are functions of the time, the interest rate, and the accumulated volatility, and they satisfy a new non-linear partial differential pricing equation. In this equation, the volatility used for pricing a claim is chosen dynamically: it is either the minimal or the maximal volatility depending on the claim's curvature with respect to both the interest rate and the accumulated volatility. We compare our model to the standard Ho-Lee model. We illustrate the effectiveness of our pricing scheme with numerical calculations for a calendar spread.

JEL Classification: E43