Marco Avellaneda and Pawel Lewicki

We consider a model of 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 may represent, for instance, the extreme values of the implied volatility of liquidly traded options observed over a certain period of time. In this model, the interest rate process consistent with no-arbitrage and with the initial term-structure of forward rates is not determined uniquely . More precisely, there exist one interest-rate process for each volatility path within the band determined by the minimal and maximal volatilities.

Due to uncertainty in the volatility , the present value of an interest rate sensitive security cannot be determined exactly unless the security is a series of discount bonds. Nevertheless, it is possible to calculate extreme values, corresponding to worst-case scenarios of future volatility for short positions (``ask price'') and long positions (``bid price'') in any security or portfolio of securities. These extreme values are functions of the time-to-maturity , the current spot rate and an additional variable: the ``accumulated variance''. We show that the extreme prices can be found by solving nonlinear partial differential equations.