Pricing Interest Rate Contingent Claims in Markets with Uncertain Volatililites
Marco Avellaneda and Pawel Lewicki
Unpublished Manuscript, 1995

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.