Pricing Interest Rate Contingent Claims in Markets with Uncertain
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Unpublished Manuscript, 1995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
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.