In the general theory for the interest rate structure
the market is assumed to be complete. On the contrary,
if the bond's dynamics is given by the following equation:
\noindent
where $W_t$ is a standard brownian motion (possibly n-dimensional)
on $\Omega$ and
$\eta\in E$ represents an additional source of randomness,
the market is incomplete even in presence of an infinite number
of assets.
In the terminology of F&"ollmer and Schweizer, the
market incompleteness is a result of insufficient information.
In this paper, the interest rate structure under stochastic volatility
is studied by using the mean-varince hedging and the local risk minimization
approach.
The set of equivalent martingale measures is characterized,
by exploiting some results of Schweizer (1996),
Rheinl&"ander and Schweizer (1997) and of Biagini, Guasoni and Pratelli
(1999).
Imposing the so-called Heath-Jarrow-Morton in order to have at least
one martingale measure, it turns out that it is sufficient to determine both
the Minimal Probability and the Variance-Optimal Measure
when the volatility matrix is invertible.
By using the explicit expressions for the two martingales measures
we find the local risk minimizing strategy and the mean-variance optimal
strategy for European options on bonds.
Several models for the stochastic volatility are analized:
the simplest is when the volatility
jumps according to a totally inaccessible stopping time and assumes
values with a certain probability
distribution.
The natural extension of the previous example it is the case when the
volatility is given by a
Markov process and then by a multivariate point process.
The mean-variance hedging and the local risk miniming strategies
are explicitely computed in the particular case of an option call
by using some results of the "change of
numéraire" technique by Geman, El Karoui and Rochet (1995).
As a final application, the previous results can be used straightforwardly
in order to price and hedge options on futures in
the case of incomplete markets.