Abstract:

Incompleteness can be introduced to any complete model in a
number of different ways. Typical examples are models with
jumps or stochastic volatility. Alternatively one can introduce,
as in Davis (2000), an additional source of uncertainty which
is correlated with the source of uncertainty used in the
construction of a nested complete model. I will use this
model set up in my presentation. There are essentially two
ways of dealing with the pricing and risk management issues
in incomplete markets. As there are an infinite number of
equivalent martingale measures one approach is to choose one
of them by some optimality criteria. For example, one could
take a martingale measure which is the closest to the historical
measure, with respect to a certain distance. Once the choice is
made this measure is used to compute the expectation of the
discounted payoff in order to identify the price. There are
numerous advantages and disadvantages of this approach. One of
the fundamental disadvantages is the price dependence on the
choice of numeraire. Similarity with the pricing in complete
markets in many other aspects are clear advantages. Alternative
approach is to derive the concept of price and risk management
from the idea of optimal investment. I will follow this approach
in my talk. I refer to Davis (2000) for the first results in
this direction as well as for the examples of situations in
which such models can be applied in practice and to Musiela
and Zariphopoulou (2001) for further analysis of this model
in which one trades one risky and one riskless asset but options
are written on the non-traded index. The main aim of the talk
is to analyse properties on the pricing and of the associated
risk management of such an approach. In particular, the
following issues will be discussed:

1. Consistency with the static no-arbitrage.

2. Independence on the choice of numeraire.

3. Marking to market versus marking to a portfolio.