A perspective on the pricing and risk management in incomplete markets
Dr Marek Musiela, BNP Paribas

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.