Pseudo--Arbitrage - A new Approach to Pricing and Hedging in Incomplete Markets

We develop a new approach to pricing and hedging contingent claims in incomplete markets. Mimicking as closely as possible in an incomplete markets framework the no--arbitrage arguments that have been developed in complete markets leads us to defining the concept of pseudo--arbitrage. Building on this concept we are able to extend the no--arbitrage idea to a world of incomplete markets in such a way that based on a concept of risk compatible with the axioms of Artzner et al. we can derive unique prices and corresponding optimal hedging strategies without invoking specific assumptions on preferences (other than monotonicity and risk aversion). Price processes of contingent claims are martingales under a unique martingale measure. A comparison to a version of the Hull and White stochastic volatility model shows that in contrast to their approach explicitly taking into account optimal hedging strategies leads to positive market prices of risk for volatility even if the latter is instantaneously uncorrelated with the stock price process. Our results are, however, in agreement with the findings of Lamoureux and Lastrapes.