Abstracts

Option hedging is one of the main problem in finance, both in discrete and continuous time. For complete markets, this problem is rather straightforward but most financial markets are incomplete, meaning that the risk-neutral probability is no longer unique and contingent claims are not all attainable. In incomplete markets, a first approach to price and hedge an option H is to use the superhedging method as introduced by Kreps (1981) and further studied by El Karoui and Quenez (1995), Karatzas (1997)...Its cost is given by the supremum of the expected values of the claim under all equivalent martingale measures. The corresponding process is a supermartingale under any equivalent martingale measure and the superhedging strategy is determined by the 'optional decomposition' of such an universal supermartingale (e.g. Kramkov, 1996). However, the cost of superhedging is too high in practice.
An alternative approach is to measure the risk by the quadratic risk criterion (see Föllmer and Schweizer, 1991; Schweizer, 1991). Usually, we obtain explicit and tractable formula. Nevertheless, one drawback of this approach is that we penalize both situations where the terminal wealth is smaller or larger than H. Other approaches are based on quantile and expected shortfall hedging. This latter criterion seems to be more interesting in practice than the superhedging (see Föllmer and Leukert, 1999, 2000).
It has been previously proved that both prices and hedging strategies associated to the locally risk-minimizing criteria are stable under weak convergence (see Prigent,1999; Prigent and Scaillet, 2002; Jacod et al., 2000). In this paper, we analyze and compare the weak convergence (for the Skorokhod topology) of main risk-minimizing option hedging strategies. Both complete and incomplete financial markets are considered. First, using the optional decomposition of supermartingales, we examine the weak convergence of superhedging prices, showing that generally it does not hold. Second, we extend previous results of Prigent (1999) and Prigent and Scaillet (2002) about the stability under convergence of hedging strategies associated to the locally quadratic risk-minimizing criteria. Finally, we examine the convergence of quantile and expected shorfall minimizing hedging strategies. It is proved that the stability under convergence is generally satisfied in the complete case. Nevertheless, for the incomplete case, usually there is no longer stability under weak convergence. We detail also several particular examples, leading for instance in continuous-time to Lévy processes dynamics and stochastic volatility models.