【Mathematical Finance】BENCHMARKED RISK MINIMIZATION

Abstract: This paper discusses the problem of hedging not perfectly replicable contingent claims using the num´eraire portfolio. The proposed concept of benchmarked risk minimization leads beyond the classical no-arbitrage paradigm. It provides in incomplete markets a generalization of the pricing under classical risk minimization, pioneered by F¨ollmer, Sondermann, and Schweizer. The latter relies on a quadratic criterion, requests square integrability of claims and gains processes, and relies on the existence of an equivalent risk-neutral probability measure. Benchmarked risk minimization avoids these restrictive assumptions and provides symmetry with respect to all primary securities. It employs the real-world probability measure and the num´eraire portfolio to identify the minimal possible price for a contingent claim. Furthermore, the resulting benchmarked (i.e., num´eraire portfolio denominated) profit and loss is only driven by uncertainty that is orthogonal to benchmarked-traded uncertainty, and forms a local martingale that starts at zero. Consequently, sufficiently different benchmarked profits and losses, when pooled, become asymptotically negligible through diversification. This property makes benchmarked risk minimization the least expensive method for pricing and hedging diversified pools of not fully replicable benchmarked contingent claims. In addition, when hedging it incorporates evolving information about nonhedgeable uncertainty, which is ignored under classical risk minimization.