Benchmarked Risk Minimization

Publisher:
Wiley: 24 months
Publication Type:
Journal Article
Citation:
Mathematical Finance, 2016, 26 (3), pp. 617 - 637
Issue Date:
2016-07
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This paper discusses the problem of hedging not perfectly replicable contingent claims using the numeraire portfolio. The proposed concept of benchmarked risk min- ´ imization leads beyond the classical no-arbitrage paradigm. It provides in incomplete markets a generalization of the pricing under classical risk minimization, pioneered by Follmer, 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 numeraire portfolio ´ to identify the minimal possible price for a contingent claim. Furthermore, the resulting benchmarked (i.e., numeraire 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.
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