西南财经大学金融研究院
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BENCHMARKED RISK MINIMIZATION
作者:
                     
                                           
                  KE DU
                                    Southwestern University of Finance and Economics, Institute of Financial Studies
                                 ECKHARD PLATEN
                              University of Technology, Sydney, Finance Discipline Group and School of Mathematical Sciences


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.