Risk-Minimization under the Benchmark Approach

We study the pricing and hedging of derivatives in incomplete financial markets. One of the most competitive approaches to tackle this problem is local risk-minimization, characterized by a quadratic hedging criterion. It minimizes the conditional variances of instantaneous cost increments. Classical local risk-minimization leads to the Föllmer-Schweizer decomposition of the discounted contingent claim that can be obtained as a Galtchouk-Kunita-Watanabe decomposition under the so-called minimal martingale measure, which has to exist.
The paper proposes to use the numéraire portfolio as numéraire or benchmark and the real world probability measure as pricing measure for the so-called benchmarked local risk-minimization. The real world pricing formula of the benchmark approach yields to a generalization of the classical Föllmer-Schweizer decomposition in terms of benchmarked quantities.
It identifies for a benchmarked contingent claim for its benchmarked hedgeable part the minimal possible price, and the benchmarked profit and loss with zero mean and minimal variance. Examples demonstrate that the proposed benchmarked local risk-minimization allows to handle under extremely weak assumptions a much richer modeling world than the classical methodology.