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Local risk-minimization under the benchmark approach

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Abstract

We study the pricing and hedging of derivatives in incomplete financial markets by considering the local risk-minimization method in the context of the benchmark approach, which will be called benchmarked local risk-minimization. We show that the proposed benchmarked local risk-minimization allows to handle under extremely weak assumptions a much richer modeling world than the classical methodology.

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Notes

  1. The space \(L^2({\fancyscript{F}}_T,{\mathbb P})\) denotes the set of all \({\fancyscript{F}}_T\)-measurable random variables \(H\) such that \(\mathbb {E} [H^2] = \int H^2\mathrm d{\mathbb P} < \infty \).

  2. If \(S^{\delta _*}\) is the numéraire portfolio, a security price process \(G = \{G_t,\ t \in [0, T]\}\) is called fair if its benchmarked value \(\hat{G} = \frac{G}{S^{\delta _*}}\) forms a \(\mathbb P\)-martingale.

  3. Here, \(a^\top \) identifies the transpose of the vector-valued process \(a\).

  4. Two square-integrable \({\mathbb P}\)-martingales \(N\) and \(O\) are called strongly orthogonal if their product \(NO\) is a \({\mathbb P}\)-martingale.

  5. The original definition of a locally risk-minimizing strategy is given in [28] and formalizes the intuitive idea that changing an optimal strategy over a small time interval increases the risk, at least asymptotically. Since it is a rather technical definition, it has been introduced the concept of a pseudo-locally risk-minimizing (or pseudo-optimal) strategy that is both easier to find and to characterize, as Proposition 1 will show in the following. Moreover, in the one-dimensional case and if \(\hat{S}\) is sufficiently well-behaved, pseudo-optimal and locally risk-minimizing strategies are the same, see Theorem 3.3 of [28].

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Acknowledgments

The authors like to thank Martin Schweizer and Wolfgang Runggaldier for valuable discussions. We also express our gratitude to an unknown referee, whose comments contributed to significant improvements of the presentation of this manuscript. The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no [228087].

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Authors and Affiliations

  1. Department of Mathematics, LMU, Theresienstraße, 39, 80333 , Munich, Germany

    Francesca Biagini

  2. Department of Mathematics and Computer Science, University of Perugia, via Vanvitelli, 1, 06123 , Perugia, Italy

    Alessandra Cretarola

  3. Finance Discipline Group, Department of Mathematical Sciences, University of Technology Sydney, PO Box 123, Broadway, NSW, 2007, Australia

    Eckhard Platen

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  1. Francesca Biagini
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  2. Alessandra Cretarola
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  3. Eckhard Platen
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Correspondence to Francesca Biagini.

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Biagini, F., Cretarola, A. & Platen, E. Local risk-minimization under the benchmark approach. Math Finan Econ 8, 109–134 (2014). https://doi.org/10.1007/s11579-014-0115-3

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