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Local risk minimization and numéraire

Part of: Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

F. Biagini  and
M. Pratelli
F. Biagini*
Affiliation:
University of Bologna
M. Pratelli*
Affiliation:
University of Pisa
*
∗ Postal address: Dipartimento di Matematica, Università di Bologna, Piazza di Porta, San Donato-40127, Bologna, Italy.
∗∗ Postal address: Dipartimento di Matematica, Università di Pisa, via Buonarroti-56100, Pisa (PI), Italy. Email address: pratelli@dm.unipi.it.
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Abstract

The ‘change of numéraire’ technique has been introduced by Geman, El Karoui and Rochet for pricing and hedging contingent claims in the case of complete markets. In this paper we study the ‘change of numéraire’ using the ‘locally risk-minimizing approach’, when the market is not complete. We prove that, if the stochastic process which represents the prices is continuous, the l.r.m. strategy is invariant by a change of numéraire (this result is false in the right-continuous case, as is shown by some counterexamples).

We also give an extension of Merton's formula to the case of stochastic volatility.

Keywords

Stochastic integralsmodels of financial marketschange of numérairelocally risk-minimizing strategiesminimal martingale measure

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1126 - 1139
Copyright
Copyright © Applied Probability Trust 1999 

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