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Local risk minimization and numéraire

  • F. Biagini (a1) and M. Pratelli (a2)

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.

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Corresponding author

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.

References

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