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Changes of numéraire, changes of probability measure and option pricing

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Hélyette Geman ,
Nicole El Karoui  and
Jean-Charles Rochet
Hélyette Geman*
Affiliation:
ESSEC, Cergy-Pontoise
Nicole El Karoui*
Affiliation:
Université Paris VI
Jean-Charles Rochet*
Affiliation:
GREMAQ, Université Toulouse I
*
∗ Postal address: Finance Department, ESSEC, Avenue Bernard Hirsch, BP105, 95021 Cergy-Pontoise Cedex, France.
∗∗ Postal address: GREMAQ, IDEI, Université Toulouse 1, Plane Anatole France, 31042 Toulouse, France.
∗∗∗ Postal address: Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, Tour 56–66, 75252 Paris Cedex 05, France.
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Abstract

The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with numéraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the numéraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself.

The key theorem of general numéraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.

Keywords

PROBABILITY MEASURE CHANGESMARTINGALESPRICES RELATIVE TO A NUMÉRAIREHEDGING PORTFOLIOFORWARD VOLATILITY

MSC classification

Secondary: 60G35: Signal detection and filtering

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 2 , June 1995 , pp. 443 - 458
Copyright
Copyright © Applied Probability Trust 1995 

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