Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-21T22:26:46.660Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ball, C. A. and Torous, W. N. (1983) Bond price dynamics and options. J. Financial and Quantitative Anal. 18(4).CrossRefGoogle Scholar
Bick, A. (1988) Producing derivative assets with forward contracts. J. Financial and Quantitative Anal. 23(2).Google Scholar
Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities. J. Political Econ. 81, 637–654.Google Scholar
Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985) A theory of the term structure of interest rates. Econometrica 53, 407.CrossRefGoogle Scholar
Cox, J. C., Ross, S. and Rubinstein, M. (1979) Option pricing: a simplified approach. J. Financial Econ. 3, 145–166.Google Scholar
Delbaen, F. and Schachermayer, W. (1992) A general version of the fundamental theorem of asset pricing. Conference INRIA-NSF on Mathematical Finance, Paris.Google Scholar
Dudley, R. M. (1977) Wiener functionals as Itô integrals. Ann. Prob. 5, 140–141.CrossRefGoogle Scholar
Duffie, D. and Huang, C. F. (1985) Implementing Arrow-Debreu equilibrium by continuous trading of few long-lived securities. Econometrica 53, 1337–1356.Google Scholar
El Karoui, N. and Rochet, J. C. (1989) A pricing formula for options on coupon-bonds. Working Paper, GREMAQ.Google Scholar
El Karoui, N. and Geman, H. (1991) A stochastic approach to the pricing of FRNs. RISK 4(3).Google Scholar
El Karoui, N. and Geman, H. (1994) A probabilistic approach to the valuation of floating rate notes with an application to interest rate swaps. Adv. Options and Futures Res. 7, 47–64.Google Scholar
Geman, H. (1989) The importance of the forward neutral probability in a stochastic approach of interest rates. Working Paper, ESSEC.Google Scholar
Geske, R. (1979) The valuation of compound-options. J. Financial Econ. 7, 63–81.CrossRefGoogle Scholar
Harrison, J. M. and Kreps, D. (1979) Martingale and arbitrage in multiperiods securities markets. J. Econ. Theory 20, 381–408.CrossRefGoogle Scholar
Harrison, J. M. and Pliska, S. R. (1981) Martingales and stochastic integrals in the theory of continuous trading. Stoch. Proc. Appl. 11, 215–260.Google Scholar
Heath, D., Jarrow, R. and Morton, A. (1987) Bond pricing and the term structure of interest rates. Working paper, Cornell University.Google Scholar
Jamshidian, F. (1989) An exact bond option formula. J. Finance 44, 205–209.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1988) Brownian Motion and Stochastic Calculus. Springer-Verlag, Berlin.Google Scholar
Margrabe, W. (1978) The value of an option to exchange one asset for another. J. Finance 33.Google Scholar
Merton, R. C. (1973) Theory of rational pricing. Bell J. Econ. Mangement Sci. 4, 141–183.CrossRefGoogle Scholar
Reiner, E. (1992) Quanto Mechanics. RISK 5(3).Google Scholar
Ross, S. A. (1978) A simple approach to the valuation of risky streams. J. Business 51, 453–475.Google Scholar
Stricker, C. (1990) Arbitrage et lois de martingale. Ann. Inst. H. Poincaré 26(3).Google Scholar