Geman ElKaroui Rochet(1995)

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Changes of Numraire, Changes of Probability Measure and Option PricingAuthor(s): Hlyette Geman, Nicole El Karoui and Jean-Charles RochetSource: Journal of Applied Probability, Vol. 32, No. 2 (Jun., 1995), pp. 443-458Published by: Applied Probability Trust

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J. Appl. Prob. 32, 443-458 (1995)Printed n Israel

c Applied Probability Trust 1995

CHANGES OF NUMERAIRE, CHANGES OF PROBABILITY MEASUREAND OPTION PRICING

HILYETTE GEMAN,* ESSEC, Cergy-PontoiseNICOLE EL KAROUI,** Universitt Paris VIJEAN-CHARLES ROCHET,*** GREMAQ, UniversitO oulouse

Abstract

The use of the risk-neutral robability measure has proved to be very powerful orcomputing he prices of contingent claims in the context of complete markets, or theprices of redundant ecurities when the assumption of complete markets s relaxed.We show here that many other probability measures can be defined n the same wayto solve different asset-pricing problems, in particular option pricing. Moreover,these probability measure changes are n fact associated with numeraire hanges; hisfeature, besides providing a financial nterpretation, ermits efficient election of thenumeraire appropriate or the pricing of a given contingent claim and also permitsexhibition of the hedging portfolio, which is in many respects more important hanthe valuation itself.

The key theorem of general numeraire change is illustrated by many examples,among which the extension to a stochastic nterest rates framework f the Margrabeformula, Geske formula, etc.

PROBABILITY MEASURE CHANGES; MARTINGALES; PRICES RELATIVE TO A NUMtRAIRE;HEDGING PORTFOLIO; FORWARD VOLATILITY

AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 90A09

SECONDARY 60G35

1. Introduction

One of the most popular echnical tools for computing asset prices is the so-called'risk-adjusted robability measure'. Elaborating n an initial dea of Arrow, Ross (1978)and Harrison and Kreps 1979) have shown that the absence of arbitrage pportunitiesimplies the existence of a probability measure Q, such that the current price of any basicsecurity s equal to the Q-expectation f its discounted uture payments. n particular,between two payment dates, the discounted price of any security s a Q-martingale.When markets are complete, i.e. when enough non-redundant ecurities are beingtraded, Q is unique.

Received 20 July 1993; revision received 4 January 1994.* Postal address: Finance Department, ESSEC, Avenue Bernard Hirsch, BP105, 95021 Cergy-

Pontoise Cedex, France.** Postal address: GREMAQ, IDEI, Universite Toulouse 1, Plane Anatole France, 31042 Toulouse,

France.*** Postal address: Laboratoire de Probabilit6s, Universit6 Paris VI, 4 Place Jussieu, Tour 56-66,

75252 Paris Cedex 05, France.

443

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444 H. GEMAN, N. EL KAROUI AND J.-C. ROCHET

By using a very simple echnical argument Theorem 1, Section 2) we prove hat manyother probability measures an be defined n a similar way, and prove equally useful nvarious kinds of option pricing problems. More specifically, f X(t) is the price process ofa non-dividend-paying ecurity at least in the relevant ime period), our main theoremstates the existence of a probability measure Qx such that the price of any security Srelative to the num raire X is a Qx-martingale. A very general numeraire changeformula s then provided and different pplications o exchange options and options onoptions in a stochastic nterest rates environment, ptions on bonds, etc. illustrate heefficiency f the right choice of numeraire. ome of the results n the paper may be foundmore or less explicitly n the existing iterature. Our goal is to emphasize he generalityand the efficiency f the numeraire hange methodology.

2. The model and the crucial heorem

We consider a stochastic ntertemporal conomy, where uncertainty s represented ya probability pace (0, F, P). The only role of the probability P is in fact to define henegligible ets. Most of our applications will be taken n a continuous-time ramework,within a bounded ime interval 0, T] but our basic argument s also valid for a discrete-time economy.

We will not completely pecify he underlying ssumptions n the economy. The flowof information accruing o all the agents n the economy s represented y a filtration(f)t E[0, T], satisfying 'the usual hypotheses', .e. the filtration (F)0o1., is rightcontinuous and

70contains all the P-null sets of ?F.

In the following, the word 'asset' represents a general financial instrument. Wedistinguish wo classes of assets. One class consists of the basic securities, which aretraded on the markets nd are the components of the portfolio defined below. The otherclass of assets o be considered s the class of derivative ecurities, lso called contingentclaims, for which the key issues are the valuation and hedging. All asset price processesare continuous F- semimartingales. he prices S,(t), - --. , S,(t) of the basic securities reobserved on the financial markets and almost surely strictly positive for all t; moregenerally, unless otherwise pecified, he price of any asset is almost surely positive.

The fundamental oncept in the pricing or hedging of contingent claims is the self-

financing eplicating ortfolio, and these self-financing ortfolios consequently deserveparticular ttention buy and hold portfolios are the simplest example of self-financingportfolios since there is no trade). More generally, hese portfolios track the targetchanges over time with no addition of money.

The financial alue V(t) of a portfolio which ncludes he quantities w,(t), . , w,(t) ofthe assets 1, 2,. - -, n is given by

(1) V(t)= X wk(t)Sk(t) and V(t)?O foralltkl1

where the processes (w,(t))r_o,...,(w,(t))to>

are adapted, i.e. the quantitiesw1(t),

?, w,(t) are chosen according o the information vailable at time t. The vector

process (w,(t)), ... ,(w,(t))to

is called the portfolio strategy.

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Changes f numraire, changes f probability easure nd option ricing 445

Definition 1. The portfolio is called self-financing f the vector stochastic ntegralST

Z._lWk(t)dSk(t) xists and

(2) dV(t)= E wk(t)dSk(t).k-I

Remark. To understand he intuition behind Equation 2), let us take the example ofa simple strategy, .e. a strategy ebalanced nly at fixed dates 0 = to< t

Geman ElKaroui Rochet(1995)

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