Inmathematical finance,Margrabe's formula^[1] is anoption pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived byWilliam Margrabe^[2] in 1978. Margrabe's paper has been cited by over 2000 subsequent articles.^[3]

SupposeS₁(t) andS₂(t) are the prices of two risky assets at timet, and that each has a constant continuous dividend yieldq_i. The option,C, that we wish to price gives the buyer the right, but not the obligation, to exchange the second asset for the first at the time of maturityT. In other words, its payoff,C(T), is max(0,S₁(T) - S₂(T)).

If the volatilities ofS_i's areσ_i, then${\displaystyle \sigma =\sqrt{{\sigma }_1^2+{\sigma }_2^2-2{\sigma }_1{\sigma }_2\rho }}$, whereρ is the Pearson's correlation coefficient of the Brownian motions of theS_i 's.

Margrabe's formula states that the fair price for the option at time 0 is:

${\displaystyle e^{-q_{1}T}{S}_1(0)N(d_1)-e^{-q_{2}T}{S}_2(0)N(d_2)}$

where:

${\displaystyle q_1,q_2}$ are the expected dividend rates of the prices${\displaystyle S_1,S_2}$ under the appropriate risk-neutral measure,

${\displaystyle N}$ denotes thecumulative distribution function for astandard normal,

${\displaystyle d_1=(\ln (S_1(0)/S_2(0))+(q_2-q_1+{\sigma }^2/2)T)/\sigma \sqrt{T}}$,

${\displaystyle d_2=d_1-\sigma \sqrt{T}}$.

Margrabe's model of the market assumes only the existence of the two risky assets, whose prices, as usual, are assumed to follow ageometric Brownian motion. The volatilities of these Brownian motions do not need to be constant, but it is important that the volatility ofS₁/S₂,σ, is constant. In particular, the model does not assume the existence of a riskless asset (such as azero-coupon bond) or any kind ofinterest rate. The model does not require an equivalent risk-neutral probability measure, but an equivalent measure under S₂.

The formula is quicklyproven by reducing the situation to one where we can apply theBlack-Scholes formula.

• First, consider both assets as priced in units ofS₂ (this is called 'usingS₂ asnumeraire'); this means that a unit of the first asset now is worthS₁/S₂ units of the second asset, and a unit of the second asset is worth 1.
• Under this change of numeraire pricing, the second asset is now a riskless asset and its dividend rateq₂ is the interest rate. The payoff of the option, repriced under this change of numeraire, is max(0,S₁(T)/S₂(T) - 1).
• So the original option has become acall option on the first asset (with its numeraire pricing) with a strike of 1 unit of the riskless asset. Note the dividend rateq₁ of the first asset remains the same even with change of pricing.
• Applying theBlack-Scholes formula with these values as the appropriate inputs, e.g. initial asset valueS₁(0)/S₂(0), interest rateq₂, volatilityσ, etc., gives us the price of the option under numeraire pricing.
• Since the resulting option price is in units ofS₂, multiplying through byS₂(0) will undo our change of numeraire, and give us the price in our original currency, which is the formula above. Alternatively, one can show it by theGirsanov theorem.

Notes

Primary reference

• William Margrabe,"The Value of an Option to Exchange One Asset for Another",Journal of Finance, Vol. 33, No. 1, (March 1978), pp. 177–186.

Discussion

• Mark Davis, Imperial College London,Multi-Asset Options
• Rolf Poulsen, University of Gothenburg,The Margrabe Formula

• Mathematical finance
• Options (finance)
• Financial models

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