Margrabe's formula

In mathematical finance, Margrabe's formula is an option pricing formula. It applies to an option to exchange one risky asset for another risky asset at maturity.

Suppose S₁(t) and S₂(t) are the prices of two risky assets at time t, and that each has a constant continuous dividend yield q_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 maturity T. In other words, its payoff, C(T), is max(0, S₁(T) - S₂(T)).

Margrabe's model of the market assumes only the existence of the two risky assets, whose prices, as usual, are assumed to follow a geometric Brownian motion. The volatilities of these Brownian motions do not need to be constant, but it is important that the volatility of S₁/S₂, σ, is constant. In particular, the model does not assume the existence of a riskless asset (such as a zero-coupon bond) or any kind of interest rate.

If the volatilities of S_i 's are σ_i, then $\textstyle\sigma = \sqrt{\sigma_1^2 + \sigma_2^2 - 2 \sigma_1\sigma_2\rho}$, where ρ is the correlation coefficient of the Brownian motions of the S_i 's.

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

$e^{-q_1 T}S_1(0) N(d_1) - e^{-q_2 T}S_2(0) N(d_2)$

where N denotes the cumulative distribution function for a standard normal, d₁ = (ln (S_1(0)/S_2(0)) + (q₂ - q₁ + σ²/2)T)/σ$\sqrt{T}$, and d₂ = d₁ - σ$\sqrt{T}$.

The formula is quickly proven by reducing the situation to one where we can apply the Black-Scholes formula. First, consider both assets as priced in units of S₂ (this is called 'using S₂ as numeraire'); this means that a unit of the first asset now has is worth S₁/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 rate q₂ 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 a call option on the first asset (with its numeraire pricing) with a strike of 1 unit of the riskless asset. Note the dividend rate q₁ of the first asset remains the same even with change of pricing.

Applying the Black-Scholes formula with these values as the appropriate inputs, e.g. initial asset value S₁(0)/S₂₍₀₎, interest rate q₂, volatility σ, etc., gives us the price of the option under numeraire pricing. Since the resulting option price is in units of S₂, multiplying through by S₂(0) will undo our change of numeraire, and give us the price in our original currency, which is the formula above.