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In mathematical finance, Margrabe's formula^{[1]} is an option pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived by William Margrabe (Phd Chicago) in 1978. Margrabe's paper has been cited by over 1500 subsequent articles.^{[2]}
Suppose S_{1}(t) and S_{2}(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, S_{1}(T) - S_{2}(T)).
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:
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 S_{1}/S_{2}, 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.
The formula is quickly proven by reducing the situation to one where we can apply the Black-Scholes formula.
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