Margrabe's formula

In mathematical finance, Margrabe's formula 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 in 1978. Margrabe's paper has been cited by over 2000 subsequent articles.

Formula

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 qi. 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)).

If the volatilities of Si's are σi, then

σ

=

σ

1

2

+

σ

2

2

−

2

σ

1

σ

2

ρ

{\displaystyle \textstyle \sigma ={\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}-2\sigma _{1}\sigma _{2}\rho }}}

, where ρ is the Pearson's correlation coefficient of the Brownian motions of the Si 's.

Margrabe's formula states that the fair 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

)

{\displaystyle e^{-q_{1}T}S_{1}(0)N(d_{1})-e^{-q_{2}T}S_{2}(0)N(d_{2})}

where:

q

1

,

q

2

{\displaystyle q_{1},q_{2}}

are the expected dividend rates of the prices

S

1

,

S

2

{\displaystyle S_{1},S_{2}}

under the appropriate risk-neutral measure,

N

{\displaystyle N}

denotes the cumulative distribution function for a standard normal,

d

1

=

(

ln

⁡

(

S

1

(

0

)

/

S

2

(

0

)

)

+

(

q

2

−

q

1

+

σ

2

/

2

)

T

)

/

σ

T

{\displaystyle d_{1}=(\ln(S_{1}(0)/S_{2}(0))+(q_{2}-q_{1}+\sigma ^{2}/2)T)/\sigma {\sqrt {T}}}

,

d

2

=

d

1

−

σ

T

{\displaystyle d_{2}=d_{1}-\sigma {\sqrt {T}}}

.

Derivation

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.