Exchange Options

An exchange option (Margrabe option) gives the holder the right, at expiry $T$, to exchange one unit of asset 2 for one unit of asset 1. Payoff: $\max (S_1(T)-S_2(T),0)$. Given:

$S_1(0)=\$100,S_2(0)=\$95,{\sigma }_1=30\%,{\sigma }_2=25\%,\rho =0.5,T=1\text{ year},$

with no dividends on either asset.

Questions

1. State the Margrabe formula for the value of this option.
2. Compute the effective volatility $\sigma$.
3. Compute $d_1$ and $d_2$.
4. Compute the option's value today.