Consider the exchange option that gives the holder the right but not the obligation to exchange risky asset S₂ for another risky asset S₁. Let the price dynamics of S₁ and S₂ under the risk neutral measure be governed by

where dZ₁ dZ₂ = ρdt. Let V (S₁,S₂,τ) denote the price function of the exchange option, whose terminal payoff takes the form

Show that the governing equation for V (S₁,S₂,τ) is given by

By taking S₂ as the numeraire and defining the similarity variables: 

show that the governing equation for W(x,τ) becomes

Verify that the solution to W(x,τ) is given by

Referring to the original option price function V (S₁,S₂,τ), we obtain

Transcribed Image Text:

d Si Si = (r-qi) dt + o; dzi, i = 1,2,