We define the geometric average of the price path of asset price S_i, i = 1, 2, during the time interval [t,t + T ] by

Consider an Asian option involving two assets whose terminal payoff is given by max(G₁(t+T )−G₂(t+T ), 0). Show that the price formula of this European Asian option at time t is given by (Boyle, 1993)

Apply the price formula of the exchange option in Problem 3.34. 

Problem 3.34. 

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:

T exp(-7 ^ In S₁ (1 + u) du). (t T Gi (t+T)= exp