We define the geometric average of the price path of asset price S i , i =

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

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

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)

where d V (S, S2, t; T) = SN (d) - SN (d), = S exp 5, exp(-( + 2)). i=1,2, o  =3+33-300102, T 12 In+T  d=d -

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

d Si Si = (rqi) dt+o; d Z, i = 1, 2,

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

V (S1, S2, 0) = max(S  S, 0).

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

at ~ 202v  + p0102S1 S2. asp + (r - 91)S1- alv OSTS2 S1 +  82021 as  + (r - 92) S2- -rV. IS2By taking S2 as the numeraire and defining the similarity variables: 

X = S1 S2 and W(x, t) = V (S1, S2, T) S

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

aW  ola 2 2  a w 22 + (91 - 92)x a W x - 92 W.Verify that the solution to W(x,τ) is given by

where W(x, t) = e-91 xN(d) - e-92 N(-d), d = In +(92-91 + 2)t OT $ In + (91-92 OT 92 +2) T d2 2 0 = 0 -

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

V (S1, S2, T)e9 S N (d)  e92 SN(-d).

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