Let S(t_i) denote the asset price at time t_i,i = 1, 2, ··· ,N, where 0 = t₀ N = T. Define the discretely monitored arithmetic average and geometric average by

Let c_A(0; X) and c_G(0; X) denote the time-0 value of the European Asian fixed strike call option with strike price X and whose underlying are A(T) and G(T), respectively. Under the usual Geometric Brownian process assumption of the asset price, show that (Nielsen and Sandmann, 2003)

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1 A(T): = - N i=1 N S(t) and G(T) = S(ti) HSG) 1/N]