6. (3pts+3pts+3pts+3pts) Consider the Bachalier model St = oWt where W is a Brownian motion. Let XT = (ST - K)+ = max{ST - K, 0} be the payoff of a European call option with strike K > 0. The price of the option is CK = E((ST - K) +). 2 2 Also, let v(x) = 2702T e 202T and V(x) = Sty(u)du. (1) Show that CK = SR xy(x)dx - K fo v(x)dx and CK = KU(K) + 02 Ty(K) - K. (2) Show that dCK = C9 = - dK K 2/(x) dx, d'CK = C(2) = 4(K). dK2 (3) By Taylor's theorem conclude that CK has the expansion CK = _ CKKK K20 Derive the formula for ck, k 2 0, in terms of v. (4) Setting A = ovT/V27 and calculate the leading terms to show that K K2 K4 CK = A - 2 4 7 A 9672 43 + O(K / A5 )