Assume (, U , P) is a probability space with a Brownian motion (W (t), t 0).

Let n(:c, t) := ]E[e"{T'*](S\""(T) K)+] where S'-'(T) = 9:30"\"2)':T"3+\"Wm'w[m. This function represents the value of a. European call option in the Black-Scholes model. From a previous HW exercise, we also have 11.01:, t) = mN(d+(:t, ) Ke"iT't3N(d_(:n, t)) This is known as the Black-Scholes formula. Here _ 1n(a:/K) + (r i o2)(T t) _ o'y'T t and N(z) = ffm :e'wndw is the cumulative distribution function of en N(0,1) random variable. ow 01 \f