Question: Let {Wt : t 2 0} be a Brownian motion on a probability space (ELF, IE) with a ltration {It : t 2 0}. Consider

Let {Wt : t 2 0} be a Brownian motion on

Let {Wt : t 2 0} be a Brownian motion on a probability space (ELF, IE") with a ltration {It : t 2 0}. Consider the BlackScholesMerton model with bank account and stock process dB; = Byrdt, Bo = 1, (83 = Stadt 'l' StUth, 80 = 80, where (1,0 > 0 and r 2 0 are constants. We denote by C(SO,K, T) and by P(So, K, T), the price at time 0 of a Call and respectively Put option on the stock St with strike K and maturity T. (a) Find a probability measure If", equivalent to P, under which as; = Strdt + 3mm, where W is a Brownian motion under If" (b) Show that e'T'St is a Pmartingale. (c) Calculate C(SO,K, T) and P090, K, T). ((1) Using Ito's formula, write the Stochastic Differential Equation veried by the process S: (i.e. St raised at the power 7) for a parameter 7 E (0, 1). (e) Give now the price of a call option and put option on the process 8: with maturity T and strike K

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