V (Black-Scholes model) Consider a continuous-time model with a risk-free asset with price (Bt)t[0,T] and a risky asset with price (St)t[0,T] satisfying

dBt = Bt r dt, B0 = 1, dSt = St

dt + St dWt , S0 = s > 0,

where r > 0, > 0, = 2/4 and (Wt)t[0,T] is a Brownian motion, with respect to a filtration F = (Ft)t[0,T] and under the statistical probability P.

1. Define the stochastic process (Yt)t[0,T] by Yt := St , for all t [0, T]. Show that (Yt)t[0,T] is a geometric Brownian motion, i.e., show that (Yt)t[0,T] solves an SDE of the form:

dYt = Yt dWt , Y0 = s,

for a constant > 0 to be determined.

2. Let Q denote the risk-neutral probability. Write the stochastic differential equation (SDE) satisfied by the process (St)t[0,T] under the probability Q.

For the following questions, let us consider a European derivative with maturity T and payoff G(ST ) = ST and denote by t(G) its arbitrage-free price at time t, for every t [0, T].

3. By applying the risk-neutral valuation formula, determine explicitly the pricing function F : [0, T] R+ R+ such that t(G) = F(t, St), for all t [0, T].

4. By relying on the result of question 3, compute explicitly the dynamic self-financing hedging portfolio = ( B, S ) for the derivative with payoff G(ST ) = ST .