We consider a continuous time market, where the interest rate r = 0, and the risky asset S = (St)ostsT follows the Black-Scholes model with initial value So = 1, drift and volatility > 0 (without any dividend), so that dS = S,dt+SdB. a) (5 points) A self-financing portfolio is given by (x,o), where a represents the initial wealth of the portfolio, and or represents the number of risky asset in the portfolio at time t. Let (II)[0.7] be the wealth process of the portfolio, write down the dynamic of II in form of dII = dt + BdB, for some (to be founded) process (a, 3). b) (5 points) There exists a unique risky-neutral probability Q, together with a Brownian motion BQ under the probability measure Q. Please give the expression of the process S as a function of (t, B). c) (40 points) We consider a derivaitive option with payoff g(ST) = log(ST) at maturity T. (5 points) Compute the value V == EQ [log(ST)]. - (25 points) Let v(t, x) = log(x) - o(T-t), compute dv, v and v, and check that v satisfies the equation + 1/10 x 82 v(t, x) = 0, (t, x) = [0,T) (0,), with terminal condition v(T, x) = log(x) for all x (0,). (10 points) Remember that S, is a function of (t, B), apply the It formula on v(t, S) to deduce that log(Sr) = Vo+dSt, where or := St Finally, deduce the (no-arbitrage) price of the derivative option with payoff g(ST) = log(ST).