2) Let (Sthejo,z be the solution of the stochastic differential equation dS: = St =dBt with Sp > 0. a) (10 marks) Show that (Stheor-] is a martingale on [0, T - =] for every E e (0,T). Hint: Consider the stopping times Tn = (1 - 7 ) T ) A if [t = [0,T] : IS:| 2n), n 21, and use Proposition 8.1. b) (5 marks) Find the value of Sy by a simple argument. c) (10 marks) Show that (Sthejoy is a strict local martingale on [0, 7]. d) (5 marks) Plot a sample graph of (So)rep,r] with 7 =1, and attach or upload it with your submission. 3) Consider the positive strict local martingale (So)rear] solution of dS, = 5/dB, with S, 3 0, where St has the probability density function So 13 v2xt 2t exp (1/T - 1/So) _ exp (1/1 + 1/Sp)? r>0. 2t te (0, T]. a) (5 marks) Plot a sample graph of (St)rep,r] with 7 =1, and attach or upload it with your submission. b) (10 marks) Compute E[Sy] and check that the condition of Question (1c) is satisfied. Hint: Use the change of variable y = 1/r. c) (10 marks) Compute the limit of E[Sy] as So tends to infinity. d) (10 marks) Compute the price E[(Sr - K)+] of a European call option with strike price K > 0 in this model, assuming a risk-free interest rate r = 0. e) (5 marks) Show that E[(ST-K)+] is bounded uniformly in So > 0 and K > 0 by a constant depending on 7 3 0