3. Let (Bt) denote a Brownian motion under the real-world measure with Bo = 0. Consider the Black-Scholes model for the stock price, d.St = 2Stdt + 4StdBt, So = 1, the savings account is given by t = 1 for all t. = (a) Write down the condition for a portfolio in this model to be self-financing. Consider the portfolio given by a = -t (units of the stock) and b Sudu (units of the savings account), determine with proof whether this portfolio is self-financing. ER State the Girsanov theorem. Using it, or otherwise, derive the expression (not the stochastic differential) for St, in terms of a Brownian motion under the equivalent martingale measure (EMM). (c) Denote by Ct the price at time t 2 of the call option on this stock with exercise price K = 1 and expiration date T = 2. By quoting an appropriate result, give the expression for Ct. Find the answer (in terms of the normal distribution function) for the case when t = 1. SSESMENT SESSMEN Y 202 ASSESS MONASH ESSMENT 3. Let (Bt) denote a Brownian motion under the real-world measure with Bo = 0. Consider the Black-Scholes model for the stock price, d.St = 2Stdt + 4StdBt, So = 1, the savings account is given by t = 1 for all t. = (a) Write down the condition for a portfolio in this model to be self-financing. Consider the portfolio given by a = -t (units of the stock) and b Sudu (units of the savings account), determine with proof whether this portfolio is self-financing. ER State the Girsanov theorem. Using it, or otherwise, derive the expression (not the stochastic differential) for St, in terms of a Brownian motion under the equivalent martingale measure (EMM). (c) Denote by Ct the price at time t 2 of the call option on this stock with exercise price K = 1 and expiration date T = 2. By quoting an appropriate result, give the expression for Ct. Find the answer (in terms of the normal distribution function) for the case when t = 1. SSESMENT SESSMEN Y 202 ASSESS MONASH ESSMENT