Question: 3 let bt denote a brownian motion under the realworld...
3. Let (Bt) denote a Brownian motion under the real-world measure with B0 = 0. Consider the Black-Scholes model for the stock price,
dSt =−2Stdt+2StdBt, S0 =1,
and the savings account is given by βt = e^(2t) .
(a) Solve the equation for the price of the stock St and show that it is not a martingale under the real-world measure.
(b) State the Girsanov theorem. Using it, or otherwise, derive the expression for St, in terms of a Brownian motion under the equivalent martingale measure (EMM).
(c) Denote by Ct the price at time t ≤ 1 of the call option on this stock
with exercise price K = 1 and expiration date T = 1. By quoting an
appropriate result, give the expression for Ct. Find the answer (in terms 1
of the normal distribution function) for the case when t = 1/2 .
(d) Write down the condition for a portfolio in this model to be self-financing. Consider the portfolio given by at = −1 (stock) and bt = Ste−2t (savings account), determine with proof whether this portfolio is self-financing.