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  3. 3 let bt denote a brownian motion under the realworld...

Question: 3 let bt denote a brownian motion under the realworld...

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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)

 

 

  1. (a)  Solve the equation for the price of the stock St and show that it is not a martingale under the real-world measure.

  2. (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).

  3. (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 = Ste2t (savings account), determine with proof whether this portfolio is self-financing. 

 

 

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