Question

3. Consider the Black-Scholes model for the price of stock S_t, dS_t = – – S_t dt + – – S_t dB_t, S_0 = 1, and savings account ?_t = e^{– – t}. (a) Show that the discounted stock price S_t/?_t is not a martingale. (b) Change the probability measure so that the discounted stock price is a martingale under the new measure, use Girsanov theorem. Derive the expression for S_t under the new measure. (c) Denote by C_t the price at time t, of the call option on this stock with exercise price K = 1 and expiration date T = 1, t ? 1. By quoting an appropriate result, give the expression for C_0, the price of this option at time zero, and show that it is equal to e^{-– –} ?_{-– –}^? (e^{– – x + – –} - 1)?(x)dx, where ?(x) = – – – – – e^{-– –}. Using the Black-Scholes formula, or otherwise, show that C_0 = ?(– –) - e^{-– –}?(– –), where ?(z) = ?_{-?}^z ?(x)dx. (d) Write down the condition for a portfolio in this model to be self-financing. Find all possible self-financing portfolios a_t, b_t with a_t = 1 for all t Consider the portfolio given by a_t = 1 (stock) and b_t (savings account). Find all possible b_t, 0 ? t ? 1, so that this portfolio is self-financing.

          3. Consider the Black-Scholes model for the price of stock S_t,
dS_t = – – S_t dt + – – S_t dB_t,
S_0 = 1, and savings account ?_t = e^{– – t}.
(a) Show that the discounted stock price S_t/?_t is not a martingale.
(b) Change the probability measure so that the discounted stock price is a martingale under the new measure, use Girsanov theorem. Derive the expression for S_t under the new measure.
(c) Denote by C_t the price at time t, of the call option on this stock with exercise price K = 1 and expiration date T = 1, t ? 1. By quoting an appropriate result, give the expression for C_0, the price of this option at time zero, and show that it is equal to
e^{-– –} ?_{-– –}^? (e^{– – x + – –} - 1)?(x)dx,
where ?(x) = – – – – – e^{-– –}. Using the Black-Scholes formula, or otherwise, show that C_0 = ?(– –) - e^{-– –}?(– –), where ?(z) = ?_{-?}^z ?(x)dx.
(d) Write down the condition for a portfolio in this model to be self-financing. Find all possible self-financing portfolios a_t, b_t with a_t = 1 for all t
Consider the portfolio given by a_t = 1 (stock) and b_t (savings account). Find all possible b_t, 0 ? t ? 1, so that this portfolio is self-financing.

3. Consider the Black-Scholes model for the price of stock St,
dSt = – – St dt + – – St dBt,
S0 = 1, and savings account ?t = e^– – t.
(a) Show that the discounted stock price St/?t is not a martingale.
(b) Change the probability measure so that the discounted stock price is a martingale under the new measure, use Girsanov theorem. Derive the expression for St under the new measure.
(c) Denote by Ct the price at time t, of the call option on this stock with exercise price K = 1 and expiration date T = 1, t ? 1. By quoting an appropriate result, give the expression for C0, the price of this option at time zero, and show that it is equal to
e^-– – ?-– –^? (e^– – x + – – - 1)?(x)dx,
where ?(x) = – – – – – e^-– –. Using the Black-Scholes formula, or otherwise, show that C0 = ?(– –) - e^-– –?(– –), where ?(z) = ?-?^z ?(x)dx.
(d) Write down the condition for a portfolio in this model to be self-financing. Find all possible self-financing portfolios at, bt with at = 1 for all t
Consider the portfolio given by at = 1 (stock) and bt (savings account). Find all possible bt, 0 ? t ? 1, so that this portfolio is self-financing.

Added by Duane E.

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