E ( ST ? Se ? (T ?t var( ST ?S 2e2 ? (T ?t [e? 2 2 2 (T ?t ?1] Since var(ST ? E[(ST ] ? [ E(ST ] , it follows that E[(ST 2 ] ? var(ST ? [ E (ST ]2 so that E[( ST 2 ] ? S 2e2 ? (T ?t [e? 2 2 (T ?t ? 1] ? S 2e2 ? (T ?t ?S 2e(2 ? ?? (T ?t In a risk-neutral world ? ? r so that ? [(S 2 ] ? S 2e(2 r ?? 2 (T ?t E T Using risk-neutral valuation, the value of the derivative security at time t is ? [(S 2 ] e ? r (T ?t E T ? S 2e(2 r ?? 2 (T ?t ? r (T ?t e ? S 2e( r ?? 2 (T ?t (b If: f ? S 2 e( r ?? 2 (T ?t 2 ?f ?? S 2 (r ?? 2 e( r ?? (T ?t ?t 2 ?f ?2 Se( r ?? (T ?t ?S 2 ?2 f ?2e( r ?? (T ?t 2 ?S The left-hand side of the Black-Scholes–Merton differential equation is: 2 2 2 ? S 2 (r ? ? 2 e( r ?? (T ?t ? 2rS 2e( r ?? (T ?t ? ? 2 S 2e( r ?? (T ?t ?rS 2e( r ?? 2 (T ?t ?rf Hence the Black-Scholes equation is satisfied. Problem 15.30. Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months. a. What is the price of the option if it is a European call? b. What is the price of the option if it is an American call? c. What is the price of the option if it is a European put? d. Verify that put–call parity holds. In this case S0 ? 30 , K ? 29 , r ? 0?05 , ? ? 0?25 and T ? 4 ? 12 d1 ? ln(30 ? 29 ? (0?05 ? 0?252 ? 2 ? 4 ? 12 ? 0?4225 0?25 0?3333
d2 ? ln(30 ? 29 ? (0?05 ? 0?252 ? 2 ? 4 ? 12 ? 0?2782 0?25 0?3333 N (0?4225 ? 0?6637? N (0?2782 ? 0?6096 N (?0?4225 ? 0?3363? N (?0?2782 ? 0?3904 a. The European call price is 30 ? 0?6637 ?
29e?0?05?4?12 ? 0?6096 ? 2?52 or $2.52. b. The American call price is the same as the European call price. It is $2.52. c. The European put price is
29e?0?05?4?12 ? 0?3904 ? 30 ? 0?3363 ? 1?05 or $1.05. d. Put-call parity states that: p ? S ? c ? Ke? rT In this case c ? 2?52 , S0 ? 30 , K ? 29 , p ? 1?05 and e? rT ? 0?9835 and it is easy to verify that the relationship is satisfied, Problem 15.31. Assume that the stock in Problem 15.30 is due to go ex-dividend in 1.5 months. The expected dividend is 50 cents. a. What is the price of the option if it is a
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