Please Solve the all the problems given below on Valuing Stock Options: The Black-Scholes-Merton Model

Problem 1
A stock price is currently $40. Assume that the expected return from the stock is 15% and its volatility is 25%. What is the probability distribution for the rate of return (with continuous compounding) earned over a one-year period?

Problem 2
A stock price has an expected return of 16% and a volatility of 35%. The current price is $38.
a) What is the probability that a European call option on the stock with an exercise price of $40 and a maturity date in six months will be exercised?
b) What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised?

Problem 3
Prove that, with the notation in the chapter, a 95% confidence interval for is between

Problem 4
A portfolio manager announces that the average of the returns realized in each of the last 10 years is 20% per annum. In what respect is this statement misleading?

Problem 5
Assume that a non-dividend-paying stock has an expected return of and a volatility of . An innovative financial institution has just announced that it will trade a derivative that pays off a dollar amount equal to

at time . The variables and denote the values of the stock price at time zero and time T.
a) Describe the payoff from this derivative.
b) Use risk-neutral valuation to calculate the price of the derivative at time zero.

Problem 6
What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is three months?

Problem 7
What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is six months?

Problem 8
A call option on a non-dividend-paying stock has a market price of . The stock price is $15, the exercise price is $13, the time to maturity is three months, and the risk-free interest rate is 5% per annum. What is the implied volatility?

Problem 9
Show that the Black-Scholes-Merton formula for a call option gives a price that tends to as .

Problem 10
Explain carefully why Black's approach to evaluating an American call option on a dividend-paying stock may give an approximate answer even when only one dividend is anticipated. Does the answer given by Black's approach understate or overstate the true option value? Explain your answer.

Problem 11
Consider an American call option on a stock. The stock price is $70, the time to maturity is eight months, the risk-free rate of interest is 10% per annum, the exercise price is $65, and the volatility is 32%. A dividend of $1 is expected after three months and again after six months. Use the results in the appendix to show that it can never be optimal to exercise the option on either of the two dividend dates. Use DerivaGem to calculate the price of the option.

Problem 12
A stock price is currently $50 and the risk-free interest rate is 5%. Use the DerivaGem software to translate the following table of European call options on the stock into a table of implied volatilities, assuming no dividends. Are the option prices consistent with the assumptions underlying Black-Scholes-Merton?

Problem 13.
Show that the Black-Scholes-Merton formulas for call and put options satisfy put-call parity.

Problem 14
Show that the probability that a European call option will be exercised in a risk-neutral world is, with the notation introduced in this chapter, . What is an expression for the value of a derivative that pays off $100 if the price of a stock at time T is greater than ?

Further Questions

Problem 15
If the volatility of a stock is 18% per annum, estimate the standard deviation of the
percentage price change in (a) one day, (b) one week, and (c) one month.

Problem 16
A stock price is currently $50. Assume that the expected return from the stock is 18% per annum and its volatility is 30% per annum. What is the probability distribution for the stock price in two years? Calculate the mean and standard deviation of the distribution. Determine the 95% confidence interval.

Problem 17
Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows:
30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0,
32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2
Estimate the stock price volatility. What is the standard error of your estimate?

Problem 18
A financial institution plans to offer a derivative that pays off a dollar amount equal to at time where is the stock price at time . Assume no dividends. Defining other variables as necessary use risk-neutral valuation to calculate the price of the derivative at time zero.
(Hint: The expected value of can be calculated from the mean and variance of given in Section 13.1.)

Problem 19
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.

Problem 20
Assume that the stock in Problem 13.26 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 European call?
b. What is the price of the option if it is a European put?
c. Use the results in the Appendix to this chapter to determine whether there are any circumstances under which the option is exercised early.

Problem 21
Consider an American call option when the stock price is $18, the exercise price is $20, the time to maturity is six months, the volatility is 30% per annum, and the risk-free interest rate is 10% per annum. Two equal dividends of 40 cents are expected during the life of the option, with ex-dividend dates at the end of two months and five months. Use Black's approximation and the DerivaGem software to value the option. Suppose now that the dividend is on each ex-dividend date. Use the results in the Appendix to determine how high can be without the American option being exercised early.