Homework Problems

1. Calculate the price of a three-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum.

2. 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 two-year period?

3. A stock price follows geometric Brownian motion with 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?

4. 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?

5. 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. 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.

6. A call option on a non-dividend-paying stock has a market price of $2.50. 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?

7. Consider an American call option on a stock. The stock price is $50, the time to maturity is 15 months, the risk-free rate of interest is 8% per annum, the exercise price is $55, and the volatility is 25%. Dividends of $1.50 are expected in 4 months and 10 months. Show that it can never be optimal to exercise the option on either of the two dividend dates. Calculate the price of the option.

8. A company’s stock price is $50 and 10 million shares are outstanding. The company is considering giving its employees three million at-the-money five-year call options. Option exercises will be handled by issuing more shares. The stock price volatility is 25%, the five-year risk-free rate is 5% and the company does not pay dividends. Estimate the cost to the company of the employee stock option issue.

9. 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.

10. 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?

11. 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. Assume that the stock 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. If the option is an American call, are there any circumstances when it will be exercised early?

12. A company has granted 500,000 options to its executives. The stock price and strike price are both $40. The options last for 12 years and vest after four years. The company decides to value the options using an expected life of five years and a volatility of 30% per annum. The company pays no dividends and the risk-free rate is 4%. What will the company report as an expense for the options on its income statement?

13. Suppose a company grants stock options that last 8 years and vest after 3 years. The stock price and strike price are both $40.00. The stock price volatility is 30%, the risk-free rate is 5%, and the company pays no dividends. What is the (risk-neutral) expected life for the employee stock option? What is the value of the option obtained by using this expected life in Black-Scholes-Merton?

14. A company has granted 2,000,000 options to its employees. The stock price and strike price are both $60. The options last for 8 years and vest after two years. The company decides to value the options using an expected life of six years and a volatility of 22% per annum. The dividend on the stock is $1, payable half way through each year, and the risk-free rate is 5%. What will the company report as an expense for the options on its income statement?

Homework Problems 1. Calculate the price of a three-month Europ...