Problem Set 6
Options, Futures and Derivative Securities
Instructions: This problem set is due on 4/14 at 11:59 pm CST and is an individual assignment. All problems must be handwritten. Scan your work and submit a PDF file. For all probability computations, use the table at the end of this assignment.
Note: All interest rates and dividend yields are expressed per year with continuous compounding.
Problems
Problem 1 What is the price of a European put option on a non-dividend-paying stock when the stock price is $88, the strike price is $90, the risk-free interest rate is 5% per annum, the volatility is 45% per annum, and the time to maturity is eight months?
Problem 2 Consider an option on a non-dividend-paying stock when the stock price is $50, the exercise price is $49, the risk-free interest rate is 4.5%, the volatility is 38% per annum, and the time to maturity is three months.
- What is the price of the option if it is a European call?
- What is the price of the option if it is an American call?
- What is the price of the option if it is a European put?
- Verify that put-call parity holds.
Problem 3 Consider a European call option expiring in 6 months and with strike price equal to $48 on a non-dividend paying stock that currently trades for $50. Interestingly, the volatility of the stock is zero. If the risk-free rate is 8% per year with continuous compounding, what is the price of the option?
Problem 4 Consider a European call option expiring in 6 months and with strike price equal to $52 on a non-dividend paying stock that currently trades for $50. Interestingly, the volatility of the stock is zero. If the risk-free rate is 5% per year with continuous compounding, what is the price of the option?
Problem 5 Consider a European put option expiring in 9 months and strike price $152 written on a non-dividend paying stock. The risk-free rate is 6% per year with continuous compounding and the stock price is $150. What is the minimum price for the put that would allow you to compute its implied volatility?
Problem 6 Suppose that the sales team of a trading desk just sold a European call option contract, that is 100 European call options, to an important client. The contract is written on a non-dividend paying stock that trades for $150, expires in two years and has a strike price of $155. The risk-free rate is 5% per year with continuous compounding. A trader of the desk estimate that the volatility of the stock returns is 55% and expected to remain constant for the life of the contract.
- How many shares of the stock does the trader need to buy/sell initially in order to hedge the exposure created by the sale of the contract?
- How many risk-free bonds with face value $155 and expiring in two years does the trader need to buy/sell in order to make sure that the strategy is self-financing?
- How many risk-free bonds with face value $100 and expiring in two years does the trader need to buy/sell in order to make sure that the strategy is self-financing?
- Why choosing a different face value for the bonds does not change the price of the call option.
Problem 7 A call option on a non-dividend-paying stock has a market price of $12.39. The stock price is $100, the exercise price is $100, the time to maturity is six months, and the risk-free interest rate is 5% per annum. What is the implied volatility of the call? Explain the method you used to find the implied volatility.
Problem 8 Calculate the value of a three-month at-the-money European call option on a stock index when the index is at 5,000, the risk-free interest rate is 5% per annum, the volatility of the index is 30% per annum, and the dividend yield on the index is 2% per annum.
Problem 9 The S&P 100 index currently stands at 2,392 and has a volatility of 50% per annum. The risk-free rate of interest is 4% per annum and the index provides a dividend yield of 1% per annum. Calculate the value of a three-month European put with strike price 2,300.
The Standard Normal Distribution
The following table reports values for \phi(z) = P(Z ≤ z), where Z \sim \mathcal{N}(0, 1).
| z | P(Z ≤ z) | z | P(Z ≤ z) | z | P(Z ≤ z) | z | P(Z ≤ z) |
|---|---|---|---|---|---|---|---|
| -2.37 | 0.0089 | -1.17 | 0.1210 | 0.03 | 0.5120 | 1.23 | 0.8907 |
| -2.32 | 0.0102 | -1.12 | 0.1314 | 0.08 | 0.5319 | 1.28 | 0.8997 |
| -2.27 | 0.0116 | -1.07 | 0.1423 | 0.13 | 0.5517 | 1.33 | 0.9082 |
| -2.22 | 0.0132 | -1.02 | 0.1539 | 0.18 | 0.5714 | 1.38 | 0.9162 |
| -2.17 | 0.0150 | -0.97 | 0.1660 | 0.23 | 0.5910 | 1.43 | 0.9236 |
| -2.12 | 0.0170 | -0.92 | 0.1788 | 0.28 | 0.6103 | 1.48 | 0.9306 |
| -2.07 | 0.0192 | -0.87 | 0.1922 | 0.33 | 0.6293 | 1.53 | 0.9370 |
| -2.02 | 0.0217 | -0.82 | 0.2061 | 0.38 | 0.6480 | 1.58 | 0.9429 |
| -1.97 | 0.0244 | -0.77 | 0.2206 | 0.43 | 0.6664 | 1.63 | 0.9484 |
| -1.92 | 0.0274 | -0.72 | 0.2358 | 0.48 | 0.6844 | 1.68 | 0.9535 |
| -1.87 | 0.0307 | -0.67 | 0.2514 | 0.53 | 0.7019 | 1.73 | 0.9582 |
| -1.82 | 0.0344 | -0.62 | 0.2676 | 0.58 | 0.7190 | 1.78 | 0.9625 |
| -1.77 | 0.0384 | -0.57 | 0.2843 | 0.63 | 0.7357 | 1.83 | 0.9664 |
| -1.72 | 0.0427 | -0.52 | 0.3015 | 0.68 | 0.7517 | 1.88 | 0.9699 |
| -1.67 | 0.0475 | -0.47 | 0.3192 | 0.73 | 0.7673 | 1.93 | 0.9732 |
| -1.62 | 0.0526 | -0.42 | 0.3372 | 0.78 | 0.7823 | 1.98 | 0.9761 |
| -1.57 | 0.0582 | -0.37 | 0.3557 | 0.83 | 0.7967 | 2.03 | 0.9788 |
| -1.52 | 0.0643 | -0.32 | 0.3745 | 0.88 | 0.8106 | 2.08 | 0.9812 |
| -1.47 | 0.0708 | -0.27 | 0.3936 | 0.93 | 0.8238 | 2.13 | 0.9834 |
| -1.42 | 0.0778 | -0.22 | 0.4129 | 0.98 | 0.8365 | 2.18 | 0.9854 |
| -1.37 | 0.0853 | -0.17 | 0.4325 | 1.03 | 0.8485 | 2.23 | 0.9871 |
| -1.32 | 0.0934 | -0.12 | 0.4522 | 1.08 | 0.8599 | 2.28 | 0.9887 |
| -1.27 | 0.1020 | -0.07 | 0.4721 | 1.13 | 0.8708 | 2.33 | 0.9901 |
| -1.22 | 0.1112 | -0.02 | 0.4920 | 1.18 | 0.8810 | 2.37 | 0.9911 |