Problem Set 2
Options, Futures and Derivative Securities
Instructions: This problem set is due on 2/5 at 11:59 pm CST and is an individual assignment. All problems must be handwritten. Scan your work and submit a PDF file.
Problem 1 A stock index currently stands at 5,000. The risk-free interest rate is 8% per year with continuous compounding and the dividend yield on the index is 4% per year. What should be the futures price for an 8-month contract?
Problem 2 The current EUR/USD exchange rate is $1.05 per €. The interest rate in USD is 5% whereas the interest rate in EUR is 3%.
- Compute the 8-month EUR/USD no-arbitrage forward price. Express your answer with four decimals.
- If you enter into a short-forward as described in a. over a notional of €10,000,000, how many dollars and euros will be exchanged in eight months from now? Be explicit about which currency you’re buying and which currency you are selling.
- If the EUR/USD forward price is $1.08 per €, is there an arbitrage opportunity? If so, explain how to exploit it. Assume that you can buy or sell a maximum of €100 million forward.
Problem 3 Consider a stock that currently trades at $120. Analysts expect the stock to pay dividends of $4 every three months. The next dividend is expected next month. The risk-free rate is 8% per year with continuous compounding and assumed the same for all maturities.
- Compute the 6-month no-arbitrage forward price for the stock.
- If the forward price is 120, is there an arbitrage opportunity? If so, explain how to exploit it.
Problem 4 The spot price of silver is $31.04 per ounce. The storage costs are $1.10 per ounce per year payable quarterly in advance. Assuming that interest rates are 8% per annum for all maturities, calculate the futures price of silver for delivery in 12 months.
Problem 5 The E-mini S&P 500 futures contract is one of the most liquid and actively traded futures in the world. The contract value is defined as $50 \times the value of the S&P 500 Index.
Say you deposit $30,000 in your margin account and sell one S&P 500 E-mini futures at $6,101.24. Complete the following table describing the evolution of your margin account.
| Day | Futures Price | Gain/Loss | Margin Account |
|---|---|---|---|
| 0 | 6,101.24 | – | 30,000.00 |
| 1 | 6,084.32 | ||
| 2 | 6,105.89 | ||
| 3 | 6,110.21 |
Problem 6 On January 24, 2025, AAPL stock closed at $222.78. A June call option on AAPL with strike $225 traded at $14.40 per share. If the stock price at maturity is $250, compute the profit per share of buying such a call option.
Problem 7 On January 24, 2025, TSLA stock closed at $406.58. A December put option on TSLA with strike $400 traded at $103.90 per share. If the stock price at maturity is $380, compute the profit per share of buying such a put option.
Problem 8 The price of a stock is $120. The price of a one-year European put option on the stock with a strike price of $90 is quoted as $21 and the price of a one-year European call option on the stock with a strike price of $150 is quoted as $15. Suppose that an investor buys 100 shares, shorts 100 calls, and buys 100 puts. Complete the following table for the strategy.
| Stock Price | 105 | 135 | 165 | 195 |
|---|---|---|---|---|
| Payoff | ||||
| Profit |
Problem 9 Suppose you think that there is a small possibility that XYZ stock might depreciate substantially in value in the next six months. Say the stock’s current price is $120, and put options expiring in six months with an exercise price of $100 are selling at a premium of $5 per share. With $20,000 to invest, you are considering investing $12,000 in the stock, $5,000 in puts, and the rest in a risk-free account. The risk-free rate is 5% per year with continuous compounding. Compute the profit of your portfolio six months from now if the price of XYZ stock at point in time is $90.