Problem Set 1
Options, Futures and Derivative Securities
Instructions: This problem set is due on 1/27 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 Consider the following risk-free securities available to buy or sell to all investors in the market.
| Security | Price (t=0) | Cash Flow (t=1) | Cash Flow (t=2) | Cash Flow (t=3) |
|---|---|---|---|---|
| A | 38 | 40 | ||
| B | ? | 30 | ||
| C | 16 | 20 | ||
| D | 298 | 120 | 120 | 120 |
- What should be the no-arbitrage price of security B?
- If security B is trading at 24, is there an arbitrage opportunity? If so, explain how to exploit it.
Problem 2 Consider the following risk-free securities available to buy or sell to all investors in the market.
| Security | Price (t=0) | Cash Flow (t=1) | Cash Flow (t=2) | Cash Flow (t=3) |
|---|---|---|---|---|
| A | 76 | 80 | ||
| B | 55 | 20 | 40 | |
| C | 78 | 20 | 40 | 50 |
| D | ? | 20 | ||
| E | ? | 100 |
- What should be the no-arbitrage price of security D?
- What should be the no-arbitrage price of security E?
Problem 3 An investor receives $1,080 in one year in return for an investment of $1,000 now. Calculate the percentage return per year with:
- Annual compounding
- Semiannual compounding
- Monthly compounding
- Continuous compounding
Problem 4 An effective annual rate (EAR) of 9% per year is equivalent to which rate expressed per year with continuous compounding?
Problem 5 You have information of cash flows and zero-coupon rates (per year with continuous compounding) for different maturities as shown below:
| Time (years) | 1 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| Zero-coupon rate (%) | 5.0 | 5.5 | 6.0 | 6.0 | 6.5 |
| Cash flow | 100 | 150 | 200 | 250 | 300 |
Compute the present value of those cash flows.
Problem 6 Suppose you enter into a 6-month forward contract on a non-dividend-paying stock when the stock price is $100, and the risk-free interest rate is 10% per year with continuous compounding.
- What is the no-arbitrage forward price?
- If the forward price is 102, is there an arbitrage opportunity? If so, explain how to exploit it.
Problem 7 You enter in a 1-year long forward contract on a non-dividend-paying stock when the stock price is $50, and the risk-free rate of interest is 10% per year with continuous compounding.
- What are the forward price and the initial value of the forward contract?
- Six months later, the price of the stock is $45, and the risk-free interest rate is still 10%.
- What are the forward price and the value of the forward contract?
- If you decide to close the forward position, how much do you need to pay or get paid?