Mock Midterm 1
Questions
Problem 1 (3 pts) 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 | ? | 10 | ||
| B | 64 | 50 | 25 | |
| C | 40 | 50 | ||
| D | 253 | 50 | 100 | 150 |
- How can you synthesize D using A, B, and C?
- What should be the no-arbitrage price of security A?
- If security A is trading at 8, is there an arbitrage opportunity? If so, explain how to exploit it. Be explicit about what you and what you sell, and in which quantities.
Problem 2 (5 pts) The current GBP/USD exchange rate is $1.26 per £. The interest rate in USD is 4% whereas the interest rate in GBP is 5%.
- Compute the 2-year GBP/USD no-arbitrage forward price. Express your answer with four decimals.
- If the GBP/USD forward price is $1.22 per £, is there an arbitrage opportunity? If so, explain how to exploit it by answering the questions below. Assume that you can buy or sell a maximum of £100 million forward.
- Do you buy or sell British pounds forward? How many?
- How many British pounds do you borrow or invest at the GBP interest rate? How much this corresponds in US dollars?
- How many US dollars do you borrow or invest at the USD interest rate?
- Suppose that one year ago, you entered into a short forward to sell £100 million in two years at the forward price described in a. Now the forward has one year to go, the GBP/USD exchange rate is $1.35 per £, and interest rates in USD and GBP have remained unchanged. If you decide to close your forward position, how much do you need to pay or get paid?
Problem 3 (2 pts) The CME crude oil futures contract is is defined over 1,000 barrels of light sweet crude oil. Say you deposit $50,000 in your margin account and buy one crude oil futures at $72.32. Complete the following table describing the evolution of your margin account.
| Day | Futures Price | Gain/Loss | Margin Account |
|---|---|---|---|
| 0 | 72.32 | – | 50,000.00 |
| 1 | 73.32 | ||
| 2 | 71.37 | ||
| 3 | 71.29 |
Problem 4 (2 pts) Stock XYZ current price is $150, and calls expiring in six months with an exercise price of $200 are selling at a premium of $10 per share. With $14,000 to invest, you are considering selling $1,000 worth of calls and buying $15,000 worth of XYZ stock. Compute the profit and the net rate of return of your portfolio six months from now if the price of XYZ stock at point in time is $180.
Problem 5 (4 pts) An 8-month European call option on a non-dividend-paying stock is currently selling for $5. The stock price is $132, the strike price is $130, and the risk-free interest rate is 4% per year. What opportunities could an arbitrageur exploit? Use the table below to indicate what you buy or sell (circle it in the table below), and compute the corresponding cash flows today and in eight months from the point of view of the arbitrageur.
| Position | t = 0 | t = 8 / 12 | |
|---|---|---|---|
| S_{t} \leq 130 | S_{t} > 130 | ||
| Buy / Sell Call | |||
| Buy / Sell Stock | |||
| Borrow / Deposit | |||
| Total Cash Flow | |||
Problem 6 (1 pts) The price of a non-dividend paying stock is $250. The risk-free rate is 5% per year with continuous compounding. Consider a European put option with strike price $280 and maturity 1 year. What should be the price of the put if the volatility of the stock returns is extremely high?
Problem 7 (3pts) On 2/5/2025 at 10:28 AM CST, AMZN stock was trading for $235.47. You got the following information on options with different strike prices and maturities written on AMZN.
| 2025-03-07 | 2025-05-16 | 2025-08-15 | 2026-01-16 | |||||
|---|---|---|---|---|---|---|---|---|
| Strike | Call | Put | Call | Put | Call | Put | Call | Put |
| 210.0 | 29.97 | 2.12 | 34.99 | 5.80 | 40.60 | 9.38 | 48.80 | 13.40 |
| 215.0 | 25.70 | 2.97 | 31.00 | 7.25 | 37.05 | 10.95 | 46.16 | 15.05 |
| 220.0 | 21.25 | 4.14 | 27.58 | 8.72 | 34.40 | 12.55 | 43.00 | 15.60 |
| 225.0 | 18.20 | 5.67 | 25.24 | 10.50 | 30.70 | 12.75 | 39.53 | 18.84 |
| 230.0 | 14.90 | 7.50 | 21.40 | 12.50 | 28.00 | 16.05 | 36.50 | 21.20 |
| 235.0 | 11.40 | 9.65 | 18.90 | 14.80 | 25.35 | 18.65 | 33.90 | 23.29 |
| 240.0 | 9.05 | 11.60 | 16.49 | 17.30 | 22.80 | 21.55 | 31.36 | 25.70 |
| 245.0 | 7.22 | 15.40 | 14.10 | 19.70 | 21.05 | 24.30 | 29.05 | 26.10 |
| 250.0 | 5.52 | 18.50 | 12.08 | 22.70 | 18.90 | 27.15 | 26.80 | 28.94 |
| 255.0 | 4.05 | 20.90 | 10.40 | 23.20 | 16.90 | 29.45 | 24.60 | 34.03 |
Draw the payoff and profit diagram as a function of the final stock price of a strategy in which you:
- Buy a Aug 25 call with strike price 235.
- Buy a Aug 25 put with strike price 235.
- Sell a Aug 25 call with strike price 250.
- Sell a Aug 25 put with strike price 220.
In your diagram, indicate the cutoff prices that lead to a gain/loss.
Formula Sheet
In the following, S denotes the stock or spot price of an asset, r is the continuously-compounded risk-free rate expressed per year and T denotes the time-to-maturity of a forward, futures or an option.
Forward Contracts
If the asset pays cash dividends the forward price is given by F = (S - D) e^{r T}, where D denotes the present value of all dividends paid during the life of the option.
If the asset pays a dividend yield \delta, the forward price is given by F = S e^{(r - \delta) T}. Note that the previous expression gives you the forward price of a non-dividend paying asset by setting \delta = 0.
The value today of a long forward position with initial forward price K is V = (F - K) e^{-r T}, where F = S e^{(r - \delta) T} is the corresponding forward price today.
Put-Call Parity
For European call and put options with strike price K and time-to-expiration T written on a non-dividend paying asset, we have that C - P = S - K e^{-r T}, where C and P denote the call and put prices.
Option Pricing Bounds
For European call and put options written on a non-divided paying stock we have that \begin{aligned} \max(S - K e^{-r T}, 0) & \leq C \leq S, \\ \max(K e^{-r T} - S, 0) & \leq P \leq K e^{-r T}. \end{aligned}