422_Exam_3_AK_sp21-1

Econ 422 University of Washington Winter 2021 Exam 3 Useful formulae: Put-call parity P = C - S 0 + PV ( X + D ) Hedge Ratio for call, binomial pricing model H = C u - C d uS 0 - dS 0 Black-Scholes-Merton formula C = SN ( d 1 ) - Xe - rT N ( d 2 ) d 1 in Black-Scholes-Merton d 1 = ln S/X + ( r + σ 2 / 2) T σ √ T d 2 in Black-Scholes-Merton d 2 = d 1 - σ √ T Spot-futures parity F 0 = S 0 (1 + r - d ) T Futures spread parity F T 1 = F T 0 (1 + r - d ) T 1 - T 0 Modern Portfolio Theory futures pricing F 0 = E ( P T ) 1 + r f 1 + k T Exchange rate futures pricing, direct quotes F 0 = E 0 1 + r foreign 1 + r domestic T

Econ 422: Exam 3 Winter 2021 You do not need to explain multiple-choice answers. All multiple-choice ques- tions worth 1 point each. 1. You discover that by fitting a cubic polynomial to a graph of stock returns you can predict whether a stock will rise or fall in value about 80% of the time. This is: (a) A violation of all three forms of the efficient market hypothesis (b) A violation of only the weak form of the efficient market hypothesis (c) A violation of only the strong form of the efficient market hypothesis (d) Not a violation of any form of the efficient market hypothesis 2. Your brother in law works as the CFO for a Fortune 500 company, and (illegally) gives you early warning that a merger between his company and a rival firm is going to be announced soon. If the strong form of the efficient market hypothesis holds, then this information should (a) Be able to earn you a profit by exploiting insider information before the public knows about it (b) Be unable to earn you a profit, as the effect of the merger should already be reflected in prices 3. Which of the following is a reason why it may be difficult for arbitrageurs to take advantage of traders’ behavioral biases? (a) It may cost a lot of money in margin requirements to hold a position against the market until the market corrects (b) Arbitrage trades only make very small profits, so investors don’t waste their time finding mispricings in the market (c) The true value of an asset is whatever the investors decide to pay for it, so prices can never be separate from values even if investors are behaviorally biased (d) All of the above 4. Consider the following options, all with the same underlying asset: • Option W : American Call on a dividend paying stock • Option X : European Call on a dividend paying stock • Option Y : American Put on a dividend paying stock • Option Z : European Put on a dividend paying stock Which pair of these options have the same value? (a) Option W and Option X (b) Option Y and Option Z (c) Option W and Option Y (d) None of the above 5. The premium on a call option is $5. The strike price of the option is $100. We can be certain that: 2

Econ 422: Exam 3 Winter 2021 You must show your work for all short-answer questions 11. Consider the two-state model of option valuation, where the underlying stock has price S 0 = 100 , the risk-free interest rate is 8%, and the stock will either increase in value to a price of 120 or increase in value to a price of 105. (a) (3 points) Construct a leveraged equity portfolio that perfectly replicates a European call with strike price 110. The call will be worth either $10 or $0 at expiration, so we should buy a share of stock and borrow enough money to pay back $105 at the end of the year. This would involve borrowing 105 1 . 08 = $97 . 22 . This will give us a portfolio worth either $15 if the stock goes up in value, or $0 if the stock goes down in value. (b) (3 points) Find the value of the call. The leveraged equity portfolio is worth exactly 1.5 times the value of the call in either state of the world, so it must cost 1.5 times as much as the call does. The leveraged equity portfolio costs 100 - 97 . 22 = $2 . 78 to enter, and so the call must cost 2 . 78 1 . 5 = $1 . 85 (c) (3 points) Instead of the call, we want to price a European put with strike price 110. Construct a perfectly hedged portfolio that involves long or short positions on a put option, shares of stock, and either borrowing or lending money. The put will be worth either $5 or $0 at expiration, while the stock is worth either $120 or $105. The hedge ratio is therefore H = 0 - 5 120 - 105 = - 1 3 . To construct a perfect hedge, we buy 1 put and 1 3 of a share of stock. (d) (3 points) Find the value of the put. The hedge will be worth $40 at expiration whether the stock goes up in value or down in value. With an 8% risk-free rate, the hedge is worth 40 1 . 08 = $37 . 04 in present value today. The cost to enter this position is P + 1 3 (100) = 37 . 04 , thus P = $3 . 70 . (e) (3 points) Could you have used put–call parity to find the value of the put from the price of the call? Why or why not? We can use put–call parity, as our options are both European options with the same strike price, on the same stock. We can also see this by applying our formula: 3 . 70 = 1 . 85 - 100 + 110 1 . 08 . 12. The current risk-free interest rate in the United States is 0.05% (be careful! This is not 5%!). The current exchange rate between the US Dollar and the Mexican Peso is 20 Pesos per Dollar. The current futures price for this exchange rate in 1 year (June 2022) is 21 Pesos per Dollar. (a) (3 points) Is the US Dollar expected to appreciate or depreciate in value against the Mexican Peso? The US Dollar currently buys 20 Pesos, and buys 21 Pesos in the future, so it is appreciating in value against the Peso, as it buys more Pesos in the future than it does today. (b) (3 points) What must the current risk-free interest rate be in Mexico if the currency markets are correctly priced 4

Econ 422: Exam 3 Winter 2021 If the markets are correctly priced, the covered interest parity relationship must hold: 21 = 20 1+ r M 1+ . 0005 . Thus, r M = 21 20 (1 . 0005) - 1 = 5 . 0525% . (c) (3 points) You discover that the actual interest rate in Mexico is 10%. Does the futures market overprice or underprice the Mexican Peso relative to the US Dollar? If the actual interest rate in Mexico is 10%, then the futures exchange rate should be 21.89 Pesos per Dollar. Instead it is only 21 Pesos per Dollar, so the futures market believes the Dollar will not be able to buy as many Pesos as it actually should be worth, or in other words, the futures market is overpricing the Peso relative to the Dollar. (d) (3 points) If we wanted to take advantage of this mispricing, should we borrow money in the US or in Mexico? If the futures market is overpricing the Peso, we want to short the Peso in the futures market, selling Pesos at 21 Pesos per Dollar in the futures market while buying them for 20 Pesos per Dollar today. To get the money to buy these Pesos, we should borrow in the US and lend in Mexico. (e) (3 points) These exchange rates are given as indirect quotes (from a US per- spective). If the exchange rates in the spot and futures markets were given as direct quotes instead (again from a US perspective), would any of the answers to the previous questions change? If so, which answers would change? If not, why not? None of the answers would change. Whether the quotes are direct or indirect does not change anything but the numbers we use for the exchange rates. All the underlying concepts remain the same. 13. So called “meme stocks” have made the news throughout the first half of 2021, as internet forum based retail investors have joined together to trade in stocks such as Gamestop and AMC, leading the stock prices of these companies to jump far above their valuations as calculated by traditional analysts. You believe that, while the prices of these meme stocks are likely to fall somewhat by the end of the year, you have also noticed that the prices of meme stocks tend to go up and down much faster and with larger swings than most other stocks. Let the initial, pre-meme price of these stocks be S 0 , the current price of these stocks to be S 1 , and your belief about the true value of these stocks to be S 2 . (a) (7 points) Describe an options strategy that you will use to take advantage of these beliefs. Be explicit: What options will you buy or write, and at what strike prices? We want to construct an options strategy that will payoff if the stock’s value is very different at expiration from a set value, because high volatility will mean that it is relatively unlikely that the price of the stock will remain what it is today. We don’t want to make a trade that relies on the price of the stock falling or rising, so we want to buy both a put and a call with strike prices X = S 1 . If the price increases, the call will pay off, while the put will expire unexercised. If the price decreases, the put will pay off, while the call will expire unexercised. As long as the price of the underlying stock moves far enough from the current price, we will make a profit, only losing if the stock stays where it is. 5