Derivatives Markets Solutions Chapters 1-10 (FM)

Chapter 1 Introduction to Derivatives Question 1.1. This problem offers different scenarios in which some companies may have an interest to hedge their exposure to temperatures that are detrimental to their business. In answering the problem, it is useful to ask the question: Which scenario hurts the company, and how can it protect itself? a) A soft drink manufacturer probably sells more drinks when it is abnormally hot. She dislikes days at which it is abnormally cold, because people are likely to drink less, and her business suffers. She will be interested in a cooling degree-day futures contract, because it will make payments when her usual business is slow. She hedges her business risk. b) A ski-resort operator may fear large losses if it is warmer than usual. It is detrimental to her business if it does not snow in the beginning of the season, or if the snow is melting too fast at the end of the season. She will be interested in a heating degree-day futures contract, because it will make payments when her usual business suffers, thus compensating the losses. c) During the summer month, an electric utility company, such as one in the South of the United States, will sell a lot of energy during days of excessive heat, because people will use their air conditioners, refrigerators and fans more often, thus consuming a lot of energy and increasing profits for the utility company. In this scenario, the utility company will have less business during relatively colder days, and the cooling degree-day futures offers a possibility to hedge such risk. Alternatively, we may think of a utility provider in the North-East during the winter months, in a region where people use many additional electric heaters. This utility provider will make more money during unusually cold days, and may be interested in a heating degree-day contract, because that contract pays off if the primary business suffers. d) An amusement park operator fears bad weather and cold days, because people will abstain from going to the amusement park during cold days. She will buy a cooling-degree futures to offset her losses from ticket sales with gains from the futures contract. Question 1.2. A variety of counter-parties are imaginable. For one, we could think about speculators who have differences in opinion and who do not believe that we will have excessive temperature variations during the life of the futures contracts. Thus, they are willing to take the opposing side, receiving a payoff if the weather is stable. Alternatively, there may be opposing hedging needs: Compare the ski-resort operator and the softdrink manufacturer. The cooling degree-day futures contract will pay off if it the weather is relatively 1

mild, and we saw that the resort operator will buy the futures contract. The buyer of the cooling degree-day futures will make a loss if the weather is cold (which means that the seller of the contract will make a gain). Since the soft drink manufacturer wants additional money if it is cold, she may be interested in taking the opposite side of the cooling degree-day futures. Question 1.3. a) Remember that the terminology bid and ask is formulated from the market makers perspective. Therefore, the price at which you can buy is called the ask price. Furthermore, you will have to pay the commission to your broker for the transaction. You pay: ($41.05 × 100) + $20 = $4,125.00 b) Similarly, you can sell at the market maker’s bid price. You will again have to pay a commission, and your broker will deduct the commission from the sales price of the shares. You receive: ($40.95 × 100) − $20 = $4,075.00 c)

Your round-trip transaction costs amount to: $4,125.00 − $4,075.00 = $50

Question 1.4. In this problem, the brokerage fee is variable, and depends on the actual dollar amount of the sale/purchase of the shares. The concept of the transaction cost remains the same: If you buy the shares, the commission is added to the amount you owe, and if you sell the shares, the commission is deducted from the proceeds of the sale. a)

($41.05 × 100) + ($41.05 × 100) × 0.003 = $4,117.315 = $4,117.32

b)

($40.95 × 100) − ($40.95 × 100) × 0.003 = $4,082.715 = $4,082.72

c)

$4,117.32 − $4,082.72 = $34.6

The variable (or proportional) brokerage fee is advantageous to us. Our round-trip transaction fees are reduced by $15.40. 2

Chapter 1 Introduction to Derivatives

Question 1.5. In answering this question it is important to remember that the market maker provides a service to the market. He stands ready to buy shares into his inventory and sell shares out of his inventory, thus providing immediacy to the market. He is remunerated for this service by earning the bid-ask spread. The market maker buys the security at a price of $100, and he sells it at a price of $100.10. If he buys 100 shares of the security and immediately sells them to another party, he is earns a spread of: 100 × ($100.12 − $100) = 100 × $0.12 = $12.00 Question 1.6. A short sale of XYZ entails borrowing shares of XYZ and then selling them, receiving cash. Therefore, initially, we will receive the proceeds from the sale of the asset, less the proportional commission charge: 300 × ($30.19) − 300 × ($30.19) × 0.005 = $9,057 × 0.995 = $9,011.72 When we close out the position, we will again incur the commission charge, which is added to the purchasing cost: 300 × ($29.875) + 300 × ($29.875) × 0.005 = $8,962.5 × 1.005 = $9,007.31 Finally, we subtract the cost of covering the short position from our initial proceeds to receive total profits: $9,011.72 − $9,007.31 = $4.41. We can see that the commission charge that we have to pay twice significantly reduces the profits we can make.

Question 1.7. a) A short sale of JKI stock entails borrowing shares of JKI and then selling them, receiving cash, and we learned that we sell assets at the bid price. Therefore, initially, we will receive the proceeds from the sale of the asset at the bid (ignoring the commissions and interest). After 180 days, we cover the short position by buying the JKI stock, and we saw that we will always buy at the ask. Therefore, we earn the following profit: 400 × ($25.125) − 400 × ($23.0625) = $10,050 − $9,225.00 = $825.00 3

b)

We have to pay the commission twice. The commission will reduce our profit: 400 × ($25.125) − 400 × ($25.125) × 0.003 − (400 × ($23.0625) + 400 × ($23.0625)) = $10,050 × 0.997 − $9,225 × 1.003 = $10,019.85 − $9252.675 = $767.175.

c) The proceeds from short sales, minus the commission charge are $10,019.85 (or $10,050 if you ignore the commission charge). Since the 6-month interest rate is given, and the period of our short sale is exactly half a year, we can directly calculate the interest we could earn (and that we now lose) on a deposit of $10,019.85: $10,019.85 × (0.03) = $300.5955 = $300.60 or, without taking into account the commission charge: $10,050.00 × (0.03) = $301.50. Question 1.8. We learned from the main text that short selling is equivalent to borrowing money, and that a short seller will often have to deposit the proceeds of the short sale with the lender as collateral. A short seller is entitled to earn interest on his collateral, and the interest rate he earns is called the short rebate in the stock market. Usually, the short rebate is close to the prevailing market interest rate. Sometimes, though, a particular stock is scarce and difficult to borrow. In this case, the short rebate is substantially less than the current market interest rate, and an equity lender can earn a nice profit in the form of the difference between the current market interest rate and the short rebate. By signing an agreement as mentioned in the problem, you give your brokerage firm the possibility to act as an equity lender, using the shares of your account. Brokers want you to sign such an agreement because they can make additional profits.

Question 1.9. If we borrow an asset from a lender, we have the obligation to make any payments to the lender that she is entitled to as a stockholder. As the lender is entitled to the dividend on the day the stock goes ex dividend, but does not receive it from the company, because we have sold her stock, we must provide the dividend. This payment is tax-deductible for us. In a perfect capital market, we would expect that the stock price falls exactly by the amount of the dividend on the ex-date. Therefore, we should not care. 4

Chapter 1 Introduction to Derivatives

However, two complications may arise. First, we may have borrowed a large amount of shares, and the increased dividend forces us to pay more to the lender, and we may not have the additional required money. On a different note, an unexpected increase of the dividend is a strong signal that the company is doing exceedingly well, and empirically, we observe sharp price increases after such announcements. As we have a short position in the stock, we make money if the stock price falls. Therefore, an unexpected increase in the dividend is very bad for our position, and we should care! Question 1.10. The following information on short interest comes from NASDAQ’s market data Internet site (http://www.marketdata.nasdaq.com). They explain: How is short interest in Nasdaq stocks calculated? Short selling is the selling of a security which the seller does not own, or any sale which is completed by the delivery of a security borrowed by the seller. Short selling is a legitimate trading strategy. Short sellers assume the risk that they will be able to buy the stock at a more favorable price than the price at which they sold short. [. . .] To calculate short interest in Nasdaq stocks, NASD member firms are instructed to report to the NASDR TS-Customer Advocacy & Quality Management Department, on a monthly basis, their short positions, for all accounts, in shares, warrants, units, ADRs, and convertible preferreds resulting from short sales. Once the short position reports are received by the Product Deployment and Support Department, the short interest is then compiled for each Nasdaq security. [. . .] The monthly short interest information does include the adjustment for stock splits. The adjustment to the short interest for stocks that split on or before the reporting settlement date will automatically be reflected in the most current reporting period. However, for stock splits that occur after the settlement date, the adjustment will be reflected in the following reporting period. You can download a monthly text file listing short interest positions for all Nasdaq issues by going to: http://www.marketdata.nasdaq.com/mr4c.html (link valid as of 05/19/2002). The following is a choice of the short interest of the first five as well as some prominent stocks of their April 2002 listing:

5

security name 02Micro International Limited 1-800 Contacts, Inc. 1-800 FLOWERS.COM, Inc. 1-800-ATTORNEY, Inc. 1st Constitution Bancorp (NJ) Intel Corporation Juniper Networks, Inc. Yahoo! Inc. priceline.com Incorporated

security symbol OIIM

current shares short 2846753

prev. month shares short 713442

change in shares short 2133311

% change in shares short 299

average daily volume 797977

CTAC FLWS

3047592 466209

2571988 532703

475604 −66494

18 −12

190341 57857

ATTY

733

576

157

27

8646

FCCY

132

0

132

0

1099

INTC JNPR YHOO PCLN

74613777 25482068 27236052 3748882

80023642 29421056 27338277 4417898

−5409865 −3938988 −102225 −669016

−7 −13 0 −15

42550432 18507315 9687090 1842726

In general, stocks that lend themselves to some speculation and stocks around corporate events (mergers and acquisition, dividend dates, etc.) with uncertain outcomes will have a particularly high short interest. It is theoretically possible to have short interest of more than 100%, because some market participants (e.g., market makers) have the ability to short sell a stock without having a locate, i.e., having someone who actually owns the stock and has agreed to lend it. Question 1.11. We are interested in borrowing the asset “money.” Therefore, we go to an owner (or, if you prefer, to, a collector) of the asset, called Bank. The Bank provides the $100 of the asset money in digital form by increasing our bank account. We sell the digital money by going to the ATM and withdrawing cash. After 90 days, we buy back the digital money for $102, by depositing cash into our bank account. The lender is repaid, and we have covered our short position. Question 1.12. We are interested in borrowing the asset “money” to buy a house. Therefore, we go to an owner of the asset, called Bank. The Bank provides the dollar amount, say $250,000, in digital form in our mortgage account. As $250,000 is a large amount of money, the bank is subject to substantial credit risk (e.g., we may lose our job) and demands a collateral. Although the money itself is not subject to large variations in price (besides inflation risk, it is difficult to imagine a reason for money to vary in value), the Bank knows that we want to buy a house, and real estate prices vary substantially. Therefore, the Bank wants more collateral than the $250,000 they are lending. In fact, as the Bank is only lending up to 80% of the value of the house, we could get a mortgage of $250,000 for a house that is worth $250,000 ÷ 0.8 = $312,500. We see that the bank factored in a haircut of $312,500 − $250,000 = $62,500 to protect itself from credit risk and adverse fluctuations in property prices. We buy back the asset money over a long horizon of time by reducing our mortgage through annuity payments. 6

Chapter 2 An Introduction to Forwards and Options Question 2.1. The payoff diagram of the stock is just a graph of the stock price as a function of the stock price:

In order to obtain the profit diagram at expiration, we have to finance the initial investment. We do so by selling a bond for $50. After one year we have to pay back: $50 × (1 + 0.1) = $55. The second figure shows the graph of the stock, of the bond to be repaid, and of the sum of the two positions, which is the profit graph. The arrows show that at a stock price of $55, the profit at expiration is indeed zero.

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Part 1 Insurance, Hedging, and Simple Strategies

Question 2.2. Since we sold the stock initially, our payoff at expiration from being short the stock is negative.

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Chapter 2 An Introduction to Forwards and Options

In order to obtain the profit diagram at expiration, we have to lend out the money we received from the short sale of the stock. We do so by buying a bond for $50. After one year we receive from the investment in the bond: $50 × (1 + 0.1) = $55. The second figure shows the graph of the sold stock, of the money we receive from the investment in the bond, and of the sum of the two positions, which is the profit graph. The arrows show that at a stock price of $55, the profit at expiration is indeed zero.

Question 2.3. The position that is the opposite of a purchased call is a written call. A seller of a call option is said to be the option writer, or to have a short position. The call option writer is the counterparty to the option buyer, and his payoffs and profits are just the opposite of those of the call option buyer. Similarly, the position that is the opposite of a purchased put option is a written put option. Again, the payoff and profit for a written put are just the opposite of those of the purchased put. It is important to note that the opposite of a purchased call is NOT the purchased put. If you do not see why, please draw a payoff diagram with a purchased call and a purchased put.

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Part 1 Insurance, Hedging, and Simple Strategies

Question 2.4. a)

The payoff to a long forward at expiration is equal to: Payoff to long forward = Spot price at expiration – forward price

Therefore, we can construct the following table: Price of asset in 6 months Agreed forward price 40 50 45 50 50 50 55 50 60 50 b)

Payoff to the long forward −10 −5 0 5 10

The payoff to a purchased call option at expiration is: Payoff to call option = max[0, spot price at expiration – strike price]

The strike is given: It is $50. Therefore, we can construct the following table: Price of asset in 6 months 40 45 50 55 60

Strike price 50 50 50 50 50

Payoff to the call option 0 0 0 5 10

c) If we compare the two contracts, we immediately see that the call option has a protection for adverse movements in the price of the asset: If the spot price is below $50, the buyer of the call option can walk away, and need not incur a loss. The buyer of the long forward incurs a loss, while he has the same payoff as the buyer of the call option if the spot price is above $50. Therefore, the call option should be more expensive. It is this attractive option to walk away that we have to pay for. Question 2.5. a)

The payoff to a short forward at expiration is equal to: Payoff to short forward = forward price – spot price at expiration

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Chapter 2 An Introduction to Forwards and Options

Therefore, we can construct the following table: Price of asset in 6 months Agreed forward price 40 50 45 50 50 50 55 50 60 50 b)

Payoff to the short forward 10 5 0 −5 −10

The payoff to a purchased put option at expiration is: Payoff to put option = max[0, strike price – spot price at expiration]

The strike is given: It is $50. Therefore, we can construct the following table: Price of asset in 6 months 40 45 50 55 60

Strike price 50 50 50 50 50

Payoff to the call option 10 5 0 0 0

c) The same logic as in question 2.4.c) applies. If we compare the two contracts, we see that the put option has a protection for increases in the price of the asset: If the spot price is above $50, the buyer of the put option can walk away, and need not incur a loss. The buyer of the short forward incurs a loss and must meet her obligations. However, she has the same payoff as the buyer of the put option if the spot price is below $50. Therefore, the put option should be more expensive. It is this attractive option to walk away if things are not as we want that we have to pay for. Question 2.6. We need to solve the following equation to determine the effective annual interest rate: $91 × (1 + r) = $100. We obtain r = 0.0989, which means that the effective annual interest rate is approximately 9.9%. Remember that when we drew profit diagrams for the forward or call option, we drew the payoff on the vertical axis, and the index price at the expiration of the contract on the horizontal axis. In this case, the particularity is that the default-free zero-coupon bond will pay exactly $100, no matter what the stock price is. Therefore, the payoff diagram is just a horizontal line, intersecting the y-axis at $100. The textbook provides the answer to the question concerning the profit diagram in the section “Zero-Coupon Bonds in Payoff and Profit Diagrams.” When we were calculating profits, we saw that we had to find the future value of the initial investment. In this case, our initial investment 11

Part 1 Insurance, Hedging, and Simple Strategies

is $91. How do we find the future value? We use the current risk-free interest rate and multiply the initial investment by it. However, as our bond is default-free, and does not bear coupons, the effective annual interest rate is exactly the 9.9% we have calculated before. Therefore, the future value of $91 is $91 × (1 + 0.0989) = $100, and our profit in six months is zero! Question 2.7. a) It does not cost anything to enter into a forward contract—we do not pay a premium. Therefore, the payoff diagram of a forward contract coincides with the profit diagram. The graphs have the following shape:

b) We have seen in question 2.1. that in order to obtain the profit diagram at expiration of a purchase of XYZ stock, we have to finance the initial investment. We did so by selling a bond for $50. After one year we had to pay back: $50 × (1 + 0.1) = $55. Therefore, our total profit at expiration from the purchase of a stock that was financed by a loan was: $ST − $55, where ST is the value of one share of XYZ at expiration. But this profit from buying the stock and financing it is the same as the profit from our long forward contract and both positions do not require any initial cash—but then, there is no advantage in investing in either instrument. c) The dividend is only paid to the owner of the stock. The owner of the long forward contract is not entitled to receive the dividend, because she only has a claim to buy the stock in the future for a given price, but she does not own it yet. Therefore, it does matter now whether we own the stock or the long forward contract. Because everything else is the same as in part a) and b), it is now beneficial to own the share: We can receive an additional payment in the form of the dividend 12

Chapter 2 An Introduction to Forwards and Options

if we own the stock at the ex-dividend date. This question hints at the very important fact that we have to be careful to take into account all the benefits and costs of an asset when we try to compare prices. We will encounter similar problems in later chapters. Question 2.8. We saw in question 2.7. b) that there is no advantage in buying either the stock or the forward contract if we can borrow to buy a stock today (so both strategies do not require any initial cash) and if the profit from this strategy is the same as the profit of a long forward contract. The profit of a long forward contract with a price for delivery of $53 is equal to: $ST − $53, where ST is the (unknown) value of one share of XYZ at expiration of the forward contract in one year. If we borrow $50 today to buy one share of XYZ stock (that costs $50), we have to repay in one year: $50 × (1 + r). Our total profit in one year from borrowing to buy one share of XYZ is therefore: $ST − $50 × (1 + r). Now we can equate the two profit equations and solve for the interest rate r: $ST − $53 ⇔ $53 $53 −1 ⇔ $50 ⇔ r

= $ST − $50 × (1 + r) = $50 × (1 + r) = r = 0.06

Therefore, the 1-year effective interest rate that is consistent with no advantage to either buying the stock or forward contract is 6 percent. Question 2.9. a) If the forward price is $1,100, then the buyer of the one-year forward contract receives at expiration after one year a profit of: $ST − $1,100, where ST is the (unknown) value of the S&R index at expiration of the forward contract in one year. Remember that it costs nothing to enter the forward contract. Let us again follow our strategy of borrowing money to finance the purchase of the index today, so that we do not need any initial cash. If we borrow $1,000 today to buy the S&R index (that costs $1,000), we have to repay in one year: $1,000 × (1 + 0.10) = $1,100. Our total profit in one year from borrowing to buy the S&R index is therefore: $ST − $1,100. The profits from the two strategies are identical. b) The forward price of $1,200 is worse for us if we want to buy a forward contract. To understand this, suppose the index after one year is $1,150. While we have already made money in part a) with a forward price of $1,100, we are still losing $50 with the new price of $1,200. As there was no advantage in buying either stock or forward at a price of $1,100, we now need to be “bribed” to enter into the forward contract. We somehow need to find an equation that makes the two strategies comparable again. Suppose that we lend some money initially together with entering

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Part 1 Insurance, Hedging, and Simple Strategies

into the forward contract so that we will receive $100 after one year. Then, the payoff from our modified forward strategy is: $ST − $1,200 + $100 = $ST − $1,100, which equals the payoff of the “borrow to buy index” strategy. We have found the future value of the premium somebody needs us to pay. We still need to find out what the premium we will receive in one year is worth today. We need to discount it: $100/ (1 + 0.10) = $90.91. c) Similarly, the forward price of $1,000 is advantageous for us. As there was no advantage in buying either stock or forward at a price of $1,100, we now need to “bribe” someone to sell this advantageous forward contract to us. We somehow need to find an equation that makes the two strategies comparable again. Suppose that we borrow some money initially together with entering into the forward contract so that we will have to pay back $100 after one year. Then, the payoff from our modified forward strategy is: $ST − $1,000 − $100 = $ST − $1,100, which equals the payoff of the “borrow to buy index” strategy. We have found the future value of the premium we need to pay. We still need to find out what this premium we have to pay in one year is worth today. We simply need to discount it: $100/(1 + 0.10) = $90.91. We should be willing to pay $90.91 to enter into the one year forward contract with a forward price of $1,000. Question 2.10. a) Figure 2.7 depicts the profit from a long call option on the S&R index with 6 months to expiration and a strike price of $1,000 if the future price of the option premium is $95.68. The profit of the long call option is: max[0, ST − $1,000] − $95.68 ⇔ max[−$95.68, ST − $1,095.68] where ST is the (unknown) value of the S&R index at expiration of the call option in six months. In order to find the S&R index price at which the call option diagram intersects the x-axis, we have to set the above equation equal to zero. We get: ST − $1,095.68 = 0 ⇔ ST = $1,095.68. This is the only solution, as the other part of the maximum function, −$95.68, is always less than zero. b) The profit of the 6 month forward contract with a forward price of $1,020 is: $ST − $1,020. In order to find the S&R index price at which the call option and the forward contract have the same profit, we need to set both parts of the maximum function of the profit of the call option equal to the profit of the forward contract and see which part permits a solution. First, we see immediately that $ST − $1,020 = $ST − $1,095.68 does not have a solution. But we can solve the other leg: $ST − $1,020 = −$95.68 ⇔ ST = $924.32, which is the value given in the exercise.

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Chapter 2 An Introduction to Forwards and Options

Question 2.11. a) Figure 2.9 depicts the profit from a long put option on the S&R index with 6 months to expiration and a strike price of $1,000 if the future value of the put premium is $75.68. The profit of the long put option is: max[0, $1, 000 − ST ] − $75.68 ⇔ max[−$75.68, $924.32 − ST ] where ST is the (unknown) value of the S&R index at expiration of the put option in six months. In order to find the S&R index price at which the put option diagram intersects the x-axis, we have to set the above equation equal to zero. We get: $924.32 − ST = 0 ⇔ ST = $924.32. This is the only solution, as the other part of the maximum function, −$75.68, is always less than zero. b) The profit of the short 6 month forward contract with a forward price of $1,020 is: $1,020 − $ST . In order to find the S&R index price at which the put option and the sold forward contract have the same profit, we need to set both parts of the maximum function of the profit of the put option equal to the profit of the forward contract and see which part permits a solution. First, we see immediately that $1,020 − $ST = $924.32 − $ST does not have a solution. But we can solve the other leg: $1,020 − ST = −$75.68 ⇔ ST = $1,095.68, which is the value given in the exercise. Question 2.12. a) Long Forward The maximum loss occurs if the stock price at expiration is zero (the stock price cannot be less than zero, because companies have limited liability). The forward then pays 0 – Forward price. The maximum gain is unlimited. The stock price at expiration could theoretically grow to infinity, there is no bound. We make a lot of money if the stock price grows to infinity (or to a very large amount). b) Short Forward The profit for a short forward contract is forward price – stock price at expiration. The maximum loss occurs if the stock price raises sharply, there is no bound to it, so it could grow to infinity. The maximum gain occurs if the stock price is zero. c) Long Call We will not exercise the call option if the stock price at expiration is less than the strike price. Consequently, the only thing we lose is the future value of the premium we paid initially to buy the option. As the stock price can grow very large (and without bound), and our payoff grows linearly in the terminal stock price once it is higher than the strike, there is no limit to our gain.

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Part 1 Insurance, Hedging, and Simple Strategies

d) Short Call We have no control over the exercise decision when we write a call. The buyer of the call option decides whether to exercise or not, and he will only exercise if he makes a profit. As we have the opposite side, we will never make any money at the expiration of the call option. Our profit is restricted to the future value of the premium, and we make this maximum profit whenever the stock price at expiration is smaller than the strike price. However, the stock price at expiration can be very large and has no bound, and as our loss grows linearly in the terminal stock price, there is no limit to our loss. e) Long Put We will not exercise the put option if the stock price at expiration is larger than the strike price. Consequently, the only thing we lose whenever the terminal stock price is larger than the strike is the future value of the premium we paid initially to buy the option. We will profit from a decline in the stock prices. However, stock prices cannot be smaller than zero, so our maximum gain is restricted to strike price less the future value of the premium and it occurs at a terminal stock price of zero. f) Short Put We have no control over the exercise decision when we write a put. The buyer of the put option decides whether to exercise or not, and he will only exercise if he makes a profit. As we have the opposite side, we will never make any money at the expiration of the put option. Our profit is restricted to the future value of the premium, and we make this maximum profit whenever the stock price at expiration is greater than the strike price. However, we lose money whenever the stock price is smaller than the strike, hence the largest loss occurs when the stock price attains its smallest possible value, zero. We lose the strike price because somebody sells us an asset for the strike that is worth nothing. We are only compensated by the future value of the premium we received. Question 2.13. In order to be able to draw profit diagrams, we need to find the future values of the call premia. They are: i)

35-strike call: $9.12 × (1 + 0.08) = $9.8496

ii)

40-strike call: $6.22 × (1 + 0.08) = $6.7176

iii)

45-strike call: $4.08 × (1 + 0.08) = $4.4064

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Chapter 2 An Introduction to Forwards and Options

We can now graph the payoff and profit diagrams for the call options. The payoff diagram looks as follows:

We get the profit diagram by deducting the option premia from the payoff graphs. The profit diagram looks as follows:

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Part 1 Insurance, Hedging, and Simple Strategies

b) Intuitively, whenever the 45-strike option pays off (i.e., has a payoff bigger than zero), the 40-strike and the 35-strike options pay off. However, there are some instances in which the 40-strike option pays off and the 45-strike options does not. Similarly, there are some instances in which the 35-strike option pays off, and neither the 40-strike nor the 45-strike pay off. Therefore, the 35-strike offers more potential than the 40- and 45-strike, and the 40-strike offers more potential than the 45-strike. We pay for these additional payoff possibilities by initially paying a higher premium. Question 2.14. In order to be able to draw profit diagrams, we need to find the future values of the put premia. They are: a)

35-strike put: $1.53 × (1 + 0.08) = $1.6524

b)

40-strike put: $3.26 × (1 + 0.08) = $3.5208

c)

45-strike put: $5.75 × (1 + 0.08) = $6.21

We get the following payoff diagrams:

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Chapter 2 An Introduction to Forwards and Options

We get the profit diagram by deducting the option premia from the payoff graphs. The profit diagram looks as follows:

Intuitively, whenever the 35-strike put option pays off (i.e., has a payoff bigger than zero), the 40-strike and the 35-strike options also pay off. However, there are some instances in which the 40-strike option pays off and the 35-strike options does not. Similarly, there are some instances in which the 45-strike option pays off, and neither the 40-strike nor the 35-strike pay off. Therefore, the 45-strike offers more potential than the 40- and 35-strike, and the 40-strike offers more potential than the 35-strike. We pay for these additional payoff possibilities by initially paying a higher premium. It makes sense that the premium is increasing in the strike price. Question 2.15. The nice thing that lead us to the notion of indifference between a forward contract and a loanfinanced stock index purchase whenever the forward price equaled the future price of the loan was that we could already tell today what we had to pay back in the future. In other words, the return on the loan, the risk-free interest rate r, was known today, and we removed uncertainty about the payment to be made. If we were to finance the purchase of the index by short selling IBM stock, we would introduce additional uncertainty, because the future value of the IBM stock is unknown. Therefore, we could not calculate today the amount to be repaid, and it would be impossible to establish an equivalence between the forward and loan-financed index purchase today. The calculation of a profit diagram would only be possible if we assumed an arbitrary value for IBM at expiration of the futures, and we would have to draw many profit diagrams with different values for IBM to get an idea of the many possible profits we could make.

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Part 1 Insurance, Hedging, and Simple Strategies

Question 2.16. The following is a copy of an Excel spreadsheet that solves the problem:

20

Chapter 3 Insurance, Collars, and Other Strategies Question 3.1. This question is a direct application of the Put-Call-Parity (equation (3.1)) of the textbook. Mimicking Table 3.1., we have: S&R Index 900.00 950.00 1000.00 1050.00 1100.00 1150.00 1200.00

S&R Put 100.00 50.00 0.00 0.00 0.00 0.00 0.00

Loan Payoff −1000.00 0.00 −1000.00 0.00 −1000.00 0.00 −1000.00 50.00 −1000.00 100.00 −1000.00 150.00 −1000.00 200.00

−(Cost + Interest) −95.68 −95.68 −95.68 −95.68 −95.68 −95.68 −95.68

Profit −95.68 −95.68 −95.68 −45.68 4.32 54.32 104.32

The payoff diagram looks as follows:

We can see from the table and from the payoff diagram that we have in fact reproduced a call with the instruments given in the exercise. The profit diagram on the next page confirms this hypothesis. 21

Part 1 Insurance, Hedging, and Simple Strategies

Question 3.2. This question constructs a position that is the opposite to the position of Table 3.1. Therefore, we should get the exact opposite results from Table 3.1. and the associated figures. Mimicking Table 3.1., we indeed have: S&R Index −900.00 −950.00 −1000.00 −1050.00 −1100.00 −1150.00 −1200.00

S&R Put −100.00 −50.00 0.00 0.00 0.00 0.00 0.00

Payoff −(Cost + Interest) −1000.00 1095.68 −1000.00 1095.68 −1000.00 1095.68 −1050.00 1095.68 −1100.00 1095.68 −1150.00 1095.68 −1200.00 1095.68

Profit 95.68 95.68 95.68 45.68 −4.32 −54.32 −104.32

On the next page, we see the corresponding payoff and profit diagrams. Please note that they match the combined payoff and profit diagrams of Figure 3.5. Only the axes have different scales.

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Chapter 3 Insurance, Collars, and Other Strategies

Payoff-diagram:

Profit diagram:

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Part 1 Insurance, Hedging, and Simple Strategies

Question 3.3. In order to be able to draw profit diagrams, we need to find the future value of the put premium, the call premium and the investment in zero-coupon bonds. We have for: the put premium: $51.777 × (1 + 0.02) = $52.81, the call premium: $120.405 × (1 + 0.02) = $122.81 and the zero-coupon bond: $931.37 × (1 + 0.02) = $950.00 Now, we can construct the payoff and profit diagrams of the aggregate position: Payoff diagram:

From this figure, we can already see that the combination of a long put and the long index looks exactly like a certain payoff of $950, plus a call with a strike price of 950. But this is the alternative given to us in the question. We have thus confirmed the equivalence of the two combined positions for the payoff diagrams. The profit diagrams on the next page confirm the equivalence of the two positions (which is again an application of the Put-Call-Parity).

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Chapter 3 Insurance, Collars, and Other Strategies

Profit Diagram for a long 950-strike put and a long index combined:

Question 3.4. This question is another application of Put-Call-Parity. Initially, we have the following cost to enter into the combined position: We receive $1,000 from the short
sale of the index, and  we have to pay the call premium. Therefore, the future value of our cost is: $120.405 − $1,000 × (1 + 0.02) = −$897.19. Note that a negative cost means that we initially have an inflow of money. Now, we can directly proceed to draw the payoff diagram:

25

Part 1 Insurance, Hedging, and Simple Strategies

We can clearly see from the figure that the payoff graph of the short index and the long call looks like a fixed obligation of $950, which is alleviated by a long put position with a strike price of 950. The following profit diagram, including the cost for the combined position, confirms this:

Question 3.5. This question is another application of Put-Call-Parity. Initially, we have the following cost to enter into the combined position: We receive $1,000 from the short
sale of the index, and  we have to pay the call premium. Therefore, the future value of our cost is: $71.802 − $1,000 × (1 + 0.02) = −$946.76. Note that a negative cost means that we initially have an inflow of money.

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Chapter 3 Insurance, Collars, and Other Strategies

Now, we can directly proceed to draw the payoff diagram:

In order to be able to compare this position to the other suggested position of the exercise, we need to find the future value of the borrowed $1,029.41. We have: $1,029.41 × (1 + 0.02) = $1,050. We can now see from the figure that the payoff graph of the short index and the long call looks like a fixed obligation of $1,050, which is exactly the future value of the borrowed amount, and a long put position with a strike price of 1,050. The following profit diagram, including the cost for the combined position we calculated above, confirms this. The profit diagram is the same as the profit diagram for a loan and a long 1,050-strike put with an initial premium of $101.214.

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Part 1 Insurance, Hedging, and Simple Strategies

Profit Diagram of going short the index and buying a 1,050-strike call:

Question 3.6. We now move from a graphical representation and verification of the Put-Call-Parity to a mathematical representation. Let us first consider the payoff of (a). If we buy the index (let us name it S), we receive at the time of expiration T of the options simply ST . The payoffs of part (b) are a little bit more complicated. If we deal with options and the maximum function, it is convenient to split the future values of the index into two regions: one where ST < K and another one where ST ≥ K. We then look at each region separately, and hope to be able to draw a conclusion when we look at the aggregate position. We have for the payoffs in (b): Instrument ST < K = 950 ST ≥ K = 950 Get repayment of loan $931.37 × 1.02 = $950 $931.37 × 1.02 = $950 Long Call Option max (ST
− 950, 0) = 0 ST − 950 0 Short Put Option − max $950 − ST , 0 = ST − $950 Total ST ST

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Chapter 3 Insurance, Collars, and Other Strategies

We now see that the total aggregate position only gives us ST , no matter what the final index value is—but this is the same payoff as in part (a). Our proof for the payoff equivalence is complete. Now let us turn to the profits. If we buy the index today, we need to finance it. Therefore, we borrow $1,000, and have to repay $1,020 after one year. The profit for part (a) is thus: ST − $1,020. The profits of the aggregate position in part (b) are the payoffs, less the future value of the call premium plus the future value of the put premium (because we have sold the put), and less the future value of the loan we gave initially. Note that we already know that a risk-less bond is canceling out of the profit calculations. We can again tabulate:

Instrument Get repayment of loan Future value of given loan Long Call Option Future value call premium Short Put Option Future value put premium Total

ST < K $931.37 × 1.02 = $950 −$950 max (ST − 950, 0) = 0 −$120.405 × 1.02 =−$122.81
− max $950 − ST , 0 = ST − $950 $51.777 × 1.02 = $52.81 ST − 1020

ST ≥ K $931.37 × 1.02 = $950 −$950 ST − 950 −$120.405 × 1.02 = −$122.81 0 $51.777 × 1.02 = $52.81 ST − 1020

Indeed, we see that the profits for part (a) and part (b) are identical as well. Question 3.7. Let us first consider the payoff of (a). If we short the index (let us name it S), we have to pay at the time of expiration T of the options: −ST . The payoffs of part (b) are more complicated. Let us look again at each region separately, and hope to be able to draw a conclusion when we look at the aggregate position. We have for the payoffs in (b): Instrument Make repayment of loan Short Call Option Long Put Option Total

ST < K ST ≥ K −$1029.41 × 1.02 = −$1050 −$1029.41 × 1.02 = −$1050 − max (ST − 1050, 0) = 0 − max (ST − 1050, 0) = 1050 − ST 
0 max $1050 − ST , 0 = $1050 − ST −ST −ST

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Part 1 Insurance, Hedging, and Simple Strategies

We see that the total aggregate position gives us −ST , no matter what the final index value is—but this is the same payoff as part (a). Our proof for the payoff equivalence is complete. Now let us turn to the profits. If we sell the index today, we receive money that we can lend out. Therefore, we can lend $1,000, and be repaid $1,020 after one year. The profit for part (a) is thus: $1,020 − ST . The profits of the aggregate position in part (b) are the payoffs, less the future value of the put premium plus the future value of the call premium (because we sold the call), and less the future value of the loan we gave initially. Note that we know already that a risk-less bond is canceling out of the profit calculations. We can again tabulate: Instrument Make repayment of loan Future value of borrowed money Short Call Option

ST < K −$1029.41 × 1.02 = −$1050 $1050

ST ≥ K −$1029.41 × 1.02 = −$1050 $1050

− max (ST − 1050, 0) = 0

Future value of premium Long Put Option

− max (ST − 1050, 0) = 1050 − ST $71.802 × 1.02 = $73.24 0

$71.802 
× 1.02 = $73.24 max $1050 − ST , 0 = $1050 − ST −$101.214 × 1.02 = −$103.24 −$101.214 × 1.02 = −$103.24 $1,020 − ST $1,020 − ST

Future value of premium Total

Indeed, we see that the profits for part (a) and part (b) are identical as well. Question 3.8. This question is a direct application of the Put-Call-Parity. We will use equation (3.1) in the following, and input the given variables:
 Call (K, t) − P ut (K, t) = P V F0,t − K
 ⇔ Call (K, t) − P ut (K, t) − P V F0,t = −P V (K) ⇔ Call (K, t) − P ut (K, t) − S0 = −P V (K) ⇔ $109.20 − $60.18 − $1,000 = − ⇔ K = $970.00

K 1.02

Question 3.9. The strategy of buying a call (or put) and selling a call (or put) at a higher strike is called call (put) bull spread. In order to draw the profit diagrams, we need to find the future value of the cost of entering in the bull spread positions. We have: 30

Chapter 3 Insurance, Collars, and Other Strategies


 Cost of call bull spread:
$120.405 − $93.809  × 1.02 = $27.13 Cost of put bull spread: $51.777 − $74.201 × 1.02 = −$22.87 The payoff diagram shows that the payoffs to the put bull spread are uniformly less than the payoffs to the call bull spread. There is a difference, because the put bull spread has a negative initial cost, i.e., we are receiving money if we enter into it. The difference is exactly $50, the value of the difference between the two strike prices. It is equivalent as well to the value of the difference of the future values of the initial premia. Yet, the higher initial cost for the call bull spread is exactly offset by the higher payoff so that the profits of both strategies are the same. It is easy to show this with equation (3.1), the put-call-parity. Payoff-Diagram:

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Part 1 Insurance, Hedging, and Simple Strategies

Profit diagram:

Question 3.10. The strategy of selling a call (or put) and buying a call (or put) at a higher strike is called call (put) bear spread. In order to draw the profit diagrams, we need to find the future value of the cost of entering in the bull spread positions. We have:
 Cost of call bear spread:
$71.802 − $120.405 × 1.02 = −$49.575 Cost of put bear spread: $101.214 − $51.777 × 1.02 = $50.426 The payoff diagram shows that the payoff to the call bear spread is uniformly less than the payoffs to the put bear spread. The difference is exactly $100, equal to the difference in strikes and as well equal to the difference in the future value of the costs of the spreads. There is a difference, because the call bear spread has a negative initial cost, i.e., we are receiving money if we enter into it. The higher initial cost for the put bear spread is exactly offset by the higher payoff so that the profits of both strategies turn out to be the same. It is easy to show this with equation (3.1), the put-call-parity.

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Chapter 3 Insurance, Collars, and Other Strategies

Payoff-Diagram:

Profit Diagram:

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Part 1 Insurance, Hedging, and Simple Strategies

Question 3.11. In order to be able to draw the profit diagram, we need to find the future value of the costs of establishing the suggested position. We need to finance the index purchase, buy the 950-strike put of the sold call. Therefore, the future value of our cost is:
and we receive the premium  $1,000 − $71.802 + $51.777 × 1.02 = $999.57. Now we can draw the profit diagram:

The net option premium cost today is: −$71.802 + $51.777 = −$20.025. We receive about $20 if we enter into this collar. If we want to construct a zero-cost collar and keep the 950-strike put, we would need to increase the strike price of the call. By increasing the strike price of the call, the buyer of the call must wait for larger increases in the underlying index before the option pays off. This makes the call option less attractive, and the buyer of the option is only willing to pay a smaller premium. We receive less money, thus pushing the net option premium towards zero. Question 3.12. Our initial cash required to put on the collar, i.e. the net option premium, is as follows: −$51.873 + $51.777 = −$0.096. Therefore, we receive only 10 cents if we enter into this collar. The position is very close to a zero-cost collar.

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Chapter 3 Insurance, Collars, and Other Strategies

The profit diagram looks as follows:

If we compare this profit diagram with the profit diagram of the previous exercise (3.11.), we see that we traded in the additional call premium (that limited our losses after index decreases) against more participation on the upside. Please note that both maximum loss and gain are higher than in question 3.11. Question 3.13. The following figure depicts the requested profit diagrams. We can see that the aggregation of the bought and sold straddle resembles a bear spread. It is bearish, because we sold the straddle with the smaller strike price.

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Part 1 Insurance, Hedging, and Simple Strategies

a)

b)

c)

Question 3.14. a) This question deals with the option trading strategy known as Box spread. We saw earlier that if we deal with options and the maximum function, it is convenient to split the future values of the index into different regions. Let us name the final value of the S&R index ST . We have two strike prices, therefore we will use three regions: One in which ST < 950, one in which 950 ≤ ST < 1,000 and another one in which ST ≥ 1,000. We then look at each region separately, and hope to be able to see that indeed when we aggregate, there is no S&R risk when we look at the aggregate position. Instrument long 950 call short 1000 call short 950 put long 1000 put Total

ST < 950 0 0 ST − $950 $1,000 − ST $50

950 ≤ ST < 1,000 ST ≥ 1,000 ST − $950 ST − $950 0 $1, 000 − ST 0 0 0 $1,000 − ST $50 $50

We see that there is no occurrence of the final index value in the row labeled total. The option position does not contain S&R price risk. 36

Chapter 3 Insurance, Collars, and Other Strategies

b) The initial cost is the sum of the long option premia less the premia we receive for the sold options. We have: Cost $120.405 − $93.809 − $51.77 + $74.201 = $49.027 c)

The payoff of the position after 6 months is $50, as we can see from the above table.

d) The implicit interest rate of the cash flows is: $50.00 ÷ $49.027 = 1.019 ∼ = 1.02. The implicit interest rate is indeed 2 percent. Question 3.15. a) Profit diagram of the 1:2
950-, 1050-strike ratio call  spread (the future value of the initial cost of which is calculated as: $120.405 − 2 × $71.802 × 1.02 = −$23.66):

b) Profit diagram of the 2:3 (the future value of the initial
950-, 1050-strike ratio call spread  cost of which is calculated as: 2 × $120.405 − 3 × $71.802 × 1.02 = $25.91.

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Part 1 Insurance, Hedging, and Simple Strategies

c) We saw that in part a), we were receiving money from our position, and in part b), we had to pay a net option premium to establish the position. This suggests that the true ratio n/m lies between 1:2 and 2:3. Indeed, we can calculate the ratio n/m as: n × $120.405 − m × $71.802 ⇔ n × $120.405 ⇔ n/m ⇔ n/m

= = = =

0 m × $71.802 $71.802/$120.405 0.596

which is approximately 3:5. Question 3.16. A bull spread or a bear spread can never have an initial premium of zero, because we are buying the same number of calls (or puts) that we are selling and the two legs of the bull and bear spreads have different strikes. A zero initial premium would mean that two calls (or puts) with different strikes have the same price—and we know by now that two instruments that have different payoff structures and the same underlying risk cannot have the same price without creating an arbitrage opportunity. A symmetric butterfly spread cannot have a premium of zero because it would violate the convexity condition of options. 38

Chapter 3 Insurance, Collars, and Other Strategies

Question 3.17. From the textbook we learn how to calculate the right ratio λ. It is equal to: λ=

K3 − K2 1050 − 1020 = 0.3 = K3 − K1 1050 − 950

In order to construct the asymmetric butterfly, for every 1020-strike call we write, we buy λ 950strike calls and 1 − λ 1050-strike calls. Since we can only buy whole units of calls, we will in this example buy three 950-strike and seven 1050-strike calls, and sell ten 1020-strike calls. The profit diagram looks as follows:

Question 3.18. The following three figures show the individual legs of each of the three suggested strategies. The last subplot shows the aggregate position. It is evident from the figures that you can indeed use all the suggested strategies to construct the same butterfly spread. Another method to show the claim of 3.18. mathematically would be to establish the equivalence by using the Put-Call-Parity on b) and c) and showing that you can write it in terms of the instruments of a).

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Part 1 Insurance, Hedging, and Simple Strategies

profit diagram part a)

profit diagram part b)

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Chapter 3 Insurance, Collars, and Other Strategies

profit diagram part c)

Question 3.19. a) We know from the Put-Call-Parity that if we buy a call and sell a put that are at the money (i.e., S(0) = K), then the call option is slightly more expensive than the put option, the difference being the value of the stock minus the present value of the strike. Therefore, we can tell that the strike price must be a bit higher than the current stock price, and more precisely, it should be equal to the forward price. b) We sold a collar with no difference in strike prices. The profit diagram will be a straight line, which means that we effectively created a long forward contract. c) Remember that you are buying at the ask and selling at the bid, and that the bid price is always smaller than the ask. Suppose we had established a zero-cost synthetic at the forward price, and now we introduce the bid-ask spread. This means that we have to pay a little more for the call, and receive a little less for the put. We are paying money for the position, and in order to correct it, we must make the put a bit more attractive, and the call less attractive. We do so by shifting the strike price to the right of the forward: Now the buyer of the call must wait a little bit longer before his call pays off, and he is only willing to buy it for less. As the opposite is true for the put, we have established that the strike must be to the right of the forward.

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Part 1 Insurance, Hedging, and Simple Strategies

d) If we are creating a synthetic short stock, we buy the put option and sell the call option. We are buying at the ask and selling at the bid, and the bid price is always smaller than the ask. Suppose we had established a zero-cost synthetic short at the forward price, and now we introduce the bid-ask spread. This means that we have to pay a little more for the put, and receive a little less for the call. We are paying money for the position, and in order to correct it, we must make the call a bit more attractive. The call gets more attractive if the strike price decreases, because the final payoff is max(S − K, 0). Therefore, we have to shift the strike price to the left of the forward price. e) No, transaction fees are not a wash, because we are paying implicitly the bid-ask spread: If we bought a stock today and held it until the expiration of the options, we would get the future stock price less the forward price (which is equivalent to the loan we got to finance the stock purchase). Now, we established in c) that the strike price is to the right of the forward price. Therefore, we will receive from the collar of part c) the stock price less something that is larger than the forward price: We make a loss compared to the self-financed outright purchase of the stock. These considerations do not yet take into account that we incur transaction costs on two instruments, compared to only one time brokerage fees if we buy the stock directly. It is thus very important to be aware of transaction costs when comparing different investment strategies. Question 3.20. Use separate cells for the strike price and the quantities you buy and sell for each strike (i.e., make use of the plus or minus sign). Then, use the maximum function to calculate payoffs and profits. The best way to solve this problem is probably to have the calculations necessary for the payoff and profit diagrams run in the background, e.g., in another auxiliary table that you are referencing to. Define the boundaries for the calculations dynamically and symmetrically around the current stock price. Then use the diagram function with the line style to draw the diagrams.

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Chapter 4 Introduction to Risk Management Question 4.1. The following table summarizes the unhedged and hedged profit calculations: Copper price in Total cost one year $0.70 $0.90 $0.80 $0.90 $0.90 $0.90 $1.00 $0.90 $1.10 $0.90 $1.20 $0.90

Unhedged profit −$0.20 −$0.10 0 $0.10 $0.20 $0.30

Profit on short forward $0.30 $0.20 $0.10 0 −$0.10 −$0.20

Net income on hedged profit $0.10 $0.10 $0.10 $0.10 $0.10 $0.10

We obtain the following profit diagram:

Question 4.2. If the forward price were $0.80 instead of $1, we would get the following table: 43

Part 1 Insurance, Hedging, and Simple Strategies

Copper price in Total cost one year $0.70 $0.90 $0.80 $0.90 $0.90 $0.90 $1.00 $0.90 $1.10 $0.90 $1.20 $0.90

Unhedged profit −$0.20 −$0.10 0 $0.10 $0.20 $0.30

Profit on short forward $0.10 $0 −$0.10 −$0.20 −$0.30 −$0.40

Net income on hedged profit −$0.10 −$0.10 −$0.10 −$0.10 −$0.10 −$0.10

Profit on short forward −$0.25 −$0.35 −$0.45 −$0.55 −$0.65 −$0.75

Net income on hedged profit −$0.45 −$0.45 −$0.45 −$0.45 −$0.45 −$0.45

With a forward price of $0.45, we have: Copper price in Total cost one year $0.70 $0.90 $0.80 $0.90 $0.90 $0.90 $1.00 $0.90 $1.10 $0.90 $1.20 $0.90

Unhedged profit −$0.20 −$0.10 0 $0.10 $0.20 $0.30

Although the copper forward price of $0.45 is below our total costs of $0.90, it is higher than the variable cost of $0.40. It still makes sense to produce copper, because even at a price of $0.45 in one year, we will be able to partially cover our fixed costs. Question 4.3. Please note that we have given the continuously compounded rate of interest as 6%. Therefore, the effective annual interest rate is exp(0.06) − 1 = 0.062. In this exercise, we need to find the future value of the put premia. For the $1-strike put, it is: $0.0376 × 1.062 = $0.04. The following table shows the profit calculations for the $1.00-strike put. The calculations for the two other puts are exactly similar. The figure on the next page compares the profit diagrams of all three possible hedging strategies. Copper price in Total cost one year $0.70 $0.80 $0.90 $1.00 $1.10 $1.20

$0.90 $0.90 $0.90 $0.90 $0.90 $0.90

Unhedged profit −$0.20 −$0.10 0 $0.10 $0.20 $0.30

Profit on long $1.00-strike put option $0.30 $0.20 $0.10 0 0 0 44

Put Net income on premium hedged profit $0.04 $0.04 $0.04 $0.04 $0.04 $0.04

$0.06 $0.06 $0.06 $0.06 $0.16 $0.26

Chapter 4 Introduction to Risk Management

Profit diagram of the different put strategies:

Question 4.4. We will explicitly calculate the profit for the $1.00-strike and show figures for all three strikes. The future value of the $1.00-strike call premium amounts to: $0.0376 × 1.062 = $0.04. Copper price in Total cost one year $0.70 $0.80 $0.90 $1.00 $1.10 $1.20

$0.90 $0.90 $0.90 $0.90 $0.90 $0.90

Unhedged profit −$0.20 −$0.10 0 $0.10 $0.20 $0.30

Profit on short $1.00-strike call option 0 0 0 0 −$0.10 −$0.20

45

Call premium received $0.04 $0.04 $0.04 $0.04 $0.04 $0.04

Net income on hedged profit −$0.16 −$0.06 $0.04 $0.14 $0.14 $0.14

Part 1 Insurance, Hedging, and Simple Strategies

We obtain the following payoff graphs:

Question 4.5. XYZ will buy collars, which means that they buy the put leg and sell the call leg. We have to compute for each case the net option premium position, and find its future value. We have for
 a) $0.0178 − $0.0376 × 1.062 = −$0.021 b) c)




 $0.0265 − $0.0274 × 1.062 = −$0.001




 $0.0665 − $0.0194 × 1.062 = $0.050

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Chapter 4 Introduction to Risk Management

a) Copper price in Total cost one year $0.70 $0.90 $0.80 $0.90 $0.90 $0.90 $1.00 $0.90 $1.10 $0.90 $1.20 $0.90

Profit on .95 put $0.25 $0.15 $0.05 $0 0 0

Profit on short $1.00 call 0 0 0 0 −$0.10 −$0.20

Net Hedged profit premium −$0.021 $0.0710 −$0.021 $0.0710 −$0.021 $0.0710 −$0.021 $0.1210 −$0.021 $0.1210 −$0.021 $0.1210

Profit on .975 put $0.275 $0.175 $0.075 $0 0 0

Profit on short $1.025 call 0 0 0 0 −$0.0750 −$0.1750

Net Hedged profit premium −$0.001 $0.0760 −$0.001 $0.0760 −$0.001 $0.0760 −$0.001 $0.1010 −$0.001 $0.1260 −$0.001 $0.1260

Profit diagram:

b) Copper price in Total cost one year $0.70 $0.90 $0.80 $0.90 $0.90 $0.90 $1.00 $0.90 $1.10 $0.90 $1.20 $0.90

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Part 1 Insurance, Hedging, and Simple Strategies

Profit diagram:

c) Copper price in Total cost one year $0.70 $0.90 $0.80 $0.90 $0.90 $0.90 $1.00 $0.90 $1.10 $0.90 $1.20 $0.90

Profit on 1.05 put $0.35 $0.25 $0.15 $0.05 0 0

Profit on short $1.05 call 0 0 0 0 −$0.050 −$0.150

Net Hedged profit premium $0.05 $0.1 $0.05 $0.1 $0.05 $0.1 $0.05 $0.1 $0.05 $0.1 $0.05 $0.1

We see that we are completely and perfectly hedged. Buying a collar where the put and call leg have equal strike prices perfectly offsets the copper price risk. Profit diagram:

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Chapter 4 Introduction to Risk Management

Question 4.6. a) Copper price in Total cost one year $0.70 $0.80 $0.90 $1.00 $1.10 $1.20

$0.90 $0.90 $0.90 $0.90 $0.90 $0.90

Profit on Profit on two long short 1.025 $0.975 puts put −$0.325 $0.55 −$0.225 $0.35 −$0.125 $0.150 −$0.025 0 0 0 0 0

Net Hedged profit premium $0.0022 $0.0022 $0.0022 $0.0022 $0.0022 $0.0022

$0.0228 $0.0228 $0.0228 $0.0728 $0.1978 $0.2978

We can see from the following profit diagram (and the above table) that in the case of a favorable increase in copper prices, the hedged profit is almost identical to the unhedged profit. Profit diagram:

49

Part 1 Insurance, Hedging, and Simple Strategies

b) Copper price in Total cost one year $0.70 $0.80 $0.90 $1.00 $1.10 $1.20

$0.90 $0.90 $0.90 $0.90 $0.90 $0.90

Profit on two short 1.034 put −$0.6680 −$0.4680 −$0.2680 −$0.0680 0 0

Profit on three long $1 puts $0.9 $0.6 $0.3 0 0 0

Net Hedged profit premium $0.0002 $0.0002 $0.0002 $0.0002 $0.0002 $0.0002

$0.0318 $0.0318 $0.0318 $0.0318 $0.1998 $0.2998

We can see from the following profit diagram (and the above table) that in the case of a favorable increase in copper prices, the hedged profit is almost identical to the unhedged profit. Profit diagram:

Question 4.7. Telco assigned a fixed revenue of $6.20 for each unit of wire. It can buy one unit of wire for $5 plus the price of copper. Therefore, Telco’s profit in one year is $6.20 less $5.00 less the price of copper after one year.

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Chapter 4 Introduction to Risk Management

Copper price in one Total cost year $0.70 $5.70 $0.80 $5.80 $0.90 $5.90 $1.00 $6.00 $1.10 $6.10 $1.20 $6.20

Unhedged profit $0.50 $0.40 $0.30 $0.20 $0.10 0

Profit on one long forward −$0.3 −$0.2 −$0.1 0 $0.10 $0.20

Hedged profit $0.20 $0.20 $0.20 $0.20 $0.20 $0.20

We obtain the following profit diagrams:

Question 4.8. In this exercise, we need to first find the future value of the call premia. For the $1-strike call, it is: $0.0376 × 1.062 = $0.04. The following table shows the profit calculations of the $1.00-strike call. The calculations for the two other calls are exactly similar. The figures on the next page compare the profit diagrams of all three possible hedging strategies.

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Part 1 Insurance, Hedging, and Simple Strategies

Copper price in Total cost one year $0.70 $5.70 $0.80 $5.80 $0.90 $5.90 $1.00 $6.00 $1.10 $6.10 $1.20 $6.20

Unhedged profit $0.50 $0.40 $0.30 $0.20 $0.10 0

Profit on long $1.00-strike call 0 0 0 0 $0.10 $0.20

Call premium $0.04 $0.04 $0.04 $0.04 $0.04 $0.04

Net income on hedged profit $0.46 $0.36 $0.26 $0.16 $0.16 $0.16

We obtain the following profit diagrams:

Question 4.9. For the $1-strike put, we receive a premium of: $0.0376 × 1.062 = $0.04. The following table shows the profit calculations of the $1.00-strike put. The calculations for the two other puts are exactly the same. The figures on the next page compare the profit diagrams of all three possible strategies.

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Chapter 4 Introduction to Risk Management

Copper price in Total cost one year $0.70 $5.70 $0.80 $5.80 $0.90 $5.90 $1.00 $6.00 $1.10 $6.10 $1.20 $6.20

Unhedged profit $0.50 $0.40 $0.30 $0.20 $0.10 0

Profit on short Received $1.00-strike call premium −$0.30 $0.04 −$0.20 $0.04 −$0.10 $0.04 0 $0.04 0 $0.04 0 $0.04

Net income on hedged profit $0.24 $0.24 $0.24 $0.24 $0.14 $0.04

We obtain the following profit diagrams:

Question 4.10. Telco will sell collars, which means that they buy the call leg and sell the put leg. We have to compute for each case the net option premium position, and find its future value. We have for:
 a) $0.0376 − $0.0178 × 1.062 = $0.021 b) c)


 $0.0274 − $0.0265 × 1.062 = $0.001


 $0.0649 − $0.0178 × 1.062 = $0.050 53

Part 1 Insurance, Hedging, and Simple Strategies

a) Copper price Total cost in one year $0.70 $0.80 $0.90 $1.00 $1.10 $1.20

$5.70 $5.80 $5.90 $6.00 $6.10 $6.20

Unhedged profit $0.50 $0.40 $0.30 $0.20 $0.10 0

Profit on Profit on long short .95 $1.00 call put −$0.25 0 −$0.15 0 −$0.05 0 $0 0 0 $0.10 0 $0.20

Net Hedged profit premium $0.021 $0.021 $0.021 $0.021 $0.021 $0.021

$0.2290 $0.2290 $0.2290 $0.1790 $0.1790 $0.1790

Profit diagram:

b) Copper price Total cost in one year $0.70 $5.70 $0.80 $5.80 $0.90 $5.90 $1.00 $6.00 $1.10 $6.10 $1.20 $6.20

Unhedged profit $0.50 $0.40 $0.30 $0.20 $0.10 0

Profit on short .95 put −$0.275 −$0.175 −$0.075 $0 0 0

54

Profit on long $1.025 call 0 0 0 0 $0.0750 $0.1750

Net Hedged profit premium $0.001 $0.2240 $0.001 $0.2240 $0.001 $0.2240 $0.001 $0.1990 $0.001 $0.1740 $0.001 $0.1740

Chapter 4 Introduction to Risk Management

Profit diagram:

c) Copper price Total cost in one year $0.70 $5.70 $0.80 $5.80 $0.90 $5.90 $1.00 $6.00 $1.10 $6.10 $1.20 $6.20

Unhedged profit $0.50 $0.40 $0.30 $0.20 $0.10 0

Profit on Profit on long short .95 put $.95 call −$0.25 0 −$0.15 0 −$0.05 0 0 $0.050 0 $0.150 0 $0.250

Net Hedged profit premium $0.05 $0.2 $0.05 $0.2 $0.05 $0.2 $0.05 $0.2 $0.05 $0.2 $0.05 $0.2

We see that we are completely and perfectly hedged. Buying a collar where the put and call leg have equal strike prices perfectly offsets the copper price risk. Profit diagram:

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Part 1 Insurance, Hedging, and Simple Strategies

Question 4.11. a) Copper price Total cost in one year $0.70 $0.80 $0.90 $1.00 $1.10 $1.20

$5.70 $5.80 $5.90 $6.00 $6.10 $6.20

Unhedged profit $0.50 $0.40 $0.30 $0.20 $0.10 0

Profit on Profit on two short 0.95 long $1.034 call calls 0 0 0 0 0 0 −$0.025 0 −$0.125 $0.13200 −$0.225 $0.3320

Net premium

Hedged profit

−$0.0015 −$0.0015 −$0.0015 −$0.0015 −$0.0015 −$0.0015

$0.5015 $0.4015 $0.3015 $0.1765 $0.1085 $0.1085

We can see from the following profit diagram (and the above table) that in the case of a favorable decrease in copper prices, the hedged profit is almost identical to the unhedged profit. Profit diagram:

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Chapter 4 Introduction to Risk Management

b) Copper price Total cost in one year $0.70 $0.80 $0.90 $1.00 $1.10 $1.20

$5.70 $5.80 $5.90 $6.00 $6.10 $6.20

Unhedged profit $0.50 $0.40 $0.30 $0.20 $0.10 0

Profit on 2 short $1 call 0 0 0 0 −$0.200 −$0.400

Profit on three long $1.034 calls 0 0 0 0 $0.1980 $0.4980

Net premium

Hedged profit

−$0.0024 −$0.0024 −$0.0024 −$0.0024 −$0.0024 −$0.0024

$0.5024 $0.4024 $0.3024 $0.2024 $0.1004 $0.1004

We can see from the following profit diagram (and the above table) that in the case of a favorable decrease in copper prices, the hedged profit is almost identical to the unhedged profit. Profit diagram:

Question 4.12. This is a very important exercise to really understand the benefits and pitfalls of hedging strategies. Wirco needs copper as an input, which means that its costs increase with the price of copper. We may therefore think that they need to hedge against increases in the copper price. However, we must not forget that the price of wire, the source of Wirco’s revenues, also depends positively on 57

Part 1 Insurance, Hedging, and Simple Strategies

the price of copper: the price Wirco can obtain for one unit of wire is $50 plus the price of copper. We will see that those copper price risks cancel each other out. Mathematically, Wirco’s cost per unit of wire: $3 + $1.50 + ST Wirco’s revenue per unit of wire: $5 + ST and ST is the price of copper after one year. Therefore, we can determine Wirco’s profits as: 
Profit = Revenue – Cost = $5 + ST − $3 + $1.50 + ST = $0.50 We see that the profits of Wirco do not depend on the price of copper. Cost and revenue copper price risk cancel each other out. If we buy in this situation a long forward contract, we do in fact introduce copper price risk! To understand this, add a long forward contract to the profit equation: 
Profit with forward: = $5 + ST − $3 + $1.50 + ST + ST − $1 = ST − $0.50 To summarize, Copper price Total cost Total revenue in one year $0.70 $5.20 $5.70 $0.80 $5.30 $5.80 $0.90 $5.40 $5.90 $1.00 $5.50 $6.00 $1.10 $5.60 $6.10 $1.20 $5.70 $6.20

Unhedged profit $0.50 $0.50 $0.50 $0.50 $0.50 $0.50

Profit on long forward −$0.30 −$0.20 −$0.10 0 $0.10 $0.20

Net income on ‘hedged’ profit $0.20 $0.30 $0.40 $0.50 $0.60 $0.70

Question 4.13. We do in fact introduce copper price risk no matter what strategy we undertake. Therefore, no matter which instrument we are using, we increase the price variability of Wirco’s profits. Although this is a simple example, it is important to keep in mind that a company’s risk management should always take place on an aggregate level, because otherwise counterbalancing positions may be hedged twice. Question 4.14. Hedging should never be thought of as a profit increasing action. A company that hedges merely shifts profits from good to bad states of the relevant price risk that the hedge seeks to diminish. The value of the reduced profits, should the gold price rise, subsidizes the payment to Golddiggers should the gold price fall. Therefore, a company may use a hedge for one of the reasons stated in the textbook; however, it is not correct to compare hedged and unhedged companies from an accounting perspective. 58

Chapter 4 Introduction to Risk Management

Question 4.15. If losses are tax deductible (and the company has additional income to which the tax credit can be applied), then each dollar of losses bears a tax credit of $0.40. Therefore,

(1) Pre-Tax Operating Income (2) Taxable Income (3) Tax @ 40% (3b) Tax Credit After-Tax Income (including Tax credit)

Price = $9 −$1 0 0 $0.40 −$0.60

Price = $11.20 $ 1.20 $1.20 $0.48 0 $0.72

In particular, this gives an expected after-tax profit of:

 E[Profit] = 0.5 × −$0.60 + 0.5 × $0.72 = $0.06 and the inefficiency is removed: We obtain the same payoffs as in the hedged case, Table 4.7. Question 4.16. a)

Expected pre-tax profit



 Firm A: E[Profit] = 0.5 ×
$1, 000 + 0.5 × −$600 = $200 
 Firm B: E[Profit] = 0.5 × $300 + 0.5 × $100 = $200 Both firms have the same pre-tax profit. b) Expected after tax profit. Firm A: bad state (1) Pre-Tax Operating Income −$600 (2) Taxable Income $0 (3) Tax @ 40% 0 (3b) Tax Credit $240 After-Tax Income (including Tax credit) −$360

good state $1,000 $ 1,000 $400 0 $600

This gives an expected after-tax profit for firm A of:

 E[Profit] = 0.5 × −$360 + 0.5 × $600 = $120

59

Part 1 Insurance, Hedging, and Simple Strategies

Firm B: (1) Pre-Tax Operating Income (2) Taxable Income (3) Tax @ 40% (3b) Tax Credit After-Tax Income (including Tax credit)

bad state $100 $100 $40 0 $60

good state $300 $300 $120 0 $180

This gives an expected after-tax profit for firm B of:

 E[Profit] = 0.5 × $60 + 0.5 × $180 = $120 If losses receive full credit for tax losses, the tax code does not have an effect on the expected after-tax profits of firms that have the same expected pre-tax profits, but different cash-flow variability. Question 4.17. a)

The pre-tax expected profits are the same as in exercise 4.16. a).

b) While the after-tax profits of company B stay the same, those of company A change, because they do not receive tax credit on the loss anymore. c)

We have for firm A: bad state good state (1) Pre-Tax Operating Income −$600 $1,000 (2) Taxable Income $0 $1,000 (3) Tax @ 40% 0 $400 (3b) Tax Credit no tax credit 0 After-Tax Income (including Tax credit) −$600 $600

And consequently, an expected after-tax return for firm A of:

 E[Profit] = 0.5 × −$600 + 0.5 × $600 = $0 Company B would not pay anything, because it makes always positive profits, which means that the lack of a tax credit does not affect them. Company A would be willing to pay the discounted difference between its after-tax profits calculated in 4.16. b), and its new after-tax profits, $0 from 4.17. It is thus willing to pay: $120 ÷ 1.1 = $109.09. Question 4.18. Auric Enterprises is using gold as an input. Therefore, it would like to hedge against price increases in gold. 60

Chapter 4 Introduction to Risk Management

a) The cost of this collar today is the premium of the purchased 440-strike call ($2.49) less the premium for the sold 400-strike put. We calculate a cost of $2.49 − $2.21 = −$0.28, which means that Auric in fact generates a revenue from entering into this collar.

b) A good starting point are the values of part a). You see that both put and call are worth approximately the same, therefore start shrinking the 440 – 400 span symmetrically until you get a difference of 30, and then do some trial and error. This should bring you the following values: The call strike is 435.52, and the put strike is 405.52. Both call and put have a premium of $3.425. Question 4.19. As we buy the call, we will buy it at the ask, which is $0.25 above the Black-Scholes price, and we sell the put at the bid, which is $0.25 below the Black-Scholes price. Our new equal premium condition is: C + $0.25 – (P − $0.25) = 0, or C + $0.50 – P = 0. Since we know that the value of a call is decreasing in the strike, and we need a Black-Scholes call price that is $.50 less valuable than the Black-Scholes put, we know that we have to look for a pair of higher strike prices. Trial and error brings us to a call strike of 436.53, and a put strike of 406.53. The Black-Scholes call premium is $3.1938, and the put has a premium of $3.6938. Question 4.20. a) Since we know that the value of a call is decreasing in the strike, and we need to sell two call options, the Black-Scholes prices of which equal the 440-strike call price, we know that we have 61

Part 1 Insurance, Hedging, and Simple Strategies

to look for a higher strike price. Trial and error results in a strike price of 448.93. The premium of the 440-strike call is $2.4944, and indeed the Black-Scholes premium of the 448.93 strike call is $1.2472. b)

Profit diagram:

Question 4.21. If you do not know how to run a regression, or if you forgot what a regression is, you may want to type the keyword “regression” in Microsoft Excel’s help menu. It will show you how to run a regression in Excel, as well as explain to you the key features of a regression. Running a regression, we obtain a constant of 2,100,000 and a coefficient on price of 100,000. Question 4.22. a)

We have the following table: Price 3 3 2 2

Quantity 1.5 0.8 1 0.6 62

Revenue 4.5 2.4 2 1.2

Chapter 4 Introduction to Risk Management

Using Excel’s function STDEVP(4.5,2.4,2,1.2), we obtain a value of 1.2194 for the standard deviation of total revenue for Scenario C. b) Using any standard software’s command (or doing it by hand!) to determine the correlation coefficient, we obtain a value of 0.7586. Question 4.23. a) Using equation (4.7) and the values of the correlation coefficient and standard deviation of the revenue we calculated in question 4.22., we obtain the following value for the variance minimizing hedge ratio: H =−

0.7586 × 1.2194 = 1.85007 0.5

It is thus optimal to short 1.85 million bushels of corn. b) If you do not know how to run a regression, or if you forgot what a regression is, you may want to type the keyword “regression” in Microsoft Excel’s help menu. It will show you how to run a regression in Excel, as well as explain to you the key features of a regression. Running such a regression, we obtain a constant of −2,100,000 and a coefficient on price of 1,850,000, thus yielding the same results as part a). c)

Price 3

Quantity 1.5m

Unhedged Revenue 4.5m

3

0.8m

2.4m

2

1m

2m

2

0.6m

1.2m

Futures Gain −0.5 × 1.85m = −0.925m −0.5 × 1.85m = −0.925m +0.5 × 1.85m = +0.925m +0.5 × 1.85m = +0.925m

Total 3.575m 1.475m 2.925m 2.125m

Using Excel’s function STDEVP(3.575,1.475,2.925,2.125), we obtain a value of 0.7945 for the standard deviation of the optimally hedged revenue for Scenario C. We see that we were able to significantly reduce the variance of our revenues. Question 4.24. a) The expected quantity of production is 0.25 × (1.5 + 0.8 + 1 + 0.6) = 0.975 million bushels of corn.

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Part 1 Insurance, Hedging, and Simple Strategies

b)

Price

Quantity

Unhedged revenue

3

1.5m

4.5m

3

0.8m

2.4m

2

1m

2m

2

0.6m

1.2m

Futures gain from shorting 0.975m contracts −0.5 × 0.975m = −0.4875m −0.5 × 0.975m = −0.4875m 0.5 × 0.975m = 0.4875m 0.5 × 0.975m = 0.4875m

Total 4.0125m 1.9125m 2.4875m 1.6875m

Using Excel’s function STDEVP(4.0125, 1.9125, 2.4875, 1.6875), we obtain a value of 0.907004 for the standard deviation of the optimally hedged revenue for Scenario C. We see that we were able to reduce the variance of our revenues, albeit to a lesser degree than with the optimally hedged portfolio. Question 4.25. a)

b)

c)

The expected quantity is: 0.5 × (0.6 + 0.934) = 0.767 million bushels. We have: Price

Quantity

Unhedged Revenue

2

0.6m

1.2m

3

0.934m

2.802m

Futures Gain from Total shorting 0.767m contracts +0.5 × 0.767m 1.5835m = 0.3835m −0.5 × 0.767m 2.4185m = −0.3835m

The minimum quantity is 0.6 million bushels. Therefore: Price

Quantity

Unhedged Revenue

2

0.6m

1.2m

3

0.934m

2.802m

Futures Gain from Total shorting 0.6m contracts +0.5 × 0.6m 1.5m = 0.3m −0.5 × 0.6m 2.502m = −0.3m

The maximum quantity is 0.934 million bushels. Therefore: Price

Quantity

Unhedged Revenue

2

0.6m

1.2m

3

0.934m

2.802m

64

Futures Gain from Total shorting 0.934m contracts +0.5 × 0.934m 1.667m = 0.467m −0.5 × 0.934m 2.335m = −0.467m

Chapter 4 Introduction to Risk Management

d) The hedge position that eliminates price variability shifts enough revenue from the good state to the bad state so that you make the same money in both states of the world (which are either a price of three or a price of two). We have to solve: 1.2m + 0.5 × X = 2.802m − 0.5 × X ⇔ X = 1.602m This leads to the following table: Price

Quantity

Unhedged Revenue

2

0.6m

1.2m

3

0.934m

2.802m

Futures Gain from Total shorting 1.602m contracts +0.5 × 1.602m 2.001m = 0.801m −0.5 × 1.602m 2.001m = −0.801m

We see again that we have to short more contracts than our maximum production is. The fact that quantity goes up when prices go up is responsible for this extensive amount of hedging.

65

Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Description Outright Sale Security Sale and Loan Sale Short Prepaid Forward Contract Short Forward Contract

Get Paid at Time 0 T

Lose Ownership of Receive Payment Security at Time of 0 S0 at time o 0 S0 erT at time T

0

T

?

T

T

? ×erT

Question 5.2. a) The owner of the stock is entitled to receive dividends. As we will get the stock only in one year, the value of the prepaid forward contract is today’s stock price, less the present value of the four dividend payments: P = $50 − F0,T

4


$1e−0.06× 12 i = $50 − $0.985 − $0.970 − $0.956 − $0.942 3

i=1

= $50 − $3.853 = $46.147 b) The forward price is equivalent to the future value of the prepaid forward. With an interest rate of 6 percent and an expiration of the forward in one year, we thus have: P F0,T = F0,T × e0.06×1 = $46.147 × e0.06×1 = $46.147 × 1.0618 = $49.00

Question 5.3. a) The owner of the stock is entitled to receive dividends. We have to offset the effect of the continuous income stream in form of the dividend yield by tailing the position: P F0,T = $50e−0.08×1 = $50 × 0.9231 = $46.1558

We see that the value is very similar to the value of the prepaid forward contract with discrete dividends we have calculated in question 5.2. In question 5.2., we received four cash dividends, 66

Chapter 5 Financial Forwards and Futures

with payments spread out through the entire year, totaling $4. This yields a total annual dividend yield of approximately $4 ÷ $50 = 0.08. b) The forward price is equivalent to the future value of the prepaid forward. With an interest rate of 6 percent and an expiration of the forward in one year we thus have: P F0,T = F0,T × e0.06×1 = $46.1558 × e0.06×1 = $46.1558 × 1.0618 = $49.01

Question 5.4. This question asks us to familiarize ourselves with the forward valuation equation. a) We plug the continuously compounded interest rate and the time to expiration in years into the valuation formula and notice that the time to expiration is 6 months, or 0.5 years. We have: F0,T = S0 × er×T = $35 × e0.05×0.5 = $35 × 1.0253 = $35.886 b)

The annualized forward premium is calculated as:     F0,T $35.50 1 1 ln = 0.0284 = annualized forward premium = ln T S0 0.5 $35

c) For the case of continuous dividends, the forward premium is simply the difference between the risk-free rate and the dividend yield:     F0,T S0 × e(r−δ)T 1 1 = ln annualized forward premium = ln T S0 T S0 1 1  (r−δ)T  = (r − δ) T = ln e T T =r −δ Therefore, we can solve: 0.0284 = 0.05 − δ ⇔ δ = 0.0216 The annualized dividend yield is 2.16 percent. Question 5.5. a) We plug the continuously compounded interest rate and the time to expiration in years into the valuation formula and notice that the time to expiration is 9 months, or 0.75 years. We have: F0,T = S0 × er×T = $1,100 × e0.05×0.75 = $1,100 × 1.0382 = $1,142.02 67

Part 2 Forwards, Futures, and Swaps

b)

We engage in a reverse cash and carry strategy. In particular, we do the following: Description Long forward, resulting from customer purchase Sell short the index Lend +S0 TOTAL

Today 0

In 9 months ST − F0,T

+S0 −S0 0

−ST S0 × erT S0 × erT − F0,T

Specifically, with the numbers of the exercise, we have: Description Today In 9 months Long forward, resulting 0 ST − $1,142.02 from customer purchase Sell short the index $1,100 −ST Lend $ 1,100 −$1,100 $1,100 × e0.05×0.75 = $1,142.02 TOTAL 0 0 Therefore, the market maker is perfectly hedged. She does not have any risk in the future, because she has successfully created a synthetic short position in the forward contract. c)

Now, we will engage in cash and carry arbitrage: Description Short forward, resulting from customer purchase Buy the index Borrow +S0 TOTAL

Today 0

In 9 months F0,T − ST

−S0 +S0 0

ST −S0 × erT F0,T − S0 × erT

Specifically, with the numbers of the exercise, we have: Description Short forward, resulting from customer purchase Buy the index Borrow $1,100 TOTAL

Today 0

In 9 months $1, 142.02 − ST

−$1,100 ST $1,100 −$1,100 × e0.05×0.75 = −$1,142.02 0 0

Again, the market maker is perfectly hedged. He does not have any index price risk in the future, because he has successfully created a synthetic long position in the forward contract that perfectly offsets his obligation from the sold forward contract.

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Chapter 5 Financial Forwards and Futures

Question 5.6. a) We plug the continuously compounded interest rate, the dividend yield and the time to expiration in years into the valuation formula and notice that the time to expiration is 9 months, or 0.75 years. We have: F0,T = S0 × e(r−δ)×T = $1,100 × e(0.05−0.015)×0.75 = $1,100 × 1.0266 = $1,129.26 b)

We engage in a reverse cash and carry strategy. In particular, we do the following: Description Today Long forward, resulting 0 from customer purchase Sell short tailed position +S0 e−δT of the index −S0 e−δT Lend S0 e−δT TOTAL 0

In 9 months ST − F0,T −ST S0 × e(r−δ)T S0 × e(r−δ)T − F0,T

Specifically, we have: Description Today Long forward, resulting 0 from customer purchase Sell short tailed position $1,100 × .9888 of the index = 1087.69 Lend $1,087.69 −$1,087.69 TOTAL

0

In 9 months ST − $1, 129.26 −ST $1,087.69 × e0.05×0.75 = $1,129.26 0

Therefore, the market maker is perfectly hedged. He does not have any risk in the future, because he has successfully created a synthetic short position in the forward contract. c) Description Short forward, resulting from customer purchase Buy tailed position in index Borrow S0 e−δT TOTAL

Today 0

In 9 months F0,T − ST

−S0 e−δT

ST

S0 e−δT 0

−S0 × e(r−δ)T F0,T − S0 × e(r−δ)T

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Part 2 Forwards, Futures, and Swaps

Specifically, we have: Description Short forward, resulting from customer purchase Buy tailed position in index Borrow $ 1,087.69

Today 0

In 9 months $1,129.26 − ST

−$1,100 × .9888 = −$1,087.69 $1,087.69

ST

TOTAL

0

−$1,087.69 × e0.05×0.75 = −$1,129.26 0

Again, the market maker is perfectly hedged. He does not have any index price risk in the future, because he has successfully created a synthetic long position in the forward contract that perfectly offsets his obligation from the sold forward contract. Question 5.7. We need to find the fair value of the forward price first. We plug the continuously compounded interest rate and the time to expiration in years into the valuation formula and notice that the time to expiration is 6 months, or 0.5 years. We have: F0,T = S0 × e(r)×T = $1,100 × e(0.05)×0.5 = $1,100 × 1.02532 = $1,127.85 a) If we observe a forward price of 1135, we know that the forward is too expensive, relative to the fair value we determined. Therefore, we will sell the forward at 1135, and create a synthetic forward for 1,127.85, make a sure profit of $7.15. As we sell the real forward, we engage in cash and carry arbitrage: Description Today In 9 months Short forward 0 $1, 135.00 − ST Buy position in index −$1,100 ST Borrow $1,100 −$1,100 $1,127.85 TOTAL 0 $7.15 This position requires no initial investment, has no index price risk, and has a strictly positive payoff. We have exploited the mispricing with a pure arbitrage strategy. b) If we observe a forward price of 1,115, we know that the forward is too cheap, relative to the fair value we have determined. Therefore, we will buy the forward at 1,115, and create a synthetic short forward for 1,127.85, make a sure profit of $12.85. As we buy the real forward, we engage in a reverse cash and carry arbitrage:

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Chapter 5 Financial Forwards and Futures

Description Today In 9 months Long forward 0 ST − $1,115.00 Short position in index $1,100 −ST Lend $1,100 −$1,100 $1,127.85 TOTAL 0 $12.85 This position requires no initial investment, has no index price risk, and has a strictly positive payoff. We have exploited the mispricing with a pure arbitrage strategy. Question 5.8. First, we need to find the fair value of the forward price. We plug the continuously compounded interest rate, the dividend yield and the time to expiration in years into the valuation formula and notice that the time to expiration is 6 months, or 0.5 years. We have: F0,T = S0 × e(r−δ)×T = $1,100 × e(0.05−0.02)×0.5 = $1,100 × 1.01511 = $1,116.62 a) If we observe a forward price of 1,120, we know that the forward is too expensive, relative to the fair value we have determined. Therefore, we will sell the forward at 1,120, and create a synthetic forward for 1,116.82, making a sure profit of $3.38. As we sell the real forward, we engage in cash and carry arbitrage: Description Today In 9 months Short forward 0 $1,120.00 − ST Buy tailed position in −$1,100 × .99 ST index = −$1,089.055 Borrow $1,089.055 $1,089.055 −$1,116.62 TOTAL 0 $3.38 This position requires no initial investment, has no index price risk, and has a strictly positive payoff. We have exploited the mispricing with a pure arbitrage strategy. b) If we observe a forward price of 1,110, we know that the forward is too cheap, relative to the fair value we have determined. Therefore, we will buy the forward at 1,110, and create a synthetic short forward for 1116.62, thus making a sure profit of $6.62. As we buy the real forward, we engage in a reverse cash and carry arbitrage: Description Today In 9 months Long forward 0 ST − $1,110.00 Sell short tailed position in $1,100 × .99 −ST index = $1,089.055 Lend $1,089.055 −$1,089.055 $1,116.62 TOTAL 0 $6.62

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Part 2 Forwards, Futures, and Swaps

This position requires no initial investment, has no index price risk, and has a strictly positive payoff. We have exploited the mispricing with a pure arbitrage strategy. Question 5.9. a) A money manager could take a large amount of money in 1982, travel back to 1981, invest it at 12.5%, and instantaneously travel forward to 1982 to reap the benefits, i.e. the accured interest. Our argument of time value of money breaks down. b) If many money managers undertook this strategy, competitive market forces would drive the interest rates down. c) Unfortunately, these arguments mean that costless and riskless time travel will not be invented. Question 5.10. a) We plug the continuously compounded interest rate, the forward price, the initial index level and the time to expiration in years into the valuation formula and solve for the dividend yield: F0,T = S0 × e(r−δ)×T F0,T

= e(r−δ)×T S0   F0,T ⇔ ln = (r − δ) × T S0   F0,T 1 ⇔ δ = r − ln T S0   1129.257 1 ln = 0.05 − 0.035 = 0.015 ⇒ δ = 0.05 − 0.75 1100 ⇔

Remark: Note that this result is consistent with exercise 5.6., in which we had the same forward prices, time to expiration etc. b)

With a dividend yield of only 0.005, the fair forward price would be: F0,T = S0 × e(r−δ)×T = 1,100 × e(0.05−0.005)×0.75 = 1,100 × 1.0343 = 1,137.759

Therefore, if we think the dividend yield is 0.005, we consider the observed forward price of 1,129.257 to be too cheap. We will therefore buy the forward and create a synthetic short forward, capturing a certain amount of $8.502. We engage in a reverse cash and carry arbitrage:

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Chapter 5 Financial Forwards and Futures

Description Today Long forward 0 Sell short tailed position in $1,100 × .99626 index = $1,095.88 Lend $1,095.88 −$1,095.88 TOTAL 0 c)

In 9 months ST − $1,129.257 −ST $1,137.759 $8.502

With a dividend yield of 0.03, the fair forward price would be: F0,T = S0 × e(r−δ)×T = 1,100 × e(0.05−0.03)×0.75 = 1,100 × 1.01511 = 1,116.62

Therefore, if we think the dividend yield is 0.03, we consider the observed forward price of 1,129.257 to be too expensive. We will therefore sell the forward and create a synthetic long forward, capturing a certain amount of $12.637. We engage in a cash and carry arbitrage: Description Today Short forward 0 Buy tailed position in −$1,100 × .97775 index = −$1,075.526 Borrow $1,075.526 $1,075.526 TOTAL 0

In 9 months $1,129.257 − ST ST $1,116.62 $12.637

Question 5.11. a) The notional value of 4 contracts is 4 × $250 × 1200 = $1,200,000, because each index point is worth $250, and we buy four contracts. b) The margin protects the counterparty against default. In our case, it is 10% of the notional value of our position, which means that we have to deposit an initial margin of: $1,200,000 × 0.10 = $120,000 Question 5.12. a) The notional value of 10 contracts is 10 × $250 × 950 = $2,375,000, because each index point is worth $250, we buy 10 contracts and the S&P 500 index level is 950. With an initial margin of 10% of the notional value, this results in an initial dollar margin of $2,375,000 × 0.10 = $237,500. b) We first obtain an approximation. Because we have a 10% initial margin, a 2% decline in the futures price will result in a 20% decline in margin. As we will receive a margin call after a 20% decline in the initial margin, the smallest futures price that avoids the maintenance margin call

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Part 2 Forwards, Futures, and Swaps

is 950 × .98 = 931. However, this calculation ignores the interest that we are able to earn in our margin account. Let us now calculate the details. We have the right to earn interest on our initial margin position. As the continuously compounded interest rate is currently 6%, after one week, our initial margin has grown to: 1

$237,500e0.06× 52 = $237,774.20 We will get a margin call if the initial margin falls by 20%. We calculate 80% of the initial margin as: $237,500 × 0.8 = $190,000 10 long S&P 500 futures contracts obligate us to pay $2,500 times the forward price at expiration of the futures contract. Therefore, we have to solve the following equation: $237,774.20 + (F1W − 950) × $2,500 ≥ $190,000 ⇔ $47774.20 ≥ − (F1W − 950) × $2,500 ⇔ 19.10968 − 950 ≥ −F1W ⇔ F1W ≥ 930.89 Therefore, the greatest S&P 500 index futures price at which we will receive a margin call is 930.88. Question 5.13. a) Description Today At expiration of the contract Long forward 0 ST − F0,T = ST − S0 erT −S0 S0 erT Lend S0 ST Total −S0 In the first row, we made use of the forward price equation if the stock does not pay dividends. We see that the total aggregate position is equivalent to the payoff of one stock. b)

In the case of discrete dividends, we have:

74

Chapter 5 Financial Forwards and Futures

Description

Today

At expiration of the contract

Long forward

0

ST − F0,T = ST − S0 erT +

Lend S0 −

n 

e−rti Dti

−S0 +

i=1

Total

n 

e−rti Dti

+S0 erT −

i=1

−S0 +

n 

n 

n 

er(T −ti ) Dti

i=1

er(T −ti ) Dti

i=1

e−rti Dti

ST

i=1

c) In the case of a continuous dividend, we have to tail the position initially. We therefore create a synthetic share at the time of expiration T of the forward contract. Description Today Long forward 0 Lend S0 e−δT −S0 e−δT Total −S0 e−δT

At expiration of the contract ST − F0,T = ST − S0 e(r−δ)T S0 e(r−δ)T ST

In the first row, we made use of the forward price equation if the stock pays a continuous dividend. We see that the total aggregate position is equivalent to the payoff of one stock at time T. Question 5.14. An arbitrageur believing that the observed forward price, F(0,T), is too low will undertake a reverse cash and carry arbitrage: Buy the forward, short sell the stock and lend out the proceeds from the short sale. The relevant prices are therefore the bid price of the stock and the lending interest rate. Also, she will incur the transaction costs twice. We have: Description Today Long forward 0 Sell short tailed position +S0b e−δT of the index Pay twice transaction cost −2 × k −S0b e−δT + 2 × k Lend S0b e−δT − 2 × k TOTAL 0

In 9 months ST − F0,T −ST b −δT

l +S0 e − 2 × k × er T b −δT

l +S0 e − 2 × k × er T − F0,T



l To avoid arbitrage, we must have S0b − 2 × k × er T − F0,T ≤ 0. This is equivalent to F0,T ≥

b l S0 − 2 × k × er T . Therefore, for any F0,T smaller than this bound, there exist arbitrage opportunities.

75

Part 2 Forwards, Futures, and Swaps

Question 5.15. a) We use the transaction cost boundary formulas that were developed in the text and in exercise 5.14. In this part, we set k equal to zero. There is no bid-ask spread. Therefore, we have F + = 800e0.055 = 800 × 1.05654 = 845.23 F − = 800e0.05 = 800 × 1.051271 = 841.02 b) Now, we will incur an additional transaction fee of $1 for going either long or short the forward contract. Stock sales or purchases are unaffected. We calculate: F + = (800 + 1) e0.055 = 801 × 1.05654 = 846.29 F − = (800 − 1) e0.05 = 799 × 1.051271 = 839.97 c) Now, we will incur an additional transaction fee of $2.40 for the purchase or sale of the index, making our total initial transaction cost $3.40. We calculate: F + = (800 + 3.40) e0.055 = 803.40 × 1.05654 = 848.82 F − = (800 − 3.40) e0.05 = 796.60 × 1.051271 = 837.44 d) We also have to take into account as well the additional cost that we incur at the time of expiration. We can calculate: F + = (800 + 3.40) e0.055 + 2.40 = 803.40 × 1.05654 = 851.22 F − = (800 − 3.40) e0.05 − 2.40 = 796.60 × 1.051271 = 835.04 e) Let us make use of the hint. In the cash and carry arbitrage, we will buy the index and have thus at expiration time ST . However, we have to pay a proportional transaction cost of 0.3% on it, so that the position is only worth 0.997 × ST . However, we need ST to set off the index price risk introduced by the short forward. Therefore, we will initially buy 1.003 units of the index, which leaves us exactly ST after transaction costs. Additionally, we incur a transaction cost of 0.003 × S0 for buying the index today, and of $1 for selling the forward contract. F + = (800 × 1.003 + 800 × 0.003 + 1) e0.055 = 805.80 × 1.05654 = 851.36 The boundary is slightly higher, because we must take into account the variable, proportional cash settlement cost we incur at expiration. The difference between part d) and part e) is the interest we have to pay on $2.40, which is $.14. In the reverse cash and carry arbitrage, we will sell the index and have to pay back at expiration −ST . However, we have to pay a proportional transaction cost of 0.3% on it, so that we have exposure of −1.003 × ST . However, we only need an exposure of −ST to set off the index price risk introduced by the long forward. Therefore, we will initially only sell 0.997 units of the index, which leaves us 76

Chapter 5 Financial Forwards and Futures

exactly with −ST after transaction costs at expiration. Additionally, we incur a transaction cost of 0.003 × S0 for buying the index today, and $1 for selling the forward contract. We have as a new lower bound: F − = (800 × 0.997 − 800 × 0.003 − 1) e0.05 = 794.20 × 1.051271 = 834.92 The boundary is slightly lower, because we forego some interest we could earn on the short sale, because we have to take into account the proportional cash settlement cost we incur at expiration. The difference between part d) and part e) is the interest we are foregoing on $2.40, which is $.12 (at the lending rate of 5%). Question 5.16. a)

The one-year futures price is determined as: F0,1 = 875e0.0475 = 875 × 1.048646 = 917.57

b) One futures contract has the value of $250 × 875 = $218,750. Therefore, the number of contracts needed to cover the exposure of $800,000 is: $800,000 ÷ $218,750 = 3.65714. Furthermore, we need to adjust for the difference in beta. Since the beta of our portfolio exceeds 1, it moves more than the index in either direction. Therefore, we must increase the number of contracts. The final hedge quantity is: 3.65714 × 1.1 = 4.02286. Therefore, we should short-sell 4.02286 S&P 500 index future contracts. As the correlation between the index and our portfolio is assumed to be one, we have no basis risk and have perfectly hedged our position and transformed it into a riskless investment. Therefore, we expect to earn the risk-free interest rate as a return over one year. Question 5.17. It is important to realize that, because we can go long or short a future, the sign of the correlation does not matter in our ranking. Suppose the correlation of our portfolio in question 5.16. with the S&P 500 is minus 1. Then we can do exactly the same calculation, but would in the end go long the futures contract. It is thus the absolute correlation coefficient that should be as close to one as possible. Therefore, the ranking is 0, 0.25, −0.5, −0.75, 0.85, −0.95, with 0 having the highest basis risk. Question 5.18. The current exchange rate is 0.02E/Y, which implies 50Y/E. The euro continuously compounded interest rate is 0.04, the yen continuously compounded interest rate 0.01. Time to expiration is 0.5 years. Plug the input variables into the formula to see that: Euro/Yen forward = 0.02e(0.04−0.01)×0.5 = 0.02 × 1.015113 = 0.020302 Yen/Euro forward = 50e(0.01−0.04)×0.5 = 50 × 0.98511 = 49.2556 77

Part 2 Forwards, Futures, and Swaps

Question 5.19. The current spot exchange rate is 0.008$/Y, the one-year continuously compounded dollar interest rate is 5%, and the one-year continuously compounded yen interest rate is 1%. This means that we can calculate the fair price of a one-year $/Yen forward to be: Dollar/Yen forward = 0.008e(0.05−0.01) = 0.008 × 1.0408108 = 0.0083265 We can see that the observed forward exchange rate of 0.0084 $/Y is too expensive, relative to the fair forward price. We therefore sell the forward and synthetically create a forward position: Description

Today in $ Sell $/Y forward 0 Buy Yen for 0.0079204 dollar −0.0079204 Lend 0.99005 Yen — Borrow 0.0079204 dollar +0.0079204 Total 0

in Yen — +0.99005 −0.99005 — 0

At expiration in $ 0.0084$ — — −0.0083265 0.0000735

of the contract in Yen −1 — 1 — 0

Therefore, this transaction earned us 0.0000735 dollars, without any exchange risk or initial investment involved. We have exploited an inherent arbitrage opportunity. With a forward exchange rate of 0.0083, the observed price is too cheap. We will buy the forward and synthetically create a short forward position. Description

Today At expiration in $ in Yen in $ Buy $/Y forward 0 — −0.0083$ Sell Yen for 0.0079204 dollar +0.0079204 −0.99005 — Borrow 0.99005 Yen — +0.99005 — Lend 0.0079204 dollar −0.0079204 — +0.0083265 Total 0 0 0.0000265

of the contract in Yen +1 — −1 — 0

Therefore, we again made an arbitrage profit of 0.0000265 dollars. Question 5.20. a) The Eurodollar futures price is 93.23. Therefore, we can use equation (5.20) of the main text to back out the three-month LIBOR rate: r91 = (100 − 93.23) ×

1 91 1 = 0.017113. × × 100 4 90

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Chapter 5 Financial Forwards and Futures

b) We will have to repay principal plus interest on the loan that we are taking from the following June to September. Because we shorted a Eurodollar futures, we are guaranteed the interest rate we calculated in part a). Therefore, we have a repayment of: $10,000,000 × (1 + r91 ) = $10,000,000 × 1.017113 = $10,171,130

79

Chapter 6 Commodity Forwards and Futures Question 6.1. The spot price of a widget is $70.00. With a continuously compounded annual risk-free rate of 5%, we can calculate the annualized lease rates according to the formula: F0,T = S0 × e(r−δl )×T F0,T

= e(r−δl )×T S0
 F0,T ⇔ ln = (r − δl ) × T S0
 F0,T 1 ⇔ δl = r − ln T S0 ⇔

Time to expiration 3 months 6 months 9 months 12 months

Forward price Annualized lease rate $70.70 0.0101987 $71.41 0.0101147 $72.13 0.0100336 $72.86 0.0099555

The lease rate is less than the risk-free interest rate. The forward curve is upward sloping, thus the prices of exercise 6.1. are an example of contango. Question 6.2. The spot price of oil is $32.00 per barrel. With a continuously compounded annual risk-free rate of 2%, we can again calculate the lease rate according to the formula:
 F0,T 1 δl = r − ln T S0 Time to expiration 3 months 6 months 9 months 12 months

Forward price Annualized lease rate $31.37 0.0995355 $30.75 0.0996918 $30.14 0.0998436 $29.54 0.0999906

80

Chapter 6 Commodity Forwards and Futures

The lease rate is higher than the risk-free interest rate. The forward curve is downward sloping, thus the prices of exercise 6.2. are an example of backwardation. Question 6.3. The question asks us to find the lease rate such that F0,T = S0 . We take our pricing formula, F0,T = S0 × e(r−δl )×T , and immediately see that the sought equality is established if e(r−δl )×T = 1, which is guaranteed for any T > 0 if and only if r = δ. If the lease rate were 3.5%, the lease rate would be higher than the risk-free interest rate. Therefore, a graph of forward prices would be downward sloping, and thus there would be backwardation. Question 6.4. a) As we need to borrow a pencil to sell it short, we must pay the lender the lease rate for the time we borrow the asset, i.e., until expiration of the contract in one year. After one year, we have to pay back one pencil, which will cost us ST , the uncertain future pencil price, plus the leasing costs: Total payment = ST + ST × e0.05 − 1 = ST e0.05 = 1.05127 × ST . It does not make sense to store pencils in equilibrium, because even if we have an active lease market for pencils, the lease rate is smaller than the risk-free interest rate. Lending money at ten percent is more profitable than lending pencils at five percent. b)

The equilibrium forward price is calculated according to our pricing formula: F0,T = S0 × e(r−δl )×T = $0.20 × e(0.10−0.05)×1 = $0.20 × 1.05127 = $0.2103,

which is the price given in the exercise. c)

Let us first look at the different arbitrage strategies we can use in each case.

c1)

Pencils can be sold short. We can engage in our usual reverse cash and carry arbitrage: Transaction Time 0 Time T = 1 Long forward 0 ST − F0,T Short-sell tailed pen- $0.19025 −ST cil position, @ 0.05 Lend short-sale −$0.19025 $0.2103 proceeds @ 0.1 Total 0 $0.2103 − F0,T

For there to be no arbitrage, F0,T ≥ $0.2103 c2) Suppose pencils cannot be sold short. Then we have no ability to create the short position necessary to offset the pencil price risk from the long forward. Consequently, we are not able to find a lower boundary for the pencil forward in this case. 81

Part 2 Forwards, Futures, and Swaps

c3)

Pencils can be loaned. We engage in a cash and carry arbitrage: Transaction Time 0 Time T = 1 Short forward 0 F0,T − ST Buy tailed pencil −$0.19025 ST position, lend @0.05 borrow @ 0.1 $0.19025 −$0.2103 Total 0 F0,T − $0.2103

For there to be no arbitrage, F0,T ≤ $0.2103 c4)

Suppose pencils cannot be loaned. Then our cash and carry arbitrage becomes: Transaction Short forward Buy pencil borrow @ 0.1 Total

Time 0 Time T = 1 0 F0,T − ST −$0.20 ST $0.20 −$0.2210 0 F0,T − $0.2210

For there to be no arbitrage, F0,T ≤ $0.2210 Therefore, we have found the following no-arbitrage regions: loan=yes, short-sale=yes loan=no, short-sale=yes loan=yes, short-sale=no loan=no, short-sale=no

Lower bound on forward F0,T ≥ $0.2103 F0,T ≥ $0.2103 — —

Upper bound on forward F0,T ≤ $0.2103 F0,T ≤ $0.2210 F0,T ≤ $0.2103 F0,T ≤ $0.2210

Question 6.5. a) The spot price of gold is $300.00 per ounce. With a continuously compounded annual riskfree rate of 5%, and at a one-year forward price of 310.686, we can calculate the lease rate according to the formula:
 
F0,T 1 310.686 δl = r − ln = 0.015 = 0.05 − ln T S0 300 b)

Suppose gold cannot be loaned. Then our cash and carry “arbitrage” is: Transaction Short forward Buy gold Borrow @ 0.05 Total

Time 0 Time T = 1 0 310.686 − ST −$300 ST $300 −$315.38 0 −4.6953 82

Chapter 6 Commodity Forwards and Futures

The forward price bears an implicit lease rate. Therefore, if we try to engage in a cash and carry arbitrage, but if we do not have access to the gold loan market, and thus to the additional revenue on our long gold position, it is not possible for us to replicate the forward price. We incur a loss. c)

If gold can be loaned, we engage in the following cash and carry arbitrage: Transaction Time 0 Time T = 1 Short forward 0 310.686 − ST Buy tailed gold −$295.5336 ST position, lend @ 0.015 Borrow @ 0.05 $295.5336 −$310.686 Total 0 0

Therefore, we now just break even: Since the forward was fairly priced, taking the implicit lease rate into account, this result should not surprise us. Question 6.6. a) The forward prices reflect a market for widgets in which seasonality is important. Let us look at two examples, one with constant demand and seasonal supply, and another one with constant supply and seasonal demand. One possible explanation might be that widgets are extremely difficult to produce and that they need one key ingredient that is only available during July/August. However, the demand for the widget is constant throughout the year. In order to be able to sell the widgets throughout the year, widgets must be stored after production in August. The forward curve reflects the ever increasing storage costs of widgets until the next production cycle arrives. Once produced, widget prices fall again to the spot price. Another story that is consistent with the observed prices of widgets is that widgets are in particularly high demand during the summer months. The storage of widgets may be costly, which means that widget producers are reluctant to build up inventory long before the summer. Storage occurs slowly over the winter months and inventories build up sharply just before the highest demand during the summer months. The forward prices reflect those storage cycle costs. b) Let us take the December 2001 forward price as a proxy for the spot price in December 2001. We can then calculate with our cash and carry arbitrage tableau: Transaction Time 0 Time T = 3/12 Short March forward 0 3.075 − ST Buy December −3.00 ST Forward (= Buy spot) Pay storage cost −0.03 Total −3.00 3.045

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Part 2 Forwards, Futures, and Swaps

We can calculate the annualized rate of return as: 3.045 = e(r)×T 3.00 
3.045 = r × 3/12 ⇔ ln 3.00 r = 0.05955 which is the prevailing risk-free interest rate of 0.06. This result seems to make sense. c) Let us again take the December 2001 forward price as a proxy for the spot price in December 2001. We can then calculate with our cash and carry arbitrage tableau: Transaction Time 0 Time T= 9/12 Short Sep forward 0 2.75 − ST Buy spot −3.00 ST Pay storage cost Sep −0.03 FV(Storage Jun) −0.0305 FV(Storage Mar) −0.0309 Total −3.00 2.6586 We can calculate the annualized rate of return as: 
2.6586 2.6586 (r)×T =e = r × 9/12 ⇔ ln 3.00 3.00 r = −0.16108 This result does not seem to make sense. We would earn a negative annualized return of 16% on such a cash and carry arbitrage. Therefore, it is likely that our naive calculations do not capture an important fact about the widget market. In particular, we will buy and hold the widget through a time where the forward curve indicates that there is a significant convenience yield attached to widgets. It is tempting, although premature, to conclude that a reverse cash and carry arbitrage may make a positive 16 % annualized return. Question 6.7. deals with this aspect. Question 6.7. a) Let us take the December 2001 forward price as a proxy for the spot price in December 2001. We can then calculate a reverse cash and carry arbitrage tableau:

84

Chapter 6 Commodity Forwards and Futures

Transaction Time 0 Time T = 3/12 Long March forward 0 ST − 3.075 Short widget +3.00 −ST Lend money −3.00 +3.045 Total

0

−0.03

We need to receive 0.03 as a compensation from the lender of the widget. This cost reflects the storage cost of the widget that the lender does not need to pay. The lease rate is negative. b) Let us take the December 2001 forward price as a proxy for the spot price in December 2001. We can then calculate a reverse cash and carry arbitrage tableau: Transaction Time 0 Long Sept forward 0 Short widget +3.00 Lend money −3.00

Time T = 9/12 ST − 3.075 −ST +3.13808

Total

0.06308

0

Although we free the lender of the widget of the burden to pay three times storage costs, we still need to pay him 0.06308. This reflects the fact that we hold the widget through the period in which widgets are valuable (as reflected by the forward contracts), and are returning it at a time it is worth less. The lender is only willing to do so if we compensate him for the opportunity cost. Question 6.8. a)

The first possibility is a simple cash and carry arbitrage: Transaction Time 0 Time T = 3/12 Short March forward 0 3.10 − ST Buy December −3.00 ST forward (= Buy spot) Borrow @ 6% +3.00 −3.045 Pay storage cost −0.03 Total 0 +0.025

The second possibility involves using the June futures contract. It is a forward cash and carry strategy: Transaction Time = T(1) = 3/12 Short March forward 3.10 − ST (1) Buy June forward 0 Lend @ 6% −3.10 Receive storage cost Total 0 85

Time = T(2) = 6/12 −ST (2) ST (2) − 3.152 3.1469 +0.03 +0.02485

Part 2 Forwards, Futures, and Swaps

We can use the June futures in our calculations and claim to receive storage costs, because it is easy to show that the value of it is reflecting the negative lease rate of the storage costs. b) It is not possible to undertake an arbitrage with the futures contracts that expire prior to September 2002. A decrease in the September futures value means that we would need to buy the September futures contract, and any arbitrage strategy would need some short position in the widget. However, the drop in the futures price in September indicates that there is a significant convenience yield factored into the futures price over the period June–September. As we have no information about it, it is not possible for us to guarantee that we find a lender of widgets at a favorable lease rate to follow through our arbitrage trading program. The decrease in the September futures may in fact reflect an increase in the opportunity costs of widget owners. Question 6.9. If the February corn forward price is $2.80, the observed forward price is too expensive relative to our theoretical price of $2.7273. We will therefore sell the February contract short, and create a synthetic long position, engaging in cash and carry arbitrage: Transaction Nov Dec Jan Short Feb forward 0 Buy spot −2.50 Borrow purchasing +2.50 cost Pay storage cost Dec, −0.05 borrow storage cost +0.05 Pay storage cost Jan, −0.05 borrow storage cost +0.05 Pay storage cost Feb Total 0 0 0

Feb 2.80 − ST ST −2.57575 −0.051005 −0.0505 −0.05 0.072745

We made an arbitrage profit of 0.07 dollar. If the February corn forward price is $2.65, the observed forward price is too low relative to our theoretical price of $2.7273. We will therefore buy the February contract, and create a synthetic short position, engaging in reverse cash and carry arbitrage:

86

Chapter 6 Commodity Forwards and Futures

Transaction Nov Dec Jan Feb Long Feb forward 0 ST − 2.65 Sell spot 2.50 −ST Lend short sale −2.50 +2.57575 proceeds Receive storage cost +0.05 +0.051005 Dec, lend them −0.05 Receive storage cost +0.05 +0.0505 Jan, lend them −0.05 Receive cost Feb +0.05 Total 0 0 0 0.077255 We made an arbitrage profit of 0.08 dollar. It is important to keep in mind that we ignored any convenience yield that there may exist to holding the corn. We assumed the convenience yield is zero. Question 6.10. Our best bet for the current spot price is the first available forward price, properly discounted by taking the interest rate and the lease rate into account, and by ignoring any storage cost and convenience yield (because we do not have any information on it): F0,T = S0 × e(r−δl )×T ⇔ S0 = F0,T × e−(r−δl )×T ⇔ S0 = 313.81 × e−(0.06−0.015)×1 = 313.81 × 0.956 = 300.0016 Question 6.11. a) Widgets do not deteriorate over time and are costless to store, therefore the lender does not save anything by lending me the widget. On the other hand, there is a constant demand and flexible production—there is no seasonality. Therefore, we should expect that the convenience yield is very close to the risk-free rate, merely compensating the lender for the opportunity cost. b) Demand varies seasonally, but the production process is flexible. Therefore, we would expect that producers anticipate the seasonality in demand, and adjust production accordingly. Again, the lease rate should not be much higher than the risk-free rate. c) Now we have the problem that the demand for widgets will spike up without an appropriate adjustment of production. Let us suppose that widget demand is high in June, July and August. Then, we will face a substantial lease rate if we were to borrow the widget in May, to return it in September: We would deprive the merchant of the widget when he would need it most (and could probably earn a significant amount of money on it), and we would return it when the season is over. We most likely pay a substantial lease rate. 87

Part 2 Forwards, Futures, and Swaps

On the other hand, suppose we want to borrow the widget in January, and return it in June. Now we return the widget precisely when the merchant needs it, and have it over a time where demand is low, and he is not likely to sell it. The lease rate is likely to be very small. However, those stylized facts are weakened by the fact that the merchant can costlessly store widgets, so the smart merchant has a larger inventory when demand is high, offsetting the above effects at a substantial amount. d) Suppose that production is very low during June, July and August. Let us think about borrowing the widget in May, and returning it in September. We do again deprive the merchant of the widget when he needs it most, because with a constant demand, less production means widgets become a comparably scarce resource and increase in price. Therefore, we pay a higher lease rate. The opposite effects can be observed for a widget-borrowing from January to June. Again, these stylized facts are offset by the above mentioned inventory considerations. e) If widgets cannot be stored, the seasonality problems become very severe, leading to larger swings in the lease rate due to the impossibility of managing inventory.

88

Chapter 7 Interest Rate Forwards and Futures Question 7.1. Using the bond valuation formulas (7.1), (7.3), (7.6) we obtain the following yields and prices: Maturity

Zero-Coupon Bond Yield

Zero Coupon Bond Price

1 2 3 4 5

0.04000 0.04500 0.04500 0.05000 0.05200

0.96154 0.91573 0.87630 0.82270 0.77611

One-Year Implied Forward Rate 0.04000 0.05003 0.04500 0.06515 0.06003

Par Coupon

Cont. Comp. Zero Yield

0.04000 0.04489 0.04492 0.04958 0.05144

0.03922 0.04402 0.04402 0.04879 0.05069

Question 7.2. The coupon bond pays a coupon of $60 each year plus the principal of $1,000 after five years. We have cash flows of [60, 60, 60, 60, 1060]. To obtain the price of the coupon bond, we multiply each cash flow by the zero-coupon bond price of that year. This yields a bond price of $1,037.25280. Question 7.3. Maturity

Zero-Coupon Bond Yield

Zero Coupon Bond Price

1 2 3 4 5

0.03000 0.03500 0.04000 0.04500 0.05000

0.97087 0.93351 0.88900 0.83856 0.78353

One-Year Par Coupon Implied Forward Rate 0.03000 0.03000 0.04002 0.03491 0.05007 0.03974 0.06014 0.04445 0.07024 0.04903

89

Cont. Comp. Zero Yield 0.02956 0.03440 0.03922 0.04402 0.04879

Part 2 Forwards, Futures, and Swaps

Question 7.4. Maturity

Zero-Coupon Bond Yield

Zero Coupon Bond Price

1 2 3 4 5

0.05000 0.04200 0.04000 0.03600 0.02900

0.95238 0.92101 0.88900 0.86808 0.86681

Maturity

Zero-Coupon Bond Yield

Zero Coupon Bond Price

1 2 3 4 5

0.07000 0.06000 0.05000 0.04500 0.04000

0.93458 0.88999 0.86384 0.83855 0.82193

One-Year Par Coupon Implied Forward Rate 0.05000 0.05000 0.03406 0.04216 0.03601 0.04018 0.02409 0.03634 0.00147 0.02962

Cont. Comp. Zero Yield

One-Year Par Coupon Implied Forward Rate 0.07000 0.07000 0.05009 0.06029 0.03028 0.05065 0.03016 0.04578 0.02022 0.04095

Cont. Comp. Zero Yield

0.04879 0.04114 0.03922 0.03537 0.02859

Question 7.5.

0.06766 0.05827 0.04879 0.04402 0.03922

Question 7.6. In order to be able to solve this problem, it is best to take equation (7.6) of the main text and solve progressively for all zero-coupon bond prices, starting with year one. This yields the series of zero-coupon bond prices from which we can proceed as usual to determine the yields. Maturity

Zero-Coupon Bond Yield

Zero Coupon Bond Price

1 2 3 4 5

0.03000 0.03500 0.04000 0.04700 0.05300

0.97087 0.93352 0.88899 0.83217 0.77245

One-Year Par Coupon Implied Forward Rate 0.03000 0.03000 0.04002 0.03491 0.05009 0.03974 0.06828 0.04629 0.07732 0.05174

90

Cont. Comp. Zero Yield 0.02956 0.03440 0.03922 0.04593 0.05164

Chapter 7 Interest Rate Forwards and Futures

Question 7.7. a) We are looking for r0 (1, 3). We will use equation (7.3) of the main text, and the known one-year and three-year zero-coupon bond prices. We have to solve the following equation:


3−1

1 + r0 (1, 3)

⇔ r0 (1, 3) b)

P (0, 1) P (0, 3)   P (0, 1) 0.943396 −1= − 1 = 0.07504 = P (0, 3) 0.816298

=

Let’s calculate the zero-coupon bond price from year 1 to 2 and from year 1 to 3, they are: P (0, 2) 0.881659 = = 0.93456 P (0, 1) 0.943396 P (0, 3) 0.816298 = = 0.86528 P0 (1, 3) = P (0, 1) 0.943396

P0 (1, 2)

=

Now, we have the relevant implied forward zero-coupon prices and can find the coupon of the par 2-year coupon bond issued at time 1 according to formula (7.6). c=

1 − P0 (1, 3) 0.13472 = = 0.074851 P0 (1, 2) + P0 (1, 3) 0.93456 + 0.86528

Question 7.8. a) We have to take into account the interest we (or our counterparty) can earn on the FRA settlement if we settle the loan on initiation day, and not on the actual repayment day. Therefore, we tail the FRA settlement by the prevailing market interest rate of 5%. The dollar settlement is:   rannually − rFRA (0.05 − 0.06) × $500,000.00 = −$4,761.905 × notional principal = 1 + rannually 1 + 0.05 b) If the FRA is settled on the date the loan is repaid (or settled in arrears), the settlement amount is determined by:   rannually − rFRA × notional principal = (0.05 − 0.06) × $500,000.00 = −$5,000 We have to pay at the settlement, because the interest rate we could borrow at is 5%, but we have agreed via the FRA to a borrowing rate of 6%. Interest rates moved in an unfavorable direction.

91

Part 2 Forwards, Futures, and Swaps

Question 7.9. a) We have to take into account the interest we (or our counterparty) can earn on the FRA settlement if we settle the loan on initiation day, and not on the actual repayment day. Therefore, we tail the FRA settlement by the prevailing market interest rate of 7.5%. The dollar settlement is:   rannually − rFRA (0.075 − 0.06) × notional principal = × $500,000.00 = $6,976.744 1 + rannually 1 + 0.075 b) If the FRA is settled on the date the loan is repaid (or settled in arrears), the settlement amount is determined by: 

 rannually − rFRA × notional principal = (0.05 − 0.06) × $500,000.00 = −$5,000

We receive money at the settlement, because our hedge pays off. The market interest rate has gone up, making borrowing more expensive. We are compensated for this loss through the insurance that the short position in the FRA provides. Question 7.10. We can find the implied forward rates using the following formula:


 1 + r0 (t, t + s) =

P (0, t) P (0, t + s)

This yields the following rates on the synthetic FRAs: 0.99009 − 1 = 0.010884 0.97943 0.99009 − 1 = 0.025734 r0 (90, 270) = 0.96525 0.99009 − 1 = 0.039596 r0 (90, 360) = 0.95238 r0 (90, 180) =

Question 7.11. We can find the implied forward rate using the following formula:


 1 + r0 (t, t + s) =

P (0, t) P (0, t + s)

With the numbers of the exercise, this yields: r0 (180, 360) =

0.97943 − 1 = 0.028403 0.95238 92

Chapter 7 Interest Rate Forwards and Futures

The following table follows the textbook in looking at forward agreements from a borrower’s perspective, i.e. a borrower goes long an FRA to hedge his position, and a lender is thus short the FRA. Transaction Enter short FRA

t =0

t = 180 −10M

Buy 9.7943M Zero Coupons maturing at time t = 180 Sell (1 + 0.028403) ∗ 10M ∗ 0.95238 Zero Coupons maturing at time t = 360 TOTAL

−9.7943M

+10M

+10M × 1.028403 ×0.95238 = 9.7943M 0

t = 360 +10M × 1.028403 = 10.28403M −10.28403M

0

0

By entering in the above mentioned positions, we are perfectly hedged against the risk of the FRA. Please note that we are making use of the fact that interest rates are perfectly predictable. Question 7.12. We can find the implied forward rate using the following formula:


 1 + r0 (t, t + s) =

P (0, t) P (0, t + s)

With the numbers of the exercise, this yields: r0 (270, 360) =

0.96525 − 1 = 0.0135135 0.95238

The following table follows the textbook in looking at forward agreements from a borrower’s perspective, i.e. a borrower goes long on an FRA to hedge his position, and a lender is thus short the FRA. Since we are the counterparty for a lender, we are in fact the borrower, and thus long the forward rate agreement. Transaction today Enter long FRAU

t =0

t = 270 10M

Sell 9.6525M Zero Coupons maturing at time t = 180 Buy (1 + 0.013514) ∗ 10M ∗ 0.95238 Zero Coupons maturing at time t = 360 TOTAL

9.6525M

−10M

−10M × 1.013514 ×0.95238 = −9.6525M 0

t = 360 −10M × 1.013514 = −10.13514M +10.13514M

0

0

By entering in the above mentioned positions, we are perfectly hedged against the risk of the FRA. Please note that we are making use of the fact that interest rates are perfectly predictable. 93

Part 2 Forwards, Futures, and Swaps

Question 7.13. First, let us calculate the value of the 3-year par coupon bond after we have held it for 2 years. After two years, the bond still entitles us to receive one coupon and the repayment of the principal. We have to discount those payments, which we receive at t = 3, with the implied forward rate from year two to three. This determines the value of the 3-year par coupon bond after 2 years. We have: B2 =

106.95485 = 99.0258 1.0800705

Furthermore, after one year, we received a coupon of 6.95485 dollars, which we reinvested at the prevailing interest rate, the implied forward rate from year one to year two, 1.0700237, and we receive a coupon of 6.95485 at the end of year two. The value of the coupons at the end of year two is: Sum of coupon values = 6.95485 × 1.0700237 + 6.95485 = 14.3967045 Therefore, our two-year gross return is: 2 − year return =

14.3967045 + 99.0258 − 1 = 1.134225 100

Finally, we annualize this return by taking its square root. This yields an annualized gross return of 1.065, which was to be shown. Question 7.14. We would like to guarantee the return of 6.5%. We receive payments 6.95485 after year one and year two, and a payment of 106.95485 after year three. If interest rates are uncertain, we face an interest rate risk for the investment of the first coupon from year one to year two and for the discounting of the final payment from year three to year one. Suppose we enter into a forward rate agreement to lend 6.95484 from year one to year two at the current forward rate from year one to year two, and we enter into a forward rate agreement to borrow 106.95485 tailed by the prevailing forward rate for year two to year three, at the prevailing forward rate. This leads to the following cash-flow table: Transaction today Buy 3-year par bond Receive first coupon Enter short FRAU Receive second coupon Enter long FRA for tailed position

t =0 −100 0 0 0 0

Receive final coupon and principal TOTAL

0 −100

t =1 6.95485 −6.95485

0 94

t =2

t =3

6.95485 × 1.0700237 6.95485 106.95485/1.0800705 −106.95485 = 99.025804 106.95485 113.4225 0

Chapter 7 Interest Rate Forwards and Futures

√ We see that we can secure the same gross return as in the previous question, 113.4225 ÷ 100 = 1.065. By entering appropriate FRAs, we secured the desired return of 6.5%. Please note that we made use of the fact that we knew that we wanted to undo the position at t = 2. Question 7.15. a) Let us follow the suggestion of the problem and buy the 2-year zero-coupon bond. We will create a synthetic lending opportunity at the zero-coupon implied forward rate of 7.00238% and we will finance it by borrowing at 6.8%, thus creating an arbitrage opportunity. In particular, we will have: t =0 t =1 t =2 −0.881659 *1.0700237 0 1.0700237 = −0.943396 Sell 1 one-year zero coupon bond +0.943396 −1 Borrow 1 from year one to year two @ 6.8% +1 −1.06800 TOTAL 0 0 0.0020237

Transaction today Buy 1.0700237 two-year zero-coupon bonds

We see that we have created something out of nothing, without any risk involved. We have found an arbitrage opportunity. b) Let us follow the suggestion of the problem and sell the 2-year zero-coupon bond. We will create a synthetic borrowing opportunity at the zero-coupon implied forward rate of 7.00238% and we will lend at 7.2%, thus creating an arbitrage opportunity. In particular, we will have: Transaction today t =0 t =1 t =2 Sell 1.0700237 two-year zero-coupon bonds 0.881659 *1.0700237 0 −1.0700237 = 0.943396 Buy 1 one-year zero coupon bond −0.943396 +1 Lend 1 from year one to year two @ 7.2% −1 +1.07200 TOTAL 0 0 0.0019763 We see that we have created something out of nothing, without any risk involved. We have indeed found an arbitrage opportunity. Question 7.16. a)

The implied LIBOR of the September Eurodollar futures of 96.4 is:

100 − 96.4 = 0.9% 400

b) As we want to borrow money, we want to buy protection against high interest rates, which means low Eurodollar future prices. We will short the Eurodollar contract. c) One Eurodollar contract is based on a $1 million 3-month deposit. As we want to hedge a loan of $50M, we will enter into 50 short contracts. 95

Part 2 Forwards, Futures, and Swaps

d) A true 3-month LIBOR of 1% means an annualized position (annualized by market conventions) of 1% ∗ 4 = 4%. Therefore, our 50 short contracts will pay:


 96.4 − (100 − 4) × 100 × $25 × 50 = $50,000

The increase in the interest rate has made our loan more expensive. The futures position that we entered to hedge the interest rate exposure, compensates for this increase. In particular, we pay $50,000,000 × 0.01 − payoff futures = $500,000 − $50,000 = $450,000, which corresponds to the 0.9% we sought to lock in. Question 7.17. aa) The interest rate is higher than the rate of the forward rate agreement, therefore the lender must pay the borrower. If the FRA is settled on day 60, the payment made by the lender to the borrower is: (r150 − rFRA ) (0.028 − 0.025) × notional principal = × $100,000,000.00 = $291,828.79 1 + r150 1 + 0.028 ab) If the FRA is settled on the date that the loan is repaid (or settled in arrears), the settlement amount is determined by: (r150 − rFRA ) × notional principal = (0.028 − 0.025) × $100,000,000.00 = $300,000.00 The lender pays the borrower, because we are in the state of the world in which the lender does not need protection: Interest rates have risen, and thus he makes a payment to bring back the interest he earns to 2.5%. ba) The interest rate is lower than the rate of the forward rate agreement, therefore the lender will receive payment from the borrower. If the FRA is settled on day 60, the payment made by the borrower to the lender is: (r150 − rFRA ) (0.022 − 0.025) × $100,000,000.00 = −$293,542.07 × notional principal = 1 + 0.022 1 + r150 bb) If the FRA is settled on the date the loan is repaid (or settled in arrears), the settlement amount is determined by: (r150 − rFRA ) × notional principal = (0.022 − 0.025) × $100,000,000.00 = −$300,000.00 The lender is paid by the borrower, because we are in the state of the world in which the lender’s protection pays off: Interest rates have gone down, and thus she is compensated for the loss in investment proceeds. The payment of the borrower brings back the interest she earns to 2.5%. 96

Chapter 7 Interest Rate Forwards and Futures

Question 7.18. a) We face the classic problem of asset mismatch. We are interested in locking in an interest rate for a 150-day investment, 60 days from now. However, while the Eurodollar futures matures 60 days from now, it secures a lending rate for 90 days. We face the problem that the 90-day and 150-day interest rates may not be perfectly correlated. (For example, the term structure could, over the next 60 days, move from upward sloping to downward sloping). b) As we want to lend money, we want to buy protection against low interest rates, which means high Eurodollar future prices. We will therefore long the Eurodollar contract. 100 − 94 c) The implied LIBOR of the September Eurodollar futures of 94 is: = 1.5%. Under 400 the assumption that the 3-month LIBOR rate and the 150-day interest rate are based on the same 150 annualized interest rate of 6%, we are able to lock in an interest rate of: 1.5% × = 2.5%. Please 90 note that this is a rather strong assumption. d) One Eurodollar futures contract is based on a $1 million 3-month deposit. As we want to hedge an investment of $100M, we will enter into 100 long contracts. Again, we are making the strong assumption that the annualized 3-month LIBOR rate and the annualized 150 day rate are identical and perfectly correlated. Question 7.19. We will calculate the necessary different bond prices according to the formula: B(y) =

n 

Ci

i=1

(1 + y)i

Ci are the coupon payments, and for i = n, it includes the payment of the principal. This yields the following bond prices, and the price difference with respect to the base case of a yield of 7%: Seven-year, 6% coupon bond: Yield 6.5% 6.75% 7% 7.25% 7.5%

Price 97.2577 95.9226 94.6107 93.3217 92.0551

Price change relative to 7% yield price 2.647 1.3119 0 −1.289 −2.5556

97

Part 2 Forwards, Futures, and Swaps

Ten-year, 8% coupon bond: Yield 7% 7.25% 7.5% 7.75% 8%

Price Price change relative to 7.5% yield price 107.0236 3.5916 105.2073 1.7753 103.4320 0 101.6966 −1.7354 100.0000 −3.432

We are now able to solve for the true hedge ratio by equating the price changes corresponding to the yield changes. We solve: price change bond1 = −N × price change bond2 This gives for: an increase of 25 basis points: N = −0.74277 a decrease of 25 basis points:

N = −0.73897

an increase of 50 basis points: N = −0.7446 a decrease of 50 basis points:

N = −0.737

We see that the hedge ratios for increases and decreases of the same number of basis points differ. Note as well that the difference between the hedge ratios increases with an increase in the basis points. This is a consequence of the convexity of the bonds. It is caused by the changes in duration as the interest rate changes. Question 7.20. We will use the Excel functions Duration and Mduration to calculate the required durations. They are of the form: MDURATION(Start Date; Terminal Date; Coupon; Yield; Frequency) DURATION(Start Date; Terminal Date; Coupon; Yield; Frequency), where frequency determines the number of coupon payments per year. In order to use the function, we have to give Excel a start date and terminal date, but we can just pick two dates that are exactly the requested number of years apart. Plugging in the values of the exercises yields: a)

Macaulay Duration = 4.59324084 Modified Duration = 4.3983078

b)

Macaulay Duration = 5.99377496 Modified Duration = 5.73566981 98

Chapter 7 Interest Rate Forwards and Futures

c) We need to find the yield to maturity of this bond first. We can do so by using the YIELD function of Excel. Plugging in the relevant values, we get: Yield = 0.07146759. Now we can again use the Mduration and Duration formulas. This yields: Macaulay Duration = 7.6955970 Modified Duration = 7.1822957 Question 7.21. b) Let us start with part b), because we already know the Excel function for the Macaulay duration, DURATION(). Using the equation with DURATION(01/01/1980; 01/01/2000; 0.06; 0.20; 2), and DURATION (01/01/1970; 01/01/2000; 0.06; 0.20; 2) yields for the 20-year bond: Macaulay Duration = 6.09533079 and for the 30-year bond: Macaulay Duration = 5.66839682 a) Now we can back out the price value of a basis point by multiplying the Macaulay duration by −B(y)/(1 + y). In order to do so, we must find the prices of the two bonds. As the yield to maturity is given, we simply have: B(y) =

n 

Ci

i=1

(1 + y)i

This yields for the two bonds: B(6, 20y) = 31.546645 B(6, 30y) = 30.229899 Therefore, we have the following PVPBs: 31.546645 = −181.4031 1.06 30.229899 PVBP(30 years) = −5.66839682 × = −161.6557 1.06 PVBP(20 years) = −6.09533079 ×

c) As we can see from the above example, this statement is not always true. The above example is an extreme one in which the yield to maturity is extremely high. Since the Macaulay duration is a weighted average of the time until the bond payments occur, with the weights being the percentage 99

Part 2 Forwards, Futures, and Swaps

of the bond price accounted for by each payment, we see that with a very high yield, the last payments get significantly more discounted than the previous ones. For a coupon bond, the last payment is the largest one, being interest plus principal. Therefore, with a high yield, the largest payment of a long term bond gets a higher discount than a bond with the same characteristics but a shorter maturity. Overall, this can make the duration of a long-term bond smaller than that of a short-term one. Question 7.22. We will exploit equation (7.13) of the main text to find the optimal hedge ratio: N =−

D1 B1 (y1 ) / (1 + y1 ) 6.631864 × 106.44/ (1.05004) 672.255918 = =− = −0.887707 D2 B2 (y2 ) / (1 + y2 ) 7.098302 × 112.29/ (1.05252) 757.2951883

Therefore, we have to short 0.887707 units of the nine-year bond for every eight-year bond to obtain a duration-matched portfolio. Question 7.23. We can verify the conversion factor for the 6-year semi-annual bond by calculating in Excel: PRICE(1/1/94, 1/1/00, 0.04, 0.06, 100, 2) = 90.045996 For the eight year bond, we calculate a conversion factor of: PRICE(1/1/92, 1/1/00, 0.055, 0.06, 100, 2) = 96.8597245 Note that those Excel bond values are for $100 par, so the conversion factors are obtained by dividing the above results by 100. Now, we are in a position to calculate the difference between the invoice price and the market price, using the futures price of 117.02. 6-year bond: 117.02 × 0.90046 − 102.48 = 2.8961 8-year bond: 117.02 × 0.96860 − 113.564 = −0.2188 We see that it is more advantageous for the short position to deliver the 6-year bond, as the owner of the short position can make a small profit by delivering the 6-year bond. He would lose money if he delivered the 8-year bond.

100

Chapter 7 Interest Rate Forwards and Futures

Question 7.24. a) Compute the convexity of a 3-year bond paying annual coupons of 4.5% and selling at par. For a par bond, the yield to maturity is equal to the coupon, or 4.5% in our case. We can calculate the convexity based on the formula:  n n (n + 1) i (i + 1) C/m M 1 + Convexity = i=1 B(y) m2 (1 + y/m)i+2 m2 (1 + y/m)n+2  4.5/1 4.5/1 2 (2 + 1) 1 1 (1 + 1) + = 1+2 2 2 100 1 1 (1 + 0.045) (1 + 0.045)2+2 104.5 3 (3 + 1) + 2 1 (1 + 0.045/1)3+2 = 10.3680 b) Compute the convexity of a 3-year 4.5% coupon bond that makes semiannual coupon payments and that currently sells at par. Now, m = 2. We have n = m ∗ T = 2 ∗ 3 = 6. The convexity is:  n n (n + 1) i (i + 1) C/m M 1 + Convexity = i=1 B(y) m2 (1 + y/m)i+2 m2 (1 + y/m)n+2  2.25/1 2.25/1 2 (2 + 1) 1 1 (1 + 1) + = 1+2 2 2 100 2 2 (1 + 0.0225) (1 + 0.0225)2+2 2.25/1 2.25/1 3 (3 + 1) 4 (4 + 1) + + 3+2 2 2 2 2 (1 + 0.0225) (1 + 0.0225)4+2 102.25 6 (6 + 1) 2.25/1 5 (5 + 1) + + 5+2 2 2 2 2 (1 + 0.0225) (1 + 0.0225/1)6+2 = 9.3302 c) Is the convexity different in the two cases? Why? Yes, the convexity for the semi-annual bond is smaller. We spread the bond payments out over more periods, which makes the bond’s duration slightly less susceptible to interest rate changes. Question 7.25. Suppose a 10-year zero coupon bond with a face value of $100 trades at $69.20205. a)

What is the yield to maturity and modified duration of the zero-coupon bond?

101

Part 2 Forwards, Futures, and Swaps

The yield of the zero-coupon bond is equal to 100 P0 (0, 10) 1/10

100 −1 = 69.20205 = 0.03750

(1 + y)10 = y

Calculate its modified duration. The modified duration of the ten-year zero coupon bond is:  n 1 n 1 C/m M i modified duration = + × i=1 m (1 + y/m)i m (1 + y/m)n B (y) 1 + y/m  1 10 100 1 × = 69.20205 1 + 0.0375 1 (1 + 0.0375)10 10 = 9.63855 = 1.0375 b) Calculate the approximate bond price change for a 50 basis point increase in the yield, based on the modified duration you calculated in part a). Also calculate the exact new bond price based on the new yield to maturity. We can use the formula in the main text: B (y + ε) = B (y) − [DMod × B (y) ε] = 69.20205 − 9.63855 × 69.20205 × 0.0050 B (0.0425) = 65.86701 New bond price: B (0.0425) = c)

100 = 66.01703 1.042410

Calculate the convexity of the 10-year zero-coupon bond.  n i (i + 1) C/m M n (n + 1) 1 + Convexity = i=1 B(y) m2 (1 + y/m)i+2 m2 (1 + y/m)n+2  10 (10 + 1) 100 1 0+ = 69.20205 12 (1 + 0.0375/1)10+2 = 102.19190

102

Chapter 7 Interest Rate Forwards and Futures

d) Now use the formula (equation 7.15) that takes into account both duration and convexity to approximate the new bond price. Compare your result to that in part b). B (y + ε) = B (y) − [DMod × B (y) ε] + 0.5 × Convexity × B (y) × ε 2 = 69.20205 − 9.63855 × 69.20205 × 0.0050 + 0.5 × 102.19190 × 69.20205 ×0.00502 B (0.0515) = 65.86701 + 0.088299 = 65.95541 The approximation is much better now compared to the result in part b), but it is still somewhat off the true price. The long time to maturity and the considerable change in basis points for this bond is responsible for the deviation.

103

Chapter 8 Swaps Question 8.1. We first solve for the present value of the cost per two barrels: $22 $23 = 41.033. + 1.06 (1.065)2 We then obtain the swap price per barrel by solving: x x = 41.033 + 1.06 (1.065)2 ⇔ x = 22.483, which was to be shown. Question 8.2. a)

We first solve for the present value of the cost per three barrels, based on the forward prices: $22 $21 $20 + = 55.3413. + 2 1.06 (1.065) (1.07)3

We then obtain the swap price per barrel by solving: x x x + + = 55.341 2 1.06 (1.065) (1.07)3 ⇔ x = 20.9519 b)

We first solve for the present value of the cost per two barrels (year 2 and year 3): $21 (1.065)2

+

$22 (1.07)3

= 36.473.

We then obtain the swap price per barrel by solving: x 2

(1.065) ⇔ x

+

x (1.07)3

= 36.473 = 21.481

104

Chapter 8 Swaps

Question 8.3. Since the dealer is paying fixed and receiving floating, she generates the cash-flows depicted in column 2. Suppose that the dealer enters into three short forward positions, one contract for each year of the active swap. Her payoffs are depicted in columns 3, and the aggregate net cash flow position is in column 4. Year Net Swap Payment Short Forwards Net Position 1 S1 − $20.9519 $20 − S1 −0.9519 2 S1 − $20.9519 $21 − S1 +0.0481 3 S1 − $20.9519 $22 − S1 +1.0481 We need to discount the net positions to year zero. We have: P V (netCF ) =

−0.9519 0.0481 1.0481 + + = 0. 1.06 (1.065)2 (1.07)3

Indeed, the present value of the net cash flow is zero. Question 8.4. The fair swap rate was determined to be $20.952. Therefore, compared to the forward curve price of $20 in one year, we are overpaying $0.952. In year two, this overpayment has increased to $0.952 × 1.070024 = 1.01866, where we used the appropriate forward rate to calculate the interest payment. In year two, we underpay by $0.048, so that our total accumulative underpayment is $0.97066. In year three, this overpayment has increased again to $0.97066 × 1.08007 = 1.048. However, in year three, we receive a fixed payment of 20.952, which underpays relative to the forward curve price of $22 by $22 − $20.952 = 1.048. Therefore, our cumulative balance is indeed zero, which was to be shown. Question 8.5. Since the dealer is paying fixed and receiving floating, she generates the cash-flows depicted in column 2. Suppose that the dealer enters into three short forward positions, one contract for each year of the active swap. Her payoffs are depicted in columns 3, and the aggregate net position is summarized in column 4. Year Net Swap Payment Short Forwards Net Position −0.952 $20 − S1 1 S1 − $20.952 2 S1 − $20.952 +0.048 $21 − S1 3 S1 − $20.952 $22 − S1 +1.048

105

Part 2 Forwards, Futures, and Swaps

We need to discount the net positions to year zero, taking into account the uniform shift of the term structure. We have: P V (netCF ) =

1.0481 0.0481 −0.9519 + = −0.0081. + 2 1.065 (1.07) (1.075)3

The present value of the net cash flow is negative: The dealer never recovers from the increased interest rate he faces on the overpayment of the first swap payment. P V (netCF ) =

1.0481 0.0481 −0.9519 + = +0.0083 + 1.055 (1.06)2 (1.065)3

The present value of the net cash flow is positive. The dealer makes money, because he gets a favorable interest rate on the loan he needs to take to finance the first overpayment. The dealer could have tried to hedge his exposure with a forward rate agreement or any other derivative protecting against interest rate risk. Question 8.6. In order to answer this question, we use equation (8.13.) of the main text. We assumed that the interest rates and the corresponding zero-coupon bonds were: Quarter 1 2 3 4 5 6 7 8

Interest rate 0.0150 0.0302 0.0457 0.0614 0.0773 0.0934 0.1098 0.1265

Zero-coupon price 0.9852 0.9707 0.9563 0.9422 0.9283 0.9145 0.9010 0.8877

Using formula 8.13., we obtain the following per barrel swap prices: 4-quarter swap price: $20.8533 8-quarter swap price: $20.4284 The total costs of prepaid 4- and 8-quarter swaps are the present values of the payment obligations. They are: 4-quarter prepaid swap price: $80.3768 8-quarter prepaid swap price: $152.9256 106

Chapter 8 Swaps

Question 8.7. Using formula 8.13., and plugging in the given zero-coupon prices and the given forward prices, we obtain the following per barrel swap prices: Quarter 1 2 3 4 5 6 7 8

Zero-bond 0.9852 0.9701 0.9546 0.9388 0.9231 0.9075 0.8919 0.8763

Swap price 21.0000 21.0496 20.9677 20.8536 20.7272 20.6110 20.5145 20.4304

Question 8.8. We use formula (8.4), and replace the forward interest rate with the forward oil prices. In particular, we calculate:
6 X=

i=3 P0 (0, ti )F0,ti


6

i=3 P0 (0, ti )

= $20.3807

Therefore, the swap price of a 4-quarter oil swap with the first settlement occurring in the third quarter is $20.3807. Question 8.9. Using the 8-quarter swap price of $20.43, we can calculate the net position by subtracting the swap price from the forward prices. The 1-quarter implied forward rate is calculated from the zero-coupon bond prices. The column implicit loan balance adds the net position of each quarter and the implicit loan balance plus interest of the previous quarter. Please note that the forward curve is inverted—we are initially loaning money because the swap price is lower than the forward price. Quarter 1 2 3 4 5 6 7 8

Net position 0.5696 0.6696 0.3696 0.0696 −0.2304 −0.4304 −0.5304 −0.6304

Implied forward rate 1.0150 1.0156 1.0162 1.0168 1.0170 1.0172 1.0175 1.0178 107

Implicit loan balance 0.5696 1.2480 1.6379 1.7350 1.5341 1.1300 0.6194 0.0000

Part 2 Forwards, Futures, and Swaps

Question 8.10. We use equation (8.6) of the main text to answer this question:
8 X=

i=1 Qti P0 (0, ti )F0,ti


8

i=1 Qti P0 (0, ti )

, whereQ = [1, 2, 1, 2, 1, 2, 1, 2]

After plugging in the relevant variables given in the exercise, we obtain a value of $20.4099 for the swap price. Question 8.11. We are now asked to invert our equations. The swap prices are given, and we want to back out the forward prices. We do so recursively. For a one-quarter swap, the swap price and the forward price are identical. Given the one-quarter forward price, we can find the second quarter forward price, etc. Doing so yields the following forward prices: Quarter 1 2 3 4 5 6 7 8

Forward price 2.25 2.60 2.20 1.90 2.20 2.50 2.15 1.80

Question 8.12. With a swap price of $2.2044, and the forward prices of question 8.11., we can calculate the implied loan amount. We can calculate the net position by subtracting the swap price from the forward prices. The 1-quarter implied forward rate is calculated from the zero-coupon bond prices. The column implicit loan balance adds the net position each quarter and the implicit loan balance plus interest of the previous quarter. Please note that the shape of the forward curve—we are initially loaning money, because the swap price is lower than the forward price.

108

Chapter 8 Swaps

Quarter 1 2 3 4 5 6 7 8

Forward price 2.25 2.60 2.20 1.90 2.20 2.50 2.15 1.80

Net balance 0.0456 0.3955 −0.0042 −0.3046 −0.0043 0.2954 −0.0543 −0.4042

Forward interest rate 1.0150 1.0156 1.0162 1.0168 1.0170 1.0172 1.0175 1.0178

Implicit loan balance 0.0456 0.4418 0.4447 0.1476 0.1458 0.4437 0.3971 0.0000

Question 8.13. From the given zero-coupon bond prices, we can calculate the one-quarter forward interest rates. They are: Quarter 1 2 3 4 5 6 7 8

Forward interest rate 1.0150 1.0156 1.0162 1.0168 1.0170 1.0172 1.0175 1.0178

Now, we can calculate the deferred swap price according to the formula:
6 X=

i=2 P0 (0, ti )r0 (ti−1 , ti )
6 i=2 P0 (0, ti )

= 1.66%

Question 8.14. From the given zero-coupon bond prices, we can calculate the one-quarter forward interest rates. They are: Quarter 1 2 3 4 5 6 7 8

Forward interest rate 1.0150 1.0156 1.0162 1.0168 1.0170 1.0172 1.0175 1.0178 109

Part 2 Forwards, Futures, and Swaps

Now, we can calculate the swap prices for 4 and 8 quarters according to the formula: X=


n

i=1 P0 (0, ti )r0 (ti−1 , ti )
n , i=1 P0 (0, ti )

where n = 4 or 8

This yields the following prices: 4-quarter fixed swap price: 1.59015% 8-quarter fixed swap price: 1.66096% Question 8.15. We can calculate the value of an 8-quarter dollar annuity that is equivalent to an 8-quarter Euro annuity by using equation 8.8. of the main text. We have:
8 X=

∗ i=1 P0 (0, ti )R F0,ti
8 i=1 P0 (0, ti ) ,

where R ∗ is the Euro annuity of 1 Euro.

Plugging in the forward price for one unit of Euros delivered at time ti , which are given in the price table, yields a dollar annuity value of $0.9277. Question 8.16. The dollar zero-coupon bond prices for the three years are: P0,1 = P0,2 = P0,3 =

1 1.06 1 (1.06)2 1 (1.06)3

= 0.9434 = 0.8900 = 0.8396

R ∗ is 0.035, the Euro-bond coupon rate. The current exchange rate is 0.9$/E. Plugging all the above variables into formula (8.9) indeed yields 0.06, the dollar coupon rate:
3 R=

∗F

0,ti /x0 + P0,3
3 i=1 P0 (0, ti )

i=1 P0 (0, ti )R





F0,tn /x0 − 1

=

0.098055 + 0.062317 = 0.060 2.673

Question 8.17. We can use the standard swap price formula for this exercise, but we must pay attention to taking the right zero-coupon bonds, and the right Euro-denominated forward interest rates. From the given Euro zero-coupon bond prices, we can calculate the one-quarter forward interest rates. They are: 110

Chapter 8 Swaps

Quarter 1 2 3 4 5 6 7 8

Euro denominated implied forward interest rate 0.0088 0.0090 0.0092 0.0095 0.0096 0.0097 0.0098 0.0100

Now, we can calculate the swap prices for 4 and 8 quarters according to the formula: X=


n

i=1 P0 (0, ti )r0 (ti−1 , ti )
n i=1 P0 (0, ti ) ,

where n = 4 or 8

This yields the following prices: 4-quarter fixed swap price: 0.91267% 8-quarter fixed swap price: 0.94572% Question 8.18. We can use equation (8.9), but there is a complication: We do not have the current spot exchange rate. However, it is possible to back it out by using the methodology of the previous chapters: We know that the following relation must hold: ∗

F0,1 = X0 e(r−r ) , where the interest rates are already on a quarterly level. We can back out the interest rates from the given zero-coupon prices. Doing so yields a current exchange rate of 0.90 $/Euro. R∗ is the 8-quarter fixed swap price payment of 0.0094572. By plugging in all the relevant variables into equation 8.9, we can indeed see that this yields a swap rate of 1.66%, which is the same rate that we calculated in exercise 8.14.

111

Chapter 9 Parity and Other Option Relationships Question 9.1. This problem requires the application of put-call-parity. We have: P (35, 0.5) = C (35, 0.5) − e−δT S0 + e−rT 35 ⇔ P (35, 0.5) = $2.27 − e−0.06×0.5 32 + e−0.04×0.5 35 = $5.523. Question 9.2. This problem requires the application of put-call-parity. We have: S0 − C (30, 0.5) + P (30, 0.5) − e−rT 30 = P V (dividends) ⇔ P V (dividends) = 32 − 4.29 + 2.64 − 29.406 = $0.944. Question 9.3. a)

We can calculate an initial investment of: −800 + 75 − 45 = −770.

This position yields $815 after one year for sure, because either the sold call commitment or the bought put cancel out the stock price. Therefore, we have a one-year rate of return of 0.05844, which is equivalent to a continuously compounded rate of: 0.05680. b) We have a risk-less position in a) that pays more than the risk-free rate. Therefore, we should borrow money at 5%, and buy a large amount of the aggregate position of a), yielding a sure return of .68%. c) An initial investment of $775.252 would yield $815 after one year, invested at the riskless rate of return of 5%. Therefore, the difference between call and put prices should be equal to $24.748. d)

Following the same argument as in c), we obtain the following results: Strike Price 780 800 820 840 112

Call-Put 58.04105 39.01646 19.99187 0.967283

Chapter 9 Parity and Other Option Relationships

Question 9.4. We can make use of the put-call-parity for currency options: +P (K, T ) = −e−rf T x0 + C (K, T ) + e−rT K ⇔ P (K, T ) = −e−0.04 0.95 + 0.0571 + e−0.06 .93 = −0.91275 + 0.0571 + 0.87584 = 0.0202. A $0.93 strike European put option has a value of $0.0202. Question 9.5. The payoff of the one-year yen-denominated put on the euro is max[0, 100 − x1Y/E], where x1 is the future uncertain Y/E exchange rate. The payoff of the corresponding one-year Euro-denominated call on yen is max[0, 1/x1 − 1/100]. Its premium is, using equation (9.7),  1 1 PY (x0 , K, T ) = x0 KCE , ,T x0 K 
PY (x0 , K, T ) 1 1 , ,T = ⇔ CE x0 K x0 K 
8.763 1 1 , ,1 = = 0.00092242 ⇔ CE 95 100 9500


Question 9.6. a)

We can use put-call-parity to determine the forward price:

+ C (K, T ) − P (K, T ) = P V (f orward price) − P V (strike) = e−rT F0,T − Ke−rT   ⇔ F0,T = erT +C (K, T ) − P (K, T ) + Ke−rT   = e0.05∗0.5 $0.0404 − $0.0141 + $0.9e−0.05∗0.5 ⇔ F0,T = $0.92697. b) Given the forward price from above and the pricing formula for the forward price, we can find the current spot rate: F0,T = x0 e(r−rf )T ⇔ x0 = F0,T e−(r−rf )T = $0.92697e−(0.05−0.035)0.5 = $0.92.

113

Part 3 Options

Question 9.7. a) We make use of the version of the put-call-parity that can be applied to currency options. We have: + P (K, T ) = −e−rf T x0 + C (K, T ) + e−rT K ⇔

P (K, T ) = −e−0.01 0.009 + 0.0006 + e−0.05 .009 = −0.00891045 + 0.0006 + 0.00856106 = 0.00025.

b) The observed option price is too high. Therefore, we sell the put option and synthetically create a long put option, perfectly offsetting the risks involved. We have: Transaction Sell Put Buy Call Sell e−rf Spot Lend PV(strike) Total

t =0 0.0004 −0.0006 0.009e−0.001 = 0.00891 −0.00856 0.00015

t = T, x < K −(K − x) = x − K 0 −x K 0

t = T, x > K 0 x−K −x K 0

We have thus demonstrated the arbitrage opportunity. c) We can use formula (9.7.) to determine the value of the corresponding Yen-denominated at-the-money put, and then use put-call-parity to figure out the price of the Yen-denominated atthe-money call option.  1 1 C$ (x0 , K, T ) = x0 KPY , ,T x0 K 
C$ (x0 , K, T ) 1 1 , ,T = ⇔ PY x0 K x0 K 
1 0.0006 1 , ,T = = 7.4074Y ⇔ PY 0.009 0.009 (0.009)2


Now, we can use put-call-parity, carefully handling the interest rates (since we are now making transactions in Yen, the $-interest rate is the foreign rate): CY (K, T ) = e−rf T x0 + PY (K, T ) − e−rT K 
1 1 1 , 1 = e−0.01 + 7.4074 − e−0.05 = 110.0055 + 7.4074 − 105.692 ⇔ C 0.009 0.009 0.009 = 11.7207. We have used the direct relationship between the yen-denominated dollar put and our answer to a) and the put-call-parity to find the answer. 114

Chapter 9 Parity and Other Option Relationships

Question 9.8. Both equations (9.13) and (9.14) are violated. We use a call bull spread and a put bear spread to profit from these arbitrage opportunities.

Transaction t =0 Buy 50 strike call −9 Sell 55 strike call +10 TOTAL +1

Expiration or Exercise ST < 50 50 ≤ ST ≤ 55 ST > 55 0 ST − 50 ST − 50 0 0 55 − ST 0 ST − 50 > 0 5>0

Expiration or Exercise Transaction t = 0 ST < 50 50 ≤ ST ≤ 55 ST > 55 Buy 55 strike put −6 55 − ST 55 − ST 0 Sell 50 strike put 7 ST − 50 0 0 TOTAL +1 5>0 55 − ST > 0 0 Please note that we initially receive money, and that at expiration the profit is non-negative. We have found arbitrage opportunities. Question 9.9. Both equations (9.15) and (9.16) of the textbook are violated. We use a call bear spread and a put bull spread to profit from these arbitrage opportunities. Expiration or Exercise Transaction t = 0 ST < 50 50 ≤ ST ≤ 55 ST > 55 Buy 55 strike call −10 0 0 ST − 55 Sell 50 strike call +16 0 50 − ST 50 − ST TOTAL +6 0 50 − ST > −5 −5 Expiration or Exercise Transaction t = 0 ST < 50 50 ≤ ST ≤ 55 ST > 55 Buy 50 strike put −7 50 − ST 0 0 Sell 55 strike put 14 ST − 55 ST − 55 0 TOTAL +7 −5 ST − 55 > −5 0 Please note that we initially receive more money than our biggest possible exposure in the future. Therefore, we have found an arbitrage possibility, independent of the prevailing interest rate.

115

Part 3 Options

Question 9.10. Both equations (9.17) and (9.18) of the textbook are violated. To see this, let us calculate the values. We have: C (K1 ) − C (K2 ) 18 − 14 = 0.8 = K2 − K 1 55 − 50

and

C (K2 ) − C (K3 ) 14 − 9.50 = 0.9, = K3 − K2 60 − 55

which violates equation (9.17) and P (K2 ) − P (K1 ) 10.75 − 7 = = 0.75 K2 − K 1 55 − 50

and

14.45 − 10.75 P (K3 ) − P (K2 ) = = 0.74, K3 − K2 60 − 55

which violates equation (9.18). We calculate lambda in order to know how many options to buy and sell when we construct the butterfly spread that exploits this form of mispricing. Because the strike prices are symmetric around 55, lambda is equal to 0.5. Therefore, we use a call and put butterfly spread to profit from these arbitrage opportunities. Transaction Buy 1 50 strike call Sell 2 55 strike calls Buy 1 60 strike call TOTAL

t = 0 ST < 50 −18 0 +28 0 −9.50 0 +0.50 0

50 ≤ ST ≤ 55 55 ≤ ST ≤ 60 ST > 60 ST − 50 ST − 50 ST − 50 0 110 − 2 × ST 110 − 2 × ST 0 0 ST − 60 ST − 50 ≥ 0 60 − ST ≥ 0 0

50 ≤ ST ≤ 55 55 ≤ ST ≤ 60 ST > 60 Transaction t =0 ST < 50 0 0 0 Buy 1 50 strike put −7 50 − ST Sell 2 55 strike puts 21.50 2 × ST − 110 2 × ST − 110 0 0 Buy 1 60 strike put −14.45 60 − ST 60 − ST 60 − ST 0 TOTAL +0.05 0 ST − 50 ≥ 0 60 − ST ≥ 0 0 Please note that we initially receive money and have non-negative future payoffs. Therefore, we have found an arbitrage possibility, independent of the prevailing interest rate. Question 9.11. Both equations (9.17) and (9.18) of the textbook are violated. To see this, let us calculate the values. We have: C (K1 ) − C (K2 ) 22 − 9 = 0.65 = 100 − 80 K2 − K1

and

116

C (K2 ) − C (K3 ) 9−5 = 0.8, = K3 − K2 105 − 100

Chapter 9 Parity and Other Option Relationships

which violates equation (9.17) and P (K2 ) − P (K1 ) 21 − 4 = = 0.85 K2 − K1 100 − 80

and

24.80 − 21 P (K3 ) − P (K2 ) = = 0.76, K3 − K2 105 − 100

which violates equation (9.18). We calculate lambda in order to know how many options to buy and sell when we construct the butterfly spread that exploits this form of mispricing. Using formula (9.19), we can calculate that lambda is equal to 0.2. To buy and sell round lots, we multiply all the option trades by 5. We use an asymmetric call and put butterfly spread to profit from these arbitrage opportunities. Transaction Buy 2 80 strike calls Sell 10 100 strike calls Buy 8 105 strike calls TOTAL

t =0 −44 +90 −40 +6

Transaction Buy 2 80 strike puts Sell 10 100 strike puts Buy 8 105 strike puts TOTAL

t =0 −8 +210 −198.4 +3.6

ST < 80 0 0 0 0

80 ≤ ST ≤ 100 2 × ST − 160 0 0 2 × ST − 160 > 0

ST < 80 160 − 2 × ST 10 × ST − 1000 840 − 8 × ST 0

100 ≤ ST ≤ 105 2 × ST − 160 1000 − 10 × ST 0 840 − 8 × ST ≥ 0

80 ≤ ST ≤ 100 0 10 × ST − 1000 840 − 8 × ST 2 × ST − 160 > 0

ST > 105 2 × ST − 160 1000 − 10 × ST 8 × ST − 840 0

100 ≤ ST ≤ 105 0 0 840 − 8 × ST 840 − 8 × ST ≥ 0

ST > 105 0 0 0 0

Please note that we initially receive money and have non-negative future payoffs. Therefore, we have found an arbitrage possibility, independent of the prevailing interest rate. Question 9.12. a) Equation (9.15) of the textbook is violated. We use a call bear spread to profit from this arbitrage opportunity. Transaction t = 0 ST Sell 90 strike call +10 0 Buy 95 strike call −4 0 TOTAL +6 0

Expiration or Exercise < 90 90 ≤ ST ≤ 95 ST > 95 90 − ST 90 − ST 0 ST − 95 90 − ST > −5 −5

Please note that we initially receive more money than our biggest possible exposure in the future. Therefore, we have found an arbitrage possibility, independent of the prevailing interest rate. b) Now, equation (9.15) is not violated anymore. However, we can still construct an arbitrage opportunity, given the information in the exercise. We continue to sell the 90-strike call and buy the 95-strike call, and we loan our initial positive net balance for two years until expiration. It is 117

Part 3 Options

important that the options be European, because otherwise we would not be able to tell whether the 90-strike call could be exercised against us sometime (note that we do not have information regarding any dividends). We have the following arbitrage table: Expiration t = T Transaction t = 0 ST < 90 90 ≤ ST ≤ 95 Sell 90 strike call +10 0 90 − ST Buy 95 strike call −5.25 0 0 Loan 4.75 −4.75 5.80 5.80 TOTAL 0 5.80 95.8 − ST > 0

ST > 95 90 − ST ST − 95 5.8 +0.8

In all possible future states, we have a strictly positive payoff. We have created something out of nothing—we demonstrated arbitrage. c)

We will first verify that equation (9.17) is violated. We have: C (K1 ) − C (K2 ) 15 − 10 = 0.5 = K2 − K1 100 − 90

and

C (K2 ) − C (K3 ) 10 − 6 = 0.8, = K3 − K2 105 − 100

which violates equation (9.17). We calculate lambda in order to know how many options to buy and sell when we construct the butterfly spread that exploits this form of mispricing. Using formula (9.19), we can calculate that lambda is equal to 1/3. To buy and sell round lots, we multiply all the option trades by 3. We use an asymmetric call and put butterfly spread to profit from these arbitrage opportunities. Transaction t =0 Buy 1 90 strike calls −15 Sell 3 100 strike calls +30 Buy 2 105 strike calls −12 TOTAL +3

ST < 90 90 ≤ ST ≤ 100 100 ≤ ST ≤ 105 ST > 105 ST − 90 ST − 90 0 ST − 90 0 0 300 − 3 × ST 300 − 3 × ST 0 0 0 2 × ST − 210 0 ST − 90 ≥ 0 210 − 2 × ST ≥ 0 0

We indeed have an arbitrage opportunity. Question 9.13. We have to think carefully about the benefits and costs associated with early exercise. For the American put option, the usual cost associated with early exercise is that we give up the interest we could earn on the strike if we continued to hold the option. However, when the interest rate is zero, no interest is lost by waiting until maturity to exercise, but we have a volatility benefit from waiting, so the American put is equivalent to a European put. We will never use the early exercise feature of the American put option.

118

Chapter 9 Parity and Other Option Relationships

For the American call option, dividends on the stock are the reason why we want to receive the stock earlier, and we benefit from waiting, because we can continue to earn interest on the strike. Now, the interest rate is zero, so we do not have this benefit associated with waiting to exercise. However, we saw that there is a second benefit to waiting: the insurance protection, which will not be affected by the zero interest rate. Finally, we will not exercise the option if it is out-of-the-money. Therefore, there may be circumstances in which we will early exercise, but we will not always early exercise. Question 9.14. This question is closely related to question 9.13. In this exercise, the strike is not cash anymore, but rather one share of Apple. In parts a) and b), there is no benefit in keeping Apple longer, because the dividend is zero. a) The underlying asset is the stock of Apple, which does not pay a dividend. Therefore, we have an American call option on a non-dividend-paying stock. It is never optimal to early exercise such an option. b) The underlying asset is the stock of Apple, and the strike consists of AOL. As AOL does not pay a dividend, the interest rate on AOL is zero. We will therefore never early exercise the put option, because we cannot receive earlier any benefits associated with holding Apple – there are none. If Apple is bankrupt, there is no loss from not early exercising, because the option is worth max[0, AOL – 0], which is equivalent to one share of AOL, because of the limited liability of stock. As AOL does not pay dividends, we are indifferent between holding the option and the stock. c) For the American call option, dividends on the stock are the reason why we want to receive the stock earlier, and now Apple pays a dividend. We usually benefit from waiting, because we can continue to earn interest on the strike. However, in this case, the dividend on AOL remains zero, so we do not have this benefit associated with waiting to exercise. Finally, we saw that there is a second benefit to waiting: the insurance protection, which will not be affected by the zero AOL dividend. Therefore, there now may be circumstances in which we will early exercise, but we will not always early exercise. For the American put option, there is no cost associated with waiting to exercise the option, because exercising gives us a share of AOL, which does not pay interest in form of a dividend. However, by early exercising we will forego the interest we could earn on Apple. Therefore, it is again never optimal to exercise the American put option early. Question 9.15. We demonstrate that these prices permit an arbitrage. We buy the cheap option, the longer lived one-and-a-half year call, and sell the expensive one, the one-year option. Let us consider the two possible scenarios. First, let us assume that St < 105.127. Then, we have the following table: 119

Part 3 Options

Transaction t =0 time = t Sell 105.127 strike call +11.924 0 Buy 107.788 strike call −11.50 0 TOTAL 0.424 0

time = T ST ≤ 107.788 — 0 0

time = T ST > 107.788 — ST − 107.788 ST − 107.788

Clearly, there is no arbitrage opportunity. However, we will need to check the other possibility as well, namely that St ≥ 105.127. This yields the following no-arbitrage table:

Transaction Sell 105.127 strike call Keep stock Loan 105.127 @ 5% for 0.5 years Buy 107.788 strike call TOTAL

t =0 +11.924 −11.50 0.424

time = t 105.127 − St +St −105.127 0 0

time = T ST ≤ 107.788 — −ST +107.788 0 107.788 − ST ≥ 0

time = T ST > 107.788 — −ST +107.788 ST − 107.788 0

We have thus shown that in all states of the world, there exist an initial positive payoff and nonnegative payoffs in the future. We have demonstrated the arbitrage opportunity. Question 9.16. Short call perspective: Transaction Sell call Buy put Buy share Borrow @ rB TOTAL

t =0 +C B −P A −S A +Ke−rB T C B − P A − S A + Ke−rB T

time = T ST ≤ K 0 K − ST ST −K 0

time = T ST > K K − ST 0 ST −K 0

In order to preclude arbitrage, we must have: C B − P A − S A + Ke−rB T ≤ 0. Long call perspective: Transaction Buy call Sell put Sell share Lend @ rL TOTAL

t =0 −C A +P B +S B −Ke−rL T −C A + P B + S B − Ke−rL T

time = T ST ≤ K 0 ST − K −ST +K 0

time = T ST > K ST − K 0 −ST +K 0

In order to preclude arbitrage, we must have: −C A + P B + S B − Ke−rL T ≤ 0. 120

Chapter 9 Parity and Other Option Relationships

Question 9.17. In this problem we consider whether parity is violated by any of the option prices in Table 9.1. Suppose that you buy at the ask and sell at the bid, and that your continuously compounded lending rate is 1.9% and your borrowing rate is 2%. Ignore transaction costs on the stock, for which the price is $84.85. Assume that IBM is expected to pay a $0.18 dividend on November 8 (prior to expiration of the November options). For each strike and expiration, what is the cost if you a) Buy the call, sell the put, short the stock, and lend the present value of the strike price plus dividend? Calls 75 80 85 90 75 80 85 90

November November November November January January January January

9.9 10.3 5.3 5.6 1.9 2.1 0.35 0.45 10.5 10.9 6.5 6.7 3.2 3.4 1.2 1.35

Puts

Expiry

0.2 0.25 0.6 0.7 2.1 2.3 5.5 5.8 0.7 0.8 1.45 1.6 3.1 3.3 6.1 6.3

11/20/04 11/20/04 11/20/04 11/20/04 1/22/2005 1/22/2005 1/22/2005 1/22/2005

Maturity (in years) 0.0986 0.0986 0.0986 0.0986 0.2712 0.2712 0.2712 0.2712

Value of position −0.2894 −0.1800 −0.1706 −0.1113 −0.1443 −0.1686 −0.1929 −0.1172

All costs are negative. The positions yield a payoff of zero in the future, independent of the future stock price. Therefore, the given prices preclude arbitrage. b) Calls 75 80 85 90 75 80 85 90

November November November November January January January January

Puts

Expiry

9.9 10.3 0.2 0.25 5.3 5.6 0.6 0.7 1.9 2.1 2.1 2.3 0.35 0.45 5.5 5.8 10.5 10.9 0.7 0.8 6.5 6.7 1.45 1.6 3.2 3.4 3.1 3.3 1.2 1.35 6.1 6.3

11/20/04 11/20/04 11/20/04 11/20/04 1/22/2005 1/22/2005 1/22/2005 1/22/2005

Maturity (in years) 0.0986 0.0986 0.0986 0.0986 0.2712 0.2712 0.2712 0.2712

Value of position −0.1680 −0.2279 −0.2377 −0.2976 −0.3760 −0.2030 −0.2301 −0.2571

Again, all costs are negative. The positions yield a payoff of zero in the future, independent of the future stock price. Therefore, the given prices preclude arbitrage. Question 9.18. Consider the January 80, 85, and 90 call option prices in Table 9.1. 121

Part 3 Options

a) Does convexity hold if you buy a butterfly spread, buying at the ask price and selling at the bid? Since the strike prices are symmetric, lambda is equal to 0.5. Therefore, to buy a long butterfly spread, we buy the 80-strike call, sell two 85 strike calls and buy one 90 strike call. C (K1 ) − C (K2 ) 6.70 − 3.20 = 0.7 = K2 − K 1 85 − 80

and

C (K2 ) − C (K3 ) 3.20 − 1.35 = 0.37 = K3 − K2 90 − 85

and

3.4 − 1.20 C (K2 ) − C (K3 ) = = 0.44 K3 − K2 90 − 85

Convexity holds. b) C (K1 ) − C (K2 ) 6.5 − 3.4 = = 0.62 K2 − K1 85 − 80 Convexity holds. c) Does convexity hold if you are a market-maker either buying or selling a butterfly, paying the bid and receiving the ask? A market maker can buy a butterfly spread at the prices we sell it for. Therefore, the above convexity conditions are the ones relevant for market makers. Convexity is not violated from a market maker’s perspective.

122

Chapter 10 Binomial Option Pricing: I Question 10.1. Using the formulas given in the main text, we calculate the following values: a)

for the European call option:

b)

for the European put option:


= 0.5


= −0.5

B = −38.4316

B = 62.4513

price = 11.5684

price = 12.4513

Question 10.2. a)

Using the formulas of the textbook, we obtain the following results:
= 0.7 B = −53.8042 price = 16.1958

b) If we observe a price of $17, then the option price is too high relative to its theoretical value. We sell the option and synthetically create a call option for $19.196. In order to do so, we buy 0.7 units of the share and borrow $53.804. These transactions yield no risk and a profit of $0.804. c) If we observe a price of $15.50, then the option price is too low relative to its theoretical value. We buy the option and synthetically create a short position in an option. In order to do so, we sell 0.7 units of the share and lend $53.8042. These transactions yield no risk and a profit of $0.696. Question 10.3. a)

Using the formulas of the textbook, we obtain the following results:
= −0.3 B = 37.470788 price = 7.4707

123

Part 3 Options

b) If we observe a price of $8, then the option price is too high relative to its theoretical value. We sell the option and synthetically create a long put option for $7.471. In order to do so, we sell 0.3 units of the share and lend $37.471. These transactions yield no risk and a profit of $0.529. c) If we observe a price of $6, then the put option price is too low relative to its theoretical value. We buy the option and synthetically create a short position in the option. In order to do so, we buy 0.3 units of the share and borrow $37.471. These transactions yield no risk and a profit of $1.471. Question 10.4. The stock prices evolve according to the following picture:

Since we have two binomial steps, and a time to expiration of one year, h is equal to 0.5. Therefore, we can calculate with the usual formulas for the respective nodes: t = 0, S = 100

t = 1, S = 80

t = 1, S = 130


= 0.691


= 0.225


=1

B = −49.127

B = −13.835

B = −91.275

price = 19.994

price = 4.165

price = 38.725

Question 10.5. S(0) = 80: t = 0, S = 80 delta 0.4651 B −28.5962 premium 8.6078

t = 1, S = 64 0 0 0

124

t = 1, S = 104 0.7731 −61.7980 18.6020

Chapter 10 Binomial Option Pricing: I

S(0) = 90: t = 0, S = 90 delta 0.5872 B −40.6180 premium 12.2266

t = 1, S = 72 0 0 0

t = 1, S = 117 0.9761 −87.7777 26.4223

t = 0, S = 110 delta 0.7772 B −57.0897 premium 28.4060

t = 1, S = 88 0.4409 −29.8229 8.9771

t = 1, S = 143 1 −91.2750 51.7250

t = 0, S = 120 delta 0.8489 B −65.0523 premium 36.8186

t = 1, S = 96 0.6208 −45.8104 13.7896

t = 1, S = 156 1 −91.2750 64.7250

S(0) = 110:

S(0) = 120:

As the initial stock price increases, the 95-strike call option is increasingly in the money. With everything else equal, it is more likely that the option finishes in the money. A hedger, e.g., a market maker, must therefore buy more and more shares initially to be able to cover the obligation she will have to meet at expiration. This number of shares in the replicating portfolio is measured by delta. The initial call delta thus increases when the initial stock price increases. Question 10.6. The stock prices evolve according to the following picture:

125

Part 3 Options

Since we have two binomial steps, and a time to expiration of one year, h is equal to 0.5. Therefore, we can calculate with the usual formulas for the respective nodes: t = 0, S = 100

t = 1, S = 80

t = 1, S = 130


= −0.3088


= −0.775


=0

B = 38.569

B = 77.4396

B=0

price = 7.6897

price = 15.4396

price = 0

Question 10.7. S(0) = 80: t = 0, S = 80 t = 1, S = 64 t = 1, S = 104 delta −0.5350 −1 −0.2269 B 59.0998 91.275 29.4770 premium 16.3039 27.275 5.8770 S(0) = 90: delta B premium

t = 0, S = 90 t = 1, S = 72 t = 1, S = 117 −0.4128 −1 −0.0239 47.0781 91.275 3.4973 9.9226 19.275 0.6973

S(0) = 110: t = 0, S = 110 t = 1, S = 88 t = 1, S = 143 delta −0.2228 −0.5591 0 B 30.6064 61.4521 0 premium 6.1022 12.2521 0 S(0) = 120: delta B premium

t = 0, S = 120 t = 1, S = 96 t = 1, S = 156 −0.1511 −0.3792 0 22.6437 45.4646 0 4.5146 9.0646 0

S(0) = 130: t = 0, S = 130 t = 1, S = 104 t = 1, S = 169 delta −0.0904 −0.2269 0 B 14.6811 29.4770 0 premium 2.9271 5.8770 0 126

Chapter 10 Binomial Option Pricing: I

As the initial stock price increases, the 95-strike put option is increasingly out of the money. With everything else equal, it is more likely that the option finishes out of the money. A hedger, e.g., a market maker, must therefore sell fewer shares initially to be able to cover the obligation she will have to meet at expiration. This number of shares in the replicating portfolio is measured by delta. The initial put delta thus tends towards zero when the initial stock price increases. Question 10.8. We must compare the results of the equivalent European put that we calculated in exercise 10.6. with the value of immediate exercise. In 10.6., we calculated: t = 1, S = 80

t = 1, S = 130


= −0.775


=0

B = 77.4396

B=0

price = 15.4396

price = 0

immediate exercise =

immediate exercise

max (95 − 80, 0) = 15

= max(95 − 130, 0) = 0

Since the value of immediate exercise is smaller than or equal to the continuation value (of the European options) at both nodes of the tree, there is no benefit to exercising the options before expiration. Therefore, we use the European option values when calculating the t = 0 option price: t = 0, S = 100
= −0.3088 B = 38.569 price = 7.6897 immediate exercise = max (95 − 100, 0) = 0 Since the option price is again higher than the value of immediate exercise (which is zero), there is no benefit to exercising the option at t = 0. Since it is never optimal to exercise earlier, the early exercise option has no value. The value of the American put option is identical to the value of the European put option. Question 10.9. a) We can calculate the option delta, B and the premium with our standard binomial pricing formulas:
=1 B = −46.296 price = 53.704 127

Part 3 Options

It is no problem to have a d that is larger than one. The only restriction that we have imposed is that d < e(r−δ)h = e(0.07696)1 = 1.08, which is respected. b) We may expect the option premium to go down drastically, because with a d equal to 0.6, the option is only slightly in the money in the down state at t = 1. However, the potential in the up state is even higher, and it is difficult to see what effect the change in u and d has on the risk-neutral probability. Let’s have a look at put-call-parity. The key is the put option. A put option with a strike of 50 never pays off, neither in a) nor in b), because in a), the lowest possible stock price is 105, and in b), it is 50. Therefore, the put option has a value of zero. But then, the put-call-parity reduces to: C = S − Ke−0.07696 = 100 − 50 × 0.926 = 53.704. Clearly, as long as the strike price is inferior to the lowest value the stock price can attain at expiration, the value of the call option is independent of u and d. Indeed, we can calculate:
=1 B = −46.296 price = 53.704 c) Again, we are tempted to think in the wrong direction. You may think that, since the call option can now expire worthless in one state of the world, it must be worth less than in part b). This is not correct. Let us use put-call-parity to see why. Now, with d = 0.4, a stock price of 40 at t = 1 is admissible, and the corresponding put option has a positive value, because it will pay off in one state of the world. We can use put-call-parity to see that: C = S − Ke−0.07696 + P = 100 − 50 × 0.926 + P = 53.704 + P > 53.704. Indeed, we can calculate:
= 0.9 B = −33.333 price = 56.6666 Question 10.10. a)

We can calculate for the different nodes of the tree: node uu node ud = du delta 1 0.8966 B −92.5001 −79.532 call premium 56.6441 15.0403 value of early exercise 54.1442 10.478

128

node dd 0 0 0 0

Chapter 10 Binomial Option Pricing: I

Using these values at the previous node and at the initial node yields: t =0 delta 0.7400 B −55.7190 call premium 18.2826 value of early exercise 5

node d node u 0.4870 0.9528 −35.3748 −83.2073 6.6897 33.1493 0 27.1250

Please note that in all instances the value of immediate exercise is smaller than the continuation value, the (European) call premium. Therefore, the value of the European call and the American call are identical. b) We can calculate similarly the binomial prices at each node in the tree. We can calculate for the different nodes of the tree: node uu node ud = du delta 0 −0.1034 B 0 12.968 put premium 0 2.0624 value of early exercise 0 0

node dd −1 92.5001 17.904 20.404

Using these values at the previous node and at the initial node yields: t =0 delta −0.26 B 31.977 put premium 5.979 value of early exercise 0

node d node u −0.513 −0.047 54.691 6.859 10.387 1.091 8.6307 0

c) From the previous tables, we can see that at the node dd, it is optimal to early exercise the American put option, because the value of early exercise exceeds the continuation value. Therefore, we must use the value of 20.404 in all relevant previous nodes when we back out the prices of the American put option. We have for nodes d and 0 (the other nodes remain unchanged): delta B put premium value of early exercise

t =0 node d −0.297 −0.594 36.374 63.005 6.678 11.709 0 8.6307

The price of the American put option is indeed 6.678.

129

Part 3 Options

Question 10.11. a)

We can calculate for the different nodes of the tree, taking into account the dividend yield: node uu node ud = du delta 0.974 0.6687 B −92.5001 −56.239 call premium 45.1773 10.635 value of early exercise 46.398 5

node dd 0 0 0 0

We can see that for the node uu, the value of early exercise exceeds the continuation value. In this case, we exercise the American option early if we are at the node uu, and the value of the American call and the European call option will differ. We have for the European call option: delta B call premium value of early exercise

t =0 0.587 −44.760 13.941 5

node d 0.354 −25.014 4.7304 0

node u 0.8124 −70.887 25.719 23.911

t =0 0.602 −46.037 14.183 5

node d 0.354 −25.014 4.7304 0

node u 0.841 −73.759 26.262 23.911

and for the American call option: delta B call premium value of early exercise b)

We can calculate similarly the binomial prices at each node in the tree for the put option: node uu node ud = du delta 0 −0.3049 B 0 36.262 put premium 0 5.767 value of early exercise 0 0

node dd −0.9737 92.500 23.639 24.278

Using those put premium values at the previous nodes and at the initial node yields: t =0 delta −0.336 B 42.936 put premium 9.326 value of early exercise 0 The price of the European put option is: 9.326. 130

node d node u −0.594 −0.136 65.052 19.179 15.068 3.05 10.9035 0

Chapter 10 Binomial Option Pricing: I

c) From the previous tables, we can see that at the node dd, it is optimal to early exercise the American put option, because the value of early exercise exceeds the continuation value. Therefore, we must use the value of 24.278 in all relevant previous nodes when we back out the prices of the American put option. We have for nodes d and 0 (the other nodes remain unchanged): delta B put premium value of early exercise

t =0 node d −0.346 −0.616 44.06 67.177 9.5046 15.406 0 10.903

The price of the American put option is 9.5046. Question 10.12. a)

We can calculate u and d as follows: u = e(r−δ)h+σ d=

√ h

√ e(r−δ)h−σ h

√ 0.25

= e(0.08)×0.25+0.3×

= 1.1853

=

= 0.8781

√ e(0.08)×0.25−0.3× 0.25

b) We need to calculate the values at the relevant nodes in order to price the European call option: delta B call premium

t =0 0.6074 −20.187 4.110

node d 0.1513 −4.5736 0.7402

node u 1 −39.208 8.204

c) We can calculate at the relevant nodes (or, equivalently, you can use put-call-parity for the European put option): European put delta B put premium

t =0 −0.3926 18.245 2.5414

node d −0.8487 34.634 4.8243

node u 0 0 0

For the American put option, we have to compare the premia at each node with the value of early exercise. We see from the following table that at the node d, it is advantageous to exercise the option early; consequently, we use the value of early exercise when we calculate the value of the put option. American put delta B put premium value of early exercise

t =0 −0.3968 18.441 2.5687 0 131

node d node u −0.8487 0 34.634 0 4.8243 0 4.8762 0

Part 3 Options

Question 10.13. a) This question deals with the important issue of rebalancing a replicating portfolio. From the previous exercise, part a), we calculate delta and B of the call option. We obtained: delta B call premium

t =0 0.6074 −20.187 4.110

node d 0.1513 −4.5736 0.7402

node u 1 −39.208 8.204

Therefore, at time t = 0, we will buy 0.6074 shares of the stock and borrow $20.187 from the bank. This will cost us $4.110, and our proceeds from the sold option are $5, which yields a total profit of $0.89. b) Suppose that in the next period, we are in state u (without loss of generality). At that point, the stock price is u × S0 = 1.1853 × 40 = 47.412. Since we assume that the call is fairly priced, we can buy a call to offset our written call for $8.204. We sell our 0.6074 shares for $28.798, and we pay back the money we borrowed, plus accrued interest: 20.187 × e0.08×0.25 = 20.5948. Thus, our total cash flow is $28.798 − $8.204 − $20.5948 ≈ 0 (small differences due to rounding). We see that we were perfectly hedged, and have no cash outflow in period 1. If the option continued to be overpriced, we would have to change the replicating portfolio according to the above table (i.e., in state u, we would buy an additional 1 − 0.6074 shares and take on an additional loan of 39.208 − 20.187 to finance it) and stick with our option until the final period. In the final period, the payoff from the option exactly offsets our obligation from the hedging position, and again, there would be no cash outflow. c) If the option were underpriced, we would liquidate our position as in part b), but could make an additional profit, because we could buy the offsetting option for less than it is worth. Even better, if we could buy more than one option at the advantageous price, we could build up another arbitrage position, entering into a position where we buy the cheap option and replicate the short position synthetically. Question 10.14. a) We can calculate the price of the call currency option in a very similar way to our previous calculations. We simply replace the dividend yield with the foreign interest rate in our formulas. Thus, we have: node uu node ud = du delta 0.9925 0.9925 B −0.8415 −0.8415 call premium 0.4734 0.1446

132

node dd 0.1964 −0.1314 0.0150

Chapter 10 Binomial Option Pricing: I

Using these call premia at all previous nodes yields: delta B call premium

t =0 0.7038 −0.5232 0.1243

node d 0.5181 −0.3703 0.0587

node u 0.9851 −0.8332 0.2544

The price of the European call option is $0.1243. b)

For the American call option, the binomial approach yields: node uu node ud = du delta 0.9925 0.9925 B −0.8415 −0.8415 call premium 0.4734 0.1446 value of early exercise 0.4748 0.1436

node dd 0.1964 −0.1314 0.0150 0

Using the maximum of the call premium and the value of early exercise at the previous nodes and at the initial node yields: delta B call premium value of early exercise

t =0 0.7056 −0.5247 0.1245 0.07

node d 0.5181 −0.3703 0.0587 0

node u 0.9894 −0.8374 0.2549 0.2540

The price of the American call option is: $0.1245. Question 10.15. a) We can calculate the price of the currency option in a very similar way to our previous calculations. We simply replace the dividend yield with the foreign interest rate in our formulas. Thus, we have: delta B put premium

node uu node ud = du 0 −0.352 0 0.419 0 0.069

node dd −0.993 0.99 0.2504

Using these put premia at all previous nodes yields: t =0 delta −0.509 B 0.605 put premium 0.13678 value of early exercise 0 133

node d −0.725 0.7870 0.1865 0.172

node u −0.207 0.273 0.0449 0

Part 3 Options

b)

For the American put option, this yields: node uu node ud = du delta 0 −0.352 B 0 0.419 put premium 0 0.069 value of early exercise 0 0.0064

node dd −0.993 0.99 0.2504 0.2548

Using the maximum of the put premium and the value of early exercise at the previous nodes and at the initial node yields: t =0 node d delta −0.455 −0.651 B 0.617 0.8043 put premium 0.1386 0.1894 value of early exercise 0.08 0.172

node u −0.181 0.273 0.045 0

The price of the American put option is: $0.1386. Question 10.16. aa) We now have to inverse the interest rates: We have a Yen-denominated option, therefore, the dollar interest rate becomes the foreign interest rate. With these changes, and equipped with an exchange rate of Y120/$ and a strike of Y120, we can proceed with our standard binomial procedure. delta B call premium

node uu node ud = du 0.9835 0.1585 −119.6007 −17.4839 9.3756 1.0391

node dd 0 0 0

Using these call premia at all previous nodes yields: t =0 delta 0.3283 B −36.6885 call premium 2.7116

node d 0.0802 −8.4614 0.5029

node u 0.5733 −66.8456 5.0702

The price of the European Yen-denominated call option is $2.7116. ab)

For the American call option, the binomial approach yields: delta B call premium value of early exercise

node uu node ud = du 0.9835 0.1585 −119.6007 −17.4839 9.3756 1.0391 11.1439 0 134

node dd 0 0 0 0

Chapter 10 Binomial Option Pricing: I

Using the maximum of the call premium and the value of early exercise at the previous nodes and at the initial node yields: delta B call premium value of early exercise b)

t =0 0.3899 −43.6568 3.1257 0

node d 0.0802 −8.4614 0.5029 0

node u 0.6949 −81.2441 5.9259 5.4483

For the Yen-denominated put option, we have: node uu node ud = du delta 0 −0.8249 B 0 102.1168 put premium 0 5.7287 value of early exercise 0 3.1577

node dd −0.9835 119.6007 17.2210 15.8997

We can clearly see that early exercise is never optimal at those stages. We can therefore calculate at the previous nodes: delta B put premium value of early exercise

t =0 −0.6229 82.1175 7.37 0

node d −0.8870 110.7413 11.602 8.2322

node u −0.3939 52.3571 2.9372 0

We can see that the American and the European put option must have the same price, since it is never optimal to exercise the American put option early. The price of the put option is 7.37. c) The benefit of early exercise for a put option is to receive the strike price earlier on and start earning interest on it. The cost associated with early exercising a put is to stop earning income on the asset we give up. In this case, the strike is 120 Yen, and the Yen interest rate is not very favorable compared to the dollar interest rate. We would give up a high yield instrument and receive a low yield instrument when we early exercise the put option. This is not beneficial, and it is reflected by the non-optimality of early exercise of the put option. For the call option, the opposite is true: When exercising the call option, we receive a dollar and give up 120 Yen. Therefore, we receive the high-yield instrument, and if the exchange rate moves in our favor, we want to exercise the option before expiration. Question 10.17. We have to pay attention when we calculate u and d. We must use the formulas given in the section options on futures contracts of the main text. In particular, we must remember that, while it is 135

Part 3 Options

possible to calculate a delta, the option price is just the value of B, because it does not cost anything to enter into a futures contract. We calculate: u = eσ d=

√ h

√ 1

= e0.1

√ e−σ h

=

= 1.1052

√ e−0.1 1

= 0.9048

Now, we are in a position to calculate the option’s delta and B, and thus the option price. We have: delta B premium

0.6914 18.5883 18.5883

This example clearly shows that the given argument is not correct. As it costs nothing to enter into the futures contract, we would not have to borrow anything if the statement was correct. We do not borrow to buy the underlying asset. Rather, we borrow exactly the right amount so that we can, together with the position in the underlying asset, replicate the payoff structure of the call option in the future (remember that we initially solved the system of two equations). Question 10.18. a) We have to use the formulas of the textbook to calculate the stock tree and the prices of the options. Remember that while it is possible to calculate a delta, the option price is just the value of B, because it does not cost anything to enter into a futures contract. In particular, this yields the following prices: For the European call and put, we have: premium = 122.9537. The prices must be equal due to put-call-parity. b) We can calculate for the American call option: premium = 124.3347 and for the American put option: premium = 124.3347. c) We have the following time 0 replicating portfolios: For the European call option: Buy 0.5371 futures contracts. Borrow 122.9537 For the European put option: Sell 0.4141 futures contracts. Borrow 122.9537 136

Chapter 10 Binomial Option Pricing: I

Question 10.19. a)

The price of a European call option with a strike of 95 is $24.0058.

b)

The price of a European put option with a strike of 95 is $14.3799.

c) Now, we have for the European call option a premium of $14.3799 and for the European put option a premium of $24.0058. Exchanging the four inputs in the formula inverts the call and put relationship. We will encounter a theoretical motivation for this fact in later chapters. Question 10.20. a)

The price of an American call option with a strike of 95 is $24.1650.

b)

The price of an American put option with a strike of 95 is $15.2593

c) Now, we have for the American 100-strike call option a premium of $15.2593 and for the European put option a premium of $24.165. Both option prices increase as we would have expected, and the relation we observed in question 10.19. continues to hold. Question 10.21. Suppose e(r−δ)h > u > d We short a tailed position of the stock and invest the proceeds at the interest rate: This yields:

short stock loan money Total

t =0 +e−δh S −e−δh S 0

period h, state = d −d × S +e(r−δ)h S >0

period h, state = u −u × S +e(r−δ)h S >0

We have shown that if e(r−δ)h > u > d, there is a true arbitrage possibility. Conversely, suppose u > d > e(r−δ)h We then buy a tailed position of the stock and borrow at the prevailing interest rate: This would yield:

buy stock borrow money Total

t =0 −e−δh S +e−δh S 0

period h, state = d +d × S −e(r−δ)h S >0

period h, state = u +u × S −e(r−δ)h S >0

We have shown that if u > d > e(r−δ)h , there is a true arbitrage possibility.

137