ActuarialBrew Exam MFE Models for Financial Economics ...actuarialbrew.com/data/documents/ActuarialBrew-MFE-Solutions-20… · Chapter 1 Underlying Assets Solution 1.01 A Chapter

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Exam MFE Solutions Overview

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Solutions to the Exam MFE Questions Overview

Solutions to Questions The Solutions to ActuarialBrew’s MFE Questions provide full solutions to the over 500 exam-style practice questions contained in the MFE Questions.

If you have access to the online normal distribution calculator or a spreadsheet when working the Questions, the SOA advises students to use 5 decimal places for both the inputs and the outputs of the normal distribution calculator. If you don’t have access to either one, we provide the old printed Normal Distribution Table at the end of the Questions for your convenience. Please note that your results from using the printed table may be different due to rounding.

Practice Questions Each solution has the following key to indicate the question’s degree of difficulty. The more boxes that are filled in, the more difficult the question: Easy: Very Difficult:

Table of Contents The MFE Questions cover the entire required course of reading for Exam MFE: Chapter 01: Underlying Assets Chapter 02: Forwards and Futures Chapter 03: Calls and Puts Chapter 04: Put-Call Parity Chapter 05: Applications of Calls and Puts Chapter 06: Put-Call Parity and Replication Chapter 07: Comparing Options Chapter 08: Binomial Trees: Part I Chapter 09: Binomial Trees: Part II Chapter 10: Lognormally Distributed Prices Chapter 11: The Black-Scholes Formula Chapter 12: The Greeks and Other Measures Chapter 13: Delta-Hedging Chapter 14: Exotic Options: Part I Chapter 15: Exotic Options: Part II Chapter 16: Monte Carlo Simulation Chapter 17: Volatility Chapter 18: The Black Model for Options on Bonds Chapter 19: Binomial Short-Rate Models

Exam MFE Solutions Overview

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Exam MFE Solutions Chapter 1 – Underlying Assets

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Solutions to the Part I Questions

Chapter 1 Underlying Assets Solution 1.01 A Chapter 1, Derivatives The owner of a catastrophe bond receives the full amount of the scheduled bond payments only if the specified catastrophe does not occur.

Solution 1.02 A Chapter 1, Derivatives Options are derivatives, but not all derivatives are options.

Solution 1.03 D Chapter 1, Uses of Derivatives If the pre-tax profit is 50,000, then the after-tax profit is: 50,000 (1 0.35) 32,500

Since the company receives no tax credit for losses, a pre-tax loss of 40,000 results in an after-tax loss of 40,000. Therefore, before the derivative is purchased, the expected value of the after-tax profit is: 0.75 32,500 0.25 ( 40,000) 14,375

After the derivative is purchased, the pre-tax profit is 35,000, and the after-tax profit is: 35,000 (1 0.35) 22,750

The difference is: 22,750 14,375 8,375

Exam MFE Solutions Chapter 1 – Underlying Assets


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Exam MFE Solutions Chapter 2 – Forwards and Futures


Chapter 2 Forwards and Futures Solution 2.01 D Chapter 2, Expected Value of Stock Price The current price is 50, and the forward price is 48:

( )( )

,( )

( )

48 50

r T tt T t

r

F S S e

e

Therefore, the dividend yield is greater than the risk-free rate. The expected price is based on the expected return, :

( )( )

( )50

T tt T tE S S e

P e

Since the risk premium is positive, we have:

( ) ( )50 50 because re e r

> 48P

Choices A, C, and E could be true, but only Choice D is certain to be true. For example, P could be 52, making Choices A and C false. Or P could be 49, making Choice E false. Based on the information provided in the question, only Choice D is certain to be true.

Solution 2.02 D Chapter 2, Forward Contract s A forward contract requires payment and delivery upon expiration of the contract. Therefore, the question is describing a one-year forward contract.

Solution 2.03 B Chapter 2, Forward Contract s A fully leveraged purchase consists of borrowing the entire stock price in order to buy the stock immediately. Therefore, the question is describing a fully leveraged purchase.

Solution 2.04 B Chapter 2, Forward Contracts The dividend payments occurring at the end of 4 months and 10 months are made before the forward contract matures, so their accumulated value must be subtracted from the accumulated value of the stock price:

( ), ,

0.04(1 0) 0.04(12 4)/12 0.04(12 10)/120,1

0,1

0,1

( ) ( )

( ) 40 1.20 1.20

( ) 41.6324 1.2324 1.2080

( )

r T tt T t t TF S e S FV Div

F S e e e

F S

F S

39.1919



Solution 2.05 B Chapter 1, Uses of Derivatives A: A farmer is hurt when the price of corn falls. A short position in corn does well when the price falls, so the position reduces risk, and Choice A is not correct. B: An electric utility is hurt by higher coal prices. A short position in coal does poorly when the price of coal rises, so Choice B increases the utility’s risk. Therefore, Choice B is likely correct. C: A railroad operator is hurt by higher fuel prices. A long position in fuel does well when fuel prices increase, so the position reduces risk, and Choice C is not correct. D: A farmer is hurt by higher fertilizer prices, because farmers have to buy fertilizer to produce their crops. A long position in fertilizer does well when fertilizer prices increase, so the position reduces risk and Choice D is not correct. E: The consulting firm is hurt when the value of the Euro falls. A short position in Euros does well when the Euro falls, so the position reduces risk and Choice E is not correct. Since Choices A, C, D, and E reduce risk and Choice B increases risk, the best answer is Choice B.

Solution 2.06 A Chapter 2, Futures Contracts Participants in the futures markets are required to post margin, and they may receive a margin call requiring additional margin.

Solution 2.07 E Chapter 2, Forward Contracts Choice A is true, because the payoffs of a short position are the opposite of the payoffs of a long position. Choice B is true, because there is no cost to enter into a forward contract. Choice C is true, because borrowing the funds to purchase the stock results in the same profit as the profit obtained by taking a long position in the forward contract:

1 1

Borrow Buy stock Long forward contract50 1.05 52.50S S

Choice D is true. If the 5% interest rate is compounded continuously, then the forward price should be:

0.0550 52.56F e The forward price of 52.50 is therefore too low. Arbitrage can be earned by buying the forward and offsetting the risk by selling the stock short and lending the proceeds. At the end of one year, the payoff of this strategy is certain to be positive:

0.051 1

Forward contract payoff Cost of repurchasing stock Risk-free lending

52.50 50 0.06S S e

Since there was no initial cost to the strategy and the payoff is positive, arbitrage profits are available. As shown above, the strategy calls for selling the stock. Since Choice D calls for selling the stock, Choice D is true. Choice E is false. If the stock pays a dividend, then the forward price should be:

0.550 1.05 1.00 1.05 51.48F



The forward price of 52.50 is therefore too high, and arbitrage profit can be earned by taking a short position in the forward contract and borrowing to purchase the stock.

0.51 1

Short forward contract payoff Stock payoff Risk-free borrowing

52.50 1.00 1.05 50 1.05 1.02S S

Since there was no initial cost to the strategy and the payoff is positive, arbitrage profits are available. As shown above, the strategy calls for taking a short position in the forward contract. Since Choice E calls for taking a long position in the forward contract, Choice E is false.

Solution 2.08 A Chapter 2, Forward and Prepaid Forward Contracts An outright purchase consists of paying the stock’s price immediately for immediate receipt of the stock: 0A S

A fully leveraged purchase consists of borrowing to obtain the funds necessary to buy the stock now. The borrowed funds are paid in the future:

0rTB S e

A prepaid forward contract calls for paying the prepaid forward price immediately for future receipt of the stock:

0, 0( )P TTC F S e S

A forward contract calls for paying the forward price in the future for future receipt of the stock:

( )0, 0( )r T

TD F S e S

Since the dividend yield is greater than the risk-free rate, we have:

( )

0 0 0 0T r T rTe S e S S e S

C < D < A < B

Solution 2.09 B Chapter 2, Forward Contracts The dividend of 3 is paid after the forward contract matures, so it does not affect the price of the forward contract. We use the formula for the price of a forward contract to solve for the stock price:

( ), ,

(1 0)0,1 0 0,1

0.50

0

( ) ( )

( ) (1 ) ( )

65 (1.05) 2(1.05)

r T tt T t t TF S e S FV Div

F S i S FV Div

SS

63.86



Solution 2.10 A Chapter 2, Forward Contracts A stock can be replicated with a forward contract and lending: Stock Forward Lending

Buying a zero-coupon bond is lending. Therefore, a long stock position is replicated by buying a forward and by buying a zero-coupon bond. This position is described by Choice A. Alternatively, the payoff to a long stock position is: TS

A long stock goes up with the stock price, so we need an instrument that goes up with the stock price. Consider the payoff of a long forward: 0,T TS F

If we lend the present value of 0,TF , then we have a payoff of:

0,TF

Combining the payoff of the long forward with the payoff from lending (i.e., buying a risk-free asset), gives us the same payoff of a long stock position:

0, 0,T T T TS F F S Thus, a long stock position is replicated with a long forward and by buying a zero-coupon bond. This position is described by Choice A.

Solution 2.11 B Chapter 2, Forward Contracts If the forward is not used, then the expected cost of the gold is calculated using the probabilities provided in the question:

Price 600 0.50 (0.3 900 0.3 1,000 0.4 1,200) 1,125

Cost 0.3 900 0.3 1,000 0.4 1,200 1,050Expected profit 75

If the forward is used, then the cost of the gold is equal to 1,000:

Price 600 0.50 1,000 1,100

Cost 1,000Expected profit 100

If the forward is not used, then the expected profit is 75, and if the forward is used, then the expected profit is 100: Increase in expected profit 100 75 25

Solution 2.12 B Chapter 2, Forward Contracts The accumulation factor for one quarter is:

0.12 0.25 0.03e e



The accumulated value of the dividends is:

0.33 12 0.030.03 11 0.03 10 11

0,3 0.031.02( ) 2.00 1.02 1.02 2.00

1 1.022.00 15.7910 31.5820

e eFV Div e ee

The forward price is the accumulated value of the stock price minus the accumulated value of the dividends:

(3 0) 0.12 30,3 0,3( ) ( ) 191.30 31.5820r

tF S e S FV Div e 242.61

Solution 2.13 A Chapter 2, Forward Contracts If arbitrage is not possible, then the forward price should be:

( )( ) (0.06 0.02)(0.5 0), ( ) 200 204.0403r T t

t T tF S e S e

Since the observed forward price of 205 is too high, arbitrage can be earned by selling the forward and synthetically purchasing the forward. The steps at time 0 result in a net cash of zero at the outset: Sell the forward at a price of 205.

Buy the prepaid forward on the stock for 0.02 0.5200e .

Borrow 0.02 0.5200e . At the end of 6 months, the cash flow is:

0.02 0.5 0.06 0.50.5 0.5205 200 205 204.0403S S e e 0.9597

This arbitrage strategy is described by Choice A.

Solution 2.14 C Chapter 2, Forward Contracts If arbitrage is not possible, then the forward price should be:

( )( ) (0.06 0.02)(0.5 0), ( ) 200 204.0403r T t

t T tF S e S e

Since the observed forward price of 204 is too low, arbitrage can be earned by buying the forward and synthetically selling the forward. The steps at time 0 result in a net cash of zero at the outset: Buy the forward at a price of 204.

Sell the prepaid forward on the stock for 0.02 0.5200e .

Lend 0.02 0.5200e . At the end of 6 months, the cash flow is:

0.02 0.5 0.06 0.50.5 0.5204 200 204 204.0403S S e e 0.0403

This arbitrage strategy is described by Choice C.



Solution 2.15 D Chapter 2, Forward and Prepaid Forward Contracts An outright purchase consists of paying the stock’s price immediately for immediate receipt of the stock: 0 100A S

A fully leveraged purchase consists of borrowing to obtain the funds necessary to buy the stock now. The borrowed funds are paid in the future:

0.050 100 105.13rTB S e e

A prepaid forward contract calls for paying the prepaid forward price immediately for future receipt of the stock:

0.020, 0( ) 100 98.02P TTC F S e S e

A forward contact calls for paying the forward price in the future for future receipt of the stock:

( ) 0.05 0.020, 0( ) 100 103.05r T

TD F S e S e

The maximum payment minus the minimum payment is:

, , , , , , 105.13 98.02Max A B C D Min A B C D B C 7.11

Solution 2.16 E Chapter 2, Forwards and Arbitrage If the forward price is high enough to allow arbitrage, then an arbitrageur will sell the forward and buy a synthetic forward. Selling the forward results in a transaction fee of 1 at time 0. An arbitrage strategy cannot have any negative cash flows, so the fee is borrowed. The time-1 payoff from selling the forward is: 1 11 1.05 1.05F S F S

Buying the synthetic forward calls for buying the stock and borrowing the price of the stock. Buying the stock results in a transaction fee of 2 at time 0, which is also borrowed. The time-1 payoff from buying the synthetic forward is: 1 1(50 2) 1.05 54.60S S

If arbitrage is not available, then the combination of selling the forward and buying the synthetic forward must result in a payoff that is less than or equal to zero:

1 11.05 54.60 0F S SF

55.65

If the forward price were to exceed 55.65, then the strategy described above would be an arbitrage strategy.

Solution 2.17 B Chapter 2, Futures Contracts For convenience, let’s calculate the margin account per unit of the index. Then, at the end, we can multiply by the number of contracts and the number of units per contract. To avoid rounding error, we keep the numbers stored in the calculator.



The initial margin and the maintenance margin per unit of the index are:

Initial margin 0.10 1,220 122Maintenance margin 0.80 Initial margin 0.80 122 97.60

The mark-to-market adjustments and the interest earned are shown below. Futures Mark to Margin Margin

Day Price Market Interest Account Deposits0 1,220.00 122.001 1,200.00 -20.0000 0.01671 102.01671 0.002 1,210.00 10.0000 0.01398 112.03069 0.00

The margin account at the end of day 1 is:

0.05/365122.00 (1,200 1,220) 102.01671e

The margin account at the end of day 2 is:

0.05/365102.01671 (1,210 1,200) 112.03069e

The calculations above are per unit of the index. To answer the question, we must multiply the margin account balance by the multiplier: 112.03069 Multiplier 112.03069 (40 500) 2,240,613.79

Solution 2.18 C Chapter 2, Futures Contracts The price of the index that triggers a margin call is not affected by the number of contracts nor the contract size, so we can focus on just one unit of the index. The initial margin and the maintenance margin per unit of the index are:


Since the investor took a long position, when the index drops to 1,200, the investor receives a negative mark-to-market adjustment: Day 1 mark-to-market adjustment New price Old Price 1,200 1,220 20

At the end of day 1, the balance in the margin account is:

0.05/365122.00 20 102.01671e If the next day’s index price, X, does not require additional margin to be added, then it must satisfy the inequality below:

0.05/365

0.05/365

Day 2 margin account Maintenance margin

(Day 1 margin account) Mark-to-market adjustment 97.60

102.01671 ( 1,200) 97.60

e

e XX

1,195.5693

Solution 2.19 B Chapter 2, Futures Contracts The price of the index that triggers a margin call is not affected by the number of contracts nor the contract size, so we can focus on just one unit of the index.



The initial margin and the maintenance margin per unit of the index are:


Since the investor took a short position, when the index drops to 1,200, the investor receives a positive mark-to-market adjustment:

Day 1 mark-to-market adjustment (New price Old Price) (1,200 1,200)20

At the end of day 1, the balance in the margin account is:

0.05/365122.00 20 142.01671e If the next day’s index price, X, does not require additional margin to be added, then it must satisfy the inequality below:

0.05/365

0.05/365

Day 2 margin account Maintenance margin

(Day 1 margin account) Mark-to-market adjustment 97.60

142.01671 (1,200 ) 97.60

e

e XX

1,244.4362

Solution 2.20 C Chapter 2, Forward Contracts The forward price is greater than 80, because the forward price is the stock price accumulated at the excess of the risk-free rate over the dividend yield:

( )( ),

2( )

( )

8080 for ( ) 0

r T tt T t

r

F S e S

F eF r

From the second line above, we see that:

2( )80 rFe The expected value of the stock’s price is equal to the current price accumulated at the expected return over the dividend yield:

( )( )

2( )

2( ) 2( ) 2( )

2( )

2( )

100 80

100 because 80

100

100100 for ( ) 0

T tt T t

r r

r

r

E S S e

e

Fe e Fe

F e

F eF r

The question says that investors require compensation for risk, which implies that the risk premium is positive: 0r Therefore, we have:

80 and 100F F 80 <



Solution 2.21 D Chapter 2, Forward Contracts and Arbitrage There are two kinds of strategies offered in the answer choices. Choices A, B, and C are identical, except that the forward prices are different. The most advantageous of the choices is Choice C, because it provides the investor with the highest forward price. Choices D and E are identical, except that the forward prices are different. The more advantageous of the two choices is Choice D, because it requires the investor to pay the lowest forward price. Because we are seeking a strategy that provides arbitrage, it will be the most advantageous of the strategies, and therefore the solution must be Choice C or Choice D. Let’s consider Choice C. The investor buys the stock for the ask price of 176, borrows the cost of the stock at an annual effective interest rate of 4.10%, and enters into a forward contract obligating the investor to receive 183.21 in exchange for a share of stock. The net cash flow to investor at the outset is zero, and at the end of 1 year, the payoff is: 1 1176 1.041 (183.21 ) 0.0060S S

Since the payoff is negative, Choice C does not describe an arbitrage strategy. Next, we consider Choice D. The investor sells the stock for the bid price of 175, lends the proceeds at an annual effective interest rate of 4.00%, and enters into a forward contract obligating the investor to pay 181.90 to receive a share of stock. The net cash flow to investor at the outset is zero, and at the end of 1 year, the payoff is: 1 1175 1.04 ( 181.90) 0.1000S S

Since the cost is zero and the payoff is positive, Choice D describes an arbitrage strategy.

Solution 2.22 D Chapter 2, Forward Contracts A stock can be replicated with a forward contract and lending: Stock Forward Lending

Rearranging, we see that a short forward can be replicated with a short stock and with lending: Forward Stock Lending

Buying a zero-coupon bond is lending. Thus, a short forward position is replicated with a short stock and by going long a zero-coupon bond. This position is described by Choice D. Alternatively, the payoff to a short forward position is: 0,T TF S

A short forward goes down with the stock price, so we need an instrument that goes down with the stock price. Consider the payoff of a short stock: TS

If we lend the present value of 0,TF , then we have a payoff of:

0,TF



Combining the payoff of the short stock with the payoff from lending (i.e., buying a risk-free asset), gives us the same payoff of a short forward position: 0, 0,T T T TS F F S

Thus, a short forward position is replicated with a short stock and by being long a zero-coupon bond. This position is described by Choice D.

Solution 2.23 B Chapter 2, Futures Contracts We assume that the futures price is equal to the forward price. Therefore, the initial futures price can be found using the initial index price:

( )( ),

(0.03 0.05)(1 0)0,1

0,1

1,400

1,372.28

r T tt T tF e S

F e

F

The futures price at the end of 6 months is based on the index price at the end of 6 months:

(0.03 0.05)(1 0.5)

0.5,1 0.5

0.5,1 0.50.9900

F e S

F S

Since the margin account does not earn interest, the total of the mark-to-market adjustments is equal to 20,000. We can find the total of these adjustments by comparing the futures price in 6 months with the initial futures price:

0.5,1 0,10.5

0.5

20,000 500

40 0.9900 1,372.28

F F

SS

1,345.67

Exam MFE Solutions Chapter 3 – Calls and Puts


Chapter 3 Calls and Puts Solution 3.01 D Chapter 3, Call and Put Payoffs The maximum loss occurs if the option expires out-of-the-money. In this event, the profit is:

0Profit Payoff (Accumulated cost of establishing the position)

0 500 500Te

The loss is the opposite of the profit: Loss 500

Solution 3.02 C Chapter 3, Call and Put Profit If the stock price increases to 120, then the call option pays off for: (0,120 70) 50Max

If the stock price decreases to 90, then the call option pays off for: (0,90 70) 20Max

The expected value of the payoff is: Expected payoff 50% 50 50% 20 35

The expected profit is:

Accumulated cost of Expected profit Expected payoff

establishing the position35 30 1.03 4.10

Solution 3.03 C Chapter 3, Call Options The accumulated values of the call option prices are:

40-strike call: 7.33 1.06 7.7745-strike call: 3.30 1.06 3.5050-strike call: 0.95 1.06 1.01

The question asks us to find the stock prices that satisfy the following:

0, 45 3.50 0, 50 1.01 0, 40 7.77Max S Max S Max S

Let’s consider the left inequality first, and let’s call it Inequality I:

Inequality I: 0, 45 3.50 0, 50 1.01 Max S Max S

The stock price can be less than or equal to 45, between 45 and 50, or greater than 50:

I. 453.50 1.01 always true

II. 45 5045 3.50 0 1.0147.49

S

SSS



III. 50

45 3.50 50 1.0148.50 51.01 never true

SS S

Considering the cases above, we observe that Inequality I is true when: 47.49S Next, we consider the second inequality, and let’s call it Inequality II:

Inequality II: 0, 50 1.01 0, 40 7.77Max S Max S


I. 401.01 7.77 never true

II. 40 500 1.01 40 7.77

46.76III. 50

50 1.01 40 7.7751.01 47.77 always true

S

SS

SS

S S

Considering the cases above, we observe that Inequality II is true when: 46.76S Both Inequality I and Inequality II are true when: 47.49 and 46.76 S S 46.76 < < 47.49S When we analyzed Inequality I, we saw it is never true when S>50. Therefore, to save a little time, we could have skipped the analysis of S>50 for Inequality II.

Solution 3.04 E Chapter 3, Call Options The accumulated values of the call option prices are:

40-strike call: 7.33 1.06 7.7745-strike call: 2.10 1.06 2.2350-strike call: 0.95 1.06 1.01


0, 45 2.23 0, 50 1.01 0, 40 7.77Max S Max S Max S


Inequality I: 0, 45 2.23 0, 50 1.01 Max S Max S




I. 452.23 1.01 always true

II. 45 5045 2.23 0 1.0146.22

III. 5045 2.23 50 1.01

47.23 51.01 never true

S

SSS

SS S

Considering the cases above, we observe that Inequality I is true when: 46.22S Next, we consider the second inequality, and let’s call it Inequality II:

Inequality II: 0, 50 1.01 0, 40 7.77Max S Max S


I. 401.01 7.77 never true

II. 40 500 1.01 40 7.77

46.76III. 50

50 1.01 40 7.7751.01 47.77 always true

S

SS

SS

S S

Considering the cases above, we observe that Inequality II is true when: 46.76S Both Inequality I and Inequality II are true when: 46.22 and 46.76S S There is no value of S that satisfies these two conditions, so the range is empty. When we analyzed Inequality I, we saw it is never true when S>50. Therefore, to save a little time, we could have skipped the analysis of S>50 for Inequality II.

Solution 3.05 D Chapter 3, Call and Put Options Choice A is not correct, because the stock price at expiration is 41, which is equal to neither Option A’s strike price of 35 nor Option B’s strike price of 45. Choice B is not correct, because the stock price of 41 at expiration is greater than the put’s strike of 35 and less than the call’s strike of 45, so neither of the options is in-the-money. Choice C is not correct, because when the stock price is 34, Option A is in-the-money. Choice D is correct, because Option A is in-the-money when the stock price is 34 but Option B is never in-the-money. Choice E is not correct, because Option A is in-the-money when the stock price is 34, since 34 is less than the Option A’s strike price of 35.



Solution 3.06 C Chapter 3, Call and Put Payoffs We can use Option A’s and Option B’s payoffs to find the final stock price:

Option B payoff 2 (Option A payoff)(0,80 ) 2 (0, 50)

80 2( 50)80 2 1003 180

60

Max S Max SS SS S

SS

We can use the final stock price to find the payoff of Option C: (0,70 ) (0,70 60)Max S Max 10

Solution 3.07 E Chapter 3, Call and Put Payoffs Let S be the price of the stock. Choice A is false, because if 100S , then it is possible that 110S and Option B is out-of-the-money. Choice B is false, because if 100S , then 110S , so Option B is out-of-the-money. Choice C is false, because if 100S , then it is possible that 110S and Option B is in-the-money. Choice D is false, because if 100S , then 110S and Option B is out-of-the-money. Choice E is true, because if 100S , then 110S , so Option B is out-of-the-money.

Solution 3.08 A Chapter 3, Put Options The put option allows the producer to sell the oil for the strike price of 80 if the price of oil falls below 80. If the price of oil is 100, then the option expires unexercised, and the producer sells the oil at the market price of 100. We can consider each of the three possibilities individually. Below, we show the sales price received for the oil plus the proceeds from the put option:

30: 30 (0,80 30) 30 50 8060: 60 (0,80 60) 60 20 80100: 100 (0,80 100) 100 0 100

MaxMaxMax

The expected value of the revenue is: Expected revenue 0.1 80 0.4 80 0.5 100 90

The expected profit is the expected revenue minus the cost of production minus the accumulated cost of the put option:

0.0590 70 15 e 4.23 Alternatively, there is a 50% probability that the price of oil will be less than 80 and a 50% probability that the price of oil will be greater than 80. If the price is less than 80, then the producer uses the put option to effectively sell the oil for a price of 80.



The expected profit is the expected revenue minus the cost of production minus the accumulated cost of the put option:

0.05 0.050.5 80 0.5 100 70 15 90 70 15e e 4.23

Solution 3.09 E Chapter 3, Call and Put Payoffs The American call can be exercised at any time, and the largest possible payoff is obtained by exercising at the largest stock price:

Option A payoff 0, 0,68 62 6Max S K Max

The Bermuda put can be exercised at any time after 6 months have elapsed, and the largest possible payoff is obtained by exercising at the lowest price during the final 6 months:

Option B payoff 0, 0,59 57 2Max K S Max

The European put can only be exercised at the end of 12 months, when the stock price was 59:

Option C payoff 0, 0,60 59 1Max K S Max

From smallest to largest, the options are ranked as follows: C < B < A

Solution 3.10 C Chapter 3, Call and Put Payoffs The American put can be exercised at any time, and the largest possible payoff is obtained by exercising at the smallest stock price:

Option A payoff 0, 0,52 46 6Max K S Max

The Bermuda put can be exercised at any time after 6 months have elapsed, and the largest possible payoff is obtained by exercising at the lowest price during the final 6 months. In this case, since the stock price did not fall below 57 during the final 6 months, the option is not exercised, and the payoff is zero:

Option B payoff 0, 0,57 57 0Max K S Max

The European call can only be exercised at the end of 12 months, when the stock price was 59:

Option C payoff 0, 0,59 58 1Max S K Max

From smallest to largest, the options are ranked as follows: B < C < A

Solution 3.11 C Chapter 2, Forwards and Arbitrage Since the at-the-money put option is purchased when the index price is 500, the strike price of the put option is 500. When the index price declines to 450, the payoff is:

Put payoff 0, 0,500 450 50Max K S Max



Since each put cost 50, the profit on one put is zero:

0

Profit Payoff (Accumulated cost of establishing the position)

50 50 Te

0

Solution 3.12 C Chapter 3, Put Options The accumulated values of the put option prices are:

40-strike put: 2.12 1.06 2.2545-strike put: 4.04 1.06 4.2850-strike put: 6.65 1.06 7.05


0,45 4.28 0,50 7.05 0,40 2.25Max S Max S Max S


Inequality I: 0,45 4.28 0,50 7.05 Max S Max S


I. 4545 4.28 50 7.0540.72 42.95 always true

II. 45 500 4.28 50 7.05

47.23III. 50

4.28 7.05 never true

SS S

SS

SS

Considering the cases above, we observe that the Inequality I is true when: 47.23S Next, we consider the second inequality, and let’s call it Inequality II:

Inequality II: 0,50 7.05 0,40 2.25Max S Max S


I. 4050 7.05 40 2.2542.95 37.75 never true

II. 40 5050 7.05 0 2.25

45.20III. 50

7.05 2.25 always true

SS S

SS

SS

Considering the cases above, we observe that Inequality II is true when: 45.20S Both Inequality I and Inequality II are true when: 47.23 and 45.20 S S 45.20 < < 47.23S



When we analyzed Inequality I, we saw it is never true when 50S . Therefore, to save a little time, we could have skipped the analysis of 50S for Inequality II.

Solution 3.13 A Chapter 3, Put and Call Payoffs The payoffs of the three positions are described below:

,I. Payoff of short forward

II. Payoff of short call option 0,

III. Payoff of long put option 0,

t T T

T

T

F S

Max S K

Max K S

To answer this question, we consider the possible future values of the underlying asset, TS . Unlimited profit can only occur if the payoff is unlimited, so we consider each position

to see if its payoff is potentially unlimited. The short forward becomes more profitable as the stock price falls, and the maximum payoff of ,t TF occurs when the stock price is zero. Therefore, the profit potential is not

unlimited. The short call becomes more profitable as the stock price falls, and the maximum payoff of 0 occurs when the stock price is K or less. Therefore, the profit potential is not unlimited. The long put option becomes more profitable as the stock price falls, and the maximum payoff of K occurs when the stock price is zero. Therefore, the profit potential is not unlimited. Since none of the positions has unlimited profit potential, the answer is Choice A.

Solution 3.14 C Chapter 3, Call Options The profit from being long the call must be equal to the profit from being short the call:

0, 750 100 1.025 0, 750 100 1.025

2 0, 750 2 100 1.025

0, 750 102.50

750 102.50

Max S Max S

Max S

Max S

SS

852.50

Alternatively, since the profit from the long call is the opposite of the profit from the short call, the only way they can be equal is if the profit from both is zero. Therefore, we can solve for the price that results in the long call option having a profit of zero:

0, 750 100 1.02 0

0, 750 102.50

750 102.50

Max S

Max S

SS

852.50



Solution 3.15 D Chapter 3, Call and Put Profit Since S K , we know that the put has a positive payoff. The payoff of a long put is: Payoff of long put (0, ) when Max K S K S S K

Therefore the payoff of the short stock and the short put is the negative of the stock’s price minus the payoff of a long put: Payoff (0, ) ( )S Max K S S K S K

The accumulated cost of the positon is negative because the stock and the put were both sold, and the cost is the negative of the proceeds received:

Accumulated cost ( ) rtK P e

The profit is the payoff minus the accumulated cost:

Profit Payoff Accumulated cost ( ) ( )rt rt

rt rt

K K P e K K P e

Pe Ke K ( 1rt rtPe K e ︶

Solution 3.16 C Section 3.04, Put and Call Payoffs Let 1S be the price of one barrel of oil at time 1. Since the cost of providing oil is 48 per barrel, the payoff from the exclusive contract to sell 1,000 barrels of oil is:

1Contract payoff 1,000 48S The accumulated cost of putting the option strategy in place is negative, because the cost of the puts is less than the amount received from selling the calls:

0.05Accumulated cost of options (2,650 5,000) 2,470.49e

The quickest way to work this problem is to observe that the put option allows the oil producer to lock in a price that is at least 45 and selling the call option means that the oil producer will not receive a price that is greater than 55. Since the lowest profit occurs at the lowest price, and the highest profit at the highest price, we have:

1 11 1

45: Minimum profit 1,000( 48) ( 2,470.49)55: Maximum profit 1,000( 48) ( 2,470.49)

S SS S

529.519,470.49

Alternatively, let’s consider the option payoffs in more detail. The option payoffs are:

1

1

Long Put payoff 1,000 0,45

Short Call payoff 1,000 0, 55

Max S

Max S

These payoffs can be simplified when we note that the put option pays off only if the price is below 45, and the call option pays off only if the price is above 55:

1 11 1

45: Long Put payoff 1,000(45 )55: Short Call payoff 1,000( 55)

S SS S



The profit depends on the price of oil:

1 1 11 1 1

1 1 1

Contract payoff + Option payoff Accum. Cost of options45: 1,000( 48) 1,000(45 ) ( 2,470.49) 529.51

45 55: 1,000( 48) 0 ( 2,470.49) 1,000( 48) 2,470.4955: 1,000( 48) 1,000( 55) ( 2,

S S SS S S

S S S

470.49) 9,470.49

For oil prices less than 45 or greater than 55, the 1S terms cancel in the formulas above.

The lowest and highest profits possible when the prices are between 45 and 55 are found by plugging the prices of 45 and 55 into the middle row above:

1 11 1

Contract payoff + Option payoff Accum. Cost of options45: 1,000( 48) 2,470.49 529.5155: 1,000( 48) 2,470.49 9,470.49

S SS S

Therefore, the lowest possible profit is 529.51 and the highest possible profit is 9,470.49. A graph is the profits is shown below:

Solution 3.17 A Chapter 3, Call and Put Options The profit from being long the two calls must be equal to the profit from being long the put. Let’s see if there is a solution when the price of the index is less than 750:

2 0, 750 100 1.025 0, 750 65 1.025

2 0 100 1.025 750 65 1.025

205 683.375888.375

Max S Max S

S

SS

We began with the assumption that the index price is less than 750, so the solution of 888.375 is not valid, and we conclude that there is no solution when the price of the index is less than 750. Let’s see if there is a solution when the price of the index is greater than 750:

2 0, 750 100 1.025 0, 750 65 1.025

2 750 100 1.025 0 65 1.025

2 1,705 66.625

Max S Max S

S

SS

819.1875

Exam MFE Solutions Chapter 4 – Put-Call Parity


Chapter 4 Put-Call Parity Solution 4.01 C Section 4.01, Put-Call Parity We can use put-call parity to answer this question:

, ,0.05

( )

60.88 400 33.76

PEur t T t T EurC PV K F P

KeK

392.00

Solution 4.02 C Section 4.01, Put-Call Parity We can use put-call parity to answer this question:

, ,0.05 0.03

( )

38 400 56

PEur t T t T EurC PV K F P

Ke eK

427.00

Solution 4.03 B Chapter 4, Put-Call Parity Put-call parity can be expressed in terms of the prepaid forward price:

0, 0,( )P

Eur T T EurC PV K F P

The prepaid forward price is the present value of the forward price:

0, 0,P rTT TF F e

We can use put-call parity to find the put price:

4 0.05 4 0.0572 350 400 Eur

Eur

e e PP

31.06

Solution 4.04 B Chapter 4, Put-Call Parity Let’s write put-call parity with a strike price of 45 and again with a strike price of 55:

0.04 0.5 0.5

00.04 0.5 0.5

0

(45) 45 (45)

(55) 55 (55)

C e S e P

C e S e P

Subtracting the second equation from the first, we have:

0.04 0.5

0.02(45) (55) (45 55) (45) (55)

3.91 10 (45) (55)(55) (45)

C C e P P

e P PP P

5.89



Solution 4.05 E Chapter 4, Put-Call Parity Put-call parity can be expressed in terms of the prepaid forward price:

0, 0,( )P

Eur T T EurC PV K F P

The prepaid forward price is the present value of the forward price:

0,0, (1 )TP

T TF

Fi

We can use put-call parity and the options on gold to find the present value of the strike price:

0,2 2

0,2

1,260321 ( ) 801.04

( ) 923.9408

PV K

PV K

We can now use put-call parity to find the price of the call option on an ounce of platinum:

0,2 2

2

1,028( ) 801.041,028923.9408 801.04

Eur

Eur

Eur

C PV K

C

C

106.50

Solution 4.06 B Chapter 4, Put-Call Parity Put-call parity gives us the following relationships:

0, 0,0.04 0.03

( ) ( ) ( )

( ) 100 ( )

PT TC K PV K F P K

C K Ke e P K

Let’s consider the answer choices that have the same strike price for the call and the put:

A: 0.03 0.04(100) (100) 100 100 0.97C P e e

C: 0.03 0.04(102) (102) 100 102 0.96C P e e

E: 0.04 0.03(102) (102) 102 100 0.96P C e e

Choice B is greater than Choice A, because the value of a put option increases as its strike price increases:

(100) (99) (100) (100) because (99) (100)(100) (99) 0.97

C P C P P PC P

Choice D is less than Choice C, because the value of a call option decreases as its strike price increases:

(103) (102) (102) (102) because (103) (102)(103) (102) 0.96

C P C P C CC P

Since Choice B has the highest value, Choice B is correct.



Solution 4.07 B Chapter 4, Put-Call Parity The key to this question is that put-call parity implies that being long a call and short a stock is the same as being long a put and borrowing cash. Since borrowing and lending cash do not affect the profit of a position, Susan’s profit is equal to the profit from purchasing a put. That implies that since Larry sold a put instead of buying a put, his profit is the opposite of Susan’s profit. Consider put-call parity at the outset and when the options mature at time T:

0 0 0 0 0 0rT rT

T T T T T T

C Ke S P C S P KeC K S P P C S K

We are given that Susan’s profit is 5:

0 0

0 0

Payoff Accumulated cost 5

5

5

rTT T

rTT T

C S C S e

C S C S e

Using put-call parity to make several substitutions, we can find the profit of a short put:

0

0

0 0 0 0 0

0 0 0 0 0

0 0

Larry's profit Payoff Accumulated cost

because

5 because 5

5 because

5

rTT

rTT T T T T

rT rT rTT T

rT rT rT rT

rT rT

P P e

C S K P e P C S K

C S e K P e C S S C e

P Ke e K P e C S P Ke

P e K K P e

5

Exam MFE Solutions Chapter 5 – Application of Calls and Puts


Chapter 5 Applications of Calls and Puts Solution 5.01 D Chapter 5, Synthetic Forwards A long synthetic forward position is created by purchasing a call and selling a put, thereby locking in a future purchase price that is equal to the strike price: Synthetic Forward = Call Put The cost of purchasing the call is offset by the funds received from selling the put. We can use put-call parity to find the cost of establishing the synthetic forward:

01,575( ) 1,500 14.421.04Eur Eur

C P S PV K

Since the cost is not zero, this synthetic forward is an off-market synthetic forward. As shown above, the call price is 14.42 less than the put price. Therefore, buying the call and selling the put provides the investor with 14.42.

Solution 5.02 C Chapter 5, Caps The trader is purchasing a call and selling the stock, which describes a cap: Cap = Call Stock The payoff of the cap is: Payoff (0,140 100) 140 100Max

The profit of the cap is:


100 (22 100) 1.06

12.3592

Solution 5.03 E Chapter 5, Covered Puts The trader is selling a stock and selling a put on the stock, which describes a covered put: Covered put = Stock Put The payoff of the covered put is: Payoff 140 (0, 100 140) 140 0 140Max

The profit of the covered put is:


140 ( 11 100) 1.06

15.2804



Solution 5.04 A Chapter 5, Call and Put Options Choice A is correct, because a covered put consists of a short position in the underlying asset and a short put: Covered put = Stock Put Choice B is not correct, because a covered call consists of a long position in an asset together with a written call:

Covered call = Stock Call Choice C is not correct, because an American option can be more valuable than a Bermudan option, because an American option can be exercised at any time during the life of the option. Choice D is not correct, because an at-the-money option has a payoff of zero if exercised immediately. Choice E is not correct, because an in-the-money would have a positive payoff but not necessarily a positive profit if exercised immediately.

Solution 5.05 A Chapter 5, Covered Put When an investor writes a covered put, the investor writes a put and sells the underlying asset: Covered put = Stock Put This position is described by Choice A.

Solution 5.06 D Chapter 5, Ratio Spreads A ratio spread is constructed using two options that have different strike prices but are the same in every other way. A quantity of one of the options is purchased and a different quantity of the other option is sold. This is described by Choice D.

Solution 5.07 D Chapter 5, Box Spreads A box spread is created by purchasing a call bull spread and a put bear spread:

1 2 2 1

Box Spread Call bull spread Put bear spread( ) ( ) ( ) ( )C K C K P K P K

A box spread is used to lend funds. To borrow funds, the borrower must enter into a short box spread:

1 2 2 1

Box Spread Call bull spread Put bear spread( ) ( ) ( ) ( )C K C K P K P K

These transactions are described by Choice D.



Solution 5.08 D Chapter 5, Call Bull Spread The investor buys the 100-strike call and sells the 120-strike call. The initial cost of this position is: (100) (120) 20 11 9C C

Therefore, the investor borrows 9 at the outset. If the index price is less than 100, then both options expire out of the money and the investor’s profit is negative. Therefore, the price at which the investor breaks even must be greater than 100. Let’s consider the case where 1100 120S . In this case, the 120-strike option’s payoff is zero and the 100-strike option is in-the-money. We can solve for the price at which the profit is zero:

1 1

1

1

0, 100 0, 120 9 1.04 0

100 0 9.36 0

Max S Max S

SS

109.36

Solution 5.09 C Chapter 5, Ratio Spread The portfolio consists of a long position in 5 35-strike puts and a short position in 3 40-strike puts: 5 (35) 3 (40)P P

Let’s consider the payoff of the portfolio under 3 possibilities. Let S be the stock price at the end of 1 year. 1. Suppose that 35S . In that case, the both options expire in-the-money, and the

payoff is: 35 Payoff 5(35 ) 3(40 ) 55 2S S S S

Within this range the maximum and minimum payoffs are:

0 Maximum payoff 55 2 55 2 0 5535 Minimum payoff 55 2 55 2 35 15

S SS S

2. Suppose that 35 40S . In that case, the 35-strike option expires out-of-the-money, and the payoff is:

35 40 Payoff 3(40 ) 3 120S S S

Within this range the maximum and minimum payoffs are:

35 Minimum payoff 3 120 3 35 120 1540 Maximum payoff 3 120 3 40 120 0

S SS S

2. Suppose that 40 S . In that case, both options expire out-of-the-money, and the payoff is zero:

40 Payoff 0S

Looking at the payoffs above, we observe that the minimum payoff is 15 and the maximum payoff is 55:

Minimum payoff 15Maximum payoff 55



The cost of creating the portfolio is negative: 5 (35) 3 (40) 5 1.50 3 3.00 1.50P P

Therefore the portfolio has 1.50 that is lent for 1 year to obtain an accumulated value of:

0.081.50 1.6249e Adding this accumulated value to the payoffs above gives us the minimum and maximum profit:

Minimum profit 15 1.62 13.38Maximum profit 55 1.62 56.62

Since the maximum loss is 13.38 and maximum profit is 56.62, Choice C is correct. Alternatively, an easier way to work this kind of problem is to consider the stock prices of 0, the lower strike price, the upper strike price, and infinity. Between those points, the payoffs either decrease, remain the same, or increase, so the maximum and minimum payoffs must occur at one of those points:

5 (35) 3 (40)

0 5 (35 0) 3 (40 0) 5535 0 3 (40 35) 1540 0

0

S P P

Looking at the payoffs above, we observe that the minimum payoff is 15 and the maximum payoff is 55:

Minimum payoff 15Maximum payoff 55

Adding the accumulated value of the funds received when constructing the portfolio gives us the minimum and maximum profit:

Minimum profit 15 1.62 13.38Maximum profit 55 1.62 56.62

Since the maximum loss is 13.38 and maximum profit is 56.62, Choice C is correct.

Solution 5.10 E Chapter 5, Spreads Strategy I is a call bear spread: (70) (50)C C

As the stock price increases from 50, the 50-strike call’s payoff increases. Since the strategy is short the 50-strike call, the payoff declines until the 70-strike call begins to make offsetting payouts at a stock price of 70.



Strategy II is a put bull spread: (50) (70)P P

As the stock price decreases from 70, the 70-strike put’s payoff increases. Since the strategy is short the 70-strike, the payoff decreases as the stock price falls until the stock price reaches 50, when the 50-strike put begins to make offsetting payouts.

Strategy III is a box spread: (50) (50) (70) (70)C P P C

We can use put-call parity to see that the box spread has a constant payoff:

(50) (50) (70) (70)

( ) (50) (70) ( ) (20)P PC P P C

F S PV PV F S PV

Since the box spread can be replicated by setting aside the present value of 20, the payoff is 20.

Only Strategy II and Strategy III have payoffs that increase or remain constant as the stock price increases.

Solution 5.11 A Chapter 5, Strangles Both a strangle and a straddle benefit from volatility, because both have payoffs that increase if the stock price moves significantly in either direction. Since a straddle requires the purchase of a put option and a call option with the same strike price, both options cannot be out-of-the-money. Since the investor wishes to use out-of-the-money options, the investor must use a strangle.

Solution 5.12 E Chapter 5, Collars Buying a 5.00-strike put gives the farmer the right to sell at a price that is at least 5.00.



The farmer does not require a selling price that is greater than 5.50, so the farmer can sell an 5.50-strike call option to reduce the cost of the strategy. Selling the 5.50 call option obligates the farmer to sell at 5.50 if the price of a bushel is greater than 5.50. Buying the 5.00-strike put and selling the 5.50-strike call is described by Choice E. Below is a graph of the resulting position:

Solution 5.13 C Chapter 5, Applications of Calls and Puts Buying a 5.50-strike call gives the producer the right to buy at a price that is no more than 5.50. The producer does not require a purchase price that is less than 5.00, so the producer can sell a 5.00-strike put option to reduce the cost of the strategy. Selling the 5.00 put option obligates the producer to buy at 5.00 if the price of a bushel is less than 5.00. Buying the 5.50-strike call and selling the 5.00-strike put is described by Choice C.

Solution 5.14 D Section 5.03, Collars The call option cannot be at the money, because a put option with the same strike price has a value that is less than the value of the call option. We can use put-call parity to show this:

( )

0 0 0 0( )

0 0 0 0

( ) ( )

( ) ( )

r T tEur Eur

r T tEur Eur

C S S e S P S

S e S P S C S

The left side of the equation above is negative, so the put option’s value is too low. Reducing the put option’s strike to be lower than 0S will reduce the put option’s value

even further, so if the strike of the call option is 0S , the put option’s strike cannot be less

than 0S . To satisfy the definition of a collar, the put option’s strike cannot be greater than the strike of the call option. Therefore, we conclude that the call option cannot be at-the-money, and Choice D is false.

Solution 5.15 E Section 5.03, Collars It may seem that Choice D is false, but if the stock pays dividends, then it is possible for an at-the-money call option to have the same price as an otherwise equivalent put option with a lower strike price.

TS

Collared CornPayoff

5.00

5.50

5.00 5.50



Choice E is false. Suppose that the strike prices of both the call and the put are equal to the prepaid forward price. In that case, from put-call parity, we have:

( )

0 0 0 0( )( )

0 0 0 0

( ) ( )

( ) ( )

T T r T t T TEur Eur

r T t T T TEur Eur

C S e S e e S e P S e

S e S e P S e C S e

The left side of the equation above is negative, so the put option’s value is too low. Reducing the put option’s strike to be lower than 0

TS e will reduce the put option’s value

even further, so if the strike of the call option is 0TS e , the put option’s strike cannot be

less than 0TS e . To satisfy the definition of a collar, the put option’s strike cannot be

greater than the strike of the call option. Therefore, we conclude that the call option cannot have a strike price that is equal to the prepaid forward price.

Solution 5.16 A Chapter 5, Collars Let’s begin by considering a long collared stock position. A collared stock consists of a long stock, long low-strike put, and short high-strike call: Collared stock (90) (110)S P C

The put option guarantees that the investor can sell the stock for at least 90, and if the stock price exceeds 110, then the call option requires the investor to give up the stock in exchange for 110. Below is a graph of the resulting position:

The question describes a short collared stock position, consisting of a short stock, a short low-strike put, and a long high-strike call. Collared stock (90) (110)S P C

TS

Collared stockPayoff

90

110

90 110



The payoff diagram of this position is the mirror image of the payoff diagram shown above:

The shape of this graph best matches Choice A. The graph in this solution does not exactly match the graph in the question because it is not drawn to the same scale as the graph in the question.

Solution 5.17 E Chapter 5, Butterfly spreads and Iron Butterflies The profit graph exhibits the characteristics of the profits of symmetric butterfly spreads and iron butterflies. We can write Choices A and B so that they are recognizable as butterfly spreads: A P(45) 2P(50) + P(55)

B C(45) 2C(50) + C(55) Choice C is recognizable as an iron butterfly: C P(45) P(50) C(50) + C(55) We use put-call parity to substitute for the stock plus the put in Choice D: S +P(45) = PV(K) + C(45) We can now recognize Choice D as risk-free lending plus a butterfly spread: D S + P(45) 2C(50) + C(55) = PV(K) + C(45) 2C(50) + C(55) Since risk-free lending does not affect the profit diagram, the profit diagram for Choice D is the profit diagram of a butterfly spread. That leaves Choice E as the most likely answer. Consider the stock prices of 55 and 56, which result in payoffs of zero from the puts and positive payoffs from the stock and the 45-strike call: E S + C(45) 2P(50) + P(55) Payoff if S is 55: 55 + 10 0 + 0 = 65

Payoff if S is 56: 56 + 11 0 + 0 = 67 Since the profit is the payoff minus the accumulated value of the cost of establishing the position, we can write the profits at the two stock prices as: Profit if S is 55: Payoff AV(Cost) = 65 AV(Cost)

Profit if S is 56: Payoff AV(Cost) = 67 AV(Cost) Since the accumulated value of the cost is the same at both stock prices, we observe that the profit is 2 higher at a stock price of 56 than it is at a stock price of 55. But the profit diagram in the question shows the profit as being the same at both stock prices.

TS

(Collared stock) Payoff

110

90

90 110



Therefore, the position described by Choice E does not produce the profit diagram shown in the question.

Solution 5.18 C Chapter 5, Collars A collar is constructed by purchasing a put option and selling a call option with a higher strike price: Collar (50) (60)P C

The investor sells the stock and writes the collar. The initial cost is negative:

Stock Collar (50) (60) (60) (50) 4.46 3.77 52.00

51.31

S P C C P S

Since the cost is negative, the investor receives 51.31 at the outset. This amount is available to be lent at the risk-free rate, and the accumulated value of 51.31 is:

0.06 151.31 54.48e The payoff is the payoff of the call option minus the payoff of the put option minus the stock price:

1 1 1Payoff 0, 60 0,50Max S Max S S

Let’s consider the possible stock prices: 1. 1 1 150 Payoff 0 (50 ) 50S S S

2. 1 1 150 60 Payoff 0 0S S S

3. 1 1 160 Payoff 60 0 60S S S The maximum of these values is 50. To find the maximum profit, we add the accumulated value of the positive cash flow received at the outset: 50 54.48 4.48 Alternatively, note that the position described in the question is the opposite of a collared stock:

1 21 2

Collared Stock ( ) ( )(Collared Stock) ( ) ( )

S P K C KS P K C K

A collared stock’s payoff graph has the following form:

The minimum payoff of the collared stock is 50, so the maximum payoff of the opposite of a collared stock is 50. Adding this maximum payoff to the accumulated proceeds received from selling a collared stock provides us with the maximum possible profit: 50 54.48 4.48

TS

Collared Stock

50 60

50

60

Payoff



A graph of the profits resulting from a short position in the collared stock is shown below:

Solution 5.19 A Chapter 5, Butterfly spreads and Iron Butterflies The profit graph exhibits the characteristics of the profits of symmetric butterfly spreads and iron butterflies. We can write Choices B so that it is recognizable as a butterfly spread:

B C(35) 2C(40) + C(45) Choice C is recognizable as an iron butterfly: C P(35) P(40) C(40) + C(45) We use put-call parity to substitute for the stock plus the put in Choice D: S +P(35) = PV(K) + C(35) We can now write Choice D as: D S + P(35) 2C(40) + C(45) = PV(K) + C(35) 2C(40) + C(45) The right side of the equation above is a butterfly spread plus risk-free lending. Since risk-free lending does not affect the profit diagram, the profit diagram for Choice D is the profit diagram of a butterfly spread. We use put-call parity to substitute for the short stock plus the call in Choice E: S +C(35) = PV(K) + P(35) Let’s write Choice E and simplify using put-call parity: E S + C(35) 2P(40) + P(45) = PV(K) + P(35) 2P(40) + P(45) The right side of the equation above is a butterfly spread plus risk-free borrowing. Since risk-free borrowing does not affect the profit diagram, the profit diagram for Choice E is the profit diagram of a butterfly spread. That leaves Choice A as the most likely answer. The cost of the strategy is negative: 0.08 1.05 2 4.11 7.09 This means that an investor that implements the strategy described in Choice A receives 7.09 at the outset. If all of the options expire out of the money, then the investor’s profit is the accumulated value of 7.09, which is clearly positive. At stock prices above 45, all three options expire out of the money, so the profit should be positive. But the profit diagram shows negative profits when the stock price is above 45, so Choice A does not produce the profit diagram shown in the question.

Solution 5.20 D Chapter 5, Straddle and Strangle A straddle consists of a long position in a put and a call, where both options have the same strike price.



The accumulated cost of the straddle is:

0.08 1 0.08(40) (40) 6.28 3.21 10.28P C e e

A straddle consists of a long position in a put and a call, where the put option has a lower strike price than the call option. The accumulated cost of the strangle is:

0.08 1 0.08(30) (50) 0.53 2.63 3.42P C e e

The strangle only outperforms the straddle if the stock price movement is small, resulting in a small payoff to the straddle and a zero payoff to the strangle. For small price movements, the payoff of the straddle does not offset the larger premium paid for the straddle. As shown below, the strangle only outperforms the straddle within the range where the strangle’s payoff is zero.

At the lowest stock price where the strangle’s profit is greater than the straddle’s profit, only the 40-strike put is in-the-money:

Straddle profit Strangle profit

40 10.28 3.4233.14T

T

SS

At the highest stock price where the strangle’s profit is greater than the straddle’s profit, only the 40-strike call is in-the-money:

Straddle profit Strangle profit

40 10.28 3.4246.86

T

T

SS

Combining these two conditions, we have: 33.14 < < 46.86TS

Exam MFE Solutions Chapter 6 – Put-Call Parity and Replication

© ActuarialBrew.com 2017 Page 6.01

Solutions to the Part II Questions Chapter 6 – Solutions

Solution 6.01 B Put-Call Parity The European call premium can be found using put-call parity:

-

- -

+ = - +

+ = - +=

0 0,0.08(0.5) 0.08(0.25)

( , ) ( ) ( , )

(48,0.5) 48 50 5 4.38(48,0.5) 3.36

rTEur T Eur

Eur

Eur

C K T Ke S PV Div P K T

C e eC

Solution 6.02 B Put-Call Parity We ignore the dividend occurring in 7 months, because it occurs after the call option expires. The European call premium can be found using put-call parity:

-

- -

+ = - +

+ = - +=

0 0,0.05(0.5) 0.05(4 /12)

( , ) ( ) ( , )

(80,0.5) 80 75 4 10.37(80,0.5) 3.41

rTEur T Eur

Eur

Eur


C e eC

Solution 6.03 C Put-Call Parity This question may appear tricky since the options are euro-denominated. There is no other currency involved though, so we can use the standard put-call parity formula.

We can use put-call parity to solve for the risk-free rate of return:

d- -

- -

-

+ = +

+ = +

==

0(1) 0.04(1)

( , ) ( , )7.53 80 75 10.07

0.932490.0699

rT TEur Eur

r

r

C K T Ke S e P K T

e e

er

Solution 6.04 A Put-Call Parity Since the options are at-the-money, the strike price is equal to the stock price.



We can use put-call parity to find the price of the stock:

-

-

- -

- -

-

-

+ = - +

= - + +

= - + +

= - + +

- += =-

0 0,

0 0,0.07(0.75) 0.07(0.5)

00.07(0.75) 0.07(0.5)

0 00.07(0.5)

0 0.07(0.75)

( , ) ( ) ( , )

( , ) ( , ) ( )

1.34 31.34 31.34 3 30.441

rTEur T Eur

rTEur Eur T


S C K T P K T Ke PV Div

S Ke e

S S e e

eSe

Solution 6.05 D Synthetic Stock We can rearrange put-call parity so that it is a guide for replicating the stock:

-= + + -0 0,( , ) ( ) ( , )rTEur T EurS C K T Ke PV Div P K T

To replicate the stock, we must purchase the call, sell the put and lend the present value of the strike plus the present value of the dividends. The present value of the strike plus the present value of the dividends is:

- - -+ = + =0.09(0.75) 0.09(0.5)0, ( ) 28 3 29.04rT TKe PV Div e e

Solution 6.06 D Synthetic T-bills As an alternative to the method below, we could solve for r and then use it to find the present value of $1,000.

We can rearrange put-call parity so that it is a guide for replicating a T-bill:

d- -= - +0 ( , ) ( , )rT T Eur EurKe S e C K T P K T

To create an asset that matures for the strike price of $50, we must purchase d- Te shares of the stock, sell a call option, and buy a put option. The cost of doing this is:

d- -

-

= - +

= - +=

00.07(0.75)

( , ) ( , )52 6.56 3.6146.3904

rT TEur EurKe S e C K T P K T

e

Since it costs $46.3904 to create an asset that is guaranteed to mature for $50, it must cost 20 times as much to create an asset that is guaranteed to mature for $1,000: ¥ =20 46.3904 927.81



Solution 6.07 A Synthetic Stock To answer this question, we don’t need to know the risk-free rate of return or the time until the options expire.

We can rearrange put-call parity so that it is a guide for replicating the stock:

-= + + -0 0,( , ) ( ) ( , )rTEur T EurS C K T Ke PV Div P K T

To replicate the stock, we must purchase the call, sell the put and lend the present value of the strike plus the present value of the dividends. We can use the equation above to find the present value of the strike plus the present value of the dividends:

-

-

= + + -

+ = - + =

0,

0,

52 6.01 ( ) 3.87

( ) 52 6.01 3.87 49.86

rTT

rTT

Ke PV Div

Ke PV Div

Solution 6.08 E Currency options The current exchange rate is in the form of euros per dollar, which is the inverse of our usual form of dollars per euro.

The exchange rate in dollars per euro is:

= =01 1.04167

0.96x

Put-call parity for currency options can be used to find the value of the call option:

--

- -

+ = +

+ = +=

00.06(1) 0.04(1)

( , ) ( , )(0.94,1) 0.94 1.04167 0.005(0.94,1) 0.12056

fr TrTEur Eur

Eur

Eur

C K T Ke x e P K T

C e eC

Solution 6.09 A Currency options We treat the Swiss franc as the base currency for the first part of this question. But then at the end we must convert its value into dollars.

With the Swiss franc as the base currency, we have the following exchange rate in terms of francs per dollar:

= =01 1.25

0.80x



Put-call parity for currency options can be used to find the value of the put option:

--

- -

+ = +

+ = +=

00.06(1) 0.04(1)

( , ) ( , )0.127 1.15 1.25 (1.15,1)

(1.15,1) 0.00904

fr TrTEur Eur

Eur

Eur

C K T Ke x e P K T

e e PP

Therefore the cost of the put option is 0.00904 Swiss francs. But the possible choices are all expressed in dollars, so we must convert the value to dollars at the current exchange rate. One Swiss franc costs $0.80, so 0.00904 Swiss francs must cost (in dollars): ¥ =0.00904 0.80 0.00723

Solution 6.10 C Currency options We treat the Iraqi dinar as the base currency.

Put-call parity for currency options can be used to find interest rate on the dinar:

--

- -

-

+ = +

+ = +

==

00.5 0.08(0.5)

0.5

( , ) ( , )13.37 198 200 1.27198 180.05789

0.19

fr TrTEur Eur

r

r

C K T Ke x e P K T

e e

er

Solution 6.11 A Options on Bonds Here’s a quick refresher on compound interest. The annual effective interest rate is denoted by i:

r

n

n i

1 i e1v

1 i1 va

i

+ =

=+-

=

The price of the bond is:

-

--

= +

-= +-

= +=

0.10(15)0 15

0.10(15)0.10(15)

0.10

120 1,000

1120 1,0001

120(7.3867) 223.13021,109.5385

B a e

e ee

Only one coupon occurs before the expiration of the options, and it occurs one year from now.



Using put-call parity, we have:

-

- -

+ = - +

+ = - +=

0 0,0.10(1.25) 0.10(1)

( , ) ( ) ( , )

150 1,000 1,109.5385 120 (1,000,1.25)(1,000,1.25) 31.54

rTEur T Eur

Eur

Eur

C K T Ke B PV Coupons P K T

e e PP

Solution 6.12 E Options on Bonds The 6-month effective interest rate is:

- =0.08(0.5) 1 4.081%e The price of the bond one month after it is issued is:

( )

[ ]

0.08(12) 0.08(1 /12)0 24 4.081%

2411.0481 0.08(12) 0.08(1 /12)

0.04(24)0.08(12) 0.08(1 /12)

0.04

0.08(1 /12)

45 1,000

145 1,000

0.0481

145 1,0001

45(15.1212) 382.8929

1,063.346

B e e

e e

e e ee

e

-

-

--

È ˘= +Î ˚È ˘-Í ˙+Í ˙Í ˙Î ˚È ˘-= +Í ˙

-Í ˙Î ˚

= +

=

a

0.08(1 /12)01,070.4587

e¥=

Two coupons occur prior to the expiration of the option, the first of which occurs in 5 months and the second of which occurs in 11 months. Using put-call parity, we have:

-

- - -

+ = - +

+ = - - +=

0 0,0.08(1) 0.08(5 /12) 0.08(11 /12)

( , ) ( ) ( , )

(950,1) 950 1,070.4587 45 45 25(950,1) 133.16

rTEur T Eur

Eur

Eur

C K T Ke B PV Coupons P K T

C e e eC

Solution 6.13 B Options on Bonds The price of the bond is equal to $1,000. Since the bond price is equal to its par value, its yield must be equal to its coupon rate. Therefore the yield is 7%. Since 7% is the only interest rate provided in the problem, we use 7% as the risk-free interest rate. The 7% interest rate is compounded twice per year since coupons are paid semi-annually. Therefore, the semiannual effective interest rate is 3.5%.



Using put-call parity for bonds, we have:

( )

-

-

= - + -

= - -

=

0 0,1.5

( ) ( , ) ( , )351.035 1,000 58.43

1.035955.83

rTT Eur EurKe B PV Coupons P K T C K T

K

K

Solution 6.14 D Exchange Options The first step is to pick one of the assets to be the underlying asset. You can choose either one. We chose Stock X below, so a call option costs $2.70. If we chose Stock Y to be the underlying asset, then the same option would be a put option.

Let’s assume that Stock X is the underlying asset and Stock Y is the strike asset. In that case, the option to give up Stock Y for Stock X is a call option: =( , ,0.5) 2.70Eur t tC X Y

We can now use put-call parity for exchange options:

- -- = -

È ˘- = - -Î ˚=

, ,0.06(2 /12) 0.03(0.5)

( , ,0.5) ( , ,0.5) ( ) ( )

2.70 ( , ,0.5) 50 3 51

( , ,0.5) 5.91

P PEur t t Eur t t t T t t T t

Eur t t

Eur t t

C X Y P X Y F X F Y

P X Y e e

P X Y

Solution 6.15 A Exchange Options We didn’t need to know the risk-free rate of return. That was included in the question as a red herring.

The put option that allows its owner to give up Stock B in exchange for Stock A has Stock B as its underlying asset: =( , ,1) 11.49Eur t tP B A

Using put-call parity, we can find the value of the call option having Stock B as its underlying asset:

-- = -

- = -=

, ,0.05(1)

( , ,1) ( , ,1) ( ) ( )

( , ,1) 11.49 67 70( , ,1) 5.22


Eur t t

Eur t t

C B A P B A F B F A

C B A eC B A

We can describe the call option as a put option by switching the underlying asset and the strike asset:

==

( , ,1) ( , ,1)( , ,1) 5.22

Eur t t Eur t t

Eur t t

C B A P A BP A B



Therefore, the value of a put option giving its owner the right to give up a share of Stock A in exchange for a share of Stock B is $5.22.

Solution 6.16 D Exchange Options Let’s choose Stock X to be the underlying asset. In that case, Option A is a put option and Option B is a call option. Using put call parity, we have:

- -

- = -

- = -

- = -=

, ,

, ,0.04(5 /12) 0.04(5 /12)

00

( , ,5 /12) ( , ,5 /12) ( ) ( )

Option B Option A ( ) ( )

7.76 4047.89


P Pt T t t T t

C S Y P S Y F X F Y

F X F Y

e Y eY

Solution 6.17 E Exchange Options Let’s establish two portfolios. Portfolio X consists of 1 share of Stock A and 2 shares of Stock B. Portfolio Y consists of 1 share of Stock C and 1 share of Stock D. The first call option described in the question has Portfolio X as its underlying asset: =( , ,1) 10Eur t tC X Y

We can find the prepaid forward prices for both portfolios:

-

-

= + =

= + =

0.04(1),

0.02(1),

( ) 40 2(50) 138.4316

( ) 60 75 133.5149

Pt T tP

t T t

F X e

F Y e

We can now use put-call parity to find the value of the corresponding put option:

- = -

- = -=

, ,( , ,1) ( , ,1) ( ) ( )10 ( , ,1) 138.4316 133.5149

( , ,1) 5.0833


Eur t t

Eur t t

C X Y P X Y F X F YP X Y

P X Y

We can describe a put option as a call option by switching the underlying asset with the strike asset:

==

( , ,1) ( , ,1)( , ,1) 5.0833

Eur t t Eur t t

Eur t t

P X Y C Y XC Y X

Therefore, the call option giving its owner the right to acquire Portfolio Y in exchange for Portfolio X has a value of $5.0833.



Solution 6.18 B Exchange Options This question is easier if we assume that Stock X is the underlying asset.

Let’s assume that Stock X is the underlying asset. The American option is then an American call option. Since Stock X does not pay dividends, the American call option is not exercised prior to maturity. Therefore the American call option has the same value as a European call option: =( , ,7 /12) 10.22Eur t tC X Y

We are asked to find the value of the corresponding European put option. We can make use of put-call parity:

-- = -

- = -=

, ,0.04(7 /12)

( , ,7 /12) ( , ,7 /12) ( ) ( )

10.22 ( , ,7 /12) 100 100( , ,7 /12) 7.91


Eur t t

Eur t t

C X Y P X Y F X F Y

P X Y eP X Y

Solution 6.19 C Options on Currencies We can use put-call parity to determine the spot exchange rate:

--

- -

+ = +

+ = +=

00.05(0.5) 0.035(0.5)

00

( , ) ( , )0.047 0.89 0.011

0.920

fr TrTEur EurC K T Ke x e P K T

e x ex

Solution 6.20 B Exchange Options Neither the time to maturity nor the risk-free rate of return is needed to answer the question. They were provided as red herrings.

The prepaid forward price of Stock A is $22. The prepaid forward price of Stock B is $26. Stock A is the underlying asset, giving rise to the following generalized form of put-call parity:

- = -

- = -- = -

0 0 0 0 0, 0,

0 0 0 00 0 0 0

( , , ) ( , , ) ( ) ( )( , , ) ( , , ) 22 26( , , ) ( , , ) 4

P PEur Eur T T

Eur Eur

Eur Eur

C A B T P A B T F A F BC A B T P A B TC A B T P A B T

Therefore the put price exceeds the call price by $4. By inspection, only Choice B has a put price that is $4 greater than the call price.



Solution 6.21 C Exchange Options, Options on Currencies To answer this question, let’s use the yen as the base currency. In that case, the U.S. Dollar and the Canadian dollars are assets with prices denominated in the base currency. The U.S. dollar is the underlying asset for the options, and the Canadian dollar is the strike asset:

- -- - - = -

- = -- =

, ,0.08(0.5) 0.07(0.5)

( , , ) ( , , ) ( ) ( )

( , ,0.5) ( , ,0.5) 120 105( , ,0.5) ( , ,0.5) 13.91

P PEur t t Eur t t t T t T

Eur t t Eur t t

Eur t t Eur t t

C S Q T t P S Q T t F S F Q

C S Q P S Q e eC S Q P S Q

Solution 6.22

A Put-Call Parity Let each dividend amount be D. The first dividend occurs at the end of 2 months, and the second dividend occurs at the end of 5 months. We can use put-call parity to find D:

0 0,0.05(0.5) 0.05(2 /12) 0.05(5 /12)

0.05(2 /12) 0.05(5 /12) 0.05(0.5)

( , ) ( ) ( , )

2.55 42 40 4.36[ ] 40 4.36 2.55 42

1.97108 0.846980.4297

rTEur T EurC K T Ke S PV Div P K T

e De De

D e e eD

D

Solution 6.23

D Early Exercise For each put option, the choice is between having the exercise value now or having a 1-year European put option. Therefore, the decision depends on whether the exercise value is greater than the value of the European put option. The value of each European put option is found using put-call parity:

d

d

- -

- -

+ = +

= + -0

0

( , ) ( , )( , ) ( , )

rT TEur Eur

rT TEur Eur

C K T Ke S e P K T

P K T C K T Ke S e

The values of each of the 1-year European put options are:

- -

- -

- -

- -

= + - =

= + - =

= + - =

= + - =

0.05(1) 0.09(1)

0.05(1) 0.09(1)

0.05(1) 0.09(1)

0.05(1) 0.09(1)

(25,1) 21.93 25 50 0.01(50,1) 3.76 50 50 5.62(75,1) 0.21 75 50 25.86(100,1) 0.01 100 50 49.44

Eur

Eur

Eur

Eur

P e e

P e e

P e e

P e e



In the third and fourth columns of the table below, we compare the exercise value with the value of the European put options. The exercise value is - 0( ,0)Max K S .

Exercise European K C Value Put

$25.00 $21.93 0 0.01 $50.00 $3.76 0 5.62 $75.00 $0.21 25 25.86 $100.00 $0.01 50 49.44

The exercise value is less than the value of the European put option when the strike price is $75 or less. When the strike price is $100, the exercise value is greater than the value of the European put option. Therefore, it is optimal to exercise the special put option with an exercise price of $100.

Solution 6.24

B Exchange Options The first page of the study note, ”Some Remarks on Derivatives Markets,” tells us that “for each share of the stock the amount of dividends paid between time t and time t dt+ is assumed to be S(t ) dtd .” Therefore, the continuously compounded dividend rate for Stock 1 is 7%, and the continuously compounded dividend rate for Stock 2 is 3%.

The claim has the following payoff at time 4:

[ ]1 2(4), (4)Max S S A portfolio consisting of a share of Stock 2 and the option to exchange Stock 2 for Stock 1 effectively gives its owner the maximum value of the two stocks. If the value of Stock 2 is greater than the value of Stock 1 at time 4, then the owner keeps Stock 2 and allows the exchange option to expire unexercised. If the value of Stock 1 is greater than the value of Stock 2, then the owner exercises the option, giving up Stock 2 for Stock 1. Since Stock 2 has a continuously compounded dividend rate of 3%, the cost now of a share of Stock 2 at time 4 is:

d-

-

=

= =

00,0.03(4)

0,4 2

( )

( ) 85 75.39

P TT

P

F S e S

F S e

The cost of an exchange option allowing its owner to exchange Stock 2 for Stock 1 at time 4 is $43. Adding the costs together, we obtain the cost of the claim: + =75.39 43 118.39

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