Derivative Instruments Paris Dauphine University - Master IEF (272)
Jérôme MATHIS
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LEDa
Exercises + Solutions Chapter 10
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Exercises + Solutions Chapter 10
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Exercise (1) Call options on a stock are available with strike prices of $15, $17 21 , and $20 and expiration dates in three months. Their prices are $4, $2, and $ 21 , respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how profit varies with stock price for the butterfly spread.
Solution (1)
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Exercises + Solutions Chapter 10
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Exercise (1) Call options on a stock are available with strike prices of $15, $17 21 , and $20 and expiration dates in three months. Their prices are $4, $2, and $ 21 , respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how profit varies with stock price for the butterfly spread.
Solution (1) An investor can create a butterfly spread by buying
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Exercises + Solutions Chapter 10
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Exercise (1) Call options on a stock are available with strike prices of $15, $17 21 , and $20 and expiration dates in three months. Their prices are $4, $2, and $ 21 , respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how profit varies with stock price for the butterfly spread.
Solution (1) An investor can create a butterfly spread by buying call options with strike prices of $15 and $20 and selling
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Exercises + Solutions Chapter 10
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Exercise (1) Call options on a stock are available with strike prices of $15, $17 21 , and $20 and expiration dates in three months. Their prices are $4, $2, and $ 21 , respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how profit varies with stock price for the butterfly spread.
Solution (1) An investor can create a butterfly spread by buying call options with strike prices of $15 and $20 and selling two call options with strike prices of $17 12 . The initial investment is
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Exercises + Solutions Chapter 10
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Exercise (1) Call options on a stock are available with strike prices of $15, $17 21 , and $20 and expiration dates in three months. Their prices are $4, $2, and $ 21 , respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how profit varies with stock price for the butterfly spread.
Solution (1) An investor can create a butterfly spread by buying call options with strike prices of $15 and $20 and selling two call options with strike prices of $17 12 . The initial investment is 4+
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1 2
2
2=
Derivative Instruments
Exercises + Solutions Chapter 10
2/9
Exercise (1) Call options on a stock are available with strike prices of $15, $17 21 , and $20 and expiration dates in three months. Their prices are $4, $2, and $ 21 , respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how profit varies with stock price for the butterfly spread.
Solution (1) An investor can create a butterfly spread by buying call options with strike prices of $15 and $20 and selling two call options with strike prices of $17 12 . The initial investment is 4+
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1 2
2
2=
Derivative Instruments
1 . 2 Exercises + Solutions Chapter 10
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Solution (1) The following table shows the variation of profit with the final stock price:
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Exercises + Solutions Chapter 10
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Solution (1) The following table shows the variation of profit with the final stock price: Stock Price, ST ST < 15
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Profit
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Exercises + Solutions Chapter 10
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Solution (1) The following table shows the variation of profit with the final stock price: Stock Price, ST ST < 15 15 < ST < 17 12
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Profit 1 2
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Exercises + Solutions Chapter 10
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Solution (1) The following table shows the variation of profit with the final stock price: Stock Price, ST ST < 15 15 < ST < 17 12 17 12 < ST < 20
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Profit 1 2
(ST
15)
1 2
Derivative Instruments
Exercises + Solutions Chapter 10
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Solution (1) The following table shows the variation of profit with the final stock price: Stock Price, ST ST < 15 15 < ST < 17 12 17 12 < ST < 20 ST > 20
Jérôme MATHIS (LEDa)
Profit 1 2
(ST (20
15) ST )
1 2 1 2
Derivative Instruments
Exercises + Solutions Chapter 10
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Solution (1) The following table shows the variation of profit with the final stock price: Stock Price, ST ST < 15 15 < ST < 17 12 17 12 < ST < 20 ST > 20
Jérôme MATHIS (LEDa)
Profit 1 2
(ST (20
15) ST ) 1 2
1 2 1 2
Derivative Instruments
Exercises + Solutions Chapter 10
3/9
Exercise (2, Done) A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3. Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?
Solution (2)
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Exercise (2, Done) A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3. Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?
Solution (2) A strangle is created by buying both options. The pattern of profits is as follows: Stock Price, ST ST < 45
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Exercises + Solutions Chapter 10
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Exercise (2, Done) A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3. Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?
Solution (2) A strangle is created by buying both options. The pattern of profits is as follows: Stock Price, ST ST < 45 45 < ST < 50
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(45
Profit ST )
5
Derivative Instruments
Exercises + Solutions Chapter 10
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Exercise (2, Done) A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3. Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?
Solution (2) A strangle is created by buying both options. The pattern of profits is as follows: Stock Price, ST ST < 45 45 < ST < 50 ST > 50 Jérôme MATHIS (LEDa)
(45
Profit ST ) 5
5
Derivative Instruments
Exercises + Solutions Chapter 10
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Exercise (2, Done) A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3. Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?
Solution (2) A strangle is created by buying both options. The pattern of profits is as follows: Stock Price, ST ST < 45 45 < ST < 50 ST > 50 Jérôme MATHIS (LEDa)
Profit ST ) 5 (ST 50)
(45
5 5
Derivative Instruments
Exercises + Solutions Chapter 10
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Exercise (3, Done) Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payofffor both spreads.
Solution (3)
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Exercises + Solutions Chapter 10
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Exercise (3, Done) Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payofffor both spreads.
Solution (3) A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows: Stock Price, ST ST 35
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Payoff
Profit
Derivative Instruments
Exercises + Solutions Chapter 10
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Exercise (3, Done) Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payofffor both spreads.
Solution (3) A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows: Stock Price, ST ST 35
Jérôme MATHIS (LEDa)
Payoff 0
Profit
Derivative Instruments
Exercises + Solutions Chapter 10
5/9
Exercise (3, Done) Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payofffor both spreads.
Solution (3) A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35 Jérôme MATHIS (LEDa)
Payoff 0
Profit 3
Derivative Instruments
Exercises + Solutions Chapter 10
5/9
Exercise (3, Done) Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payofffor both spreads.
Solution (3) A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35 Jérôme MATHIS (LEDa)
Payoff 0 ST 35
Profit 3
Derivative Instruments
Exercises + Solutions Chapter 10
5/9
Exercise (3, Done) Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payofffor both spreads.
Solution (3) A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35 ST < 30 Jérôme MATHIS (LEDa)
Payoff 0 ST 35
Profit 3 ST 32 Derivative Instruments
Exercises + Solutions Chapter 10
5/9
Exercise (3, Done) Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payofffor both spreads.
Solution (3) A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35 ST < 30 Jérôme MATHIS (LEDa)
Payoff 0 ST 35 5
Profit 3 ST 32 Derivative Instruments
Exercises + Solutions Chapter 10
5/9
Exercise (3, Done) Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payofffor both spreads.
Solution (3) A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an initial cash inflow of $3. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35 ST < 30 Jérôme MATHIS (LEDa)
Payoff 0 ST 35 5
Profit 3 ST 32 2 Derivative Instruments
Exercises + Solutions Chapter 10
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Solution (3)
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Solution (3) A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows: Stock Price, ST ST 35
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Payoff
Profit
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Solution (3) A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows: Stock Price, ST ST 35
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Payoff 0
Profit
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Exercises + Solutions Chapter 10
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Solution (3) A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35
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Payoff 0
Profit 3
Derivative Instruments
Exercises + Solutions Chapter 10
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Solution (3) A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35
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Payoff 0 35 ST
Profit 3
Derivative Instruments
Exercises + Solutions Chapter 10
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Solution (3) A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35 ST < 30
Jérôme MATHIS (LEDa)
Payoff 0 35 ST
Profit 3 32 ST
Derivative Instruments
Exercises + Solutions Chapter 10
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Solution (3) A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35 ST < 30
Jérôme MATHIS (LEDa)
Payoff 0 35 ST 5
Profit 3 32 ST
Derivative Instruments
Exercises + Solutions Chapter 10
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Solution (3) A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially. The outcome is as follows: Stock Price, ST ST 35 30 ST < 35 ST < 30
Jérôme MATHIS (LEDa)
Payoff 0 35 ST 5
Profit 3 32 ST 2
Derivative Instruments
Exercises + Solutions Chapter 10
6/9
Exercise (4) Use put-call parity to show that the cost of a butterfly spread created from European puts is identical to the cost of a butterfly spread created from European calls.
Solution (4)
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Exercises + Solutions Chapter 10
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Exercise (4) Use put-call parity to show that the cost of a butterfly spread created from European puts is identical to the cost of a butterfly spread created from European calls.
Solution (4) Define c1 , c2 and c3 as the prices of calls with strike prices K1 , K2 , and K3 . Define p1 , p2 and p3 as the prices of puts with strike prices K1 , K2 , and K3 . With the usual notation
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Exercise (4) Use put-call parity to show that the cost of a butterfly spread created from European puts is identical to the cost of a butterfly spread created from European calls.
Solution (4) Define c1 , c2 and c3 as the prices of calls with strike prices K1 , K2 , and K3 . Define p1 , p2 and p3 as the prices of puts with strike prices K1 , K2 , and K3 . With the usual notation ci + Ki e
Jérôme MATHIS (LEDa)
rT
=
Derivative Instruments
Exercises + Solutions Chapter 10
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Exercise (4) Use put-call parity to show that the cost of a butterfly spread created from European puts is identical to the cost of a butterfly spread created from European calls.
Solution (4) Define c1 , c2 and c3 as the prices of calls with strike prices K1 , K2 , and K3 . Define p1 , p2 and p3 as the prices of puts with strike prices K1 , K2 , and K3 . With the usual notation ci + Ki e
rT
= pi + S
with i 2 f1, 2, 3g. Jérôme MATHIS (LEDa)
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Exercises + Solutions Chapter 10
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Solution (4)
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Solution (4) Hence c1 + c3
Jérôme MATHIS (LEDa)
2c2 + (K1 + K3
2K2 )e
Derivative Instruments
rT
=
Exercises + Solutions Chapter 10
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Solution (4) Hence c1 + c3
Jérôme MATHIS (LEDa)
2c2 + (K1 + K3
2K2 )e
Derivative Instruments
rT
= p1 + p3
2p2
Exercises + Solutions Chapter 10
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Solution (4) Hence c1 + c3 Because K2
2c2 + (K1 + K3
2K2 )e
rT
= p1 + p3
2p2
K1 =
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Solution (4) Hence c1 + c3 Because K2
2c2 + (K1 + K3
K1 = K3
Jérôme MATHIS (LEDa)
2K2 )e
rT
= p1 + p3
K2 , it follows that K1 + K3
Derivative Instruments
2p2
2K2 =
Exercises + Solutions Chapter 10
8/9
Solution (4) Hence c1 + c3 Because K2
2c2 + (K1 + K3
K1 = K3
Jérôme MATHIS (LEDa)
2K2 )e
rT
= p1 + p3
K2 , it follows that K1 + K3
Derivative Instruments
2p2
2K2 = 0 and
Exercises + Solutions Chapter 10
8/9
Solution (4) Hence c1 + c3 Because K2
2c2 + (K1 + K3
K1 = K3
rT
= p1 + p3
K2 , it follows that K1 + K3
c1 + c3
Jérôme MATHIS (LEDa)
2K2 )e
2p2
2K2 = 0 and
2c2 =
Derivative Instruments
Exercises + Solutions Chapter 10
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Solution (4) Hence c1 + c3 Because K2
2c2 + (K1 + K3
K1 = K3
rT
= p1 + p3
K2 , it follows that K1 + K3
c1 + c3
Jérôme MATHIS (LEDa)
2K2 )e
2c2 = p1 + p3
Derivative Instruments
2p2
2K2 = 0 and
2p2
Exercises + Solutions Chapter 10
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Solution (4) Hence c1 + c3 Because K2
2c2 + (K1 + K3
K1 = K3
2K2 )e
rT
= p1 + p3
K2 , it follows that K1 + K3
c1 + c3
2c2 = p1 + p3
2p2
2K2 = 0 and
2p2
The cost of a butterfly spread created using European calls is therefore exactly the same as the cost of a butterfly spread created using European puts.
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Exercise (5) A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
Solution (5)
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Exercise (5) A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
Solution (5) A straddle is created by buying both the call and the put. This strategy costs
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Exercises + Solutions Chapter 10
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Exercise (5) A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
Solution (5) A straddle is created by buying both the call and the put. This strategy costs $10. The profit/loss is shown in the following table: Stock Price, ST ST > 60
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Payoff
Profit
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Exercises + Solutions Chapter 10
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Exercise (5) A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
Solution (5) A straddle is created by buying both the call and the put. This strategy costs $10. The profit/loss is shown in the following table: Stock Price, ST ST > 60
Jérôme MATHIS (LEDa)
Payoff ST 60
Profit
Derivative Instruments
Exercises + Solutions Chapter 10
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Exercise (5) A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
Solution (5) A straddle is created by buying both the call and the put. This strategy costs $10. The profit/loss is shown in the following table: Stock Price, ST ST > 60 ST 60
Jérôme MATHIS (LEDa)
Payoff ST 60
Profit ST 70
Derivative Instruments
Exercises + Solutions Chapter 10
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Exercise (5) A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
Solution (5) A straddle is created by buying both the call and the put. This strategy costs $10. The profit/loss is shown in the following table: Stock Price, ST ST > 60 ST 60
Jérôme MATHIS (LEDa)
Payoff ST 60 60 ST
Profit ST 70
Derivative Instruments
Exercises + Solutions Chapter 10
9/9
Exercise (5) A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
Solution (5) A straddle is created by buying both the call and the put. This strategy costs $10. The profit/loss is shown in the following table: Stock Price, ST ST > 60 ST 60
Payoff ST 60 60 ST
Profit ST 70 50 ST
This shows that the straddle will lead to a loss if the final stock price is between Jérôme MATHIS (LEDa)
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Exercises + Solutions Chapter 10
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Exercise (5) A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
Solution (5) A straddle is created by buying both the call and the put. This strategy costs $10. The profit/loss is shown in the following table: Stock Price, ST ST > 60 ST 60
Payoff ST 60 60 ST
Profit ST 70 50 ST
This shows that the straddle will lead to a loss if the final stock price is between $50 and $70. Jérôme MATHIS (LEDa)
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