Unformatted text preview: 5. 59 Part 3 Options Therefore, we use a call and put butterfly spread to profit from these arbitrage opportunities.
Transaction
Buy 1 50 strike call
Sell 2 55 strike calls
buy 1 60 strike call
TOTAL t=0
18
+28
9.50
+0.50 ST < 50
0
0
0
0 Transaction
Buy 1 50 strike put
Sell 2 55 strike puts
buy 1 60 strike put
TOTAL t=0
7
21.50
14.45
+0.05 ST < 50
50 − ST 50 ≤ ST ≤ 55
0 2 × ST − 110 2 × ST − 110
60 − ST
ST − 50 ≥ 0 50 ≤ ST ≤ 55
ST − 50
0
0
ST − 50 ≥ 0 60 − ST
0 55 ≤ ST ≤ 60
ST − 50
110 − 2 × ST
0
60 − ST ≥ 0 ST > 60
ST − 50
110 − 2 × ST
ST − 60
0 55 ≤ ST ≤ 60
0
0
60 − ST
60 − ST ≥ 0 ST > 60
0
0
0
0 Please note that we initially receive money and have nonnegative future payoffs. Therefore, we
have found an arbitrage possibility, independent of the prevailing interest rate.
Question 9.12. a)
Equations (9.15) of the textbook is violated. We use a call bear spread to profit from this
arbitrage opportunity. Transaction
Sell 90 strike call
Buy 95 strike call
TOTAL t=0
+10
4
+6 Expiration or Exercise
ST < 90
90 ≤ ST ≤ 95 ST > 95
0
90 − ST
90 − ST
0
0
ST − 95
0
90 − ST > −5  5 Please note that we initially receive more money than our biggest possible exposure in the future.
Therefore, we have found an arbitrage possibility, independent of the prevailing interest rate.
b)
Now, equation (9.15) is not violated anymore. However, we can still construct an
arbitrage opportunity, given the information in the exercise. We continue to sell the 90strike call
and buy the 95strike call, and we loan our initial positive net balance for two years until
expiration. It is important that the options be European, because otherwise we would not be able
to tell whether the 90strike call could be exercised against us sometime (note that we do not
have information regarding any dividends). 60 Chapter 9 Parity and Other Option Relationships We have the following arbitrage table: Transaction
Sell 90 strike call
Buy 95 strike call
Loan 4.75
TOTAL Expiration t=T
t=0
ST < 90
90 ≤ ST ≤ 95
+10 0
90 − ST
5.25 0
0
4.75 5.80
5.80
0
5.80
95.8 − ST > 0 ST > 95
90 − ST
ST − 95
5.8
+ 0 .8 In all possible future states, we have a strictly positive payoff. We have created something out of
nothing – we demonstrated arbitrage.
c) We will first verify that equation (9.17) is violated. We have:
C (K1 ) − C (K 2 ) 15 − 10
=
= 0 .5
K 2 − K1
100 − 90 and C (K 2 ) − C (K 3 )
10 − 6
=
= 0 .8 ,
K3 − K2
105 − 100 which violates equation 9.17.
We calculate lambda in order to know how many options to buy and sell when we construct the
butterfly spread that exploits this form of mispricing. Using formula (9.19), we can calculate that
lambda is equal to 1/3. To buy and sell round lots, we multiply all the option trades by 3.
We use an asymmetric call and put butterfly spread to profit from these arbitrage opportunities.
Transaction
Buy 1 90 strike calls
Sell 3 100 strike calls
buy 2 105 strike calls
TOTAL t=0
15
+30
12
+3 ST < 90
0
0
0
0 90 ≤ ST ≤ 100
ST − 90
0
0 100 ≤ ST ≤ 105
ST − 90
300 − 3 × ST
0 ST − 90 ≥ 0 210 − 2 × S T ≥ 0 ST > 105
ST − 90
300 − 3 × ST
2 × ST − 210
0 We indeed have an arbitrage opportunity.
Question 9.14. This question is closely related to question 9.13. In this exercise, the strike is not cash anymore,
but rather one share of Apple. In parts a) and b), there is no benefit in keeping Apple longer,
because the dividend is zero.
a)
The underlying asset is the stock of Apple, which does not pay a dividend. Therefore, we
have an American call option on a nondividendpaying stock. It is never optimal to early
exercise such an option. 61 Part 3 Options b)
The underlying asset is the stock of Apple, and the strike consists of AOL. As AOL does
not pay a dividend, the interest rate on AOL is zero. We will therefore never early exercise the
put option, because we cannot receive earlier any benefits associated with holding Apple – there
are none. If Apple is bankrupt, there is no loss from not early exercising, because the option is
worth max[0, AOL – 0], which is equivalent to one share of AOL, because of the limited liability
of stock. As AOL does not pay dividends, we are indifferent between holding the option and the
stock.
c)
For the American call option, dividends on the stock are the reason why we want to
receive the stock earlier, and now Apple pays a dividend. We usually benefit from waiting,
because we can continue to earn interest on the strike. However, in this case, the dividend on
AOL remains zero, so we do not have this benefit associated with waiting to exercise. Finally,
we saw that there is a second benefit to waiting: the insurance protection, which will not be
affected by the zero AOL dividend. Therefore, there now may be circumstances in which we will
early exercise, but we will not always early exercise.
For the American put option, there is no cost associated with w...
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 Spring '14
 NguyenThiMaiLan
 Derivatives, simple strategies

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