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Unformatted text preview: Problem 10.4. Cali options on a stock are available with strike prices of $15, $17%, and $20 and
expiration dates in three months. Their prices are $4, $2, and $%, 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. 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 $175 The initial investment
is 4 + % —— 2 x 2 2 $%. The following table shows the variation of proﬁt with the ﬁnal stock
price: Stock Price Proﬁt
ST
ST < 15 mg
15 < ST < 17% (ST 15) %
17% < ST < 20 (20 ~ ST) — a ST > 20 mg Problem 10.10. 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 payoff for both spreads, 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: gthck Price Payoff _ Proﬁ?
a 2 35— _ o 3
30 S ST < 35 ST — 35 ST — 32
ST < 30 —5 —2 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 Payoff Proﬁt
ST 2 35 0 ~—~3
30 _<_’ ST < 35 35 w ST 32 — ST ST<30 5 2 Problem 10.15.
How can a forward contract on a stock with a particular delivery price and delivery date be created from options? Suppose that the delivery price is K and the delivery date is T. The forward contract
is created by buying a European call and selling a European put when both options have
strike price K and exercise date T. This portfolio provides a payoff of ST —— K under all
circumstances where ST is the stock price at time T. Suppose that E; is the forward price.
If K 2 F0, the forward contract that is created has zero value. This shows that the price
of a call equals the price of a put when the strike price is Fe. Problem 10.16.
“A box spread comprises four options. Two can be combined to create a long for ward position and two can be combined to create a short forward position.” Explain this
statement. A box spread is a bull spread created using calls and a bear spread created using
puts. With the notation in the text it consists of a) a long call with strike K1, b) a short
call with strike K2, c) a long put with strike K2, and d) a short put with strike K1. a)
and d) give a long forward contract with delivery price K1; b) and c) give a short forward
contract with delivery price K2. The two forward contracts taken together give the payoff
of K2 ~~ K1. Problem 11.9.
A stock price is currently $504, It is known that at the end of two months it will be either $53 or $48. The risk—free interest rate is 10% per annum with continuous compoundu
ing, What is the value of a twomonth European call option with a strike price of $49? Use no—arbitrage arguments. At the end of two months the value of the option will be either $4 (if the stock price
is $53) or $0 (if the stock price is $48). Consider a portfolio consisting of: +A : shares
—1 : option The value of the portfolio is either 48A or 53A «— 4 in two months. If 48A : 53A 1— 4 i.e., A208 the value of the portfolio is certain to be 38.4. For this value of A the portfolio is therefore
riskless. The current value of the portfolio is: 0.8 x 50 ~ f where f is the value of the option. Since the portfolio must earn the riskfree rate of
interest
(0.8 x. 50 — f)e0 10W” = 38.4 i.e.,
f : 2.23 The value of the option is therefore $2.23.
This can also be calculated directly from equations (11.2) and (11.3). it 2 1.06, d : SO that O 10 2/12
6 x 4 06.96
7» ____ : 0. 681
1.06 — 0.96 5 p: and
f : e'UWZ/l2 x 0.568] X 4 = 2.23 Problem 11.12. A stock price is currently $50. Over each of the next two threemonth periods it is
expected to go up by 6% or down by 5%. The risk—free interest rate is 5% per annum with continuous compounding. What is the value of a six—month European call option with a
strike price of $51 ? A tree describing the behavior of the stock price is shown in Figure 811.3. The risk—
neutral probability of an up move7 p, is given by 60 05x3/12 __ 0.95 F 1.06 — 0.95 : 05689 p: There is a payoff from the option of 56.18 — 51 = 5.18 for the highest ﬁnal node (which
corresponds to two up moves) zero in all other cases. The value of the option is therefore 5.18 x 0.56892 x e") 05W” = 1.635 This can also be calculated by working back through the tree as indicated in Figure 811.3.
The value of the call option is the lower number at each node in the ﬁgure. 56.18
5.18 Figure 811.3 Tree for Problem 11.12 Problem 1 1 . 1 3. For the situation considered in Problem 11.12, what is the value of a sixmonth Euro
pean put option with astrike price 0135951? Verify that the European call and European put prices satisfy puticall parity. If the put option were American, would it ever be optimal
to exercise it early at any of the nodes on the tree? The tree for valuing the put option is shown in Figure 811.4. We get a payoff of
51 — 50.35 = 0.65 if the middle ﬁnal node is reached and a payoff of 51 —— 45.125 = 5.875 if
the lowest ﬁnal node is reached. The value of the option is therefore (0.65 x 2 x 0.5689 x 0.4311 + 5.875 x 0.43112)e_0 05W” = 1.376 This can also be calculated by working back through the tree as indicated in Figure 811.4.
The value of the put plus the stock price is from Problem 11.12 1.376 + 50 : 51.376
The value of the call plus the present value of the strike price is 1.635 + 516—005x6/ 12 : 51.376 This veriﬁes that put—call parity holds To test whether it worth exercising the option early we compare the value calculated
for the option at each node with the payoff from immediate exercise. At node C the payoff
from immediate exercise is 51 ~~ 47 .5 2 3.5. Because this is greater than 2.8664, the option
should be exercised at this node. The option should not be exercised at either node A or node B. 56.18
53 50,35 5
0 065 1 .376 2 86
, 45.125
5.875 Figure 811.4 Tree for Problem 11.13 ...
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