37.
The two possible stock prices and the corresponding put values are:
uS
0
= 120
⇒
P
u
= 0
dS
0
= 80
⇒
P
d
= 20
The hedge ratio is:
2
1
80
120
20
0
dS
uS
P
P
H
0
0
d
u
−
=
−
−
=
−
−
=
Form a riskless portfolio by buying one share of stock and buying two puts.
The
cost of the portfolio is: S + 2P = 100 + 2P
The payoff for the riskless portfolio equals $120:
Riskless
Portfolio
S = 80
S = 120
Buy 1 share
80
120
Buy 2 puts
40
0
Total
120
120
Therefore, find the value of the put by solving:
$100 + 2P = $120/1.10
⇒
P = $4.545
According to putcall parity: P + S = C + PV(X)
Our estimates of option value satisfy this relationship:
$4.545 + $100 = $13.636 + $100/1.10 = $104.545
2115
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If we assume that the only possible exercise date is just prior to the exdividend
date, then the relevant parameters for the BlackScholes formula are:
S
0
= 60
r = 0.5% per month
X = 55
σ
= 7%
T = 2 months
In this case: C = $6.04
If instead, one commits to foregoing early exercise, then we reduce the stock price
by the present value of the dividends.
Therefore, we use the following parameters:
S
0
= 60 – 2e
−
(0.005
×
2)
= 58.02
r = 0.5% per month
X = 55
σ
= 7%
T = 3 months
In this case, C = $5.05
The pseudoAmerican option value is the higher of these two values: $6.04
39.
a.
(i) Index increases to 701.
The combined portfolio will suffer a loss.
The
written calls expire in the money; the protective put purchased expires
worthless.
Let’s analyze the outcome on a pershare basis.
The payout for
each call option is $26, for a total cash outflow of $52.
The stock is worth
$701.
The portfolio will thus be worth: $701
−
$52 = $649
The net cost of the portfolio when the option positions are established is:
$668 + $8.05 (put)
−
[2
×
$4.30] (calls written) = $667.45
(ii) Index remains at 668.
Both options expire out of the money.
The portfolio
will thus be worth $668 (per share), compared to an initial cost 30 days earlier of
$667.45.
The portfolio experiences a very small gain of $0.55.
(iii) Index declines to 635.
The calls expire worthless.
The portfolio will be
worth $665, the exercise price of the protective put.
This represents a very small
loss of $2.45 compared to the initial cost 30 days earlier of $667.45.
b.
(i) Index increases to 701.
The delta of the call approaches 1.0 as the stock goes
deep into the money, while expiration of the call approaches and exercise
becomes essentially certain.
The put delta approaches zero.
(ii) Index remains at 668.
Both options expire out of the money.
Delta of
each approaches zero as expiration approaches and it becomes certain that the
options will not be exercised.
(iii) Index declines to 635.
The call is out of the money as expiration
approaches.
Delta approaches zero.
Conversely, the delta of the put
approaches
−
1.0 as exercise becomes certain.
2116
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 Spring '13
 Ohk
 hedge ratio, Riskless Portfolio

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