23.
a.
The delta of the collar is calculated as follows:
Position
Delta
Buy stock
1.0
Buy put, X = $45
N(d
1
) – 1 = –0.40
Write call, X = $55
–N(d
1
) = –0.35
Total
0.25
If the stock price increases by $1, then the value of the collar increases by
$0.25.
The stock will be worth $1 more, the loss on the purchased put will be
$0.40, and the call written represents a
liability
that increases by $0.35.
b.
If S becomes very large, then the delta of the collar approaches zero.
Both
N(d
1
) terms approach 1.
Intuitively, for very large stock prices, the value of
the portfolio is simply the (present value of the) exercise price of the call, and
is unaffected by small changes in the stock price.
As S approaches zero, the delta also approaches zero: both N(d
1
) terms
approach 0.
For very small stock prices, the value of the portfolio is simply
the (present value of the) exercise price of the put, and is unaffected by small
changes in the stock price.
24.
Statement a: The hedge ratio (determining the number of futures contracts to sell)
ought to be adjusted by the beta of the equity portfolio, which is 1.20.
The correct
hedge ratio would be:
400
,
2
2
.
1
000
,
2
β
2,000
β
500
100
$
million
100
$
=
×
=
×
=
×
×
Statement b: The portfolio will be hedged, and should therefore earn the risk-free
rate, not zero, as the consultant claims.
Given a futures price of 100 and an equity
price of 100, the rate of return over the 3-month period is:
(100
−
99)/99 = 1.01% = approximately 4.1% annualized
25.
Put
X
Delta
A
10
−
0.1
B
20
−
0.5
C
30
−
0.9
26.
a.
Choice A: Calls have higher elasticity than shares.
For equal dollar investments,
a call’s capital gain potential is greater than that of the underlying stock.
b.
Choice B: Calls have hedge ratios less than 1.0, so the shares have higher
profit potential.
For an equal number of shares controlled, the dollar exposure
of the shares is greater than that of the calls, and the profit potential is
therefore greater.
21-6