119
Answers to Questions and Problems
1. Consider Call A, with:
X
$70;
r
0.06;
T
t
90 days;
0.4; and
S
$60. Compute the price,
DELTA, GAMMA, THETA, VEGA, and RHO for this call.
2. Consider Put A, with:
X
$70;
r
0.06;
T
t
90 days;
0.4; and
S
$60. Compute the price,
DELTA, GAMMA, THETA, VEGA, and RHO for this put.
3. Consider a straddle comprised of Call A and Put A. Compute the price, DELTA, GAMMA, THETA,
VEGA, and RHO for this straddle.
RHO
3.5985
13.4083
9.8098
VEGA
9.9144
9.9144
19.8288
THETA
8.9173
4.47790
13.6963
GAMMA
0.0279
0.0279
0.0558
DELTA
0.2735
0.7265
0.4530
price
c
p
$12.61
RHO
13.4083
VEGA
9.9144
THETA
4.7790
GAMMA
.0279
DELTA
.7265
p
$10.79
RHO
3.5985
VEGA
9.9144
THETA
8.9173
GAMMA
.0279
DELTA
.2735
c
$1.82
14
Option
Sensitivities
and Option
Hedging
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120
CHAPTER
14
OPTION SENSITIVITIES AND OPTION HEDGING
4.
Consider Call A. Assuming the current stock price is $60, create a DELTAneutral portfolio consisting of a
short position of one call and the necessary number of shares. What is the value of this portfolio for a sudden
change in the stock price to $55 or $65?
As we saw for this call, DELTA
0.2735. The DELTAneutral portfolio, given a short call component, is
0.2735 shares
1 call, costs:
.2735 ($60)
$1.82
$14.59
If the stock price goes to $55, the call price is $.77, and the portfolio will be worth:
.2735 ($55)
$.77
$14.27
With a stock price of $65, the call is worth $3.55, and the portfolio value is:
.2735 ($65)
$3.55
$14.23
Notice that the portfolio values are lower for both stock prices of $55 and $65, reflecting the negative
GAMMA of the portfolio.
5.
Consider Call A and Put A from above. Assume that you create a portfolio that is short one call and long one
put. What is the DELTA of this portfolio? Can you find the DELTA without computing? Explain. Assume
that a share of stock is added to the short call/long put portfolio. What is the DELTA of the entire position?
The DELTA of the portfolio is
1.0
0.2735
0.7265. This is necessarily true, because the DELTA of
the call is
N
(
d
1
), the DELTA of the put is
N
(
d
2
), and
N
(
d
1
)
N
(
d
2
)
1.0. If a long share of stock is
added to the portfolio, the DELTA will be zero, because the DELTA of a share is always 1.0.
6.
What is the GAMMA of a share of stock if the stock price is $55 and a call on the stock with
X
$50 has a
price
c
$7 while a put with
X
$50 has a price
p
$4? Explain.
The GAMMA of a share of stock is always zero. All other information in the question is irrelevant. The
GAMMA of a share is always zero because the DELTA of a share is always 1.0. As GAMMA measures how
DELTA changes, there is nothing to measure for a stock, since the DELTA is always 1.0.
7.
Consider Call B written on the same stock as Call A with:
X
$50;
r
0.06;
T
t
90 days;
0.4;
and
S
$60. Form a bull spread with calls from these two instruments. What is the price of the spread?
What is its DELTA? What will the price of the spread be at expiration if the terminal stock price is $60?
From this information, can you tell whether THETA is positive or negative for the spread? Explain.
As observed in problem 1, for Call A,
c
$1.82, DELTA
0.2735, and THETA
8.9173. For Call B,
c
$11.64, DELTA
0.8625, and THETA
7.7191. The long bull spread with calls consists of buying
the call with the lower exercise price (Call B) and selling the call with the higher exercise price (Call A). The
spread costs $11.64
$1.82
$9.82. The DELTA of the spread equals DELTA
B
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 Spring '08
 Danısoglu
 Hedging, Strike price

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