Chapter Seven
Option Greeks
Answers to Problems and Questions
1.
Delta is the hedge ratio, a measure of option sensitivity, and sometimes approximately the likelihood of the option ending in the money.
2.
As time passes, the delta of an in-the-money option approaches one. Call option deltas are less than one, so an in-the-money call delta must rise with the passage of time, everything else being equal.
3.
With a striking price of zero, the call option would behave exactly like the stock. Its delta would therefore be 1.0.
4.
Some strategies are constructed based upon a neutral outlook on the market.
If the position delta is not zero, then the position has an initial bullish or bearish bias.
5.
With European options, the sum of the call delta and the put delta is one.
This is approximately the case with American options. As time passes, the call delta will approach 0.5 and the put delta will approach –0.5. The sum of the two (the position delta) will not change, however.
6.
In some circumstances delta is a useful measure of the approximate likelihood that the underlying asset will finish in-the-money. This means that profit and loss diagrams can be annotated with the likelihood of reaching various profit levels, especially those that are substantially in- or out-of-the-money.
7.
Long options have negative deltas by convention simply because the time remaining with a particular option can only decrease. Time can pass, but you cannot go back in time.
8.
The lower the striking price, the higher the delta for a call option. A bull spread involves buying the low striking price (with a higher delta) and writing the high striking price (with a lower delta). This means the position delta for a vertical bull spread will always be positive.
9.
With little time remaining until expiration, the likelihood increases that an in-the-money option will remain in-the-money (and act like the stock), and vice-versa with an out-of-the-money option.
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Chapter Seven. Option Greeks
10.
Option prices are primarily determined by the value of the underlying security and its proximity to the option striking price. The striking price cannot be less than zero. An option with a striking price of zero behaves like the underlying security and will move dollar for dollar with it. Its delta is therefore 1.0. An option with an infinite striking price would have a value of zero. Its value would never change, and its delta would therefore be zero.
11.
Some strategies are neither bullish nor bearish. They should be established so that their sensitivity to changes in the underlying asset is not biased in either direction.
12.
A gamma near zero means that delta does not change rapidly. This reduces the frequency with which option positions need to be restored as time passes and market conditions change.
13.
A negative gamma can result in large losses from sharp market movements.
Negative gammas come from short option positions that can have large potential losses.
14.
Many professional spreaders do not construct spreads in one for one fashion. They might, for instance, buy ten MAR 50 calls and write 30
MAR 60 calls. Someone who does this may well be bullish, but convinced that the 60 calls will not finish in-the-money. Such a spreader might well write a quantity of out-of-the-money options so as to achieve approximate delta neutrality.
15.
Risk may increase if you achieve delta neutrality by writing options rather than buying them.
16.
Delta is the N(d
1
) function from the Black-Scholes model. This expression contains the interest rate, and it enters the expression with a positive sign.
If interest rates go up, N(d
1
) goes up.
17.
Gamma neutrality reduces the amount of time that must be spent maintaining delta neutrality and reduces the size of the adjustments that must be made.
18.
Delta measures the directional market and corresponds to being bullish or bearish. Gamma measures the speed market and is associated with how rapidly you anticipate market prices to change.
19.
Write puts: delta is positive and gamma is negative.
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Chapter Seven. Option Greeks
20.
Position risk refers to the possibility of loss due to large “doomsday” price movements in the market. Position risk stems mostly from short option positions and short stock positions.
21.
Delta, gamma, and theta are the three most important, with their importance probably in the order listed. These three statistics often influence an option trader’s choice of a particular contract.
22.
Using the CBOE options calculator for American options, the call delta is
0.568.
23.
5000/.568 = 8803

88 contracts
24.
The delta of the 50 call is 0.282. The portfolio position delta is
Stock: 5000 x 1.0 = 5,000
Calls -15 x 100 x 0.282 = - 423
4,577
Writing 15 $50 calls lowers the position delta from 5,000 to 4,577. This is an 8.5% reduction.
25.
Using the CBOE options calculator, the theoretical value of each option is
$2.74 for the call and $2.19 for the put. The option deltas are 0.568 for the call and –0.447 for the put. Set up the simultaneous equations, one for the cost and one for the delta:
Cost: $2.74C + 2.19P = $4000
Delta: 0.568 C – 0.447 P = 0
C = .447/.568 P = .787 P
2.74 (.787 P) + 2.19 P = 4000
4.574 P = 4000
P = 874.51

9 contracts
C = .787 (874.51) = 688.24

7 contracts
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Chapter Seven. Option Greeks
Check:
Delta:
Calls: 7 x 100 x 0.568 = 397.6
Puts: 9 x 100 x -.447 = -402.3
4.7
Cost:
Calls: 7 x 100 x $2.74 = $1,918
Puts: 9 x 100 x $2.19 = $1,971
26.
Find new delta values:
$3,889
Call delta = 0.747
Put delta = -0.259
Position delta:
Calls: 7 x 100 x 0.747 = 522.9
Puts: 9 x 100 x –0.259 = -233.1
289.8
Because the stock went up, the calls will become more valuable. One approach to return to delta neutrality would be to sell enough of the call positions (taking a gain) to get back to zero. To eliminate 289.8 delta points, sell 289.8/.747 = 387.95 calls, or about 4 contracts.
27.
Using the CBOE options calculator, the theoretical value of each option is
$0.96 for the $50 call and $2.19 for the $45 put. The option deltas are
0.282 for the call and –0.447 for the put. Set up the simultaneous equations, one for the cost and one for the delta:
Option Cost: - $0.96 C + $2.19 P ≈ $0
Delta: 10,000 - 0.282 C – 0.447 P = 0
From (1), C = 2.19/.96 P = 2.281 P
Substitute (3) into (2):
10,000 – (0.282)(2.281 P) – 0.447 P = 0
10,000 – 1.090 P = 0
P = 9,174

92 contracts
(1)
(2)
(3)
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Chapter Seven. Option Greeks
From (3), C = 2.281 (9,174) = 20,926 = 209 contracts
Check:
Delta:
Stock: 10,000 x 1.0 = 10,000
Calls: - 209 x 100 x 0.282 = - 5,894
Puts: 92 x 100 x -.447 = - 4,112
Cost:
6
Calls: -209 x 100 x $0.96 = ($20,064)
Puts: 92 x 100 x $2.19 = $20,148
$84
28.
Using the CBOE options calculator for American options,
Quantity
-15
-25
10
50
Contract
40 call
45 call
40 put
35 put
Total
Delta
-1265
-1420
-160
-125
-2970
Gamma
-57
-158
39
45
-131
Theta/day
17
38
-6
-5
49
29.
The delta of the $40 puts is –0.16. Hedging half the position means reducing the position delta from 4,000 to 2,000. To eliminate 2,000 delta points, you need 2,000/.16 = 12,500 options, or 125 contracts .
30.
Call total: -40
Put total: +60
Stock: + 8
If the market goes way up, the puts go to zero and you are left with 32 naked calls. This is no good; losses will be extreme. If the market goes way down, the calls to go zero, the stock goes to “zero” and you are left with long puts. This is okay.
31.
The delta of the portfolio is
(-1000 x .5) + (-500 x .8) + (-2000 x -.4) + (-500 x .7) = -450
The gamma of the portfolio is
(-1000 x 2.2) + (-500 x .6) + (-2000 x 1.3) + (500 x 1.8) = -6,000
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Chapter Seven. Option Greeks
The vega of the portfolio is
(-1000 x 1.8) + (-500 x .2) + (-2000 x .7) + (500 x 1.4) = -4,000 a.
To get the portfolio gamma neutral, you need to add 6,000 gamma points. The “traded option” has a gamma of 1.5. Therefore, buying
4,000 of these would do the trick: 4,000 x 1.5 = 6,000. This also adds 2,400 delta points to the portfolio, making the position delta –450
+ (4,000 x .6) = 1,950.
Given the above, in order to also make the portfolio delta neutral you need a short position of ¥1,950. b.
A long position in 5,000 traded options will give a vega neutral portfolio. This would add (5,000 x .8) = 4,000 vega points to the portfolio’s existing –4,000, making the position vega zero. This, however, will increase the position delta by (5,000 x .6) = 3,000, making the new position delta (-450) + (3,000) = 2,550. Hence, in addition to the 5,000 options, you need a short position of ¥ 2,550
.
29