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Chapter 7. Option Greeks

Chapter Seven

Option Greeks

Multiple Choice

1. Jones and Smith each own 100 shares of ZYX stock (current! sein" for #$0%. Jones writes a&'Y $ ca) whie Smith writes a &'Y 70 ca. *hese are the on! ZYX option positions thetwo peope ha+e. ,hich of the foowin" statements is most correct-a. Jones has a hi"her position deta in ZYX. . Smith has a hi"her position deta in ZYX.c. Jones and Smith ha+e the same position deta in ZYX.d. You cannot determine which person has the hi"hest position deta in ZYX without more

information aout ZYX stock.'/S,2 3

4. Jones and Smith each own 100 shares of ZYX stock (current! sein" for #$0%. Jones writes

a &'Y $ ca) whie Smith writes a &'Y 70 put. *hese are the on! ZYX option positionsthe two peope ha+e. ,hich of the foowin" statements is most correct-a. Jones has a hi"her position deta in ZYX. . Smith has a hi"her position deta in ZYX.c. Jones and Smith ha+e the same position deta in ZYX.d. You cannot determine which person has the hi"hest position deta in ZYX without more


5. Jones and Smith each own 100 shares of ZYX stock (current! sein" for #$0%. Joneswrites a &'Y $ put) whie Smith writes a &'Y 70 put. *hese are the on! ZYX option

positions the two peope ha+e. ,hich of the foowin" statements is most correct-a. Jones has a hi"her position deta in ZYX. . Smith has a hi"her position deta in ZYX.c. Jones and Smith ha+e the same position deta in ZYX.d. You cannot determine which person has the hi"hest position deta in ZYX without more


6. ,hich of the foowin" statements is true-a. *heta is positi+e for a option positions.1. Gamma is awa!s opposite in si"n to theta.

4. eta is positi+e for a ca option positions.5. Common stock has a deta of 8ero.'/S,2 3

9:




. *he i""est sin"e disad+anta"e of writin" puts isa. positi+e "amma.1. the risk of a stae market.4. hi"h transaction costs.5. the risk of a fain" market.

'/S,2

$. ,hich of the foowin" proa! has the owest deta-a. write in;the;mone! cas1. write puts4. u! cas and write cas with a hi"her strikin" price5. u! out;of;the;mone! puts and write puts with a hi"her strikin" price'/S,2 '

7. ,hich of the foowin" statements is true-a. Stock has a positi+e "amma.

1. *ime +aue can on! increase.4. *he deta of an in;the;mone! ca increases with time.5. Stock has a ne"ati+e theta.'/S,2 C

:. ,hich of the foowin" strate"ies proa! has the hi"hest deta-a. 3u! a ca) write a ca with a hi"her strikin" price.1. 3u! a ca) write a put.4. ,rite a ca) write a put.5. ,rite a ca) u! a put.'/S,2 3

9. ' person e<pects the market to decine sharp! in the near future. *his person wantsa. positi+e deta) positi+e "amma.1. positi+e deta) ne"ati+e "amma.4. ne"ati+e deta) positi+e "amma.5. ne"ati+e deta) ne"ati+e "amma.'/S,2

10. ,hich of the foowin" concei+a! coud e a deta neutra position-a. on" '= 60 cas1. on" '= 60 cas) on" &'Y 6 cas4. on" '= 60 cas) short &'Y 6 puts5. on" '= 60 cas) short &'Y 6 cas'/S,2

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11. ,hich of the foowin" statements is true re"ardin" an in;the;mone! ca option) e+er!thin" ese ein" e>ua-a. 's time passes its deta wi approach one. . 's time passes its +aue wi approach 8ero.c. 's time passes its theta wi remain constant.d. 's time passes its intrinsic +aue wi decine.

'/S,2 '

14. eta is thea. theoretica +aue of an option. . e<pected chan"e in the option +aue as the under!in" asset price chan"es.c. intrinsic +aue of the option.d. infuence of di+idends on the option +aue'/S,2 3

15. eta enaes the portfoio mana"er to determinea. the numer of options necessar! to mimic the returns of the under!in" securit!.

. the numer of options necessar! to reduce the risk of the under!in" portfoio ! haf.c. the numer of options necessar! to doue the portfoio return per unit of risk d. the standard de+iation of the portfoio returns.'/S,2 '

16. ?or a ca option) deta is awa!sa. "reater than one. . ess than one.c. ess than one and "reater than 8ero.d. ess than or e>ua to 8ero.'/S,2 '

1. ?or a ca option) deta @@@@@ as the strikin" price @@@@@.a. increases) increases . decreases) increasesc. increases) approaches the stock priced. decreases) approaches the stock price'/S,2 3

1$. ?or at;the;mone! uropean puts and cas on the same stock)a. the put deta is awa!s ess than the ca deta. . the put deta is awa!s e>ua to the ca deta.

c. the put deta is awa!s "reater than the ca deta.d. the put deta is awa!s ess than or e>ua to the ca deta.'/S,2 '

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17. ' portfoio contains 10)000 shares of XYZ stockA the portfoio mana"erwrites 10 XYZ cas. Bf the ca deta is 0.6) what is the position deta-a. 6 . 6c. 9)6

d. Cannot e determined'/S,2 C

1:. ' portfoio contains 1)000 shares of XYZ stockA the portfoio mana"er u!s10 XYZ puts. Bf the put deta is 0.440) what is the position deta-a. 440 . 7:0c. 10)440d. Cannot e determined'/S,2 3

19. 'n '3C JD/ 6) uropean st!e ca has a deta of 0.66A what is the detaof an '3C JD/ 6 put-a. 0.66 . 0.c. 0.d. 0.66'/S,2 C

40. ' of the foowin" wi ower position deta e<cepta. u!in" puts . u!in" casc. writin" casd. sein" stock '/S,2 3

True/False

1. Ee"a measures the sensiti+it! of the option premium to chan"es in the anticipated +oatiit!of the under!in" asset.'/S,2 *

4. Ee"a is the most inear of the option pricin" mode deri+ati+es.'/S,2 *

101




Short Answer/Problem

Dse F 5) S F 56) and H F 40 in =roems 1 and 4.

1. Dsin" the C3O Options Cacuator) prepare a deta tae for 90;da!) 'merican

st!e options with strikin" prices of 50) 5) and 5$0.

'/S,2 Striking Price

350 355 360

Call Delta 0.594 0.540 0.485 Put Delta − 0.415 − 0.471 − 0.528

4. epeat =roem 1 for uropean e<ercise st!e.

'/S,2

Striking Price350 355 360

Call Delta 0.594 0.538 0.482 Put Delta − 0.406 − 0.462 − 0.518

5. ?i in the anks with the appropriate etter2

'/S,2

=ositi+e deta) positi+e"amma B a% on" put

=ositi+e deta) ne"ati+e"amma C % on" ca

/e"ati+e deta) positi+e"amma A c% short put

/e"ati+e deta) ne"ati+e"amma D d% short ca

6. Suppose !ou u! 4 90;da! 50 ca contracts and write 4 90;da! 5$0 ca

contracts. ,hat is !our a. position deta- . position "amma-c. position theta-

104




'/S,S2a. Position delta:

350 Calls: 25 x 100 x 0.594 = 1,485.0

360 Calls: − 25 x 100 x 0.485 = − 1,212.5

272.5

b. Position gamma: (sing t!e C"#$ o%tions &al&lato'

350 Calls: 25 x 100 x 0.011 = 27.5

360 Calls: − 25 x 100 x 0.011 = − 27.5

0

&. Position t!eta: (sing t!e C"#$ o%tions &al&lato'

350 Calls: 25 x 100 x ( − 0.092 = − 230

360 Calls: − 25 x 100 x ( − 0.092 = 230

0

. 'fter 1 da!s) the stock price has increased ! 5. Dpdate the foowin" position statistics.a. =osition deta . =osition "ammac. =osition theta

)e* de'i+ati+es2

O ption Delta Gamma Theta

350 &all 0.709 0.01−

0.093360 &all 0.599 0.011 − 0.103

'/S,S2a. Position delta:

350 Calls: 25 x 100 x 0.709 = 1,772.5

360 Calls: 25 x 100 x 0.599 = − 1,497.5

275

b. Position gamma: (sing t!e C"#$ o%tions &al&lato'

350 Calls: 25 x 100 x 0.010 = 25.0

360 Calls: 25 x 100 x 0.011 = − 27.5

− 2.5

&. Position t!eta: (sing t!e C"#$ o%tions &al&lato'

350 Calls: 25 x 100 x (0.093 = − 232.5

360 Calls: 25 x 100 x (0.103 = 257.5

105




25

$. Suppose someone owns )000 shares of a #$0 stock. *he! aso ha+e written 4of the #70 cas) 4 of the #$ cas) and ha+e purchased 7 of the #$0 puts.

,rite a para"raph aout the %osition 'is- . *hat is) e<pain the conse>uences of a ar"e

market mo+ement up or down.

'/S,2 / t!e ma'-et de&lines sbstantiall, t!e &alls go to e'o, *!i&! is o-a. !e sto&- is %'ote&ted b t!e %ts, %ls t!e'e a'e an additional 25 long %t &ont'a&ts. !is %o't/olio *old bene/it sbstantiall /'om a la'ge de&line in t!e sto&- %'i&e.

/ t!e ma'-et ad+an&es, t!e sto&- *ill be &alled a*a and t!e %ts *ill ex%i'e *o't!less.!e'e *ill not be a disast'os loss, bt t!e'e *ill be no %'o/it eit!e'.

7. Suppose !ou wanted to spend #40)000 on a deta neutra 5 stradde with 90;

da! options. (See the initia conditions ao+e.% 3ased on 3ack;Schoes +aues) how man! put and ca contracts woud !ou u!-

'/S,2 !e t!eo'eti&al &all and %t +ales a'e 14.84 and 13.46, 'es%e&ti+el.

1 .540C .471P = 02 14.874 13.46P = 20,0003 /'om 1, C = 0.8722P 4 sbstitte 3 into 2 14.84(.8722P 13.46P = 200005 P = 757.486 bstitte 5 into 1, C = 660.67

onding to *!ole &ont'a&ts, P = an! C = 7. !is o+e's!oots t!e 20,000 ma'- some*!at, so it ma be bette' to 'ond one o' bot! &ont'a&ts do*n.

:. Bn =roem 7) suppose that after two da!s the stock has risen 7I:. as the position deta increased) decreased) or remained the same-

'/S,2 'ising nde'ling sto&- %'i&e *ill &ase bot! t!e %t and t!e &all delta toin&'ease. (!e %assage o/ time *ill !a+e a modest e//e&t o/ delta, bt t!e sto&- &!ange *ill o+e'*!elm t!e time e//e&t. !e %osition delta *ill in&'ease.

9. ' portfoio contains 14 contracts of XYZ JD/ 0 ca options (uropean st!e%)which se for #5 and ha+e a deta of 0.40. ow &an! JD/ 0 put contracts (sein" for #4%must !ou u! to e appro<imate! deta neutra-

'/S,2Call deltas = 125 x 100 x 0.520 = 6,500 Pt delta = 0.4806,5000.480 = 13,542 "35 contract#

106




10. ' portfoio contains 4)00 shares of '3C common stock. You cacuate thefoowin" information2

Option Price Delta Theta

JDK 60 ca 1 5I6 0.56 0.05JDK 5 put 7I: ;0.514 0.04

You decide to write the JDK 60 cas and u! the JDK 5 puts so as to e appro<imate! detaneutra '/ so as to ha+e essentia! no cash outfow. ow ma! of each option woud !ou u! or write- (You are not concerned aout whether the cas are co+ered or not.%

'/S,S2

− .534C − .312P 2500 = 0

1.75C − .875P = 0

1.75C = .875P C = 0.5P

− .534P (.5P − .312P 2500 = 0

− .579P = − 2500

P = 4317.79C = 2158.89

P = $3 contract# an! C = 22 contract#

11. 'nother portfoio contains 14)000 shares of '3C stock. ow man! JDK 60 cacontracts must the portfoio mana"er write to remo+e haf of the market risk of '3C stock-

'/S,2 o need to 'emo+e 6,000 deltas. 6000.534 = 11,236 ""2 contract#

14. ?i in the tae with L) ;) 0) or inconcusi+e2

Delta Gamma Theta

Bu !"" shares

Bu !"" shares#

write ! call

Bu !"" shares#

write ! put

$rite ! call# write

! put

Bu !"" shares#

bu ! put

10




'/S,2

Delta Gamma ThetaBu !"" shares 0 0

Bu !"" shares#

write ! call

Bu !"" shares#

write ! put

$rite ! call# write

! put

n&on&lsi+e

Bu !"" shares#

bu ! put

15. ' J'/ :0 XYZ uropean ca has a deta of 0.0 and a "amma of 0.01:0. ,hat are a% the position deta) and % the position "amma of a on" stradde composed of one ca contractand one put contract- (/o information is missin" from this proem.%

'/S,S2

a. 100 x 0.550 = 55.0

100 x (.550 1 = − 45.0

"0.0

b. 100 x 0.0180 = 1.80 100 x 0.0180 = 1.80

3.60

16. ' 5;month ca has a deta of 0.6. ,hat is the appro<imate deta of a on" straddecomposed of ten of these ca contracts and twe+e put contracts-

'/S,2Calls: 10 x 100 x 0.545 = 545

Pts: 12 x 100 x (.545 1 = − 546

− "

1. ' ca option is current! in;the;mone!. <pain how its deta wi chan"e if a. time passes . the under!in" asset decines in +auec. the stock spits two for one

'/S,S2a. t 'ises and a%%'oa&!es 1.0.

10$




b. t de&lines.&. )o &!ange

1$. You hear someone sa!) M,e do not write unco+ered options) so our "amma is awa!s positi+e.N <pain this statement) and state whether !ou a"ree or disa"ree with it.

'/S,2 !is statement is not t'e. / o *'ite a &o+e'ed &all o *ill still !a+e anegati+e gamma. !o't o%tions al*as !a+e a negati+e gamma.

17. =ortfoio ' and portfoio 3 oth ha+e a positi+e deta and a ne"ati+e "amma) ut one is >uiteconser+ati+e whie the other is >uite specuati+e. &ake up an e<ampe showin" how thiscoud e the case.

'/S,S2;'ite a %t gamma < 0, delta 0 !is is s%e&lati+e.

;'ite a &o+e'ed &all

gamma <0, delta 0 !is is &onse'+ati+e.

1:. State whether the foowin" statement is true or fase) and e<pain2 MBn the on" run)continuous! writin" co+ered cas reduces oth the e<pected return and the risk of a stock position.N

'/S,2 !is is t'e. o simltaneosl 'ed&e do*nside 'is- and +olatilit, %ls o /o'/eit la'ge %ositi+e 'et'ns.

107

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