Chapter8:HowTradersManageTheirRisks

8.1交易組合價值減少10500美元。

8.2當波動率變化2%時，交易組合價格增長200X2=400美元。

8.3兩種情形均為0.5*30*4=60美元

8.41000份期權短頭寸的Delta等于-700,可以通過買入700份股票的形式使

交易組合達到Delta中性。

8.5Theta為-100的含義是指在股價與波動率沒有變化的情況下，期權價格每天

下降100美元。假如交易員認為股價及隱含波動率在將來不會改變，交易員可以

賣出期權，并且Theta值越高越好。

8.6當?個期權承約人的Gamma絕對值較大，Ganmia本身為負，并且Delta等于

0,在市場變化率較大的情況下，期權承約人會有較大損失。

8.8看漲及看跌期權的多頭頭寸都具備正的Gamma,由圖6.9可以看出，當Gamma

為正時，對沖人在股票價格變化較大時會有收益，而在股票價格變化較小時會有

損失，因此對沖人在（b）情形收益更好，當交易組合包含期權的空頭頭寸時，

對沖人在（a）情形收益會更好。

8.9Delta的數值說明當歐元匯率增長0.01時，銀行交易價格會增加

0.01*30000=300美元，Gamma的數值說明，當歐元價格增長0.01時，銀行交易

組合的Delta會下降0.01*80000=800美元；

為了做到Delta中性,我們應該賣出30000歐元；

當匯率增長到0.93時，我們期望交易組合的Delta下降為（0.93-0.9）

*80000=24000,組合價值變為27600。為了維持Delta中性,銀行應該對2400

數量歐元空頭頭寸進行平倉，這樣可以保證歐元凈空頭頭寸為27600o

當一個交易組合的Delta為中性，同時Gamma為負時資產價格有一個較大變動時

會引起損失。因此銀行可能會蒙受損失。

8.15.

Thegammaandvegaofadelta-neutralportfolioare50per$per$and25per%,

respectively.Estimatewhathappenstothevalueoftheportfoliowhenthereisashock

tothemarketcausingtheunderlyingassetpricetodecreaseby$3anditsvolatilityto

increaseby4%.

Withthenotaiionofthetext,theincreaseinthevalueoftheportfoliois

0.5xgammcK(A5):+vegaxAc

Thisis

0.5x50x32+25x4=325

Theresultshouldbeanincreaseinthevalueoftheportfolioof$325.

8.16.

Consideraone-yearEuropeancalloptiononastockwhenthestockpriceis$30,the

strikepriceis$30,therisk-freerateis5%,andthevolatilityis25%perannum.Use

theDerivaGemsoftwaretocalculatetheprice,delta,gamma,vega,theta,andrhoof

theoption.Verifythatdeltaiscorrectbychangingthestockpriceto$30.1and

recomputingtheoptionprice.Verifythatgammaiscorrectbyrecomputingthedelta

forthesituationwherethestockpriceis$30.J.Cany

outsimilarcalculationstoverifythatvega,theta,andrhoarecorrect.

Theprice,delta,gamma,vega,theta,andrhooftheoptionare3.7008,0.6274,0.050,

0.1135,-0.00596,and0.1512.Whenthestockpriceincreasesto30.1,theoption

priceincreasesto3.7638.Thechangeintheoptionpriceis3.76383.7008=0.0630.

Deltapredictsachangeintheoptionpriceof0.6274x0.1=0.0627whichisvery

close.Whenthestockpriceincreasesto30.1,deltaincreasesto0.6324.Thesizeof

theincreaseindeltais0.6324-0.6274=0.005.Gammapredictsanincreaseof0.050

x().1=0.005whichis(tothreedecimalplaces)thesame.Whenthevolatility

increasesfrom25%to26%,theoptionpriceincreasesby0.1136from3.7008to

3.8144.Thisisconsistentwiththevegavalueof().1135.Whenthetimetomaturityis

changedfrom1to1-1/365theoptionpricereducesby0.006from3.7008to3.6948.

Thisisconsistentwithathetaof-0.00596.Finally,whentheinterestrateincreases

from5%to6%,thevalueoftheoptionincreasesby0.1527from3.7008to3.8535.

Thisisconsistentwitharhoof0.1512.

8.17.

Afinancialinstitutionhasthefollowingportfolioofover-the-counteroptions

onsterling:

TypePositionDeltaofGammaofVeguof

OptionOptionOption

Call-1,0000.502.21.8

Call-5000.800.60.2

Put-2,000-0.401.30.7

Call-5000.701.81.4

Atradedoptionisavailablewithadeltaof0.6,agammaof1.5,andavega

of0.8.

(a)Whatpositioninthetradedoptionandinsterlingwouldmaketheportfolio

bothgammaneutralanddeltaneutral?

(b)Whatpositioninthetradedoptionandinsterlingwouldmaketheportfolio

bothveganeutralanddeltaneutral?

Thedeltaoftheportfoliois

-1,000x0.50-500x0.80-2,000x(-0.40)-500x0.70=-450

Thegammaoftheportfoliois

-l,000x2.2-500x0.6-2,000x1.3-500x1.8=-6,000

Thevegaoftheportfoliois

-l,000x1.8-500x0.2-2,000x0.7-500x1.4=-4,000

(a)Alongpositionin4,000tradedoptionswillgiveagamma-neutralportfoliosince

thelongpositionhasagammaof4,000x1.5=+6,000.Thedeltaofthewhole

portfolio(includingtradedoptions)isthen:

4,000x0.6-450=1,950

Hence,inadditiontothe4,000tradedoptions,ashortpositionin￡1,950isnecessary

sothattheportfolioisbothgammaanddeltaneutral.

(b)Alongpositionin5,000tradedoptionswillgiveavega-neutralportfoliosincethe

longpositionhasavegaof5,000x0.8=+4,000.Thedeltaofthewholeportfolio

(includingtradedoptions)isthen

5,000x0.6-450=2,550

Hence,inadditiontothe5,000tradedoptions,ashortpositionin￡2,550isnecessary

sothattheportfolioisbothvegaanddeltaneutral.

8.18.

ConsideragainthesituationinProblem8.17.Supposethatasecondtradedoption

withadeltaof0.1,agammaof0.5,andavegaof0.6isavailable.Howcouldthe

portfoliobemadedelta,gamma,andveganeutral?

LetH'Ibethepositioninthefirsttradedoptionandw」bethepositioninthesecond

tradedoption.Werequire:

6,()00=1.5卬?+0.5卬2

4,000=0.8K,I+0.6卬2

ThesolutiontotheseequationscaneasilybeseentobeW|=3,200,卬2=2,400.The

wholeportfoliothenhasadeltaof

-450+3,200x0.6+2,400xO.l=1,710

Thereforetheportfoliocanbemadedelta,gammaandveganeutralbytakingalong

positionin3,200ofthefirsltradedoption,alongpositionin2,400ofthesecond

tradedoptionandashortpositionin￡1,710.

8.19.(SpreadsheetProvided)

ReproduceTable8.2.(InTable8.2,thestockpositionisroundedtothenearest100

shares.)Calculatethegammaandthetaofthepositioneachweek.Usingthe

DerivaGemApplicationsBuilderstocalculatethechangeinthevalueofthe

portfolioeachweek(beforetherebalancingattheendoftheweek)andcheck

whetherequation(8.2)isapproximatelysatisfied.(Note:DerivaGemproducesa

valueoftheta“percalendarday,"Thethetainequation8.2is“peryear.")

Considerthefirstweek.Theportfolioconsistsofashortpositionin100,000options

andalongpositionin52,200shares.Thevalueoftheoptionchangesfrom$240,053

atthebeginningoftheweekto$188,760attheendoftheweekforagainof$51,293.

Thevalueoftheshareschangefrom52,200x49二82,557,800to52,200x48.12=

$2,511,864foralossof$45,936.Thenetgainis51,293-45,936=$5,357.The

gammaandtheta(peryear)oftheportfolioare-6,554.4and430,533sothatequation

(8.2)predictsthegainas

430,533X1/52+0.5x6,554.4x(48.12-49)2=5,742

Theresultsforall20weeksareshowninthefollowingtable.

WeekActualGain($)PredictedGain($)

15,3575,742

25,6896,093

3-19,742-21,084

41,9411,572

53,7063,652

69,3209,191

76,249