Applications and Mathematical Derivation of Options Greeks From First Principle

7/27/2019 Applications and Mathematical Derivation of Options Greeks From First Principle

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ApplicationsandMathematicalDerivationofOptions

Greeksfromfirstprinciple

Hong-YiChen,RutgersUniversity,USA

Cheng-FewLee,RutgersUniversity,USA

WeikangShih,RutgersUniversity,USA

Abstract

Inthischapter,weintroducethedefinitionsofGreekletters.WealsoprovidethederivationsofGreeklettersforcallandputoptionsonbothdividends-payingstockand non-dividends stock. Then we discuss some applications of Greek letters.Finally,weshowtherelationshipbetweenGreekletters,oneoftheexamplescanbeseenfromtheBlack-Scholespartialdifferentialequation.Keywords

Greekletters,Delta,Theta,Gamma,Vega,Rho,Black-Scholesoptionpricingmodel,Black-Scholespartialdifferentialequation

30.1Introduction

Greek lettersaredefined asthe sensitivities of the option pricetoa single-unitchangeinthevalueofeitherastatevariableoraparameter.Suchsensitivitiescanrepresent the different dimensions to the risk inan option. Financial institutionswhoselloptiontotheirclientscanmanagetheirriskbyGreeklettersanalysis.

Inthischapter,wewilldiscussthedefinitionsandderivationsofGreekletters.WealsospecificallyderiveGreeklettersforcall(put)optionsonnon-dividendstockanddividends-payingstock.SomeexamplesareprovidedtoexplaintheapplicationofGreekletters.Finally,wewilldescribetherelationshipbetweenGreeklettersandtheimplicationindeltaneutralportfolio.

30.2Delta( )

The delta of an option, , is defined as the rate of change of the option pricerespectedtotherateofchangeofunderlyingassetprice:

S

=

where is the option price and S is underlying asset price. We next show thederivationofdeltaforvariouskindsofstockoption.

30.2.1DerivationofDeltaforDifferentKindsofStockOptions


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FromBlack-Scholesoptionpricingmodel,weknowthepriceofcalloptiononanon-dividendstockcanbewrittenas:

( ) ( )21

dNXedNSCr

tt

= (30.1)

andthepriceofputoptiononanon-dividendstockcanbewrittenas:

( ) ( )rt 2 t 1P Xe N d S N d

= (30.2)

where

s

str

X

S

d

++

=

2ln

2

1

2

t s

2 1 s

s

Sln r

X 2d d

+

= =

tT=

( )N isthecumulativedensityfunctionofnormaldistribution.

( ) ( )

==

1

2

12

1

2

1du

d

dueduufdN

First,wecalculate ( )( )

2

1

1

1

2

1

2

1d

e

d

dNdN

=

=

(30

( )( )

( )

rt

d

rX

Sd

d

d

d

d

eX

Se

eee

eee

e

e

d

dNdN

sst

s

s

s

=

=

=

=

=

=

++

2

22

ln

2

22

2

2

2

2

2

2

1

22

2

1

2

1

2

1

2

1

2

2

2

1

2

1

2

1

2

1

2

1

(30.4)


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Eq.(30.2) andEq.(30.3)willbeusedrepetitively indetermining followingGreekletterswhentheunderlyingassetisanon-dividendpayingstock.

ForaEuropeancalloptiononanon-dividendstock,deltacanbeshownas

1N(d ) = (30.5)

ThederivationofEq.(30.5)isinthefollowing:

( )( ) ( )

( )( ) ( )

( )

( )

( )

2 2

1 1

2 2

1 1

1 2rt

1 t

t t t

1 2r1 2

1 t

1 t 2 t

d d

r rt2 2

1 t

t s t s

d d

2 2

1 t t

t s t s

1

N d N dCN d S Xe

S S S

N d N dd dN d S Xe

d S d S

S1 1 1 1N d S e Xe e e

X2 S 2 S

1 1

N d S e S eS 2 S 2

N d

= = +

= +

= +

= +

=

ForaEuropeanputoptiononanon-dividendstock,deltacanbeshownas

1N(d ) 1 = (30.6)

ThederivationofEq.(30.6)is

( )( )

( )

( )( )

( )

( )

( )

( )

2 21 1

2 21 1

2 1rt

1 tt t t

2 1r 2 11 t

2 t 1 t

d d

r rt2 21 t

t s t s

d d

2 2t 1 t

t s t s

1

N d N dPXe N d S

S S S

(1 N d ) (1 N d )d dXe (1 N d ) S

d S d S

S1 1 1 1Xe e e (1 N d ) S e

X2 S 2 S

1 1S e N d 1 S e

S 2 S 2

N d 1

= =

=

= +

= +

=

Iftheunderlyingassetisadividend-payingstockprovidingadividendyieldatrateq,Black-Scholes formulas for the prices ofaEuropeancall option onadividend-payingstockandaEuropeanputoptiononadividend-payingstockare

( ) ( )q rt t 1 2C S e N d Xe N d

= (30.7)


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and

( ) ( )r qt 2 t 1P Xe N d S e N d

= (30.8)

where

2

t s

1

s

Sln r qX 2

d

+ +

=

2

t s

2 1 s

s

Sln r q

X 2d d

+

= =

To make the following derivations more easily, we calculate Eq. (30.9) and Eq.(30.10)inadvance.

( )2

11 21

1

1N (d )2

d

N d ed

= =

(30.9)

( )

( )

22

2

1 s

22s1

1 s

222 t ss1

21

2

2

2

d

2

d

2

d

d2 2

Sd ln r q

X 22 2

d

( r q )t2

N dN (d )

d

1e

2

1e

2

1e e e

2

1e e e

2

S1e e

X2

+ +

=

=

=

=

=

=

(30.10)

ForaEuropeancalloptiononadividend-payingstock,deltacanbeshownasq

1e N(d ) = (30.11)

Thederivationof(30.11)is


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( )( ) ( )

( )( ) ( )

( )

( )

2 21 1

21

1 2q q rt1 t

t t t

1 2q q r1 21 t

1 t 2 t

d dq q r (r q)t2 2

1 t

t s t s

d

q q q21 t t

t s t s

N d N dCe N d S e Xe

S S S

N d N dd de N d S e Xe

d S d S

S1 1 1 1e N d S e e Xe e e

X2 S 2 S

1 1e N d S e e S e e

S 2 S 2

= = +

= +

= +

= +

( )

21d

2

q

1e N d

=

ForaEuropeancalloptiononadividend-payingstock,deltacanbeshownas

[ ]q 1e N(d ) 1

= (30.12)


( )( )

( )

( )( )

( )

( )2 21 1

21

2 1r q qt1 t

t t t

2 1r q q2 11 t

2 t 1 t

d d

r (r q) q qt2 21 t

t s t s

d

q 2

tt s

N d N dPXe e N d S e

S S S

(1 N d ) (1 N d )d dXe e (1 N d ) S e

d S d S

S1 1 1 1Xe e e e (1 N d ) S e e

X2 S 2 S

1

S e e eS 2

= =

=

= +

= +

( )( )

21d

q q 2

1 tt s

q

1

1

(N d 1) S e eS 2

e (N d 1)

+

=

30.2.2ApplicationofDelta

Figure30.1showstherelationshipbetweenthepriceofacalloptionandthepriceofitsunderlyingasset.ThedeltaofthiscalloptionistheslopeofthelineatthepointofAcorrespondingtocurrentpriceoftheunderlyingasset.


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Figure30.1

Bycalculatingdeltaratio,afinancialinstitutionthatsellsoptiontoaclientcanmakeadeltaneutralpositiontohedgetheriskofchangesoftheunderlyingassetprice.Supposethatthecurrentstockpriceis$100,thecalloptionpriceonstockis$10,and the current deltaof the call option is0.4. A financial institutionsold 10calloption to its client, so that the client has right to buy 1,000 shares at time tomaturity.Toconstructadeltahedgeposition,thefinancialinstitutionshouldbuy0.4x1,000=400sharesofstock.Ifthestockpricegoesupto$1,theoptionpricewillgoupby$0.4.Inthissituation,thefinancialinstitutionhasa$400($1x400shares)gaininitsstockposition,anda$400($0.4x1,000shares)lossinitsoptionposition.The totalpayoffof the financial institution iszero.Ontheotherhand, ifthestockpricegoesdownby$1,theoptionpricewillgodownby$0.4.Thetotalpayoffofthe

financialinstitutionisalsozero.

However, the relationship between option price and stockprice is not linear, sodeltachangesoverdifferentstockprice.Ifaninvestorwantstoremainhisportfolioindeltaneutral,heshouldadjusthishedgedratioperiodically.Themorefrequentlyadjustmenthedoes,thebetterdelta-hedginghegets.

Figure30.2exhibitsthechangeindeltaaffectsthedelta-hedges.Iftheunderlyingstockhasapriceequalto$20,thentheinvestorwhousesonlydeltaasriskmeasurewillconsiderthathisportfoliohasnorisk.However,astheunderlyingstockpriceschanges,eitherupordown,thedeltachangesaswellandthushewillhavetousedifferentdeltahedging.Deltameasurecanbecombinedwithotherriskmeasuresto

yieldbetterriskmeasurement.Wewilldiscussitfurtherinthefollowingsections.


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Figure30.2

30.3Theta( )

The theta of an option, , is defined as the rate of change of the option pricerespectedtothepassageoftime:

t

=

where istheoptionpriceandt isthepassageoftime.

If T t = ,theta( )canalsobedefinedasminusonetimingtherateofchangeofthe option price respected to time to maturity. The derivation of suchtransformationiseasyandstraightforward:

( 1)t t

= = =

where T t = istimetomaturity.Forthederivationofthetaforvariouskindsofstockoption,weusethedefinitionofnegativedifferentialontimetomaturity.

30.3.1DerivationofThetaforDifferentKindsofStockOption

ForaEuropeancalloptiononanon-dividendstock,thetacanbewrittenas:

rt s1 2

SN (d ) rX e N(d )

2

=

(30.13)



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( )

21

21

r rt 1 2t 2

r r1 1 2 2t 2

1 2

2 2ts s

d

r2t 23

2s ss

d

r rt2

C N(d ) N(d )S r X e N(d ) Xe

N(d ) d N(d ) dS rX e N(d ) Xe

d d

slnr r1 X2 2S e rX e N(d )2 22

S1Xe e e

X2

= = + +

= +

+ + =

+

21

21

2t s

32

s ss

2 2ts s

d

r2t 23

2s ss

2t s

d

2t 3

2s ss

d

t

sln r

r X 2

22

slnr r

1 X2 2S e rX e N(d )2 22

sln r

1 r X 2S e2 22

1S e

2

+

+ +

=

+

+

=

21

2

s

r22

s

rt s1 2

2 rX e N(d )

SN (d ) rX e N(d )

2

=

ForaEuropeanputoptiononanon-dividendstock,thetacanbeshownas

rt s1 2

SN (d ) rX e N( d )

2

= +

(30.14)



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( )

21

r rt 2 12 t

r r 2 2 1 12 t

2 1

2t s

d

r r rt22 3

2s s

P N( d ) N( d )r X e N( d ) Xe S

(1 N(d )) d (1 N(d )) d( r)X e (1 N(d )) Xe S

d d

sln rS1 r X

( r)X e (1 N(d )) Xe e eX2 2

= = +

= +

+ = +

21

21

21

s

2 2ts s

d

2t 3

2s ss

2t s

d

r 22 t 3

2s ss

2ts s

d

2t 3

2s s

2

2

slnr r

1 X2 2S e2 22

sln r

1 r X 2rX e (1 N(d )) S e2 22

slnr r

1 X2S e2 2

+ +

+

= +

+ +

21

2

s

2

sd

r 22 t

s

r t s2 1

r t s2 1

2

2

1 2rX e (1 N(d )) S e2

SrX e (1 N(d )) N (d )

2

SrX e N( d ) N (d )

2

=

=

=

ForaEuropeancalloptiononadividend-payingstock,thetacanbeshownas

q

q rt st 1 1 2

S eq S e N(d ) N (d ) rX e N(d )

2

=

(30.15)



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( )

21

q q r r t 1 2t 1 t 2

q q r r 1 1 2 2t 1 t 2

1 2

2ts

d

q q 2t 1 t 3

2s s

C N(d ) N(d )q S e N(d ) S e r X e N(d ) Xe

N(d ) d N(d ) dq S e N(d ) S e rX e N(d ) Xe

d d

slnr q r q1 X2q S e N(d ) S e e2 2

= = + +

= +

+ +

=

21

2

1

2

s

r

2

s

2t s

d

r (r q)t23

2s ss

2 2ts s

d

q q 2t 1 t 3

2s ss

2 rX e N(d )2

sln r q

S1 r q X 2Xe e eX2 22

slnr r

1 X2 2q S e N(d ) S e e2 22

+ +

+ +

=

21

21

r

2

2t s

d

q 2t 3

2s ss

2

sd

q q r2t 1 t 2

s

qq rt s

t 1 1 2

rX e N(d )

sln r

1 r X 2S e e2 22

1 2q S e N(d ) S e e rX e N(d )2

S eq S e N(d ) N (d ) rX e N(d )

2

+

+

=

=

ForaEuropeancalloptiononadividend-payingstock,thetacanbeshownas

q

r q t s2 t 1 1

S erX e N( d ) qS e N( d ) N (d )

2

=

(30.16)



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( )

21

r r q qt 2 12 t 1 t

r r q q2 2 1 12 t 1 t

2 1

d

r r (r q)t22

P N( d ) N( d )r X e N( d ) Xe ( q)S e N( d ) S e

(1 N(d )) d (1 N(d )) drX e (1 N(d )) Xe qS e N( d ) S e

d d

S1rX e (1 N(d )) Xe e e

X2

= = + +

= +

= +

21

2

1

2t s

32

s ss

2 2ts s

d

q q 2t 1 t 3

2s ss

td

r q 22 t 3

2s s

sln r qr q X 2

22

slnr q r q

1 X2 2qS e N( d ) S e e2 22

sln

1 r q XrX e (1 N(d )) S e e2 2

+

+ +

= +

21

21

2

s

s

2 2ts s

d

q q 2t 1 t 3

2s ss

2

sd

r q q 22 t 1 t

s

r q

2 t

r q2

2

slnr q r q

1 X2 2qS e N( d ) S e e2 22

1 2rX e (1 N(d )) qS e N( d ) S e e2

rX e (1 N(d )) qS e

+

+ +

=

= q

t s1 1

qr q t s

2 t 1 1

S eN( d ) N (d )

2

S erX e N( d ) qS e N( d ) N (d )

2

=

30.3.2ApplicationofTheta( )

Thevalueofoptionisthecombinationoftimevalueandstockvalue.Whentimepasses,thetimevalueoftheoptiondecreases.Thus,therateofchangeoftheoptionpricewithrespectivetothepassageoftime,theta,isusuallynegative.

Becausethepassageoftimeonanoptionisnotuncertain,wedonotneedtomakeathetahedgeportfolio against the effect of the passageof time.However,we stillregard theta as a useful parameter, because it is a proxy of gamma in the deltaneutralportfolio.Forthespecificdetail,wewilldiscussinthefollowingsections.


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30.4Gamma( )

Thegammaofanoption, ,isdefinedastherateofchangeofdeltarespectedtotherateofchangeofunderlyingassetprice::

2

2

S S

= =

where istheoptionpriceandSistheunderlyingassetprice.

Becausetheoptionisnotlinearlydependentonitsunderlyingasset,delta-neutralhedgestrategyisusefulonlywhenthemovementofunderlyingassetpriceissmall.Oncetheunderlying asset pricemoveswider, gamma-neutral hedgeisnecessary.Wenextshowthederivationofgammaforvariouskindsofstockoption.

30.4.1DerivationofGammaforDifferentKindsofStockOption

ForaEuropeancalloptiononanon-dividendstock,gammacanbeshownas

( )1t s

1N d

S

=

(30.17)


( )

( )

( )

t

2

tt

2

t t

1 1

1 t

t

1

s

1

t s

C

SC

S S

N d d

d S

1

SN d

1N d

S

= =

=

=

=

ForaEuropeanputoptiononanon-dividendstock,gammacanbeshownas

( )1t s

1

N dS

=

(30.18)



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( )

( )

( )

t

2tt

2

t t

1 1

1 t

t1

s

1

t s

P

SP

S S

(N d 1) d

d S

1

SN d

1N d

S

= =

=

=

=

ForaEuropeancalloptiononadividend-payingstock,gammacanbeshownas

( )q

1t s

eN d

S

=

(30.19)


( )

( )

( )

( )

t

2tt

2

t t

q

1

t

1q 1

1 t

q t1

s

q

1

t s

C

SC

S S

e N(d )

S

N d d

e d S

1

Se N d

eN d

S

= =

=

=

=

=

ForaEuropeancalloptiononadividend-payingstock,gammacanbeshownas

( )q

1

t s

eN d

S

=

(30.20)



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( )( )

( )

( )

( )

t

2tt

2

t t

q

1

t

1q 1

1 t

q t1

s

q

1

t s

P

SP

S S

e (N d 1)

S

(N d 1) de

d S

1

Se N d

eN d

S

= =

=

=

=

=

30.4.2ApplicationofGamma( )

Onecanusedeltaandgammatogethertocalculatethechangesoftheoptionduetochanges in the underlying stock price. This change can be approximated by thefollowingrelations.

21

change in option value change in stock price (change in stock price)2

+

Fromtheaboverelation,onecanobservethatthegammamakesthecorrectionforthefactthattheoptionvalueisnotalinearfunctionofunderlyingstockprice.ThisapproximationcomesfromtheTaylorseriesexpansionneartheinitialstockprice.Ifwe let Vbe option value, S be stockprice, andS0 be initial stockprice, then theTaylorseriesexpansionaroundS0yieldsthefollowing.

220 0 0

0 0 0 02

220 0

0 0 02

( ) ( ) ( )1 1( ) ( ) ( ) ( ) ( )

2! 2!

( ) ( )1( ) ( ) ( ) ( )

2!

n

n

n

V S V S V S V S V S S S S S S S

S S S

V S V S V S S S S S o S

S S

+ + + +

+ + +

L

Ifweonlyconsiderthefirstthreeterms,theapproximationisthen,

2

20 00 0 02

2

0 0

( ) ( )1( ) ( ) ( ) ( )2!

1( ) ( )

2

V S V S V S V S S S S S

S S

S S S S

+

+

.

Forexample,ifaportfolioofoptionshasadeltaequalto$10000andagammaequalto$5000,thechangeintheportfoliovalueifthestockpricedropto$34from$35isapproximately,


15/27

21change in portfolio value ($10000) ($34 $35) ($5000) ($34 $35)

2

$7500

+

Theaboveanalysiscanalsobeappliedtomeasurethepricesensitivityofinterest

raterelatedassetsorportfoliotointerestratechanges.Hereweintroduce ModifiedDuration and Convexity as risk measure corresponding to the above delta andgamma. Modified durationmeasures the percentage change in asset or portfoliovalueresultingfromapercentagechangeininterestrate.

Change in priceModified Duration Price

Change in interest rate

/ P

=

=

Usingthemodifiedduration,

Change in Portfolio Value Change in interest rate

( P) Change in interest rateDuration

=

=

wecancalculatethevaluechangesoftheportfolio.Theaboverelationcorrespondstothepreviousdiscussionofdeltameasure.Wewanttoknowhowthepriceoftheportfoliochangesgivenachangeininterestrate.Similartodelta,modifieddurationonlyshowthefirstorderapproximationofthechangesinvalue.Inordertoaccountforthenonlinearrelationbetweentheinterestrateandportfoliovalue,weneedasecondorderapproximationsimilarto thegammameasurebefore,thisisthentheconvexitymeasure.Convexityistheinterestrategammadividedbyprice,

/ PConvexity =

andthismeasurecapturesthenonlinearpartofthepricechangesduetointerestrate changes. Using the modified duration and convexity together allow us todevelopfirstaswellassecondorderapproximationofthepricechangessimilartopreviousdiscussion.

2

Change in Portfolio Value P (change in rate)

1P (change in rate)

2

Duration

Convexity

+

Asaresult,(-durationxP)and(convexityxP)actlikethedeltaandgammameasurerespectively inthepreviousdiscussion.This showsthat theseGreeks can alsobeappliedinmeasuringriskininterestraterelatedassetsorportfolio.

Nextwediscusshowtomakeaportfoliogammaneutral.Supposethegammaofa

delta-neutralportfoliois ,thegammaoftheoptioninthisportfolioiso

,ando

isthenumberofoptionsaddedtothedelta-neutralportfolio.Then,thegammaofthisnewportfoliois

o o

+


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Tomakeagamma-neutralportfolio,weshouldtrade *o o

/ = options.Because

the position of option changes, the new portfolio is not in the delta-neutral.Weshouldchangethepositionoftheunderlyingassettomaintaindelta-neutral.

Forexample,thedeltaandgammaofaparticularcalloptionare0.7and1.2.Adelta-

neutral portfolio has a gamma of -2,400. To make a delta-neutral and gamma-neutralportfolio,weshould add a longpositionof 2,400/1.2=2,000shares and ashortpositionof2,000x0.7=1,400sharesintheoriginalportfolio.

30.5Vega( )

The vega of an option, , is defined as the rate of change of the option pricerespectedtothevolatilityoftheunderlyingasset:

=

where istheoptionpriceand isvolatilityofthestockprice.Wenextshowthederivationofvegaforvariouskindsofstockoption.

30.5.1DerivationofVegaforDifferentKindsofStockOption

ForaEuropeancalloptiononanon-dividendstock,vegacanbeshownas

( )t 1S N d = (30.21)



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21

21

rt 1 2t

s s s

r1 1 2 2t

1 s 2 s

123

2 t s2 2sd

2t 2

s

2

t s

d

r rt2

C N(d ) N(d )S Xe

N(d ) d N(d ) dS Xe

d dS

ln rX 21

S e2

Sln r

X 2S1Xe e e

X2

= =

=

+ +

=

+ +

( )

2 21 1

21

1

2

2

s

1 12 23

2 t s t s2 2 2sd d

2 2t t2 2

s s

3d 2 2s2

t 2

s

t 1

S Sln r ln r

X 2 X 21 1S e S e

2 2

1S e

2

S N d

+ + + +

=

=

=

ForaEuropeanputoptiononanon-dividendstock,vegacanbeshownas

( )t 1S N d = (30.22)



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21

21

rt 2 1t

s s s

r 2 2 1 1t

2 s 1 s

12

t s 2d

r rt22

s

23

2 t s2sd

2t

P N( d ) N( d )Xe S

(1 N(d )) d (1 N(d )) dXe S

d d

Sln rX 2S1

Xe e eX2

Sln r

X 21S e

2

= =

=

+ +

=

+ +

+

2 2

1 1

21

1

2

2

s

1 12 23

2t s t s2 2 2s

d d2 2

t t2 2

s s

3d 2 2s2

t 2

s

t

S Sln r ln r

X 2 X 21 1S e S e

2 2

1S e

2

S N d

+ + + +

= +

=

= ( )1

ForaEuropeancalloptiononadividend-payingstock,vegacanbeshownas

( )qt 1S e N d = (30.23)



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21

21

q rt 1 2t

s s s

q r1 1 2 2t

1 s 2 s

123

2 t s2 2sd

q 2t 2

s

t

d

r (r q )t2

C N(d ) N(d )S e Xe

N(d ) d N(d ) dS e Xe

d d

Sln r qX 21

S e e2

Sln r

XS1Xe e e

X2

= =

=

+ +

=

+

2 2

1 1

21

12

s 2

2

s

1 2 23

2 t s t s2 2 s

d dq q2 2

t t2 2

s s

3d 2 2s2

t

q2

S Sln r q ln r q

X 2 X 21 1S e e S e e

2 2

1S e

2

+

+ + + +

=

=

( )

2

s

q

t 1S e N d

=

ForaEuropeancalloptiononadividend-payingstock,vegacanbeshownas

( )qt 1S e N d = (30.24)



20/27

21

21

r qt 2 1t

s s s

r q2 2 1 1t

2 s 1 s

12

t s 2d

r (r q)t22

s

2

sd

q 2t

P N( d ) N( d )Xe S e

(1 N(d )) d (1 N(d )) dXe S e

d d

Sln r qX 2S1

Xe e eX2

1S e e

2

= =

=

+ +

=

+

2 2

1 1

123

t s2 2

2

s

1 2 23

2t s t s2 2 s

d dq q2 2

t t2 2

s s

q

t

Sln r q

X 2

S Sln r q ln r q

X 2 X 21 1S e e S e e

2 2

S e

+ +

+ + + +

= +

=

( )

21

3d 2 2s2

2

s

q

t 1

1e

2

S e N d

=

30.5.2ApplicationofVega( )

Supposeadelta-neutralandgamma-neutralportfoliohasavegaequalto andthevegaofaparticularoptionis

o .Similartogamma,wecanaddapositionof

o/

inoption tomake a vega-neutral portfolio. Tomaintain delta-neutral,we shouldchangetheunderlyingassetposition.However,whenwechangetheoptionposition,the new portfolio is not gamma-neutral. Generally, a portfolio with one optioncannotmaintainitsgamma-neutralandvega-neutralatthesametime.Ifwewantaportfoliotobebothgamma-neutralandvega-neutral,weshouldincludeatleasttwokindofoptiononthesameunderlyingassetinourportfolio.

Forexample,adelta-neutralandgamma-neutralportfoliocontainsoptionA,optionB,andunderlyingasset.Thegammaandvegaofthisportfolioare-3,200and-2,500,

respectively.OptionAhasadeltaof0.3,gammaof1.2,andvegaof1.5.OptionBhasadeltaof0.4,gammaof1.6andvegaof0.8.Thenewportfoliowillbebothgamma-

neutralandvega-neutralwhenaddingA

ofoptionAandB ofoptionBintothe

originalportfolio.

A B

Gamma Neutral: 3200 1.2 1.6 0 + + =

A BVega Neutral: 2500 1.5 0.8 0 + + =


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Fromtwoequationsshownabove,wecangetthesolutionthatA

=1000andB =

1250.Thedeltaofnewportfoliois1000x.3+1250x0.4=800.Tomaintaindelta-neutral,weneedtoshort800sharesoftheunderlyingasset.

30.6Rho( )

Therhoofanoptionisdefinedastherateofchangeoftheoptionpricerespectedtotheinterestrate:

rhor

=

where istheoptionpriceandrisinterestrate.Therhoforanordinarystockcalloptionshouldbepositivebecausehigherinterestratereducesthepresentvalueofthestrikepricewhichinturnincreasesthevalueofthecalloption.Similarly,therhoofanordinaryputoptionshouldbenegativebythesamereasoning.Wenextshow

thederivationofrhoforvariouskindsofstockoption.

30.6.1DerivationofRhoforDifferentKindsofstockoption

ForaEuropeancalloptiononanon-dividendstock,rhocanbeshownas

r

2rho X e N(d )

= (30.25)


( )

2 21 1

21

r rt 1 2t 2

r r1 1 2 2t 2

1 2

d d

r r rt2 2t 2

s s

d

2t

s

C N(d ) N(d )rho S X e N(d ) Xe

r r r

N(d ) d N(d ) dS X e N(d ) Xed r d r

S1 1S e X e N(d ) Xe e e

X2 2

1S e

2

= =

= +

= +

=

21d

r 22 t

s

r

2

1X e N(d ) S e

2

X e N(d )

+

=

ForaEuropeanputoptiononanon-dividendstock,rhocanbeshownas

r

2rho X e N( d ) = (30.26)



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( )

2 2

1 1

r rt 2 12 t

r r 2 2 1 12 t

2 1

d d

r r rt2 22 t

s s

P N( d ) N( d )rho X e N( d ) Xe S

r r r

(1 N(d )) d (1 N(d )) dX e (1 N(d )) Xe S

d r d r

S1 1X e (1 N(d )) Xe e e S eX2 2

= = +

= +

= + 2 21 1d d

r 2 22 t t

s s

r

2

1 1X e (1 N(d )) S e S e

2 2

X e N( d )

= +

=

ForaEuropeancalloptiononadividend-payingstock,rhocanbeshownas

r

2rho X e N(d ) = (30.27)


( )

2 21 1

q r rt 1 2t 2

q r r1 1 2 2t 2

1 2

d d

q r r (r q)t2 2t 2

s s

t

C N(d ) N(d )rho S e X e N(d ) Xe

r r r

N(d ) d N(d ) dS e X e N(d ) Xe

d r d r

S1 1S e e X e N(d ) Xe e e

X2 2

S e

= =

= +

= +

=

2 21 1d d

q r q2 22 t

s s

r

2

1 1e X e N(d ) S e e

2 2X e N(d )

+

=

ForaEuropeanputoptiononadividend-payingstock,rhocanbeshownas

r

2rho X e N( d ) = (30.28)



23/27

( )

2

1

r r qt 2 12 t

r r q2 2 1 12 t

2 1

d

r r ( r q) qt22 t

s

P N( d ) N( d )rho X e N( d ) Xe S e

r r r

(1 N(d )) d (1 N(d )) dX e (1 N(d )) Xe S e

d r d r

S1 1X e (1 N(d )) Xe e e S e eX2 2

= = +

= +

= +

2

1

2 21 1

d

2

s

d d

r q q2 22 t t

s s

r

2

1 1X e (1 N(d )) S e e S e e

2 2

X e N( d )

= +

=

30.6.2ApplicationofRho()

Assumethataninvestorwouldliketoseehowinterestratechangesaffectthevalue

ofa3-monthEuropeanput option she holdswith the following information. Thecurrent stock price is $65 and the strike price is $58. The interest rate and thevolatility of the stock is 5% and 30% per annum respectively. The rho of thisEuropeanputcanbecalculatedasfollowing.

2

r (0.05)(0.25)

put 2

1ln(65 58) [0.05 (0.3) ](0.25)

2Rho X e N(d ) ($58)(0.25)e N( ) 3.168(0.3) 0.25

+

= =

Thiscalculationindicatesthatgiven1%changeincreaseininterestrate,sayfrom5% to 6%, the value of this European call option will decrease 0.03168 (0.01 x3.168). This simple example can be further applied to stocks that pay dividendsusingthederivationresultsshownpreviously.

30.7DerivationofSensitivityforStockOptionsRespectivewithExercisePrice

ForaEuropeancalloptiononanon-dividendstock,thesensitivitycanbeshownas

rt2

Ce N(d )

X

=

(30.29)



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)(

2

1)(

2

1

1121)(11

21

)()(

)(

)()(

)(

2

22

2

22

2

2

2

22

1

1

1

22

1

21

21

2

1

2

1

dNe

X

SedNe

X

Se

Xe

X

SeXedNeX

eS

X

d

d

dNXedNe

X

d

d

dNS

X

dNXedNe

X

dNS

X

C

r

t

d

s

rt

d

s

s

rt

d

rr

s

d

t

rr

t

rr

t

t

=

=

=

=

=

ForaEuropeanputoptiononanon-dividendstock,thesensitivitycanbeshownas

rt

2

Pe N( d )

X

=

(30.30)


2 21 1

r rt 2 12 t

r r 2 2 1 12 t

2 1

d d

r r rt2 22 t

s s

r

2

P N( d ) N( d )e N( d ) Xe S

X X X

(1 N(d )) d (1 N(d )) de (1 N(d )) Xe S

d X d X

S1 1 1 1 1 1e (1 N(d )) Xe e e S e

X X X2 2

e (1 N(d

= +

= +

= +

=

2 21 1d d

t t2 2

s s

r

2

S S1 1)) e e

X X2 2

e N( d )

+

=

ForaEuropeancalloptiononadividend-payingstock,thesensitivitycanbeshownas

rt2

Ce N(d )

X

=

(30.31)



25/27

2 21 1

21

q r rt 1 2t 2

q r r1 1 2 2t 2

1 2

d d

q r r (r q )t2 2t 2

s s

d

2

s

C N(d ) N(d )S e e N(d ) Xe

X X X

N(d ) d N(d ) dS e e N(d ) Xe

d X d X

S1 1 1 1 1 1S e e e N(d ) Xe e eX X X2 2

S1e

2

=

=

=

=

21dq q

rt t22

s

r

2

e S e1e N(d ) e

X X2

e N(d )

=

ForaEuropeanputoptiononadividend-payingstock,thesensitivitycanbeshownas

rt2P e N( d )

X

=

(30.32)


2 21 1

r r qt 2 12 t

r r q2 2 1 12 t

2 1

d d

r r (r q) qt2 22 t

s s

P N( d ) N( d )e N( d ) Xe S e

X X X

(1 N(d )) d (1 N(d )) de (1 N(d )) Xe S e

d X d X

S1 1 1 1 1 1e (1 N(d )) Xe e e S e e

X X X2 2

= +

= +

= +

2 21 1d dq q

r t t2 22

s s

r

2

S e S e1 1e (1 N(d )) e e

X X2 2

e N( d )

= +

=

30.8RelationshipbetweenDelta,Theta,andGamma

So far, the discussion has introduced the derivation and application of eachindividualGreeksandhowtheycanbeappliedinportfoliomanagement.Inpractice,the interactionor trade-off between these parameters is of concern aswell. For

example,recallthepartialdifferentialequationfor theBlack-Scholesformulawithnon-dividendpayingstockcanbewrittenas

2

2 2

2

1

2rS S r

t S S

+ + =

Where isthevalueofthederivativesecuritycontingentonstockprice, Sisthepriceofstock,ristheriskfreerate,and isthevolatilityofthestockprice,and tis


26/27

thetimetoexpirationofthederivative.Giventheearlierderivation,wecanrewritetheBlack-ScholesPDEas

2 21

2rS S r + + =

Thisrelationgivesusthetrade-offbetweendelta,gamma,andtheta.Forexample,suppose there are two delta neutral( 0) = portfolios, one with positive gamma

( 0) > andtheotheronewithnegativegamma ( 0) < andtheybothhavevalueof

$1 ( 1) = .Thetrade-offcanbewrittenas

2 21

2S r+ =

For the first portfolio, if gamma ispositive and large, then theta isnegative andlarge.Whengammaispositive,changeinstockpricesresultinhighervalueoftheoption.Thismeansthatwhenthereisnochangeinstockprices,thevalueoftheoptiondeclinesasweapproachtheexpirationdate.Asaresult,thethetaisnegative.Ontheotherhand,whengammaisnegativeandlarge,changeinstockpricesresultinlower option value. Thismeansthatwhen there isno stockprice changes, thevalueoftheoption increases asweapproach the expirationand thetais positive.Thisgivesusatrade-offbetweengammaandthetaandtheycanbeusedasproxyforeachotherinadeltaneutralportfolio.

30.9Conclusion

Inthischapterwehaveshownthederivationofthesensitivitiesoftheoptionpricetothechangeinthevalueofstatevariablesorparameters.ThefirstGreekisdelta(

)whichistherateofchangeofoptionpricetochangeinpriceofunderlyingasset.Oncethedeltaiscalculated,thenextstepistherateofchangeofdeltawithrespecttounderlyingassetpricewhichgivesusgamma( ).Anothertworiskmeasuresaretheta( )andrho( ),theymeasurethechangeinoptionvaluewithrespectto

passingtimeandinterestraterespectively.Finally,onecanalsomeasurethechangeinoptionvaluewithrespecttothevolatilityoftheunderlyingassetandthisgivesusthevega(v ).

Therelationshipbetweentheseriskmeasuresareshown,oneoftheexamplecanbeseen from the Black-Scholes partial differential equation. Furthermore, theapplications of these Greeks letter in the portfolio management have also been

discussed.Riskmanagementisoneoftheimportanttopicsinfinancetoday,bothforacademicsandpractitioners.Giventherecentcreditcrisis,onecanobservethatitiscrucialtoproperlymeasuretheriskrelatedtotheevermorecomplicatedfinancialassets.

References

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Boyle, P. P., & Emanuel, D. (1980). Discretely adjusted option hedges. Journal of

FinancialEconomics,8(3),259-282.Duffie,D.,DynamicAssetPricingTheory,PrincetonUniversityPress,2001.

Fabozzi,F.J.,FixedIncomeAnalysis,2ndEdition,Wiley,2007.Figlewski, S. (1989). Options arbitrage in imperfect markets. Journal of Finance,

44(5),1289-1311.Galai,D.(1983).Thecomponentsofthereturnfromhedgingoptionsagainststocks.

JournalofBusiness,56(1),45-54.Hull,J.,Options,Futures,andOtherDerivatives6thEdition,Pearson,2006.

Hull,J.,&White,A.(1987).Hedgingtherisksfromwritingforeigncurrencyoptions. JournalofInternationalMoneyandFinance,6(2),131-152.

Karatzas, I. and Shreve, S.E., Brownian Motion and Stochastic Calculus, Springer,

2000.Klebaner,F.C.,IntroductiontoStochasticCalculuswithApplications,ImperialCollege

Press,2005.McDonlad,R.L.,DerivativesMarkets2ndEdition,Addison-Wesley,2005.

Shreve, S.E., Stochastic Calculus for Finance II: Continuous Time Model, Springer,2004.

Tuckman,B., FixedIncomeSecurities:ToolsforToday'sMarkets, 2ndEdition,Wiley,2002.

Applications and Mathematical Derivation of Options Greeks From First Principle

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