Solution manual for an introduction to the mathematics of financial derivatives second edition

a Pa o diagramat expiration:20 short stock short call combined position written call short stock short stock + short call FIGURE 0.1 Pa o diagramforbothashortsaleofstockandanat-the-mo

1 (a) Pa odiagramat expiration:

20

short stock short call combined position

written call

short stock

short stock + short call

FIGURE 0.1 Pa odiagramforbothashortsaleofstockandanat-the-moneycall

1 2 3 4 5 6 7 8 9 10 0

0.5 1 1.5 2 2.5 3

3.5

long put long call put + call

FIGURE 0.2 Pa odiagramforalongputwithstrikeK

1andalongcallwithstrikeK

2,K

1

<K

2

2

>K

1

short stock short call combined position short stock

written call

short stock + short call

FIGURE 0.4 Pre-maturitypayodiagramforbothashortsaleofstockandanat-the-moneycall

Pa odiagrambeforeexpiration:

0 0.5 1 1.5 2 2.5 3 3.5

4

long put long call put + call

FIGURE 0.5 Pre-maturitypayodiagramforalong putwithstrikeK1andalongcallwithstrikeK2,K1<K2

0 1 2 3 4 5 6 7 8 9 10

−6

−4

−2 0 2 4

18theUSDLiborrateat12months

and18monthsrespectively Thecash wsaregivenb

12months 18months 24months

Floatingleg +N +N

L

12

2+N 1+

wherethe1inthe24monthscolumnrepresentsthenotional amount

(b) If onehad a rate obligation and wished to pay a xed rate, , then enter into two FRA

contractsatratewithmaturity18and24months Forexample,at18months,ifthe rate

wereabo e,thentheFRAwouldbein-the-moneyb precisely theamountrequiredto osetthe

higher ratepayment Therefore,thetotalpaymentisat therate

(c) Ifonehada rateobligationandwishedtopaya xedrate,aswapisnotnecessaryaslong

asthe appropriate interest rate options are a ailable A long position in an interest rate cap at

rate and ashort position in an interest rate orat rate ,bothmaturing on the rate

paymentdate, ensurethata xedrateofispaid Ifthe rate,sayr ,isabo eatexpiry,

anet paymentat rate  isrequired after taking into accountthe valueof the cap,N (r

T

)

Ifthe rateis below at expiry, sayr , then a paymentat rater must be made onthe

rateobligation However,theshort position in the orrequires anadditionalpaymentof

N( r ) Theresultis atotalpaymentatpreciselyrate

annualinsurancecostfor1tonofwheat,andristhesimpleinterestrate IfF

t

>(S

t+c+s)(1+r),

thenconstructthefollowingarbitrageportfolio

Shortfutures 0 F

tS

t+c+s)(1+r)

Buywheatandpaystorage,insurancecosts (S

t+c+s) S

T

t(S

t+c+s)(1+r)>0

Thus, F

t

 (S

t+c+s)(1+r) If F

t

< (S

t+c+s)(1+r), one cannot immediately reverse the

holdingsin theabo eportfolioto createanotherarbitrage portfolio A problemarisessincewheat

isnottypicallyheld as aninvestmentasset If onesellswheat, itis notreasonableto assumethat

one is entitled to receive the storage and insurance Therefore, a weaker condition ensues with

heldforinvestmentsuchasgold,onecouldselltheassetandsaveonthestorageandinsurancecosts

Theseassetsproduceanexactrelationship,F

t

=(S

t+c+s)(1+r) Holdinganassetsuchaswheat

hasvaluesinceitmaybeconsumed Forinstance,alargebakeryrequireswheatforproductionand

maintainsan inventory These companies would be reluctant to substitute a futures contractfor

theactualunderlying Hence,thepriceof afutures is allowedto belessthan (S

t+c+s)(1+r)

However,ifF

t

<S

t(1+r),thenconstructthefollowingarbitrageportfolio

Position Pa oatt Pa oatT

TF

S

t(1+r)F

t

(S

t+c+s)(1+r)

(b) F

t

=$1;500<$1;543:50=(1;470)(1+:05)=S

t(1+r) Thisviolatestheabo einequality Totake

advantageofthisarbitrageopportunity,followthesecondarbitragestrategyoutlinedabo e

1 = (1+r
)

u+(1+r
)

t+1d

andafterdividingthesecondequationb S

t

1 = (1+r
)

u+(1+r
)

d

1 =

Su

t+1

S

tu+Sd

t+1

S

td

Substituteinthevaluesforr,S

u

t+1, Sd

t+1andexpressthese equationsas

d

t+1+~d

Sd

t+1

=1

1:0125(:3917320+:6083260)

= 280

t

(e) Only the up state is relevant for pricing the call option asthe call expires worthless if the stock

decreasesto$260nextperiod Thecallpriceequals

Thesamepricecalculatedinpart(b)

(f) No,dierentmartingalemeasures(i.e dierentriskneutralprobabilities~

u

and~d

(g) Adierentnormalization(numeraireasset)is used An analoguetopart(f)wouldbeastatement

assertingthat thearbitrage-freeoptionpriceisindependentofthenumeraireasset

(h) The risk premium incorporated in the option's price satis es: (1+r+ riskpremiumforC

t) =

This risk premium is usually notcalculated in thereal world Oneuses risk

-neutralprobabilitiesforcallpricing, E

Inan

incompletemarket,theremayexistriskpremiumswhich requireexplicitcalculation

2 (a) Assumetherisk-freeinterestrateriszeroandconsider thesystemofequationsgivenb

4A

2withtheproperties

1:

1+

2

=1

(1+r)

=1 assumedr=0

2:

1,

2

>0

such that therighthand sideof this systemofequations is positive, thenthe\current prices"are

arbitrage -free Inthis particular case, since nocurrentpricesare speci ed, there are anin nite

numberofpossible

1and

2solutionsin whichbothstatepricesarepositive,sumtothediscount

factor,andgeneratepositivevaluesforA

0,B

0,andC

0

(b) Ifnosuchsolutionexists,thenatleastoneofthecurrentprices(A

0,B

0,orC

0

isnon-positive In

thiscase,onewould\buy"theassetforthenon- positivepriceand beassuredofpositivepayos

inallfuturestatesoftheworld Hence,anarbitragepro texists

2

4A

497:5

72:0

126:03

1+61 Thisvalue

ofKwasgeneratedb theequation

0=

1(83 K)+(1

1)(61 K)

where

1

= ~

1sincer =0 In general, risk - neutralprobabilities and not stateprices are used

Alternatively, ifthecontractwasstruckon apreviousdate with apreviouslyspeci edstrike price

K,it'scurrentvalueisgivenb theexpectedpayoundertherisk-neutralmeasure

F

0

=p(83~ K)+(1 p)(61~ K)

Notethatin generalonedoesnotdiscountthepayowhenpricingafutures' contract

(e) Theput optiononassetC onlydependsonthe rststateasthe $92payo islessthan thestrike

price Itspriceisthereforethediscountedpayoin the ...

Thiswouldcontradictproperty(1)abo e Therefore,

i(t)>0foralli andforallt

Thesecond property requiresthe sumof thestatepricesacrossthe number ofstatesto equalthe

discountfactorforaxedt Thisassertionfollowsfrom...

\average" ;of theprevioustwointegralsandthereforehasanexpectedvalueequalto zero Overall,

thisexampleshouldnotformtheimpressionthatthechoiceofintegrandisirrelevant Forinstance,

considertheintegral...

rateobligation However,theshort position in the orrequires anadditionalpaymentof

N( r ) Theresultis atotalpaymentatpreciselyrate

annualinsurancecostfor1tonofwheat,andristhesimpleinterestrate