8/3/2019 Introductory Derivatives

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NicholasBucheleresFinance380

ForwardArbitrage(w/Dividend)

F = So

1+ rEAY( )

T

1+ dEAY( )

T

= S

o

e(r )(T)

e(d)(T)

rEAY=DomesticRiskFree,dEAY=ForeignRiskFree

$Goal

F PV(S0)

=Units

*

*Discountquantitysoldifassetreturnsdividend*

PV UnitsSold( ) =UnitsSold

(r )(T)

InvestinTreasuriestofinancelongs

PV UnitsBought( ) $$( )[ ] = $X(r)(T)

SyntheticBorrowingForeignAmount*Rate=$AmountCalculateU.S.BorrowingCost

$Amount(1+ rEAY

)T = FV($Amount) ShortForeignForwardtoRepayLoan

(qShort)(F)=

FV($Amount)

qShort =FV($Amount)

F

LowerCostLoan:CompareRates

qInitial(1+ r)T= qShort rSynthetic rDirect

N-PeriodBinomialPricingUsebackwardinductiontopricetodaysoptionfromfinal

outcomeswhoseoptionpriceequalsSminusX.

(1+ rPeriod)p= rEAY P =

N

T

h = e()(

T*)

1h = (h% +1)

l= e()(T*) 1

l= (1 l%)

T* = TN

= stdevannual

CCreturns =

stdevreturns

Length

Length ={(weekly) 152 ;(monthly)1

12 ;etc}

Compoundedweeklymeans

1

52inthedenominator

C= S+B

=Ch ClSh Sl

B =ShCl SlCh

(1+ rperiod)(Sh Sl

EuropeanCallPortfoliow/oDividendsatt=0Longsharesofunderlying;borrowBatreporate.Put-CallParitybasedontheLawofOnePricePut=(owning-1sharesofstock)+(investingPV(X)+BDollarsrisk-free).

Call=(owningsharesofstock)+(investingBdollarsrisk-

free)

Black-Scholes

d1=

lnSX( )+ rCC/year *TYears( )+

2*TYears( )

2

TYears

d2= d

1 T

Years

EuropeanPutw/oDividends

$Put= S N(d1) 1[ ] Xe(r)(T)

N d2( ) 1[ ] ReplicatingPortfolio

SellSharesofUnderlyingStock

= N(d1) 1[ ] InvestBdollarsinTreasuries

B = Xe(r)(T)

N d2( ) 1[ ]

EuropeanCallw/oDividends

$Call= SN(d1) Xe(r )(T)

N(d2) ReplicatingPortfolio

BuySharesofUnderlyingStock

= N(d1)

BorrowBdollarsRiskFree

B = Xe(r

)(T

)N(d2) EuropeanOptionsw/KnownCashDividend

ModifyassetpriceandassetvolatilitybysubtractingpresentvaldividendsbeforecalculatingBlack-Scholesvalues.

SS PV(Divs)

= (Div1)e(

rcc )(TYear )+ (Div

2)e(

rcc )(TYear

S PV(Divs) = S (PVDiv1 + PVDiv

SModified = S (PVDiv1 + PVDiv2 )

S

S PV(Divs)

Modified =

S

SModified

EuropeanOptionsw/DividendsPaidasPerfectlyPredictableNumberofUnitsoftheUnderlying(EAY)

SS

(1+ dEAY)T

NoChange

Put-CallParity:EuropeanOptionsonDividend-PayingSto

Case1CashlevelofFutureDividendsisKnowforCertain

p = c S+ PV(x)+ PV(Divs),X= Strike

Case2TheStockPaysaKnowEAYDividendYieldofdpeyear:

p = c S(1+ d)

T

+ PV(X),X= Strike

AmericanOptionsonDividend-PayingStockMustuseN-PeriodTree,Black-Scholescannotbeadjusted.

1) Calculatevalueofoptionifnotexercisedatthisnod2) Calculatevalueoftheoptionifexercised(S-Xorzer

C= S+ BOR

S X =CSubtractcashdividendfromSatNodesofDividendyie

3) Atthisnode,valueoftheoptionisthegreaterofthtwo.

Startattheendandworktowardthefrontbackwardinduct

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ConversionsContinuouslyCompounded(CC)

Xe(rCC )(T) PV

CC(X) = Xe

(rCC )(T)

AnnualPercentageRate(APR)

X(1+rAPRKFreq

)(KFreq )(T)

PVAPR (X) = X(1+rAPRKFreq

)(KFreq )(T)

EffectiveAnnualYield(EAY)

X(1+ rEAY

)TYears PV

EAY(X) = X(1+ r

EAY)T

Years

OpenInterestSumofonlynetlongpositions.

DollarGain/Return=(ChangeinPrice)(Quantity)InitialDollarInvestmentisonlytheamountofmarginone

neededtoputdown.

ReturnOnInvestment

=

DollarGain

InitialDollarInvestmnet

ToBorrow$XFromSpotMarket

Short$Xworthofspot

$X

So=#ofUnitsandgolongsame

numberofforwardunits.Willrepay

#ofUnits($ST F) ,whichwillbemorethan$Xifforwardstradeatapremium.

ToLend$XToSpotMarketBuy$Xoftheunderlyingonspotmarketandshortthesame

amountofforwards.Willreceive

#ofUnits(F $ST) ,whichwillbemorethan$Xifforwardstradeatpremium.

DeterminingMarketsExpectationofFutureDividend

F = So

(1+r )T

(1+d)T( )

ExpertiseTransfer1) Createnewstockfund

2) Shortfuturesofexpertise3) Longfuturesofbroadermarket

#ofContracts =PortfolioSize

MarketValueOfPosition

PortfolioBeta

New = Old +Value

Index(PositionNeeded)

ValueStock(Portfolio )

ExpectedSpotPrice

F= So(1+ rEAY)T FV(NetInterimBenefitso,T)Solvefor

FV(NIB),thenplugthatintoExpectedSpotPriceformula

EST = So(1+ rreturn )T FV(NIB) andsolveforExpected

Spot.Backwardation=Spot>Future&Contango=Spot