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1 . 0 1Recitation

5:Options

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LearningObjectives ev ewo oncep soPayoffprofileoPutcallparityoValuationofoptionso

Binomialtree

oPayoffreplicationoArborealCorporation

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Review:elementsofacall/putoption oCall:holderhastherightbutnottheobligationtobuyoPut:holderhastherightbutnottheobligationtosell uant tyo t eun er y ngasset:oUsuallyoneshareofstockwithcurrentpriceS

Expirationdate(T)Style:oEuropean:canonlybeexercisedatToAmerican:canbeexercisedatanytimebetween0andT.

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Review:payoffprofile

Payoff Payoff

Long

45 45

K AssetPrice K AssetPrice

Short

K 45 AssetPrice 45

K AssetPrice

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Review:payoffprofile payoffsoftheindividualcomponents:

1put 1call Straddle

+ =K1 K1 K1

1call@K1 2call@K1 1call@K3 Butterflyspread

+ + =K K K K K K

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lio

1

Portfo + =K

lio

2

Portf + =

Review:putcallparity

1call@K Bondw/FV=K

K1put@K 1stock

K

K K

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Review:putcallparity havethesamecost:

K

C PS TThisistheputcallparity.

price(K).

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Review:valueofanoptionValueofput

Strike rice K) Decrease IncreasePriceofunderlyingasset(S) Increase Decrease

ceofoptions

Maturity(T) Increase IncreasePriceofunderlyingasset

Pri

Valueofcall

Volatilityofthe Increase Increaseunderlyingasset()

Interestrate(r) Increase Decrease

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S C

Review:binomialtree period,wecanpriceoptionsgiventheunderlyingassetandariskfreeasset(bond)byreplication:

UnderlyingAsset Bond CallS B C

C1p Sd

S B/(1+r)B Cd

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Review:binomialtree

CF at t = 0 CF at t=1 CF at t=1(upstate) (downstate)

Asharesofunderlyingasset AxS A xSu AxSdBond(FV=B) B/(1+r) B B

u Replication = C = Cu = Cd

= u

u

oB=Cu A xSuoC=AxS+B/(1+r)

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Review:binomialtree probability,q:

S 1 S1 r

Then,CqCu 1 q Cd

1 rNote:qisnotrelatedtothestateprobabilityp.In

fact isnotusedinthe ricin ofC.2010/YichuanLiu 11

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Example1:payoffreplication

usingonlycallandputoptions?a)

b)

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10 15 25 30

8 12

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Example1:payoffreplication

a) Long1call(K=10)Short1call(K=15) or 1ca =25 Long1call(K=30)

b) Long1put(K=8)Short1call(K=8)Long2calls(K=12) =

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Example2:ArborealCorporation Attheendof3monthsitwillbeeither$120or$90.The3monthspotrateis2%.Whatisthevalueofa3monthEuropeancalloptionwithastrikepriceof$110?

Stock Call$120 $10

$90 $0

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Example2:ArborealCorporation oLong1/3stock:costs$34oShortbondwithFV=30:costs $30/(1+2%)= $29.41 epr ceo t eca must e

C3429.41$4.59 , probability: 120q901 q

102q0.4681 2%

Thepriceofthecallisthen10 0.468 0 1 0.468 .

1 2%2010

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