Ch7 : Options : General Properties

17 . I0 . 2019

17.10.2019

⑦ Imo

i. European call option

- noiovononndsaianooiinqioosesmrnioexercise price

oiesnoarefesog omarion,X,

ndanosihrstrike price

as Ioana goosander, Texpiry timeexercise time

2. European put option expiration date .

3.

American call optionput }9rdnr%

-

t s y

a④ option niosnuoirs ( index)

efoio European call option niobic SET 50 Index

X = 800,T

SCT) = 815 a :9rdnd'815-800 = 15

Taiga 15*of ,pushed = 15 # $100 ¥1500 '

evooaourrnvaoi option= g

SH) - X th SCT) > X

Option 's payoff O 09 CSCTIEX )

for European call option

notation# = { X ; xx

O j fo am

⇒ +

←"""

i

call is payoff = ( SCT) - X )'t

put is payoff = ( X -

Gasol option 880 -Muhanna@ ionE

E,P

european Iaa European ,".' E

"

't"

American fallAmerican put

r : interest rate ndbhmrviovni.su, undo

TCidsnawrovo ,

continuous compounding e

s⑧ Bein 22 Iman 2562* call option on Rolls - Royce stock• X = 220 pence

• T ai'

22 W'd . 2562

µ,#t!!!?,

• Elo) = 19.5 pence 1-1O T

e r = 5.23 %

idinirisio sit > Xtc er

At time -1,

iirlei.no) iirlei.no )call (sits -Xlt- CEE

"

Een- ( SCT) -XI

put ( x - sent - peer#

peek - ( x - softcall payoff payoff npntpayott^ b

- gain ←paystt

/ s.

set' sets- - -- - - - - - - - - tf

'

pEerT+' ¥--- . - u.

Put - Call Parity-g.um portfolio Nido: nowVEN

ahart puts• write and selling one put option• buying one call option

} × , T( long calls

theorem ( Put - Call Parity )

oharvqirnlsidmrohufowoo.loinnwosrnkrof.ie imgur.roEuropean call no : put to rimshot ( X ) no : on

,Option IT)

sign CE - PE = Sco) - X e-rt

Igoe ① sound CE - PE > Scot - X e-rt

oiwvnoinnoqnrrnoiisr ( Arbitrage Strategy ) asf

N 6207 O,

• buy one share ,5101

• buy one put option , PE

• write and sell call option ,CE

• invest the sum,CE - PE- 510) (or borrow

,if -0 )

oineomaom.SI r

ta not = Sco) + PE - CE t CE -PE -510) = O

N bool T,

• shoulder (CE - p go, )erT• moai laundromat put option on SCT) E X

. Noor call option wrong

VITI = ( CE- PE - Stoller 't X

utter ( CE - PE -Sco )) + XE"

f: Epe-sioux e-"70 )

Vitti"

> o :.

VITI 70 c.t .d .

No - Arbitrageprinciple .

② road CE - PE ( Sco ) - XE"

groin nognrnviion.hr dooror 6007 O

• sell short one share,Slo ) ;

•write and sell put option , PE

• buy call option , CE

• invest the sum Scott PE - CE ( or borrow, it

Gw rate r

VIO ) = - Slo ) -peedE

+ (Slo ) tp -CE) = O

or 6907 T,

• close position in risk - free asset. ,@OHP

"-E)e"

• buy share , X , ainmdronr call on; sets > x

• close position in stock

VIT) = ( Scott PE- E) em - X

VIT ) e-rt

= Slot TPE - CE -Xjr'

f :ocscoi-xe-ICE.pt/VlTIe-rT) o .

.

.

VIT) 70 c. 4. d.

No - Arbitrage . pi-

#

* aiqu Insinuates

CE -PE = Sco ) -dig - Xe"

theorem ( put- Call Parity Estimates)groin American call soo : put mid rimsoonr

'

X ro :

on goodoptionof v T shrinesriv arid option eggnogTroi

skid min water era *Ya-annotator

Scot - Xe-rt3 CA - PA 7, Scot - X-

①#②

① Nand Sco) - X e-RTS DA - PA

Wood" o O C dit - PA -Sco ) + X e-rt

or t -- O

• write and sell a call,

CAsit f. buy a put , pit } Hot -- o• buy a share , SLO)

• invest CA - PA- Sco )

-odnirsrohor call or ion t f T 130 SIT) ) X

• shoutJu

VH1 = X + ( CA- PA - Sco ) ) e't

= Her't CA- pit -scale't> o Cournoyer)

8- o @A-PA -Sco) + Xe

'"

!.

Viti 70 ( an:p's + X↳c.t.d.ro -hor born T

-

siisiisnrsronrcall, rironrput ,

X

-showier

VIT) -- X + ( CA- PA -slope" ? lxe-rttct-pt.su, )e" >o

C.f.d. No-A

② farad CA - PAS Sio) - XWmd o X L Scot - CA + PA

soohsnoynrminsnilr door

At time 0,

• write and sell a put , PAX? {. buy a call , CA

÷:::O::c:::i.

":"} ""

At t E T

* didntVohor put ,Ivo X Bodo share ⇒Jason :

set) Cx

• no visa

. 1h50 call option

VH1 = ( PA- CA -ish) ert - X ) Xert- X 70

(: X C PA- d- + say l - ie" 71 )r> o

,t >o

C.t . d .No-Arbitrage .

At time T.

cdnlddmrikhnr put )•ironor call

,

dm X lot share ⇒ Jason :

• now 152

VITI = ( PA - CA+ scope" - X ) X e"- X ) o

c. td .

No - Arbitrage .

*

1. 008 5.14,7.6 7

.Nun 7.8

, 7,16

2. 82 5. 1827.10 2.Od 5.16

,7.7

3. ohhh 7.5

3. brat 7.3

÷ :O: ::i:ii.is/::.:::n::.........6. bio 77g. aiiuu 7. IT

7. Hdd 7. It

8. mo 7.14,5. Iz

7 . 184%77 7. Is

24.10.2019

7.3Boundsonoptionpri.cc#

Es c't

,peepA

7.3.IEuropeanoptionsfc.EC5101

Good d⇐

71510)at time 0

,

• write and sell call option , CE- buy stock , SCO)• invest the balance

,KE -Scot

,without risk

.

VIO 't = - CE + Slo) t ( CE-Slo ) ) -

- O

At tim T, XLSIT)

• call ndnugp exercise adroit on stock ministry x)

• now 156

VITI = mints ITI,x ) + ( CE- Slo)) ET 70 c. td

.

V YO O

.

.

.

CE L Slot

on put - call parity ; CE - pe = Sco ) - X e-N

CE = Sco ) -Xe-rTtpm w

Zo %

.

"

. Scot - x e-" f CE Lao)

I CE - Sco ) = -X e-" Tt PE L O

- RT

PE L X ean Ot CE = 510) - X e-

RT+ PE

- SCO ) + X e-RTE PE L X e-

RT

imgur.es?n.a:::::ginicgns.i::i:ICA 7, CE

no - dividend,CED Scot - XE

"(Prop . 7.3 )

& CAD Scot - XE"

> Scot - X

(A) Slo ) - X if r > O

• ion T , CA , CE

an CA 7, CEgawd CA > CE my c. td

.

IN CA E

at time 0 A

same {• write and sell American call

,C

X,T

. buy European call , CE

• invest,

CA- CE,

without risk

viol = -dat CE t ( CA- CE ) = 0

At time#E T

• air call airman exercise or ion t E T ; SCH > X

* Gargis ammo ,X

• B X Noigu (without risk)

At time T,

988nF European call into Bogie lionnoon :• exercise European call , X ; Stl > x

htt = ( I- E) e"+ X e-"T- t)

- X >o- -v v c.t- d .

a- O

→ on American call andmush Ysidroor↳ tsioros exercise European cue J KT) -- KA-E) e'To

c.t.

d.

7.3.3Americanoptionsf-S.CO) t X f PA (on prop . 7.3 )

pAL X

dwarf PA XXat time -0

,

• write and sell American put , PA• invest , PA ,

without risk

not = - PA -1pA = O

-mi put an exercise ar t ET

•oiosve share

,X,

mushroom Sct )

Viti = ④Alert - X t SH) 70- c.t . d .

O"

- m put Maison exercise,VCT ) = pAert yo

}

For dividend - paying stock,

7.4Variablesdeterminingoptionpn.ee#• X

• Tyfour woneullnv

• 5101, rdiv

• r

7.4.IEuropeanoptionf.DEpenitence on the Strike Price .

j CE

f p E

yw x'sx"

on put - call parity

Ecx' ) - PEC x 's = Slot - X'

e-"-①

E- Cx" ) - PEN" ) -

- Scot - X"

e-"-@

① -⑧ i E- (x 's - PE ( x'

) - EIx"

, + PECX"

) = (x"-x'te"

( Evil - c "1) tfpecx't- PM) -- Ix" '

le"5o

T.ir⇒ Evil - Ecx") L Cx

"-x' le"

"" peu" , - pea'

) ( (x"- f)E"

Reimark

ggLipschitz constant .

-RT

OL e Ll

(Ecx'

)

Elx")

Lipschitz condition

DEI function Ht, y) satisfies a Lipschitz condition

if a constant L > o with

I fit , y, ) - fit.nl shyly, -921Lipschitz constant .

①

① sarod CE ( 2X 't Ci - a ) x") > a Cx'

) +Ci - 2) EH"

)

TW X = 2X'

+ ( i -d) X"

CE (x ) 7h CEN'

) ta -ACEH"

)

at time o,

• write and sell call with strike price X. CECH

• buy a call with strike price X'

,dc

and Ci-al - n x"

,a -HEH

")

• invest the balance,Ecxl - faced) ta -a)Eai'D,

without risk

viol = -Eniac think # Ecxl - faced, ta -a)Eai'D= O

- sin call ofmush an exercise or cool Tt

• a'intosin ( SIT) - X ) ; SIT) Y X

T Tgwonw strike price .Bom

• exercise at ero : (x"

) ; SH xx'

Tahir also - x'

)t+ G -s) (Stl - x

"

)t

VCT) =L (SCT) - x'It ( t -L) (SCT) - X")+- Cst ) -x)

t

tfEun -K CEH 's th-a) C"D)em > o# c.td .

Jo

#

Eun

• Dependence on the Underlying Asset Price

Baron option aiosdoouiir x Gav,Six Slo)

hrdnourenv CE( x,-1 )

= (HCT) -x )'

eroaouunv PE = ( X - x SCT) )"

( X , -1 )

& Ecs ),

PECS)

q CEj PE

¥ In s'

as"

,

s'

- x 's"',s"- x'¥7 Ecs's - Ecs 's >a

road CEO 's > Ecs"

)

t -- o,

not = - Ecs's + Ecs 's + ( Ecs 's - Ecs 's) --

O

t -- T'. call ndmuhighhrrnir , -

x'SIT) + X j x'sits>X

• exercise call inwidow,

- X + x"SIT) ; x'

'SHI> X

Htt = - x' SCT)##x" SCT) -1µs's - Ecs 's )em > o#T c.t.cl

.

>oZo

S - as'

th -d)s"

gets , sacks ) th-2) CECI)

7.4.2AmericanOption#• Dependence on the strike Price

t.CAFPA

E → c. f. d .

VIO )

htt

VITI

• Dependence on the Underlying Asset Price.

In CA

f p 't

• Dependence on the Expiry Time

7.STimevalueofoptionsfcalld.tl,t

• in the money ( ITM) if SHI > X

•at - ATM) if SH) -- X

• out of- IOTM) it SHKX

④. ITM

,Seti CX

• ATM,Siti - X

• OTM,Siti > X

SH) -425.23

I 0

y①I O

O I

is# If Hsin'sCIA - fsH

PEH - f x - SHHt

Bourn European call option ,nor .SE X Tor intrinsic value = ( S -X)

t

Els ) www.grfarldasnsr S (on prop. 7.9)

rid time value iafrnhradrintsvoiau - ①

& Ecs" ) - Ecs's E s"

- S'

for ILS"

(on prop . 7. lo)

%"Ecs", -Ecs 's E s "- X - s 't X -- Is

"-x) - Cs

'-x)

( Ecs") - Cs

"- xt E Ecs 's - CS

'-XI

→-time rater ,S time rate , s

'

,X

'

Ls"

drdo IN time value bdonhrotoa lato SXX - @

an ① vote@ odors , time value Norisringand S -- X .

*

Mid final.

moraine + present nvm 10 5/5son present / ' O' µ÷9o"M"Gov

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