FIN Ch 9 and 10 Lecture

8/19/2019 FIN Ch 9 and 10 Lecture

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FIN 413: Option mechanics andproperties

Relates to Hull Ch. 9 and 10

Text Ch. 9 provides some institutional detail that we will not cover in class


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Terminology(continued)

Fundamentals of Futures and Options Markets, 7th Ed, Ch 9, Copyright© John C. Hull !"!#

Option class The type of option on a security (ie call or put)

Option series Option of same class !ith same stri"e an#

e$piration #ate Intrinsic %alue The ma$imum of &ero or the amount an option

is in the money

Time %alue 'ierence et!een an option*s current price an# its

intrinsic %alue


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Fundamentals of Futures and Options Markets, 7th Ed, Ch 9, Copyright© John C. Hull !"!

Dividends & Stock Splits

Suppose you o!n options !ith a stri"eprice of K to uy (or sell) N shares+ o a#ustments are ma#e to the option terms

for cash #i%i#en#s .hen there is an n-for-m stoc" split,

the stri"e price is re#uce# to mK /n the no of shares that can e ou/ht

(or sol#) is increase# to nN /m Stoc" #i%i#en#s are han#le# in a manner

similar to stoc" splits

$


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Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!

Notation

c : 0uropean calloption price

p +0uropean putoption price

S 0 + Stoc" priceto#ay

K + Stri"e price

T +1ife of option in

years

σ+ 2olatility ofreturns

C + American 3alloption price

P +American 4ut optionprice

S T +Stoc" price atoption maturity

D + 4resent %alue of#i%i#en#s #urin/

option*s life r +5is"-free rate for

maturity T !ith contcomp

%


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American vs European Options

An American option is !orth atleast as much as thecorrespon#in/ 0uropean option

C ≥ c P ≥ p

7


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Calls: An Arbitrage Opportunity?

Suppose that

c = 7 S 0 = 89T = : r = :9;

K = : D = 9

Is there an aritra/e opportunity

Call pri-e too lo

". /uy -all, 0n1est 21 of of for " yr #, "e3p4!."5 6 "+.9

. hort sto-k

#. 08 "yr if t "

Call is orthless 'uy sto-k in market for st" -o1er short -ash in deposit"7e3p4."5 6 ".79

8et profit 6 ".79:st ;!.79

0F < ; " /uy sto-k for 4"5 ,-o1er short , -ash in deposit ".79 6!.79


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The lower bound on price of European callon non-dividend paying asset

3onsi#er t!o portfolios+ 4ortfolio A consists of

a 0uropean call !ith stri"e price K that e$pires at time T

A ris"-free in%estment that !ill e !orth K at time T

4ortfolio consists of one share of the un#erlyin/asset

The %alue of the t!o portfolios at time T 4ort A+

2alue of call at time T+ Max(ST – K,0) 2alue of ris"-free in%estment at time T+ K

Total %alue+ Max(ST ,K)

4ort + 2alue is ST


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Completing the arbitrage argument

At time T, Port A is always worth at leastas much as Port B.

ecause 4ort A is al!ays !orth at least asmuch as 4ort at time T, this relationship

must hol# at all times efore time T If this isnot true, then one coul# create an aritra/e

2alues of the 8 portfolios at times efore T+ 4ort A+

0uropean call %alue+ c 5is"-Free in%estment that !ill e !orth K at time T+ Ke-rT

4ort + A share of the un#erlyin/ asset is !orth S


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Lower Bound for European Call OptionPrices; No Dividends (Equation 10.4, page 233)

Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!""

The aritra/e ar/ument means2alue of 4ort A ≥ 2alue of 4ort

Thus, c + Ke-rT ≥ S

c ≥ S –Ke-rT


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Puts: An Arbitrage Opportunity?

Suppose that

p = : S 0 = 7?T = 9@ r =@;

K = 9 D = 9

Is there an aritra/eopportunity

"


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The lower bound on price of European puton non-dividend paying asset

3onsi#er t!o portfolios+ 4ortfolio 3 consists of

a 0uropean put !ith stri"e price K that e$pires at time T

A share of stoc"

4ortfolio ' consists of a ris"-free in%estment that!ill e !orth K at time T

The %alue of the t!o portfolios at time T 4ort 3+

2alue of call at time T+ Max(K-ST ,0) 2alue of ris"-free in%estment at time T+ ST

Total %alue+ Max(ST ,K)

4ort '+ 2alue is K


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Completing the arbitrage argument

At time T, Port C is always worth at leastas much as Port D.

ecause 4ort 3 is al!ays !orth at least as muchas 4ort ' at time T, this relationship must hol#

at all times efore time T If this is not true, thenone coul# create an aritra/e

2alues of the 8 portfolios at times efore T+ 4ort 3+

0uropean call %alue+ p Current stock price S

5is"-Free in%estment that !ill e !orth K at time T+ S

4ort ' + The ris"-free #eposit is !orth Ke-rT


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Lower Bound for European Put Prices; No Dividends(Equation 10.5, page 235)

Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!"%

The aritra/e ar/ument means2alue of 4ort 3 ≥ 2alue of 4ort '

Thus, p + S ≥ Ke-rT

p ≥ Ke –rT – S


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Put-Call Parity; No Dividends

1et*s reconsi#er 8 of our earlier portfolios+

4ortfolio A+ 0uropean call on a stoc" B &ero-coupon on# that pays K at time T

4ortfolio 3+ 0uropean put on the stoc" B thestoc"

Our earlier #iscussion sho!e# that at time

T oth of these portfolios !ill e !orthMax(ST ,K)

"+


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The Put-Call Parity Result(Equation 10.6,page 236)

oth are !orth ma$(S T , K ) at the maturity ofthe options

They must therefore e !orth the same to#ay

This means that

c + Ke -rT = p + S 0

Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!"7


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Arbitrage Opportunities and Put-Call Parity

Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!"

Suppose thatc = 7 S

0= 7:

T = 98@ r = :9;

K =79 D = 9

.hat are the aritra/e possiilities !hen

p = 88@ (!e*ll !orth throu/h this one)

p = : C


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Early Exercise

Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!"9

Dsually there is some chance that anAmerican option !ill e e$ercise# early

An e$ception is an American call on a non-dividend payin/ stoc"

This shoul# ne%er e e$ercise# earlyIn other !or#s, the option is !orth more ali%ethan #ea#


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For an American call option+S0 = 100;T= 0.25;K = 60;D = 0

Shoul# you e$ercise imme#iately

.hat shoul# you #o if+

Eou !ant to hol# the stoc" for the ne$t 7 months

Eou #o not feel that the stoc" is !orth hol#in/ forthe ne$t 7 months

An Extreme Situation

!


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Reasons For Not Exercising a Call Early(No Dividends)

Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!"

o income is sacrice# Eou #elay payin/ the stri"e price

Gol#in/ the call pro%i#es insurance a/ainststoc" price fallin/ elo! stri"e price


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Bounds for European or American CallOptions (No Dividends)



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Should Puts Ever Be Exercised Early ?

Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!#

Are there any a#%anta/es to e$ercisin/ anAmerican put !hen+

S 0= 9H T = 98@H r =:9; K = :99H D = 9

.hat aout !henS 0= 9H T = 98@H r =:9; K = :99H D = 9


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Bounds for European and American Put Options(No Dividends)

Fundamentals of Futures and Options Markets, 7th Ed, Ch "!,Copyright © John C. Hull !"!

$

h f d d d


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The Impact of Dividends on Lower Bounds to OptionPrices(Equations 10.8 and 10.9, pages 243-244)

%

>e-ommended e3er-ise? &erify that these

'ounds are true.

rT Ke DS c −−−≥0

0S Ke D p rT −+≥ −


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Extensions of Put-Call Parity

0uropean optionsH D > 0

c + D + Ke -rT = p + S 0

0Juation :9:9 p 8

0uropean option !ith an asset payin/ a continuous yiel# at rate J

American optionsH D = 0

S 0 - K < C - P < S 0 - Ke -rT

0Juation :9? p 87

American optionsH D > 0

S 0 - D - K < C - P < S 0 - Ke -rT

0Juation :9:: p 8

+


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Exploring early exercise of Americanoptions

.e "no! that put-call parity for 0uropeanoptions on #i%i#en# payin/ assets has thefollo!in/ form+

.hen

1et*s !rite , !here #isc(K) #enotes the si&e ofthe #iscount


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Early exercise on calls

1et*s use the last relationship an# re-!rite putcall parity

5eco/ni&e that, if the option !as American, itcoul# e e$ercise# at any time to reali&e its

intrinsic %alue Thus, an American option !oul# e e$ercise#

early if the option*s 0uropean counterpart hasnegative time value

Intrinsic %alue Time %alue


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Early exercise of calls 0uropean call option time %alue is

.hen is this ne/ati%e p is the %alue of a 0uropean put option an# is al!ays positi%e .

'isc(K) is also al!ays positi%e

Thus, the only time that the time %alue can e ne/ati%e is if

the present %alue of the #i%i#en# stream #ue efore thee$piration #ate, D, is /reater than p an# #isc(K) This is more

li"ely !hen+ D is lar/e

p is small (hi/h S or lo! K ein/ main #ri%ers of this, althou/h thereare other factors)

r is lo!, meanin/ #isc(K) is small

.hen D = 9, time %alue is ne%er ne/ati%e, so it is ne%erappropriate to e$ercise an American call on a non-#i%i#en#payin/ stoc" early


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Early exercise of puts

Dsin/ same lo/ic, !e can re-arran/e put-callparity as follo!s+

5eco/ni&e that, if the option !as American, itcoul# e e$ercise# at any time to reali&e itsintrinsic %alue

Thus, an American option !oul# e e$ercise#early if the option*s 0uropean counterpart hasnegative time value

Intrinsic %alue Time %alue


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Early exercise of puts

0uropean put option time %alue is

.hen is this ne/ati%e c is the %alue of a 0uropean call option an# is

al!ays positi%e.

'isc(K) is also al!ays positi%e

Thus, the only time that the time %alue can ene/ati%e is if #isc(K) is /reater than the sum of theeuropean call %alue an# the 42 of #i%i#en#s This ismore li"ely !hen+

r is hi/h, meanin/ #isc(K) is lar/e D is small

c is small (lo! S or hi/h K ein/ main #ri%ers of this,althou/h there are other factors)


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More fun with Put-Call Parity

3onsi#er /eneral 4ut-3all 4arity forcontinuously yiel#in/ asset+

5earran/e this+

Thin" this throu/h+ .hat is the left-han#si#e

ote the consistency !ith 3hapter @results

Documents

FIN Ch 9 and 10 Lecture