fin ch 9 and 10 lecture
TRANSCRIPT
-
8/19/2019 FIN Ch 9 and 10 Lecture
1/32
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
2/32
-
8/19/2019 FIN Ch 9 and 10 Lecture
3/32
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
4/32
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
$
-
8/19/2019 FIN Ch 9 and 10 Lecture
5/32
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
%
-
8/19/2019 FIN Ch 9 and 10 Lecture
6/32
-
8/19/2019 FIN Ch 9 and 10 Lecture
7/32
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!
American vs European Options
An American option is !orth atleast as much as thecorrespon#in/ 0uropean option
C ≥ c P ≥ p
7
-
8/19/2019 FIN Ch 9 and 10 Lecture
8/32
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
9/32
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$pires at time T
A ris"-free in%estment that !ill e !orth K at time T
4ortfolio consists of one share of the un#erlyin/asset
The %alue of the t!o portfolios at time T 4ort A+
2alue of call at time T+ Max(ST – K,0) 2alue of ris"-free in%estment at time T+ K
Total %alue+ Max(ST ,K)
4ort + 2alue is ST
-
8/19/2019 FIN Ch 9 and 10 Lecture
10/32
Completing the arbitrage argument
At time T, Port A is always worth at leastas much as Port B.
ecause 4ort A is al!ays !orth at least asmuch as 4ort at time T, this relationship
must hol# at all times efore time T If this isnot true, then one coul# create an aritra/e
2alues of the 8 portfolios at times efore T+ 4ort A+
0uropean call %alue+ c 5is"-Free in%estment that !ill e !orth K at time T+ Ke-rT
4ort + A share of the un#erlyin/ asset is !orth S
-
8/19/2019 FIN Ch 9 and 10 Lecture
11/32
Lower Bound for European Call OptionPrices; No Dividends (Equation 10.4, page 233)
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!""
The aritra/e ar/ument means2alue of 4ort A ≥ 2alue of 4ort
Thus, c + Ke-rT ≥ S
c ≥ S –Ke-rT
-
8/19/2019 FIN Ch 9 and 10 Lecture
12/32
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!
Puts: An Arbitrage Opportunity?
Suppose that
p = : S 0 = 7?T = 9@ r =@;
K = 9 D = 9
Is there an aritra/eopportunity
"
-
8/19/2019 FIN Ch 9 and 10 Lecture
13/32
The lower bound on price of European puton non-dividend paying asset
3onsi#er t!o portfolios+ 4ortfolio 3 consists of
a 0uropean put !ith stri"e price K that e$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
-
8/19/2019 FIN Ch 9 and 10 Lecture
14/32
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
15/32
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
16/32
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!
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)
"+
-
8/19/2019 FIN Ch 9 and 10 Lecture
17/32
The Put-Call Parity Result(Equation 10.6,page 236)
oth are !orth ma$(S T , K ) at the maturity ofthe options
They must therefore e !orth the same to#ay
This means that
c + Ke -rT = p + S 0
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!"7
-
8/19/2019 FIN Ch 9 and 10 Lecture
18/32
Arbitrage Opportunities and Put-Call Parity
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!"
Suppose thatc = 7 S
0= 7:
T = 98@ r = :9;
K =79 D = 9
.hat are the aritra/e possiilities !hen
p = 88@ (!e*ll !orth throu/h this one)
p = : C
-
8/19/2019 FIN Ch 9 and 10 Lecture
19/32
Early Exercise
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!"9
Dsually there is some chance that anAmerican option !ill e e$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#
-
8/19/2019 FIN Ch 9 and 10 Lecture
20/32
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!
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$t 7 months
An Extreme Situation
!
-
8/19/2019 FIN Ch 9 and 10 Lecture
21/32
Reasons For Not Exercising a Call Early(No Dividends)
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!"
o income is sacrice# Eou #elay payin/ the stri"e price
Gol#in/ the call pro%i#es insurance a/ainststoc" price fallin/ elo! stri"e price
-
8/19/2019 FIN Ch 9 and 10 Lecture
22/32
Bounds for European or American CallOptions (No Dividends)
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!
-
8/19/2019 FIN Ch 9 and 10 Lecture
23/32
Should Puts Ever Be Exercised Early ?
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!#
Are there any a#%anta/es to e$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
-
8/19/2019 FIN Ch 9 and 10 Lecture
24/32
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
25/32
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!
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 −+≥ −
-
8/19/2019 FIN Ch 9 and 10 Lecture
26/32
Fundamentals of Futures and Options Markets, 7th Ed, Ch "!, Copyright© John C. Hull !"!
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
+
-
8/19/2019 FIN Ch 9 and 10 Lecture
27/32
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
28/32
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
29/32
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
30/32
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
-
8/19/2019 FIN Ch 9 and 10 Lecture
31/32
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)
-
8/19/2019 FIN Ch 9 and 10 Lecture
32/32
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