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HEDGING STRATEGIES USING OPTIONS
PROTECTIVE PUTS
Someone who owns shares of stock has a longposition in the
security. In the investment business the term !"on#$ simp"y means
ownin# somethin#. It has nothin# to %o with time span.
&i#. '(.' is a profit an% "oss %ia#ram for the purchase of )IP
common stock at Rs.*'( per share. The stock price on e+piration
ST is %enote% on hori,onta" a+is an% profit -"oss on the vertica"
a+is. The ma+imum "oss occurs if the stock %ec"ines to ,ero whi"e
the potentia" profit is un"imite%. I#norin# commissions %ivi%en%s
an% opportunity costs the strate#y !breaks even$ if the stock priceis unchan#e% at a specific future time.
Investors occasiona""y anticipate a %ec"ine in the va"ue of an
investment but cannot convenient"y se"" it because of ta+
consi%erations or other reasons. In such a situation the investor
mi#ht consi%er usin# a protective put.
/ protective put is not a specia" kin% of put option0 it is a
%escriptive term #iven to a "on# stock position combine% with a
'
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"on# put position. If someone owns shares of )ip an% buys a )ip
put -re#ar%"ess of strikin# price or e+piration the put is a
protective put.
Fig.10.11 2on# Stock Position
Rs.
Profit
orLoss
0 ST610
610
&i#. '(.3 shows the profits an% "osses associate% with
various stock prices if someone buys a )ip Oct. Rs. 45( put at a
price of Rs.3'. To he"p in constructin# the combine% %ia#ram a
profit an% "oss worksheet "ike Tab"e '(.' he"ps.
3
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Fig.10.2: 2on# Put Position
Rs.
569
Profit Or
loss
0 ST569 590
21
The worksheet shows that the ma+imum "oss is Rs.6' an%
that it occurs at a"" stock prices of Rs.45( or be"ow. The strate#y
appears to break even at a stock price somewhere between Rs.45(an% Rs.7((. 8y checkin# a few more prices it can be foun% that
the break9even point is Rs.*:'.
:
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Table 10.1 : Protective Put )orksheet
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
Stock Price at Option Expiration0 90 190 290 390 490 590 690 90
Lon! Stock" Rs.610 #610 #520 #410 #320 #290 #120 #20 $%0 $1%0
Lon! Rs.590P&t " Rs.21 569 49 39 29 19 9 #21 #21 # 21
'''''''''''''''''''''''''''''''''''''''''''''''''''''
(et # 41 # 41 # 41 # 41 # 41 # 41 #41 $59 $159
/t this price the va"ue of the stock has risen by Rs.3'. The put
e+pires out9of9the9money so it is worth"ess. /t Rs.*:' the stock
rose e+act"y enou#h to offset the cost of the put. The ma+imum
#ain is un"imite% because the stock can rise to any va"ue. &i#. '(.:
shows the combine% positions.
In many respects a protective put is "ike a co""ision insurance
po"icy on an automobi"e. / car is va"uab"e an% its owner suffers if
it is %ama#e% in an acci%ent. To protect a#ainst this potentia" for
"oss peop"e buy insurance fu""y e+pectin# to !"ose$ a"" the money
they pay for it.
6
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Fig. 10.3: Protective Put
Rs.
Profit or Loss
0 ST
590 631
41
PROTECTIVE C/22S
The previous section showe% how put options can provi%e a
he%#e a#ainst "osses from fa""in# security prices. The same thin#
can be %one with ca"" options to provi%e a he%#e a#ainst "osses
resu"tin# from risin# security prices.
Investors make money when they se"" an asset for more than
they pay for it.
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first -or openin# transaction is a sa"e0 the secon% -or c"osin#
transaction is a purchase. Short se""ers borrow shares from their
brokers se"" them hope to buy i%entica" shares in the future at a
"ower price an% then return the borrowe% shares. C"osin# out a
short position is ca""e% !coverin# the short position$. Short se""ers
make a profit if security prices %ec"ine.
)hen &i#. '(.' -2on# Stock Position is rotate% it becomes
the %ia#ram for a short stock position as in &i#. '(.6.
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)ax. loss is &nli*ite+
It is usefu" to compare the profit an% "oss %ia#ram for a shortsa"e with that of a "on# put position. In buyin# a put the ma+imum
"oss is the option premium yet the profit potentia" is very simi"ar
to the more risky strate#y of se""in# short. =any informe%
in%ivi%ua" investors who are bearish fin% the purchase of a put
vast"y preferab"e to a short sa"e of the stock for this reason. In
a%%ition buyin# a put re>uires "ess capita" than the hefty mar#in
re>uirements necessary to open a short account with a brokera#e
firm.
/"thou#h short positions are %an#erous for the typica"
investor professiona" tra%ers he%#e fun%s an% portfo"io mana#ers
fre>uent"y use them. On the e+chan#e f"oor members routine"y
tra%e usin# short sa"es meanin# that they c"ose out the short
position before the market c"oses. They avoi% mar#in re>uirements
because the position is not he"% overni#ht an% the risk of a%verse
price movements stemmin# from overni#ht news is e"iminate%.
Even for such we""9financia" peop"e positione% on the front "ines of
the market p"ace however the potentia" for "ar#e "osses is
noteworthy.
7
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One way to he%#e this risk is to a%% a "on# ca"" position to the
short stock position. Combinin# the profit an% "oss %ia#rams for a
"on# ca"" an% a short stock position resu"ts in a p"ot "ike &i#.'(.4.
This fi#ure comes from a short sa"e of )IP ? Rs.*'( an% the
purchase of an Oct Rs.7(( ca"" ? 64. The important feature of
&i#.'(.4 is that there is no potentia" for un"imite%"osses.
Fig.10.5:Short Stock P"us 2on# Ca""
Rs.
565
Profitor 00
loss0 ST
565
#135
The most that the short se""er can "ose in this situation is
Rs.':4. )hi"e the stock price can keep risin# -resu"tin# in
increasin# "osses on the short position for every rupee the stock
rises above Rs.5( the ca"" is worth a rupee more. @ere the rupee
#ain on the ca"" e+act"y cance"s the rupee "oss on the short position.
A
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Table 10.2: Profit an% 2oss )orksheet
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;( '(( :(( 4(( 4*4 7(( 5((
S,ort stock" Rs.610 610 510 310 110 $45 #90 #290
Lon! call Rs.00" Rs.45 #45 #45 #45 #45 #45 #45 $155
''''''''''''''''''''''''''''''''''''''''''
(et 565 465 265 65 0 #135 #135
/nother way to "ook at a situation "ike this is via a profit an%
"oss worksheet "ike that in Tab"e '(.3. This presents the same
information as the &i#.'(.4. /t a stock price of ,ero the investor
#ains Rs.*'( on the short stock position. The e+pirin# ca"" is out9
of9the money so it is worth"ess an% the premium of Rs.64 is "ost.
The net #ain is Rs.4*4. /t a stock price of Rs.7(( the option is at9
the9money0 at any hi#her stock price the option wi"" be in9the
money. The "oss in the short position an% the #ain in the "on# ca""
position e+act"y cance" at any stock price above Rs.7(( so the
ma+imum "oss on the combine% position occurs at Rs.7((.
'(.: COVEREB C/22S
Sometimes an investor owns stock an% writes a ca"" a#ainst it #ivin# someone e"se the
ri#ht to buy the shares. Such a ca"" is a covere% ca"". &i#. '(.* shows the profit or "oss possibi"ities
for an October **( covere% ca"" on )IP stock. This #raph incorporates the profits an% "oss
%ia#ram for a "on# position in the stock an% a short position in the October **( ca"". Option writer
5
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#ets the premium ri#ht away an% he keeps it no matter what happens to the stock price. &i#.'(.*
shows that if the stock price were to %ec"ine the ca"" premium cushions the "oss by Rs.4(. Even if
the stock were to %rop to ,ero he keeps the option premium so his net "oss is on"y Rs.4*(
-instea% of Rs.*'( as in &i#.'(.'
Fig.10.6: Covere% Ca""
Rs.
100 Profit
or loss0 ST 560 660
560
Sometimes an investor owns stock an% fears a market %ownturn. @e mi#ht consi%er usin#
ca""s to provi%e some cushion a#ainst "osses from fa""in# market.
In &i#.'(.* the Rs.4( premium receive% from writin# the ca"" means that no actua" cash
"oss occurs unti" )IP stock fa""s be"ow the current price -Rs.*'( minus the premium receive%
-Rs.4( or Rs.4*(. )hi"e this strate#y provi%es some %ownsi%e protection it is not a particu"ar"y
effective he%#e. In #enera" an in%ivi%ua" who nee% protection a#ainst fa""in# stock prices is better
off buyin# put options.
'(.6 SPRE/BS
'(
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/ sprea% tra%in# strate#y invo"ves simu"taneous purchase an% sa"e of option contracts in
which there is an anticipate% re"ationship between the assets un%er"yin# the options.
1. Bull Spreads
One of the most popu"ar types of sprea%s is a bu"" sprea%. 8u"" sprea% strate#y is use% by
a tra%er anticipatin# increase in the price of the un%er"yin# asset. It can be create% by buyin# a
ca"" option on a stock with a certain strike price an% se""in# a ca"" option on the same stock with a
hi#her strike price. 8oth the options have the same e+piration %ate. The strate#y is i""ustrate% in
&i#. '(.7.
Fig. 10.7 : 8u"" Sprea%
Rs.
Profit
orloss
0 -1 -2 ST
The profits from the two options positions taken separate"y are shown by the %ashe% "ines an%
from the who"e strate#y is in%icate% by the so"i% "ine.
Since a ca"" price a"ways %ecreases as the strike price increases the va"ue of the option
so"% is a"ways "ess than the va"ue of option bou#ht. / bu"" sprea% when create% from ca""s
therefore re>uires an initia" investment. Suppose that ' is the strike price of the ca"" option
bou#ht 3 is the strike price of the ca"" option so"% an% S T is the stock price on e+piration %ate of
the options. Tab"e '(.: shows the tota" payoff that wi"" be rea"ise% from a bu"" sprea% in %ifferent
circumstances.
''
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Tab"e A.: Payoff from a 8u"" Sprea%
Stock Paoff fro* Paoff fro* TotalPrice lon! call option s,ort call option
ST -2 ST / -1 -2 / ST -2/-1
-1ST-2 ST / -1 0 ST#-1
ST -1 0 0 0
If the stock price %oes we"" an% is #reater than 3 the payoff is the %ifference between
the two strike prices 3 9 ' . If the stock price on the e+piration %ate "ie between the two strike
prices the payoff is STD ' . If the stock price on the e+piration %ate is be"ow ' the payoff is
,ero. The net profit is ca"cu"ate% by subtractin# the initia" investment from the payoff.
/ bu"" sprea% strate#y "imits the tra%ers upsi%e as we"" as %ownsi%e risk. It means that
the tra%er has a ca"" option with a strike price e>ua" to 'an% has chosen to #ive some upsi%e
potentia" by se""in# a ca"" option with strike price 3 -3 F '. In return for #ivin# the upsi%e
potentia" the tra%er #ets the price of the option with strike price 3 .
Example
/ tra%er buys for Rs.'4 a ca"" option on a stock with a strike price of Rs.'4( an% se""s for Rs.4 a
ca"" with a strike price of Rs.'74 on the same stock. The payoff from the bu"" sprea% is Rs.34 if
the stock price is above Rs.'74 an% ,ero if it is be"ow Rs.'4(. If the stock price is between Rs.'4(
an% Rs.'74 the payoff is the amount by which the stock price e+cee%s Rs.'4(. The cost of the
strate#y is Rs.'4 D 4 G Rs.'(. The profit is therefore1
;;;;;;;;;;;;;;;;;;;;;;;;;''''''''''''''''''
Stock Price ST Profit loss''''''''''''''''''''''''''''''''''''''
ST 15 15
150ST 15 ST / 160
ST 150 #10''''''''''''''''''''''''''''''''''''''
'3
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8u"" sprea% can a"so be create% by buyin# a put with a "ow strike price an% se""in# a put
with a hi#h strike price as i""ustrate% in &i#.'(.A. Un"ike the bu"" sprea% create% from ca""s bu""
sprea%s create% from puts invo"ve a positive cash f"ow to the tra%er upfront an% a payoff that is
either ne#ative or ,ero.
Fig.10.8: 8u"" sprea% usin# put options
Rs.
Profit or loss
0 ST
-1 -2
2. Bear Spreads
/ bu"" sprea% strate#y is use% by a tra%er anticipatin# that the stock price swi"" %ec"ine.
2ike a bu"" sprea% a bear sprea% can be create% by buyin# a ca"" with one strike price an% se""in#
a ca"" with another strike price. @owever in the case of a bear sprea% the strike price of the
option purchase% is #reater than the strike price of the option so"%. In &i#.'(.5 the profit from the
sprea% is shown by the so"i% "ine.
Fig.10.9:8ear Sprea% create% usin# ca"" option
Rs.
':
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Profit orloss 0
-1 -2 ST
Tab"e '(.6 shows the payoff that wi"" be rea"ise% from a bear sprea% in %ifferent
circumstances. If the stock price is #reater than 3 the payoff is ne#ative. If the stock price is
"ess than ' the payoff is ,ero. If the stock price is between ' an% 3 the payoff is D -STD ' .
The profit is ca"cu"ate% by a%%in# the initia" cash f"ow to the payoff.
Table 10.4 Payofff from a 8ear sprea%
Stock Paoff fro* Paoff fro* TotalPrice lon! call option s,ort call option
ST -2 ST / -2 -1 / ST #-2/-1
-1ST-2 0 -1 # ST #ST#-1
ST -1 0 0 0
'(.4 CO=8I
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Suppose the investor buys a
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112
&rom the above e+amp"e it is c"ear that it is better to buy a stra%%"e when it is "ike"y that a
stock wi"" move sharp"y one way or the other. If the investor writes both a put an% a ca"" with the
same strikin# price the same e+piration %ate on the same un%er"yin# security he is short a
stra%%"e. The stra%%"e buyer wants the stock price to move si#nificant"y in one %irection or the
other. The stra%%"e writer wants Hust the opposite1 "itt"e movement in the stock price.
&i#.'(.'' is the %ia#ram of a short stra%%"e. The ma+imum #ain in this strate#y occurs
when the options finish at9the9money an% therefore e+pire worth"ess. 2osses are potentia""y
un"imite% on the upsi%e because the short ca"" is uncovere%.
2. Sra!gles
Stran#"es are simi"ar to stra%%"es e+cept the puts an% ca""s have %ifferent strikin# prices.
The motivation for buyin# a stran#"e is simi"ar to the motivation for buyin# a stra%%"e1 the
investor e+pects a sharp price movement either up or %own in the un%er"yin# stock.
/ stran#"e has two strikin# prices. )ith a long strangle the most popu"ar version
invo"ves buyin# a put with a "ower strikin# price than the ca""s strikin# price. 8y %oin# so the
profit an% "oss characteristics are simi"ar to those of the "on# stra%%"e but the ma+imum "oss is
sma""er.
Fig.10.11: Short stra%%"e
Rs.
Profit
orloss
112 162% 1%52 ST
140
'*
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162%Losses arepotentiall
&nli*ite+
&i#.'(.'3 Shows "on# stran#"e constructe% by buyin# an Infosys
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31
'(.* BE2T/ @EBI
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In this e+amp"e the %e"ta of the investors option position is (.*+-9'((( G 9*((. In other
wor%s the investor "oses *(( S. The %e"ta of the stock is by %efinition '.( an% the "on# position
in *(( shares has a %e"ta of J*((. The %e"ta of the investors overa"" position is therefore ,ero.
The %e"ta of the asset position offsets the %e"ta of the option position. The construction of a
risk"ess he%#e is sometimes referre% to as delta hedging. The %e"ta of a ca"" option is positive
whereas the %e"ta of a put option is ne#ative.
'(.7 @EBI
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Options un"ike forwar%s re>uire the payment of a premium upfront. That is the reason
why few treasurers use these instruments. Premiums are fat because the banks that write such
options are face% with a paucity of a"ternatives to he%#e their e+posure.
Capita" account convertibi"ity wi"" a""ow corporates an% banks to access the forei#n
e+chan#e an% money markets abroa% increasin# the he%#in# possibi"ities. &urther as a proper
rupee yie"% curve be#ins to take shape the banks can price such options more appropriate"y.
'(.A @EBI
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-3 =I8OR is A.34L
'(.34 5.6'L worst case
*9month 5.34 Time
=I8OR
A.6'L 8est case
A.34
It can be observe% that if =I8OR touches the peak of '(.34L the treasurer wi"" pay an
effective rate of 5.6'L.
'(.5 I22USTR/TIVE PRO82E=S
"r#blem 1
/n investor has purchase% shares of TT2 2t%. at Rs.66( per share. /nticipatin# %ec"ine in
the share price he bou#ht a put option on TT2 stock at Rs.64 per share with a strike price of
Rs.634 per share. Show the payoffs an% %raw a #raph in%icatin# the resu"ts if the possib"e price
ran#e of stock on e+piration %ate is
Rs.,ero Rs.:4( Rs.4*(
Rs. '4( Rs.66(
Rs. 3'( Rs.4((
So"ution1
Protective put payoffs
Stock price at option e+piration -Rs.
3'
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( '4( 3'( :4( 66( 4(( 4*(
2on# stock at Rs.66( 966( 935( 93:( 95( ( *( '*(
2on# Rs.634 put at Rs.64 J:A( ':( '7( :( 964 964 964
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Rs.64A Rs.74(
S#lui#!
Payoff )orksheet
Stock price at option e+piration -Rs.
( 3(( 64A 46( *4( 74(
2on# stock at Rs.66( 4'( :'( 43 9:( 9'6( 936(
2on# Rs.634 ca"" ? Rs.43 943 943 943 943 4A 9'4A
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-a )hat is the cost of the strate#y
-b )hat is the net payoff for each of the possib"e price ran#e
S#lui#!
-a The cost of the strate#y
Rs.63 D 3* G '*
-b Payoff from a 8u"" sprea%
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;Stock Payoff from Payoff from uote% on a stock e+chan#e as1
Option Price Stock Price
3A :4(
4( :'(
44 35(
Construct a suitab"e sprea% strate#y from the view point of a tra%er who is anticipatin#
%ec"ine in the stock price an% workout the payoffs from the strate#y if the possib"e price of stock
on the e+piration %ate is in the ran#e of1
Rs.34( Rs.:(( Rs.6((
Rs.'7( Rs.:6(
S#lui#!
36
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/ bear sprea% strate#y can be create% by buyin# ca"" with a hi#her strike price i.e.
Rs.:4( an% se""in# a ca"" with "ower strike price i.e. Rs.35( to "imit the %ownsi%e risk.
Payoff from a 8ear Sprea%
''''''''''''''''''''''''''''''''''''''''Stock Payoff from Payoff from
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35
Profit-"oss
35 40 450 ST
#5
"r#blem 6
/n option has a %e"ta of (.*. The option price is Rs.'( an% the stock price is Rs.'((. The
investor has written a ca"" option -i.e. ob"i#e% to se"" on 3((( shares. The investor wishes to
he%#e his position.
-a @ow many shares shou"% he buy to he%#e his position
-b Show the resu"t if stock price #oes up by Rs.'(.
-c If after few %ays the %e"ta increases to (.*4 how many a%%itiona" shares to be
bou#ht
S#lui#!
-a
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'(.'( SU==/RK
@e%#in# is the act of transferrin# unwante% risk to someone who is wi""in# to bear it.
Options can be use% to he%#e a#ainst "osses resu"tin# from a%verse price movements. /
protective put is a "on# put position he"% in conHunction with a "on# position in the un%er"yin#
stock. This is "ike an insurance po"icy on stock. The most common use of stock options by both
in%ivi%ua"s an% institutions is writin# covere% ca""s which is the writin# of ca"" options a#ainst
stock a"rea%y owne%. This strate#y has risk an% return characteristics simi"ar to that of writin# put
options which as a strate#y much "ess fre>uent"y use%.
Options sprea%s are strate#ies in which someone is simu"taneous"y "on# an% short options
of the same type but with %ifferent strike prices. The popu"ar sprea%s are bu"" sprea%s an% bear
sprea%s. / bu"" sprea% can be create% by buyin# a ca"" -put with a "ow strike price an% se""in# a
ca"" -put with a hi#her strike price. / bear sprea% can be create% by buyin# a ca"" -put with a
hi#h strike price an% se""in# a ca"" -put with a "ow strike price.
Option combinations are strate#ies in which investor is simu"taneous"y "on# or short
options of %ifferent types. The best9known combination is a stra%%"e which is a "on# ca"" position
an% a "on# put position on the same un%er"yin# asset where the two options have the same
strikin# price. / stra%%"e is appropriate when anticipate% price movements are hi#h in either
%irection in the un%er"yin# security. Stran#"es are "ike stra%%"es e+cept that the two options have
%ifferent strike prices.
=ost options tra%ers use more sophisticate% he%#in# schemes such as %e"ta he%#in#. This
provi%es protection from sma"" chan#es in the price of the un%er"yin# asset in the ne+t sma""
interva" of time. The %e"ta he%#e is the number of units of the stock investor shou"% ho"% for each
option shorte% in or%er to create a risk"ess he%#e.
37
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$e%ie& 'uesi#!s a!d "r#blems
'. )hat is meant by a protective put )hat position in ca"" options is e>uiva"ent to a protective
put I""ustrate with a suitab"e e+amp"e.
3. E+p"ain two ways in which a bear sprea% can be create%. I""ustrate with a suitab"e e+amp"e.
:. )hat is the %ifference between a stran#"e an% a stra%%"e I""ustrate with a suitab"e e+amp"e.
6. E+p"ain how an a##ressive bear sprea% can be create% usin# put options.
4. Befine %e"ta he%#in#. E+p"ain how option %e"ta is compute%.
*. Suppose that options on a stock with strike prices Rs.'4( an% Rs.'74 cost Rs.3( an% Rs.:4
respective"y. @ow can the options be use% to create -a a bu"" sprea% an% -b a bear sprea%
Construct a tab"e that shows the profit an% "oss payoffs for both sprea%s.
7. / ca"" option with a strike price of Rs.4(( costs Rs.3(. / put option with a strike price of
Rs.64( costs Rs.:(. E+p"ain how a stran#"e can be crate% from these two options. )hat is
the pattern of profits from the stran#"e
A. The treasurer of a "ar#e manufacturin# company p"ans to borrow Rs.'(( mi""ion after *
months for a perio% of * months. The current =I8OR is A.5(L. The companys banker is
>uotin# an in%ication rate of (.3( pc per annum. It is be"ieve% that the un%er"yin# si+9month
interest rate cou"% increase or %ecrease by ' pc in the si+9month perio% before the =I8OR is
re9fi+e%. Biscuss how the IRO he"ps mana#e interest rate e+posure in the present case. &or
this purpose you can assume that interest rate reaches its e+treme point in either case.
()'s
'. The strate#y use% to #uar% a#ainst the risk of "on# position in the security is ca""e%1
/. Protective ca"" 8. Protective put
C Covere% ca"" B. Stra%%"e
3. The strate#y use% to #uar% a#ainst the risk of short position in the security is ca""e%1
/. Protective ca"" 8. Protective put
C Covere% ca"" B. Stra%%"e
:. / tra%er has a short position in stock at Rs.*'( an% buys a ca"" on stock at Rs.64 with a strike
va"ue of Rs.7((. If the stock price at e+piration is Rs.4(( the net profit -"oss wi"" be1
/. Rs.''( 8. 9 Rs. *4
C 9 Rs.''( B. Rs. *4
6. In >uestion : above what is the break9even "eve" of e+piration price
3A
8/12/2019 Hedging Options
29/32
8/12/2019 Hedging Options
30/32
C Positive B. ero or ne#ative
':. / tra%er ho"%s a ca"" an% put option on the same stock -with a strike va"ue of Rs.'76( at
Rs.A( an% Rs.:3 respective"y. If the price of un%er"yin# stock is more than Rs.'76( at
e+piration the net pay off wi"" be
/. uestion '3 above the ma+imum "oss occurs when the stock price is1
/. Rs. '76( 8. Rs. ,ero
C Rs. 6((( B.
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31/32
-
8/12/2019 Hedging Options
32/32
:. Since the market is e+pecte% to be bearish a%amba wants to enter into a bearish money
sprea%. @ow can this be accomp"ishe% usin# put options an% what wou"% be the #ain
from this money sprea% transaction if the in%e+ is at 453( on =arch :( th
6. @ow can a%amba use a stra%%"e strate#y an% what wou"% be the #ain if the in%e+ is at
453( on =arch :(th.
:3