Hedging 2004

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Hedging risk withHedging risk with

DerivativesDerivatives Review of equity optionsReview of equity options

Review of nancial futuresReview of nancial futures

Using options and futures to hedgeUsing options and futures to hedge

portfolio riskportfolio risk

Introduction to Hedge FundsIntroduction to Hedge Funds

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Options -- Contract Options -- Contract

Calls and PutsCalls and Puts

Underlying Security !u"#er of Units$Underlying Security !u"#er of Units$

%&ercise or Strike Price%&ercise or Strike Price

%&piration date%&piration date

'ption Pre"iu"'ption Pre"iu"

("erican) %uropean) (sian) etc*("erican) %uropean) (sian) etc*

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Options -- MarketsOptions -- Markets

+ ,uyer - + Seller writer$ . + Contract+ ,uyer - + Seller writer$ . + Contract

%&a"ples of Price /uotations%&a"ples of Price /uotations

Pre"iu" . Intrinsic 0alue - 1i"e Pre"Pre"iu" . Intrinsic 0alue - 1i"e Pre"

'ptions availa#le on'ptions availa#le on %quities%quities

IndiciesIndicies

Foreign CurrenciesForeign Currencies

FuturesFutures

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Options -- BasicOptions -- Basic

StrategiesStrategies ,uy Call,uy Call

Sell write$ CallSell write$ Call

,uy Put,uy Put

Sell write$ PutSell write$ Put

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Options -- AdvancedOptions -- Advanced

StrategiesStrategies StraddleStraddle

Strips and StrapsStrips and Straps

0ertical Spreads0ertical Spreads

,ullish,ullish

,earish,earish

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Options - Determinants ofOptions - Determinants of

ValueValue 0alue of Underlying (sset0alue of Underlying (sset

%&ercise Price%&ercise Price

1i"e to %&piration 1i"e to %&piration

0'2(1I2I130'2(1I2I13

Interest RatesInterest Rates 4ividends4ividends

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Options -- Black ScholesOptions -- Black Scholes

Option ricing ModelOption ricing Model C = SN(d1) - Xe-rTN(d2)

ln(S/X) +(r+

2/2)Td1 = ---------------------------

1/2

d2 = d1 - 1/2

Put-Call Parity: P = C + Xe-rT - S

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!utures Contract !utures Contract

(gree"ent to "ake sell$ or take #uy$ delivery of(gree"ent to "ake sell$ or take #uy$ delivery ofa prespecied quantity of an asset at an agreeda prespecied quantity of an asset at an agreedupon price at a specic future date*upon price at a specic future date*

e&* S5P 677 Inde& Futures8e&* S5P 677 Inde& Futures8 Price8 ++9:*+7; 4elilvery "onth8 <unePrice8 ++9:*+7; 4elilvery "onth8 <une

,uyer agrees to purchase a portfolio representing the S5P 677 or its,uyer agrees to purchase a portfolio representing the S5P 677 or itscash equivalent$ for =++9:*+7 & 967 . =9>+)696 on 1hursday prior tocash equivalent$ for =++9:*+7 & 967 . =9>+)696 on 1hursday prior to??rdrd Friday in <une* ,uyer is locking in the purchase price for theFriday in <une* ,uyer is locking in the purchase price for theportfolio*$portfolio*$

Seller agrees to deliver the portfolio descri#ed a#ove*Seller agrees to deliver the portfolio descri#ed a#ove*

!ote8 since this is a cash settled contract) if the price was +++:*+7 on!ote8 since this is a cash settled contract) if the price was +++:*+7 onthe delivery date) the #uyer would pay the seller =9)677 . +7 & 967$*the delivery date) the #uyer would pay the seller =9)677 . +7 & 967$*If the price was ++?:*+7) the seller would pay the #uyer =9)677If the price was ++?:*+7) the seller would pay the #uyer =9)677

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!utures Contract" Marking!utures Contract" Marking

to Market to Market @arking to "arket8@arking to "arket8

Price of Futures contract is reset every dayPrice of Futures contract is reset every day

AainsB2osses versus previous day areAainsB2osses versus previous day areposted to #uyer and seller "argin accountsposted to #uyer and seller "argin accounts

Futures . a #undle of consecutive +dayFutures . a #undle of consecutive +day

forward contractsforward contracts

If futures held to e&piration) eDectiveIf futures held to e&piration) eDectivedelivery price is sa"e as when contractdelivery price is sa"e as when contract

initiatedinitiated

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!utures Contract" Marking!utures Contract" Marking

to Market e#ample $C%to Market e#ample $C%contract&contract&

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Speculators often sell inde& futuresSpeculators often sell inde& futures

when they e&pect the underlyingwhen they e&pect the underlying

inde& to depreciate) and vice versa*inde& to depreciate) and vice versa*

'nde# !utures Market 'nde# !utures Market

1. Contract to sell

S&P @ 1126.1($281,525) onJune 17.

April

2. !u" S&P @ 11#6.1

($276,525) on spotar%et an eli'er@ 1126.1

June 17

. Proit * $5,###.

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Inde& futures "ay #e sold #yInde& futures "ay #e sold #y

investors to hedge risk associatedinvestors to hedge risk associated

with securities held*with securities held*


1.Contract to sell

S&P @ 1126.1($281,525) on June17.

April

2. +ar%et alls to

11#6.1.ain *$5###

June 17

. ain osets(appro-.) loss o$5### on securitiesel

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@ost inde& futures contracts are@ost inde& futures contracts are

closed out #efore their settle"entclosed out #efore their settle"ent

dates EE$*dates EE$* ,rokers who fulll orders to #uy or,rokers who fulll orders to #uy or

sell futures contracts earn asell futures contracts earn a

transaction or #rokerage fee in thetransaction or #rokerage fee in thefor" of the #idBask spread*for" of the #idBask spread*


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Hedging with DerivativesHedging with Derivatives

,asic option strategies,asic option strategies

Covered callCovered call

Protective putProtective put

Synthetic shortSynthetic short

,asic futures strategies,asic futures strategies

Using interest rate futures to reduceUsing interest rate futures to reduce

riskrisk

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Covered CallCovered Call

Sell call on stock you own* 2ong stock) shortSell call on stock you own* 2ong stock) shortcall$call$

Aood8Aood8 (s value of stock falls) loss is partially oDset #y(s value of stock falls) loss is partially oDset #y

pre"iu" received on calls sold*pre"iu" received on calls sold*

%ssentially costless since hedge generates a cash inGow%ssentially costless since hedge generates a cash inGow

,ad8,ad8 @a&i"u" inGow fro" call . pre"iu"; Hedge is less@a&i"u" inGow fro" call . pre"iu"; Hedge is less

eDective for large drop in stock priceeDective for large drop in stock price

If stock price rises) call will #e e&ercised; InvestorIf stock price rises) call will #e e&ercised; Investortransfers gains on stock to holder of call*transfers gains on stock to holder of call*

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rotective ut rotective ut

,uy put on stock you own* 2ong stock) long,uy put on stock you own* 2ong stock) longput$put$

Aood8Aood8 (s value of stock falls) loss is partially oDset #y gain(s value of stock falls) loss is partially oDset #y gain

in value of put* Aain fro" put continues to grow asin value of put* Aain fro" put continues to grow asstock price falls*stock price falls*

If stock price rises) "a&i"u" loss on put . pre"iu";If stock price rises) "a&i"u" loss on put . pre"iu";

Investor keeps all stock gains less &ed put pre"iu"*Investor keeps all stock gains less &ed put pre"iu"*

,ad8,ad8 @ore e&pensive to hedge with put@ore e&pensive to hedge with put

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S(nthetic Short S(nthetic Short

Sell call and #uy put on stock you own* 2ong stock)Sell call and #uy put on stock you own* 2ong stock)short call) long put$short call) long put$

Aood8Aood8 (s value of stock falls) loss is oDset #y gain in value of put*(s value of stock falls) loss is oDset #y gain in value of put*

Aain fro" put continues to grow as stock price falls*Aain fro" put continues to grow as stock price falls* If stock price rises) gain is oDset #y loss on call* 2oss fro"If stock price rises) gain is oDset #y loss on call* 2oss fro"

call continues to grow as stock price rises*call continues to grow as stock price rises* 0ery eDective hedging device0ery eDective hedging device Can #e selfnancing pre"iu" received on put sold oDsetsCan #e selfnancing pre"iu" received on put sold oDsets

pre"iu" paid on call purchased$pre"iu" paid on call purchased$

,ad8,ad8 'ften "ore e&pensive than si"ply shorting the stock itself*'ften "ore e&pensive than si"ply shorting the stock itself*

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Delta Hedging withDelta Hedging with

OptionsOptions Call 4elta .Call 4elta . ∆∆CC.. δδCBCBδδSS

Fro" ,lackScholes "odel)Fro" ,lackScholes "odel)∆∆CC . !d. !d++$$

%&*8%&*8 If S.*E) J.6) r.+*:)If S.*E) J.6) r.+*:) σσ .?>*).?>*)

t.7*+6>E yrs*t.7*+6>E yrs*

1hen) C . *7 and !d 1hen) C . *7 and !d++$ . 7*6+E$ . 7*6+E

If S increases #y =+) C increases #y =7*6+EIf S increases #y =+) C increases #y =7*6+E

Hedge Ratio . H . +BHedge Ratio . H . +B∆∆CC = 1/0.5197 = 1.924= 1/0.5197 = 1.924

Sell +*E9 calls per share of stock held to hedgeKSell +*E9 calls per share of stock held to hedgeK

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)#ample of Call Hedge * Held to)#ample of Call Hedge * Held to

)#piration+ , share stock)#piration+ , share stock

position position

IBM Profit S Profit C Combined

90 15,510 -20,140 -4,630

85 10,510 -10,640 -130

80 5,510 -1,140 4,370

75 510 8,360 8,870

74.49 0 8,360 8,360

70 -4,490 8,360 3,870

65 -9,490 8,360 -1,130

60 -14,490 8,360 -6,130

55 -19,490 8,360 -11,130

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Delta Hedging - utsDelta Hedging - uts

Put 4elta .Put 4elta . ∆∆PP.. δδPBPBδδSS Fro" ,lackScholes "odel and PutCall Parity)Fro" ,lackScholes "odel and PutCall Parity)

∆∆PP.. ∆∆CC L + .!dL + .!d++$ +$ +

%&*8%&*8 If S.*E) J.6) r.+*:)If S.*E) J.6) r.+*:) σσ .?>*).?>*)

t.7*+6>E yrs*t.7*+6>E yrs*

1hen) C . *7) P . *+) !d 1hen) C . *7) P . *+) !d++$ . 7*6+E)$ . 7*6+E)

and !dand !d++$ + . 7*>7?$ + . 7*>7?

If S increases #y =+) P decreases #y =7*>7?If S increases #y =+) P decreases #y =7*>7?

Hedge Ratio . H . +BHedge Ratio . H . +B∆ = 1/0.4803 = 2.082∆ = 1/0.4803 = 2.082

,uy 9*7>9 puts per share of stock held to hedgeK,uy 9*7>9 puts per share of stock held to hedgeK

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)#ample of ut Hedge * Held to)#ample of ut Hedge * Held to

)#piration+ , share stock)#piration+ , share stock

position positionIBM Profit S Profit P Combined

90 15,510 -9,891 5,619

85 10,510 -9,891 61980 5,510 -9,891 -4,381

75 510 -9,891 -9,381

74.49 0 -8,820 -8,820

70 -4,490 609 -3,88165 -9,490 11,109 1,619

60 -14,490 21,609 7,119

55 -19,490 32,109 12,619

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OptionsOptions 4elta changes over ti"eK4elta changes over ti"eK

S changesS changes

1i"e declines 1i"e declines

'ther factors r)'ther factors r) σσ$ "ay change$ "ay change

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.rue Delta Hedging * Ad/ust hedge when S.rue Delta Hedging * Ad/ust hedge when S

changeschanges

Scenarios + 5 98Scenarios + 5 98

I,@ stock drops #y =+ to =?*E ..M 2oss of =+777I,@ stock drops #y =+ to =?*E ..M 2oss of =+777

Call options also drop #y =7*6+E ..M Aain ofCall options also drop #y =7*6+E ..M Aain of

=+7?*E=+7?*E ..M!et change =?*E..M!et change =?*E

I,@ stock rises #y =+ to =6*E ..M Aain of =+777I,@ stock rises #y =+ to =6*E ..M Aain of =+777 Call options also rise #y =7*6+E? ..M 2oss ofCall options also rise #y =7*6+E? ..M 2oss of

=+7?*E=+7?*E

..M !et change =?*E$..M !et change =?*E$

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.rue Delta Hedging * Ad/ust hedge when t.rue Delta Hedging * Ad/ust hedge when t

changeschanges

Scenario ?8Scenario ?8 'ne week passes) I,@ stock at =+*E ..M 2oss of =?777'ne week passes) I,@ stock at =+*E ..M 2oss of =?777 Call options now worth =9*? ..M Aain of =?+?Call options now worth =9*? ..M Aain of =?+?

..M!et change =+?..M!et change =+?

!ew call delta . 7*79E!ew call delta . 7*79E !ew hedge ratio . +B7*79E . 9*>9 ..M Sell 6 "ore!ew hedge ratio . +B7*79E . 9*>9 ..M Sell 6 "ore

contractsKcontractsK

Scenario 8Scenario 8 'ne week passes) I,@ stock at =*E ..M Aain of =?777'ne week passes) I,@ stock at =*E ..M Aain of =?777 Call options now worth =6*>9 ..M 2oss of =9:E>Call options now worth =6*>9 ..M 2oss of =9:E>

..M !et change =?79$..M !et change =?79$ !ew call delta . 7*:9?>!ew call delta . 7*:9?> !ew hedge ratio . +B7*:9?> . +*:7? ..M ,uy ? contractsK!ew hedge ratio . +B7*:9?> . +*:7? ..M ,uy ? contractsK

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.rue Delta Hedging * Ad/ust hedge when S.rue Delta Hedging * Ad/ust hedge when S

changeschanges

Scenarios + 5 98Scenarios + 5 98

I,@ stock drops #y =+ to =?*E ..M 2oss of =+777I,@ stock drops #y =+ to =?*E ..M 2oss of =+777 Put options also rise #y =7*>7? ..M Aain of =+77>*:?Put options also rise #y =7*>7? ..M Aain of =+77>*:?

..M!et change =>*:? ..M!et change =>*:?

I,@ stock rises #y =+ to =6*E ..M Aain of =+777I,@ stock rises #y =+ to =6*E ..M Aain of =+777

Put options also fall #y =7*>7? ..M 2oss of =+77>*:?Put options also fall #y =7*>7? ..M 2oss of =+77>*:?

..M !et change =>*:?$..M !et change =>*:?$

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.rue Delta Hedging * Ad/ust hedge when t.rue Delta Hedging * Ad/ust hedge when t

changeschanges

Scenario ?8Scenario ?8 'ne week passes) I,@ stock at =+*E ..M 2oss of =?777'ne week passes) I,@ stock at =+*E ..M 2oss of =?777 Put options now worth =:*7: ..M Aain of =9>?6Put options now worth =:*7: ..M Aain of =9>?6

..M!et change =+:6$..M!et change =+:6$

!ew put delta . 7*79> L + . 7*6E9!ew put delta . 7*79> L + . 7*6E9 !ew hedge ratio . +B7*6E9 . +*: ..M Sell contractsK!ew hedge ratio . +B7*6E9 . +*: ..M Sell contractsK

Scenario 8Scenario 8 'ne week passes) I,@ stock at =*E ..M Aain of =?777'ne week passes) I,@ stock at =*E ..M Aain of =?777

Put options now worth =?*+6 ..M 2oss of =?9:Put options now worth =?*+6 ..M 2oss of =?9:..M !et change =9:$..M !et change =9:$ !ew put delta . 7*:9?> L + . 7*?:9!ew put delta . 7*:9?> L + . 7*?:9 !ew hedge ratio . +B7*?:9 . 9*:6> ..M ,uy 6 "ore!ew hedge ratio . +B7*?:9 . 9*:6> ..M ,uy 6 "ore

contractsKcontractsK

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optionsoptions 4elta represents response of call or put$ price with4elta represents response of call or put$ price with

change in the stock pricechange in the stock price

4elta changes as stock price) ti"e to e&piration)4elta changes as stock price) ti"e to e&piration)interest rates) volatility changeinterest rates) volatility change

It is too e&pensive to hedge individual stockIt is too e&pensive to hedge individual stockpositions with "atching options* It is "orepositions with "atching options* It is "oreco""on to hedge a portfolio with inde& optionsco""on to hedge a portfolio with inde& optionscross hedging$cross hedging$

@ost "anagers "onitor delta itself to decide when@ost "anagers "onitor delta itself to decide whento re#alance*to re#alance*

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A .rue rotective ut A .rue rotective ut

Puts can #e used to #uild a GoorPuts can #e used to #uild a Goor

under the value of a long positionunder the value of a long position

,uy + put per long share,uy + put per long share %&*8 2ong +777 shares of I,@ at%&*8 2ong +777 shares of I,@ at

=*E=*E

,uy +777 puts at =*+,uy +777 puts at =*+ PutsPuts guaranteeguarantee a value of =6 per sharea value of =6 per share

1his is 1his is insuranceinsurance) not a hedgeK) not a hedgeK

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A .rue rotective ut A .rue rotective ut

IBM Profit S Profit P Combined

90 15,510 -4,710 10,800

85 10,510 -4,710 5,80080 5,510 -4,710 800

75 510 -4,710 -4,200

74.49 0 -4,200 -4,200

70 -4,490 290 -4,20065 -9,490 5,290 -4,200

60 -14,490 10,290 -4,200

55 -19,490 15,290 -4,200

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Hedging with !utures $e#ample from Ma(Hedging with !utures $e#ample from Ma(

0,&0,&

1here are futures on the S5P677* Suppose 1here are futures on the S5P677* SupposeI have a portfolio that is currently worthI have a portfolio that is currently worth=+)++):9* 1he portfolio has a #eta of +*?*=+)++):9* 1he portfolio has a #eta of +*?*

<une S5P677 futures are at +?7*7 <une S5P677 futures are at +?7*7 ..M contract is worth 677 & +?7*7 ...M contract is worth 677 & +?7*7 .

=+6)?67=+6)?67

Hedge ratio .Hedge ratio . 0alue of portfolio B 0alue of Futures contract$Portfolio ,eta$0alue of portfolio B 0alue of Futures contract$Portfolio ,eta$

. +)++):9B+6)?67$+*?$ . 9*7?+ ..M Sell 9. +)++):9B+6)?67$+*?$ . 9*7?+ ..M Sell 9Contracts KContracts K

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Hedging with !utures $e#ample from Ma(Hedging with !utures $e#ample from Ma(

0,&0,&

S&P Sot in !"ne # $%n'e Port. in !"ne # $%n'e Profit Portfo(io Profit )"t"re* Combined

1573.75 10# 1,262,969 13.0# 145,297 -143,050 +2,247

1502.25 5# 1,190,321 6.5# 72,649 -71,550 1,099

1430.7 0# 1,117,672 0.0# 0 0 0

1359.15 -5# 1,045,023 -6.5# -72,649 71,550 -1,099

1287.65 -10# 972,375 -13.0# -145,297 143,050 -2,247

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Ad/usting S(stematic 1isk with Ad/usting S(stematic 1isk with

!utures!utures

P@ "ay choose to adNust syste"atic e&posureP@ "ay choose to adNust syste"atic e&posureup or down to reGectup or down to reGect investor desiresinvestor desires

e&pectations of "arket "ove"entse&pectations of "arket "ove"ents

(#out inde& futures8(#out inde& futures8 Represents contract to "akeBtake delivery of aRepresents contract to "akeBtake delivery of a

portfolio represented #y the inde&portfolio represented #y the inde&

Since inde& itself "ay #e noninvesta#le) "ost inde&Since inde& itself "ay #e noninvesta#le) "ost inde&futures contracts are cashsettledfutures contracts are cashsettled e&a"ple8e&a"ple8

S5P677 futures C@% contract value . 967 & inde&S5P677 futures C@% contract value . 967 & inde& Initial "argin8 =:O for spec) =9*6O for hedgers*Initial "argin8 =:O for spec) =9*6O for hedgers*

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!utures!utures

I have an =++ "illion stock portfolio withI have an =++ "illion stock portfolio with ββ.+*76* I want.+*76* I wantto increaseto increase ββ to +*9*to +*9*

0alue of Futures . +?+*67 & 967 . =?9>):960alue of Futures . +?+*67 & 967 . =?9>):96

ββf f . +*7*. +*7* 1arget 1arget ββ . contri#ution fro" portfolio - contri#ution. contri#ution fro" portfolio - contri#ution

fro" futuresfro" futures +*9 . +*7$+*76$ - F & ?9>):96$B=++)777)777Q+*7$+*9 . +*7$+*76$ - F & ?9>):96$B=++)777)777Q+*7$ F . F . ββ 1 1 ssββss$0$0ssB0B0FF$$ F . 6*79 .M #uy 6 contractsF . 6*79 .M #uy 6 contracts hat have we donehat have we done

Used futures contracts to leverage holdings and increaseUsed futures contracts to leverage holdings and increasee&posure to "arket riske&posure to "arket risk

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!utures!utures

Suppose targetSuppose target ββ . *E7. *E7

7*E7 . +*7$+*76$ - F & ?9>):96$B7*E7 . +*7$+*76$ - F & ?9>):96$B=++)777)777Q+*7$=++)777)777Q+*7$

F . *E7 +*76$??*9>$+*7$ . 6*79F . *E7 +*76$??*9>$+*7$ . 6*79

contracts sell$contracts sell$

e have shorted futures to reduce syste"atice have shorted futures to reduce syste"atic

e&posure*e&posure*

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Hedging with 'nterest 1ateHedging with 'nterest 1ate

!utures!utures How do you reduce duration for a #ond portfolioHow do you reduce duration for a #ond portfolio

Sell high 4) #uy low 4Sell high 4) #uy low 4 Sell #onds) #uy 1#illsSell #onds) #uy 1#ills Sell interest rate futuresSell interest rate futures

Interest rate futures8 agree"ent to "akeBtake delivery ofInterest rate futures8 agree"ent to "akeBtake delivery ofa &ed inco"e asset on a particular date for an agreeda &ed inco"e asset on a particular date for an agreedupon priceupon price

e&8 Sept 1#ond futures contracte&8 Sept 1#ond futures contract

=+77O F0 US 1reas #onds with +6years to "aturity and=+77O F0 US 1reas #onds with +6years to "aturity and> coupon what if they donTt e&ist$> coupon what if they donTt e&ist$

Price8 EE9 . EE 9B?9 of =+77)777 . =EE>)?*67Price8 EE9 . EE 9B?9 of =+77)777 . =EE>)?*67 1ick . =?+*96$ 4 . >*: years1ick . =?+*96$ 4 . >*: years

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Hedging with 'nterest 1ateHedging with 'nterest 1ate

!utures!utures I own an =++)777)777 face value portfolio of high gradeI own an =++)777)777 face value portfolio of high grade

US corporate #onds with an aggregate value of +7+7>US corporate #onds with an aggregate value of +7+7>or =++)+?)677$ and a duration of * years*or =++)+?)677$ and a duration of * years*

I e&pect rates to rise* How can I i""unie "y portfolioI e&pect rates to rise* How can I i""unie "y portfolio

1arget 4 . contri#ution of #ond port - contri#ution of 1arget 4 . contri#ution of #ond port - contri#ution offut*fut*

7 . +*7$*$ - F & EE>)?*67$B++)+?)677Q>*:$7 . +*7$*$ - F & EE>)?*67$B++)+?)677Q>*:$ F . 7*7 +*7$*$$++)+?)677BEE>)?*67$B>*:F . 7*7 +*7$*$$++)+?)677BEE>)?*67$B>*: F . E*E contracts .M short +7 1#ond futures contractsF . E*E contracts .M short +7 1#ond futures contracts

1his is the weighted average duration approach 1his is the weighted average duration approach

Hedging 2004

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