option wk3

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Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.1

Introduction

Chapter 1


The Nature of Derivatives

A derivative is an instrument whose value depends on the values of other more basic underlying variables

2


Examples of Derivatives

• Futures Contracts• Forward Contracts• Swaps• Options


Ways Derivatives are Used

To hedge risksTo speculate (take a view on the future direction of the market)To lock in an arbitrage profitTo change the nature of a liabilityTo change the nature of an investment without incurring the costs of selling one portfolio and buying another

3


Futures Contracts

A futures contract is an agreement to buy or sell an asset at a certain time in the future for a certain priceBy contrast in a spot contract there is an agreement to buy or sell the asset immediately (or within a very short period of time)


Exchanges Trading Futures

Chicago Board of TradeChicago Mercantile ExchangeEuronext Eurex BM&F (Sao Paulo, Brazil)and many more (see list at end of book)

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Futures Price

The futures prices for a particular contract is the price at which you agree to buy or sellIt is determined by supply and demand in the same way as a spot price


Electronic Trading

Traditionally futures contracts have been traded using the open outcry system where traders physically meet on the floor of the exchangeIncreasingly this is being replaced by electronic trading where a computer matches buyers and sellers

5


Examples of Futures Contracts

Agreement to:buy 100 oz. of gold @ US$600/oz. in December (NYMEX) sell £62,500 @ 1.9800 US$/£ in March (CME)sell 1,000 bbl. of oil @ US$65/bbl. in April (NYMEX)


Terminology

The party that has agreed to buy has a long positionThe party that has agreed to sell has a short position

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Example

January: an investor enters into a long futures contract on COMEX to buy 100 oz of gold @ $600 in April

April: the price of gold $615 per oz

What is the investor’s profit?


Over-the Counter Markets

The over-the counter market is an important alternative to exchangesIt is a telephone and computer-linked network of dealers who do not physically meetTrades are usually between financial institutions, corporate treasurers, and fund managers

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Size of OTC and Exchange Markets(Figure 1.2, Page 4)

Source: Bank for International Settlements. Chart shows total principal amounts for OTC market and value of underlying assets for exchange market


Forward Contracts

Forward contracts are similar to futures except that they trade in the over-the-counter marketForward contracts are popular on currencies and interest rates

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Foreign Exchange Quotes for GBP on July 14, 2006 (See page 5)

Bid OfferSpot 1.8360 1.8364

1-month forward 1.8372 1.8377




Options

A call option is an option to buy a certain asset by a certain date for a certain price (the strike price)A put option is an option to sell a certain asset by a certain date for a certain price (the strike price)

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American vs European Options

An American option can be exercised at any time during its lifeA European option can be exercised only at maturity


Intel Option Prices (Sept 12, 2006; Stock Price=19.56); See page 6

Calls PutsStrike Price Oct Jan Apr Oct Jan Apr

($) 2006 2007 2007 2006 2007 2007$15.00 4.650 4.950 5.150 0.025 0.150 0.275$17.50 2.300 2.775 3.150 0.125 0.475 0.725$20.00 0.575 1.175 1.650 0.875 1.375 1.700$22.50 0.075 0.375 0.725 2.950 3.100 3.300$25.00 0.025 0.125 0.275 5.450 5.450 5.450

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Exchanges Trading Options

Chicago Board Options ExchangeAmerican Stock ExchangePhiladelphia Stock ExchangeInternational Securities ExchangeEurex (Europe)and many more (see list at end of book)


Options vs Futures/Forwards

A futures/forward contract gives the holder the obligation to buy or sell at a certain priceAn option gives the holder the right to buy or sell at a certain price

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Three Reasons for Trading Derivatives:Hedging, Speculation, and Arbitrage

Hedge funds trade derivatives for all three reasons (See Business Snapshot 1.1)

When a trader has a mandate to use derivatives for hedging or arbitrage, but then switches to speculation, large losses can result. (See Barings, Business Snapshot 1.2)



Hedging Examples (Example 1.1 and 1.2, page 11)

A US company will pay £10 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contractAn investor owns 1,000 Microsoft shares currently worth $28 per share. A two-month put with a strike price of $27.50 costs $1. The investor decides to hedge by buying 10 contracts

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Value of Microsoft Shares with and without Hedging (Fig 1.4, page 12)

20,000

25,000

30,000

35,000

40,000

20 25 30 35 40

Stock Price ($)

Value of Holding ($)

No HedgingHedging


Speculation Example (pages 14)

An investor with $2,000 to invest feels that Amazon.com’s stock price will increase over the next 2 months. The current stock price is $20 and the price of a 2-month call option with a strike of $22.50 is $1What are the alternative strategies?

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Arbitrage Example (pages 15-16)

A stock price is quoted as £100 in London and $182 in New YorkThe current exchange rate is 1.8500What is the arbitrage opportunity?


1. Gold: An Arbitrage Opportunity?

Suppose that:The spot price of gold is US$600The quoted 1-year futures price of gold is US$650The 1-year US$ interest rate is 5% per annumNo income or storage costs for gold

Is there an arbitrage opportunity?

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2. Gold: Another Arbitrage Opportunity?

Suppose that:The spot price of gold is US$600The quoted 1-year futures price of gold is US$590The 1-year US$ interest rate is 5% per annumNo income or storage costs for gold



The Futures Price of Gold

If the spot price of gold is S & the futures price is for a contract deliverable in T years is F, then

F = S (1+r )T

where r is the 1-year (domestic currency) risk-free rate of interest.In our examples, S=600, T=1, and r=0.05 so that

F = 600(1+0.05) = 630

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1. Oil: An Arbitrage Opportunity?

Suppose that:The spot price of oil is US$70The quoted 1-year futures price of oil is US$80The 1-year US$ interest rate is 5% per annumThe storage costs of oil are 2% per annum



2. Oil: Another Arbitrage Opportunity?

Suppose that:The spot price of oil is US$70The quoted 1-year futures price of oil is US$65The 1-year US$ interest rate is 5% per annumThe storage costs of oil are 2% per annum


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Hedging Strategies Using Futures

Chapter 3


Long & Short Hedges

A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the priceA short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price

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Arguments in Favor of Hedging

Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables


Arguments against Hedging

Shareholders are usually well diversified and can make their own hedging decisionsIt may increase risk to hedge when competitors do notExplaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult

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Convergence of Futures to Spot(Hedge initiated at time t1 and closed out at time t2)

Time

Spot Price

FuturesPrice

t1 t2


Basis Risk

Basis is the difference between spot & futuresBasis risk arises because of the uncertainty about the basis when the hedge is closed out

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Long Hedge

Suppose thatF1 : Initial Futures PriceF2 : Final Futures PriceS2 : Final Asset Price

You hedge the future purchase of an asset by entering into a long futures contractCost of Asset=S2 – (F2 – F1) = F1 + Basis


Short Hedge

Suppose thatF1 : Initial Futures PriceF2 : Final Futures PriceS2 : Final Asset Price

You hedge the future sale of an asset by entering into a short futures contractPrice Realized=S2+ (F1 – F2) = F1 + Basis

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Choice of Contract

Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedgeWhen there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. There are then 2 components to basis


Optimal Hedge Ratio

Proportion of the exposure that should optimally be hedged is

where σS is the standard deviation of ΔS, the change in the spot price during the hedging period, σF is the standard deviation of ΔF, the change in the futures price during the hedging periodρ is the coefficient of correlation between ΔS and ΔF.

h S

F

= ρσσ

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Hedging Using Index Futures(Page 62)

To hedge the risk in a portfolio the number of contracts that should be shorted is

where P is the value of the portfolio, β is its beta, and F is the current value of one futures (=futures price times contract size)

FP

β


Reasons for Hedging an Equity Portfolio

Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.)Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outpeform the market.)

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Example

Futures price of S&P 500 is 1,000Size of portfolio is $5 millionBeta of portfolio is 1.5One contract is on $250 times the index

What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?


Changing Beta

What position is necessary to reduce the beta of the portfolio to 0.75?What position is necessary to increase the beta of the portfolio to 2.0?

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Rolling The Hedge Forward

We can use a series of futures contracts to increase the life of a hedgeEach time we switch from 1 futures contract to another we incur a type of basis risk

Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Global Edition.Copyright © John C. Hull 2010

Futures Options

Chapter 16

46

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Background

The commodity Futures Trading Commission authorized the trading of options on futures on an experimental basis in 1982.Permanent trading wa approved in 1987.The popularity of the contract with investors has grown very fast.

Expiration Months

Futures options are referred to by the delivery month of the underlying futures contract---not by the expiration month of the option.Most futures options are American.The expiration date of a futures option contract is usually on, or a few days before, the earliest delivery date of the underlying futures contract.

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Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010

Mechanics of Call Futures Options

When a call futures option is exercised the holder acquires

1. A long position in the futures2. A cash amount equal to the excess of the

futures price at previous settlement over the strike price

49


Mechanics of Put Futures Option

When a put futures option is exercised the holder acquires

1. A short position in the futures2. A cash amount equal to the excess of the strike

price over the futures price at previous settlement

50

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The Payoffs

If the futures position is closed out immediately:

Payoff from call = (previous Settlement price – K)+(F –previous settlement price – F)=F-K

Payoff from put = K – F

where F is futures price at time of exercise

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Example 16.1 Mechanics of call futures options

An investor buys a July call futures option contract on gold. The contract size is 100 ounces. The strike price is 900.The investor exercises when the July gold futures price is 940 and the most recent settlement price is 938.The outcome

1. The investor receives a cash amount equal to (938-900)*100=$3,800

2. The investor receives a long futures contract.3. The investor closes out the long futures contract

immediately for a gain of (940-938)*100=$2004. Total payoff = $4,000

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Example 16.2 Mechanics of put futures options

An investor buys a September put futures option contract on corn. The contract size is 5,000 bushels. The strike price is 300 cents.The investor exercises when the September corn futures price is 280 and the most recent settlement price is 279.The outcome

1. The investor receives a cash amount equal to (3.00-2.79)*5,000=$1,050

2. The investor receives a short futures contract.3. The investor closes out the short futures contract

immediately for a loss of (2.79-2.80)*5,000= -$504. Total payoff = $1,000


Potential Advantages of FuturesOptions over Spot Options

Futures contract may be easier to trade than underlying assetExercise of the option does not lead to delivery of the underlying asset Futures options and futures usually trade in adjacent pits at exchangeFutures options may entail lower transactions costs

54

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European Spot Options and European Futures Options

The payoff from a European call option with strike price K on the spot price of an asset is

Max (ST - K,0), where ST is the spot price at the option’s maturity.The payoff from a European all option with the same strike price on the futures price of the asset is

Max (FT - K, 0), where FT is the futures price at the option’s maturity.

If the futures contract matures at the same time as the option, then FT = ST, and the two options are equivalent.

55


Put-Call Parity for European Futures Options (Equation 16.1, page 347)

Consider the following two portfolios:A. a European call futures option plus Ke-rT of cashB. a European put futures option plus long futures

contract plus cash equal to F0e-rT

56

29



57

FT > K FT =< K

Portfolio A Call futures option FT − K 0

Ke-rT K K

Total FT KPortfolio B Put futures option

Long futures F0e-rT

0(FT − F0 )

F0

K− FT(FT − F0 )

F0

Total FT K


Since portfolio A and portfolio B worth the same at time T, they should worth the same today.The value of portfolio A today is

c+Ke-rT

The value of portfolio B today is p+F0 e-rT

Hencec+Ke-rT=p+F0 e-rT or c+(K-F0)e-rT=p

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Bounds for European Futures Options

c+(K-F0)e-rT=p

Because the price of a put or a call cannot be negative, it follows from the above equation that

c > (F0 – K)e-rT

p > (K – F0)e-rT

59


The Bounds for American Futures Options

F0 e-rT – K < C – P < F0 – Ke-rT

Because American futures options can be exercised at any time, we must have the following

C > (F0 – K)

P > (K – F0)

60

31


Futures Price = $33Option Price = $4

Futures Price = $28Option Price = $0

Futures price = $30Option Price=?

Binomial Tree Example

A 1-month call option on futures has a strike price of 29.

61


Consider the Portfolio: long Δ futuresshort 1 call option

Portfolio is riskless when 3Δ – 4 = –2Δ or Δ= 0.8

3Δ – 4

-2Δ

Setting Up a Riskless Portfolio

62

32


Valuing the Portfolio( Risk-Free Rate is 6% )

The riskless portfolio is: long 0.8 futuresshort 1 call option

The value of the portfolio in 1 month is –1.6

The value of the portfolio today is –1.6e – 0.06/12 = –1.592

63


Valuing the Option

The portfolio that is long 0.8 futuresshort 1 option

is worth –1.592The value of the futures is zeroThe value of the option must therefore be 1.592

64

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Generalization of Binomial Tree Example (Figure 16.2, page 349)

A derivative lasts for time T and is dependent on a futures

F0dƒd

F0uƒuF0

ƒ

65


Generalization(continued)

Consider the portfolio that is long Δ futures and short 1 derivative

The portfolio is riskless when

Δ =−−

ƒ u dfF u F d0 0

F0u Δ − F0 Δ – ƒu

F0d Δ− F0Δ – ƒd

66

34



Value of the portfolio at time T is F0u Δ –F0Δ – ƒu

Value of portfolio today is – ƒHence

ƒ = – [F0u Δ –F0Δ – ƒu]e-rT

67



Substituting for Δ we obtainƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where

dudp

−−

=1

68

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Numerical Example

Consider previously example. u=33/30=1.1, d= 28/30=0.9333, 4=0.06, T=1/12,

fu = 4, and fd = 0.

69

[ ] 592.106.044.0

4.09333.01.1

9333.01

12/106.0 =×+×=

=−

−=

×−ef

p


Growth Rates For Futures Prices

A futures contract requires no initial investment.In a risk-neutral world the expected profit from holding a position in an investment that costs zero to set up must be zero.The expected growth rate of the futures price is therefore zero.The futures price can therefore be treated like a stock paying a dividend yield of r.This is consistent with the results we have presented so far (put-call parity, bounds, binomial trees).

70

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Valuing European Futures Options

We can use the formula for an option on a stock paying a continuous yield

Set S0 = current futures price (F0)Set q = domestic risk-free rate (r )

Setting q = r ensures that the expected growth of F in a risk-neutral world is zero

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A European Call and Put on a stock paying a dividend yield at rate q

The formulas is

TdT

TqrKSd

TTqrKSd

dNeSdNKep

dNKedNeScqTrT

rTqT

σσ

σ

σσ

−=−−+

=

+−+=

−−−=

−=−−

−−

10

2

01

102

210

)2/2()/ln(

)2/2()/ln( where

)( )(

)( )(

72

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Using Forward Prices

Define F0 as the forward price of the index for a contract with maturity T.

TdT

TKSd

TTKSd

dNeFdNKep

dNKedNeFc

eSF

rTrT

rTrT

Tqr

σσ

σ

σσ

−=−

=

+=

−−−=

−=

=

−−

−−

−

10

2

01

102

210

)(00

2/2)/ln(

2/2)/ln( where

)( )(

)( )(

73


Black’s Model Compare equations on p.28 and equations on p.29, one clue could be derived.The two sets of equations are identical when we set q = r.Fischer Black was the first to show that European futures options can be valued using equations on p.28 with q = r and S0 replaced by F0.

74

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Black’s Model

The formulas for European options on futures are known as Black’s model

[ ][ ]

TdT

TKFd

TTKF

d

dNFdNKep

dNKdNFecrT

rT

σ−=σ

σ−=

σ

σ+=

−−−=

−=−

−

10

2

01

102

210

2/2)/ln(

2/2)/ln(

)()(

)()(

where

75


How Black’s Model is Used in Practice

European futures options and spot options are equivalent when future contract matures at the same time as the option.This enables Black’s model to be used to value a European option on the spot price of an asset

76

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Valuation of a European futures option

Consider a European put futures option on crude oil. The time to the option’s maturity is four months, the current futures prices is $60, the exercise price is $60, the risk-free interest rate is 9% per annum, and the volatility of the futures price is 25% per annum. The put price is given by

77


Valuation of a European futures option (continued)

The put price p is $3.35.

78

( ) 35.34712.0605288.060

5288.0)(4712.0)(

07216.012/425.007216.0

07216.012/425.0

)12/4()2/25.0()60/60ln()2/()/ln(

12/409.02

1

12

220

1

=×−×=

=−=−

−=−=−=

=×+

=+

=

×−ep

dNdN

TddT

TKFd

σσ

σ

40

Valuing a spot option using futures prices

Consider a 6-month European call option on spot gold6-month futures price is 930, 6-month risk-free rate is 5%, strike price is 900, and volatility of futures price is 20%Value of option is given by Black’s model with F0=930, K=900, r=0.05, T=0.5, and s=0.2It is 44.19

Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 79

Valuing a spot option using futures prices (continued)

Fundamentals of Futures and Options Markets, 7th Ed, Ch 16, Copyright © John C. Hull 2010 80

( ) 19.44$)1611.0(900)3026.0(930

1611.05.02.03026.0

3026.05.02.0

5.0)2/2.0()900/930ln()2/()/ln(

5.005.0

12

220

1

=×−×=

=−=−=

=×+

=+

=

×− NNec

TddT

TKFd

σσ

σ

41


American Futures Option Prices vs American Spot Option Prices

If futures prices are higher than spot prices (normal market), an American call on futures is worth more than a similar American call on spot. An American put on futures is worth less than a similar American put on spot.When futures prices are lower than spot prices (inverted market) the reverse is true.

81

Futures Style Options (page 353-54)

A futures-style option is a futures contract on the option payoffSome exchanges trade these in preference to regular futures optionsThe futures price for a call futures-style option is

The futures price for a put futures-style option is


)()( 210 dKNdNF −

)()( 102 dNFdKN −−−

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Put-Call Parity Results: Summary

:Futures

:exchange Foreign

:Indices

:Stock Paying dNondividen

rTrT

TrrT

qTrT

rT

eFpeKc

eSpeKc

eSpeKc

SpeKc

f

−−

−−

−−

−

+=+

+=+

+=+

+=+

0

0

0

0

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Summary of Key Results from Chapters 15 and 16

We can treat stock indices, currencies, & futures like a stock paying a continuous dividend yield of q

For stock indices, q = average dividend yield on the index over the option lifeFor currencies, q = rƒ

For futures, q = r84

43


Mechanics of Options Markets

Chapter 8


Types of Options

A call is an option to buyA put is an option to sellA European option can be exercised only at the end of its lifeAn American option can be exercised at any time

44


Option Positions

Long callLong putShort callShort put


Long Call (Figure 8.1, Page 186)

Profit from buying one European call option: option price = $5, strike price = $100.

30

20

10

0-5

70 80 90 100

110 120 130

Profit ($)

Terminalstock price ($)

45


Short Call (Figure 8.3, page 188)

Profit from writing one European call option: option price = $5, strike price = $100

-30

-20

-10

05

70 80 90 100

110 120 130

Profit ($)



Long Put (Figure 8.2, page 188)

Profit from buying a European put option: option price = $7, strike price = $70

30

20

10

0-7

70605040 80 90 100

Profit ($)


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Short Put (Figure 8.4, page 189)

Profit from writing a European put option: option price = $7, strike price = $70

-30

-20

-10

70

70

605040

80 90 100

Profit ($)Terminal

stock price ($)


Payoffs from OptionsWhat is the Option Position in Each Case?

K = Strike price, ST = Price of asset at maturity

Payoff Payoff

ST STKK

Payoff Payoff

ST STKK

47


Assets UnderlyingExchange-Traded OptionsPage 190-191

StocksForeign CurrencyStock IndicesFutures


Specification ofExchange-Traded Options

Expiration dateStrike priceEuropean or AmericanCall or Put (option class)

48


Terminology

Moneyness :At-the-money optionIn-the-money optionOut-of-the-money option


Terminology(continued)

Option classOption seriesIntrinsic valueTime value

49


Dividends & Stock Splits (Page 193-194)

Suppose you own N options with a strike price of K :

No adjustments are made to the option terms for cash dividendsWhen there is an n-for-m stock split,

the strike price is reduced to mK/nthe no. of options is increased to nN/m

Stock dividends are handled in a manner similar to stock splits


Dividends & Stock Splits(continued)

Consider a call option to buy 100 shares for $20/shareHow should terms be adjusted:

for a 2-for-1 stock split?for a 5% stock dividend?

50


Market Makers

Most exchanges use market makers to facilitate options tradingA market maker quotes both bid and ask prices when requestedThe market maker does not know whether the individual requesting the quotes wants to buy or sell


Margins (Page 197-198)

Margins are required when options are soldFor example when a naked call option is written the margin is the greater of:1 A total of 100% of the proceeds of the sale plus

20% of the underlying share price less the amount (if any) by which the option is out of the money

2 A total of 100% of the proceeds of the sale plus 10% of the underlying share price

51


Warrants

Warrants are options that are issued (or written) by a corporation or a financial institutionThe number of warrants outstanding is determined by the size of the original issue & changes only when they are exercised or when they expire


Warrants(continued)

Warrants are traded in the same way as stocks The issuer settles up with the holder when a warrant is exercisedWhen call warrants are issued by a corporation on its own stock, exercise will lead to new treasury stock being issued

52


Executive Stock Options

Option issued by a company to executivesWhen the option is exercised the company issues more stock Usually at-the-money when issued


Executive Stock Options continued

They become vested after a period of time (usually 1 to 4 years)They cannot be soldThey often last for as long as 10 or 15 yearsAccounting standards are changing to require the expensing of executive stock options

53


Convertible Bonds

Convertible bonds are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined exchange ratio


Convertible Bonds(continued)

Very often a convertible is callableThe call provision is a way in which the issuer can force conversion at a time earlier than the holder might otherwise choose

option wk3

Documents

cash amount

recent settlement

previous settlement

underlying

european call

final asset

month call

underlying