11/20/2015 Nattawoot Koowattanatianchai 1 Derivatives Analysis - - PowerPoint PPT Presentation

11/20/2015 Nattawoot Koowattanatianchai 1

11/20/2015 Nattawoot Koowattanatianchai 2

Derivatives Analysis

Nattawoot Koowattanatianchai

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Lecture 3

Option contracts

Discussion topics

 Option contracts

 Basic definitions and illustration

• f option contracts

 Types of options  Principles of option pricing  Discrete-time option pricing: The

Binomial Model

 Continuous-time option pricing:

The Black-Scholes-Merton model

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Readings

 CFA Program Curriculum 2015 -

Level II – Volume 6: Derivatives and Portfolio Management.

 Reading 49

 Don M. Chance and Robert

Brooks, An Introduction to Derivatives and Risk Management, 9th Edition, 2013, Thomson.

 Chapters 3-5

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Option contracts

 Definition

 A contract that gives its

holder the right, not the

• bligation, to buy or sell an

underlying asset at a fixed price by a certain time in the future. The party granting the right is called

• ption seller (or the short or
• ption writer)

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Option contracts

 Parties in an option contract

 The long (also called option buyer or option

holder) holds the right to buy/sell the underlying.

 The short (also called option seller or option

writer) grants the right to the long party.

 Call

 An option granting the right to buy the underlying.

 Put

 An option granting the right to sell the underlying.

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Option contracts

 Option price

 To obtain the right to buy/sell

the underlying, the option buyer pays the seller a sum of money, commonly referred to as the option price (or the

• ption premium or just the

premium).

 The money is paid when the

• ption contract is initiated.

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Basic characteristics

 Exercise price (also called strike price,

striking price, or strike)

 It is the fixed price at which the option holder can

buy or sell the underlying.

 Exercise (or exercising) the option

 Use of the right to buy or sell the underlying.

 Expiration date

 When the expiration date arrives, an option that is

not exercised simply expires.

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Basic characteristics

 Exercising a call

 The buyer pays the exercise

price and receives either the underlying or an equivalent cash settlement.

 The seller, who receives the

exercise price from the buyer and delivers the underlying, or alternatively, pays an equivalent cash settlement.

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Basic characteristics

 Exercising a put

 The buyer delivers the stock

and receives the exercise price

• r an equivalent cash

settlement.

 The seller receives the

underlying and must pay the exercise price or the equivalent cash settlement.

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Basic characteristics

 Cash settlement

 The option holder exercising a call receives the

difference between the market value of the underlying and the exercise price from the seller in cash.

 The option holder exercising a put receives the

difference between the exercise price and the market value of the underlying in cash.

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Basic characteristics

 European-style exercise

 The option can be

exercised only on its expiration day.

 American-style exercise

 The option can be

exercised on any day through the expiration day.

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Basic characteristics

 Exchange-listed, standardized options

 The exchange specifies a designated number of

units of the underlying, and other terms of an

• ption contract (e.g., expiration dates, exercise

prices, minimum price quotation unit, exercising style, settlement style, and contract size), with the exception of price that will be negotiated by two parties.

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Basic characteristics

 Exchange-listed, standardized options

 Standardized options are traded on exchanges.

 Some exchanges have pit trading, whereby parties meet

in the pit and arrange a transaction.

 Some exchanges use electronic trading, in which

transactions are conducted through computers.

 Transactions are guaranteed by the clearinghouse, i.e.,

the clearing house will step in and fulfill the obligation if the seller reneges at exercise.

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Basic characteristics

 Exchange-listed, standardized options

 The majority of trading occurs in options that are

close to being at-the-money. Options that are far in-the-money or far out-of-the-money, called deep-in-the-money and deep-out-of-the-money

• ptions, are usually not very actively traded and

are often not even listed for trading.

 Most exchange-listed options have fairly short-

term expirations, usually the current month, the next month, and perhaps one or two other months.

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Basic characteristics

 Exchange-listed, standardized options

 Defaults are rare.

 When the buyer purchases the option, the premium,

which one might think would go to the seller, instead goes to the clearinghouse, which maintains it in the margin account. In addition, the seller must post some margin money, which is based on a formula that reflects whether the seller has a position that hedges the risk and whether the option is in- or out-of-the-money. Although defaults are rare, the clearinghouse has always been successful in paying when the seller defaults.

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Basic characteristics

 Exchange-listed, standardized options

 On the expiration day

 In-the-money options are always exercised, assuming

they are in-the-money by more than the transaction cost

• f buying or selling the underlying or arranging a cash

settlement when exercising.

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Basic characteristics

 Over-the-counter options

 An over-the-counter option is created off of an

exchange by any two parties who agree to trade.

 The buyer is subject to the possibility of the writer

defaulting, but not the other way around.

 Brokers in the market attempt to match buyers of

• ptions with sellers, thereby earning a

commission.

 Dealers offer to take either side of the option

transaction, usually laying off (hedging) the risk in another transaction.

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Basic characteristics

 Over-the-counter options

 Over-the-counter options markets are essentially

• unregulated. There are no guarantees that the

seller will perform; hence, the buyer faces credit

• risk. As such, option buyers must scrutinize

sellers’ credit risk and may require some risk reduction measures, such as collateral.

 Contracts can be customized on all terms, such

as price, exercise price, time to expiration, deification of the underlying, settlement or delivery, size of the contract, etc.

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Examples of options

 Consider some calls and puts on SUNW. The

date is 13 June and SUNW is selling for $16.25. Here are closing prices of four American options:

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Exer erci cise se price ice July ly ca calls lls October

• ber

calls ls July ly puts ts October

• ber

puts 15.00 2.35 3.30 0.90 1.85 17.50 1.00 2.15 2.15 3.20

Examples of options

 July 15 call

 This option permits the holder to buy SUNW at a

price of $15 a share any time through 20 July.

 To obtain this option, one would pay a price of

$2.35.

 The seller received $2.35 on 13 June and must be

ready to sell SUNW to the buyer for $15 during the period through 20 July.

 The option holder has no reason to exercise the

• ption right now.

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Examples of options

 July 17.50 call

 This call is cheaper than the July 15 call.  The cheaper price comes from the fact that July

17.50 call is less likely to be exercised, because the stock has a higher hurdle to clear.

 A buyer is not willing to pay as much and a seller

is more willing to take less for an option that is less likely to be exercised.

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Examples of options

 October calls

 For any exercise price, October calls would be

more expensive than the July calls because they allow a longer period for the stock to make the move that the buyer wants.

 October options are more likely to be exercised

than July options; therefore, a buyer would be willing to pay more and the seller would demand more for the October calls.

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Examples of options

 October 17.50 put

 This option costs $3.20 and allows the buyer to

sell SUNW at a price of $17.50 any time up through 18 October.

 The buyer has no reason to exercise the option

right now, because it would mean he would be buying the option for $3.20 and selling a stock worth $16.25 for $17.50. In effect, the buyer would part with $19.45 and obtain only $17.50.

 The buyer of a put obviously must be anticipating

that the stock will fall before the expiration day.

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Examples of options

 October 15 put

 For any exercise price, October calls would be

more expensive than the July calls because they allow a longer period for the stock to make the move that the buyer wants.

 October options are more likely to be exercised

than July options; therefore, a buyer would be willing to pay more and the seller would demand more for the October calls.

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Examples of options

 If at expiration, the stock is at 16.

 Calls with an exercise price of 15 would be

exercised.

 Actual delivery: The seller delivers the stock and the

buyer pays the seller; through the clearinghouse, $15 per share.

 Cash settlement: The seller pays the buyer $1.

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Concept of moneyness

 Moneyness of options

 In-the-money options are those in which

exercising the option would produce a cash inflow that exceeds the cash outflow.

 Calls are in-the-money when the exercise price exceeds

the value of the underlying.

 Puts are in-the-money when the exercise price exceeds

the value of the underlying.

 At-the-money options are those in which

exercising the option would produce a zero cash flow.

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Concept of moneyness

 Moneyness of options

 Out-of-the-money options are those in which

exercising the option would produce a cash

• utflow that exceeds the cash inflow.

 At-the-money options can effectively be viewed as

• ut-of-the-money options because their exercise

would not bring in more money than is paid out.

 One would not necessarily exercise an in-the-

money option, but one would never exercise an

• ut-of-the-money option.

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Types of financial options

Financial options Stock options Index options Bond options Interest rate

• ptions

Currency

• ptions

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Stock and index options

 Stock options

 Options on individual stocks, also called equity

• ptions, are among the most popular.

 Exchanged-listed options are available on most

widely traded stocks, and an option on any stock can potentially be created on the over-the-counter market.

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Stock and index options

 Index options

 An index option is an option on a stock index. A

stock index is just an artificial portfolio of stocks.

 Example

 Consider options on the S&P 500 Index, which trade on

the Chicago Board Options Exchange and have a designated index contract multiplier of 250. On 13 June

• f a given year, the S&P 500 closed at 1,241.60. A call
• ption with an exercise price of $1,250 expiring on 20

July was selling for $28. The option is European style and settles in cash.

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Stock and index options

 Index options

 Example

 The underlying, the S&P 500, is treated as though it

were a share of stock worth $1,241.60, which can be bought, using the call option, for $1,250 on 20 July.

 At expiration, if the option is in-the-money, the buyer

exercises it and the writer pays the buyer the $250 contract multiplier times the difference between the index value at expiration and $1,250.

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Bond options

 Options on bonds are primarily traded in the

• ver-the-counter markets. They are almost

always options on government bonds.

 Example

 Consider a US T-bond maturing in 27 years. The

bond has a coupon of 5.50%, a yield of 5.75%, and is selling for $0.9659 per $1 par.

 An over-the-counter call option on this bond with

an exercise price of $0.98 per $1 par covers $5 million face value of bonds.

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Bond options

 Example

 Suppose the buyer exercises the call when the

bond price is at $0.995.

 The option is in-the-money by $0.995 - $0.98 = $0.015

per $1 par. The buyer would assume the risk of the seller defaulting.

 Delivery: The seller would deliver $5 million face value

• f bonds, which would be worth $5,000,000($0.995) =

$4,975,000. The buyer would pay $5,000,000($0.98) = $4,900,000.

 Cash settlement: The seller pays the buyer

0.015($5,000,000) = $75,000.

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Interest rate options

 Definition

 An option in which the underlying is an interest

• rate. It has an exercise rate (or strike rate), which

is expressed on an order of magnitude of an interest rate. At expiration, the option payoff is based on the difference between the underlying rate in the market and the exercise rate. Whereas an FRA is a commitment to make one interest payment and receive another at a future date, an interest rate option is the right to make one interest payment and receive another.

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Interest rate options

 Interest rate call

 An option in which the holder has the right to

make a known interest payment and receive an unknown interest payment.

 The underlying is the unknown interest rate.  If the unknown underlying rate turns out to be

higher than the exercise rate at expiration, the

• ption is in-the-money and is exercised;
• therwise, the option simply expires.

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Interest rate options

 Interest rate put

 An option in which the holder has the right to

make an unknown interest payment and receive a known interest payment.

 If the unknown underlying rate turns out to be

lower than the exercise rate at expiration, the

• ption is in-the-money and is exercised;
• therwise, the option simply expires.

 Interest rate options are settled in cash and most

tend to be European style.

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Interest rate options

 Example

 Consider an option expiring in 90 days on 180-day

• Libor. The option buyer chooses an exercise rate
• f 5.5% and a notional principal of $10 million.

 On the expiration day, suppose that 180-day Libor

is 6%.

 The call option is in-the-money. The payoff to the holder

• f the option is:

 This money is not paid at expiration but will be paid 180

days later

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Interest rate options

 Payoff of an interest rate call  Payoff of an interest rate put

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Interest rate options

 Hedging using interest rate options

 Borrowers often use interest rate call options to

hedge the risk of rising rates on floating-rate loans.

 Lenders often use interest rate put options to

hedge the risk of falling rates on floating-rate loans.

 Since floating-rate loans usually involve multiple

interest payments, both borrowers and lenders need options expiring on each rate reset date to hedge.

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Interest rate options

 Hedging using interest rate options

 A combination of interest rate calls is referred to

as “an interest rate cap” or sometimes just “a cap”.

 A series of call options on an interest rate, with each

• ption expiring at the date on which the floating loan rate

will be reset, and with each option having the same exercise rate.

 Each component call option is called “a caplet”.  The price of an interest rate cap is the sum of the prices

• f the options that make up the cap.

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Interest rate options

 Hedging using interest rate options

 A combination of interest rate puts is referred to

as “an interest rate floor” or sometimes just “a floor”.

 A series of put options on an interest rate, with each

• ption expiring at the date on which the floating loan rate

will be reset, and with each option having the same exercise rate.

 Each component call option is called “a floorlet”  The price of an interest rate floor is the sum of the prices

• f the options that make up the floor.

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Interest rate options

 Hedging using interest rate options

 A combination of caps and floors is called “an

interest rate collar”.

 A combination of a long cap and a short floor or a short

cap and a long floor.

 Example

 Consider a borrower in a floating rate loan who wants to

hedge the risk of rising interest rates but is concerned about the requirement that this hedge must have a cash

• utlay up front: the option premium.

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Interest rate options

 Hedging using interest rate options

 Example

 A collar, which adds a short floor to a long cap, is a way

• f reducing and even eliminating the up-front cost of the
• cap. The sale of the floor brings in cash that reduces the

cost of the cap.

 It is possible to set the exercise rates such that the price

received for the sale of the floor precisely offsets the price paid for the cap, thereby completely eliminating the up-front cost.

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Interest rate options

 Hedging using interest rate options

 Example

 Although the cap allows the borrower to be paid when

rates are high, the sale of the floor requires the borrower to pay the counterparty when rates are low. Thus, the cost of protection against rising rates is the loss of the advantage of falling rates.

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Currency options

 Definition

 A currency option allows the holder to buy (if a

call) or sell (if a put) an underlying currency at a fixed exercise rate, expressed as an exchange rate.

 Hedging using currency options

 Many companies, knowing that they will need to

convert a currency X at a future date into a currency Y, will buy a call option on currency Y specified in terms of currency X.

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Currency options

 Example

 A US comp-any will be needing €50 million for an

expansion project in three months. Thus, it will be buying euros and is exposed to the risk of the euro rising against the dollar. Even though it has that concern, it would also like to benefit if the euro weakens against the dollar. Thus, it might buy a call option on the euro.

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Currency options

 Example

 Suppose this call specifies an exercise rate of

$0.90. So the company pays cash up front for the right to buy €50 million at a rate of $0.90 per euro.

 If the option expires with the euro above 0.90, the

company buys euros at $0.90 and avoid any additional cost over $0.90.

 If the option expires with the euro below $0.90,

the company does not exercise the option and buys euros at the market rate.

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Currency options

 Example

 Alternative outcomes

 Dollar expires below €1.1111, that is €1 > $0.90  Company sells $45 million (€50 million × $0.90) at €1.1111,

equivalent to buying €50 million.

 Dollar expires above €1.1111, that is, €1 < $0.90  Company sells sufficient dollars to buy €50 million at the

market rate.

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Currency options

 Example

 The call on the euro can be viewed as a put on

the dollar.

 A call to buy €50 million at an exercise price of $0.90 is

also a put to sell €50 million × $0.90 = $45 million at an exercise price of 1/$0.90, or €1.1111.

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Options on futures

 Definition

 Options in which the underlying is a futures

contract.

 A call option on futures gives the holder the right to enter

into a long futures contract at a fixed futures price.

 A put option on futures gives the holder the right to enter

into a short futures contract at a fixed futures price.

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Options on futures

 Example

 Consider an option on the Eurodollar futures

contract trading at the Chicago Mercantile

• Exchange. On 13 June of a particular year, an
• ption expiring on 13 July was based on the July

Eurodollar futures contract. That futures contract expires on 16 July, a few days after the option

• expires. The call option with exercise price of

95.75 had a price of $4.60. The underlying futures price was 96.21 (based on a discount rate of 100 – 96.21 = 3.79). The contract size is $1 million.

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Options on futures

 Example

 The buyer of this call option on a futures would

pay 0.046($1,000,000) = $46,000 and would

• btain the right to buy the July futures contract at

a price of 95.75. On the contract initial date, the

• ption was in the money by 96.21 – 95.75 = 0.46

per $100 face value.

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Options on futures

 Example

 Suppose that on the expiration date, the futures

price is 96.00.

 The holder of the call would exercise it and obtain a long

futures position at a price of 95.75. The price of the underlying futures is 96.00, so the margin account is immediately marked to market with a credit of 0.25 or $625. The party on the short side of the contract is immediately set up with a short futures contract at the price of 95.75. That party will be charged the $625 gain that the long has made.

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Options on futures

 Example

 Suppose that on the expiration date, the futures

price is 96.00.

 If the contract is in-the-money by 96 – 95.85 = $0.25 per

$100 par, it is in-the-money by 025/100 = 0.0025, or 0.25% of the face value. Because the face value is $1 million, the contract is in the money by (0.0025)(90/360)($1,000,000) = $625.

 Alternatively, the futures price at 95.75 is 1 –

(0.0425)(90/360) = $0.989375 per $1 par, or $989,375. At 96, the futures price is 1 – (0.04)(90/360) = $0.99 per $1 par, or $990,000. The difference is $625.

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Options on futures

 Example

 Suppose that on the expiration date, the futures

price is 96.00.

 So, exercising this option is like entering into a futures

contract at a price of $989,375 and having the price immediately go to $990,000, a gain of $625. The call holder must deposit money to meet the Eurodollar futures margin, but the exercise of the option gives him $625. In other words, assuming he meets the minimum initial margin requirement, he is immediately credited with $625 more.

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Other types of options

 Commodity options

 Options in which the asset underlying the option is

a commodity, such as oil, gold, wheat, or soybeans.

 There are exchange-traded as well as over-the-

counter commodity options. Over-the-counter

• ptions on oil are widely used.

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Other types of options

 Options on electricity, various

sources of energy, and weather

 Electricity is not considered a

storable asset because it is produced and almost immediately consumed, but it is nonetheless an asset and certainly has a volatile price. Consequently, it is ideally suited for options and other derivatives trading.

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Other types of options

 Options on electricity, various sources of

energy, and weather

 Weather is hardly an asset but simply a random

factor that exerts an enormous influence on economic activity. The need to hedge against and speculate on the weather has created a market in which measures of weather activity, such as economic losses from storms or average temperature or rainfall, are structured into a derivative instrument.

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Other types of options

 Options on electricity, various sources of

energy, and weather

 Example

 Consider a company that generates considerable

revenue from outdoor summer activities, provided that it does not rain. Obviously a certain amount of rain will

• ccur, but the more rain, the greater the losses for the
• company. It could buy a call option on the amount of

rainfall with the exercise price stated as a quantity of

• rainfall. If actual rainfall exceeds the exercise price, the

company exercises the option and receives the amount

• f money related to the excess of the rainfall amount
• ver the exercise price.

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Other types of options

 Real options

 A real option is an option associated with the

flexibility inherent in capital investment projects. For example, companies may invest in new projects that have the option to defer the full investment, expand, or contract the project at a later date, or even terminate the project. These

• ptions do not trade in markets the same way as

financial and commodity options, and they must be evaluated much more carefully.

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Principles of option pricing

 Pricing and valuation of options

 Recall that the value of a contract is what

someone must pay to buy into it or what someone would receive to sell out of it. A forward or futures contract has a zero value at the start, but the value turns positive or negative as prices or rates change.

 The forward or futures price is the price that the

parties agree will be paid on the future date to buy and sell the underlying.

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Principles of option pricing

 Pricing and valuation of options

 An option has a positive value at the start. The

buyer must pay money and the seller receives money to initiate the contract. Prior to expiration, the option always has a positive value to the buyer and negative value to the seller.

 In a forward or futures contract, the two parties

agree on the fixed price the buyer will pay the

• seller. This fixed price is set such that the buyer

and seller do not exchange any money.

11/20/2015 Nattawoot Koowattanatianchai 65

Principles of option pricing

 Pricing and valuation of options

 The corresponding fixed price at which a call

holder can buy the underlying or a put holder can sell the underlying is the exercise price. It, too, is negotiated between buyer and seller but still results in the buyer paying the seller money up front in the form of an option premium or price.

 Forward or futures price corresponds more to the

exercise price of an option. In the option market, the option price is the option value. We do not distinguish between these two words.

11/20/2015 Nattawoot Koowattanatianchai 66

Principles of option pricing

 Notation

 S0, ST = price of the underlying asset at time 0

(today) and time T (expiration)

 X = exercise price  r = risk-free rate  T = time to expiration, equal to number of days to

expiration divided by 365

11/20/2015 Nattawoot Koowattanatianchai 67

Principles of option pricing

 Notation

 c0, cT = price of European call today and at

expiration

 C0, CT = price of American call today and at

expiration

 p0, pT = price of European put today and at

expiration

 P0, PT = price of American put today and at

expiration

11/20/2015 Nattawoot Koowattanatianchai 68

Principles of option pricing

 Payoff values

 At expiration, a call option is worth either zero or

the difference between the underlying price and the exercise price, whichever is greater:

 cT = Max(0, ST – X)  CT = Max(0, ST – X)  At expiration, a European option and an American

• ption have the same payoff because they are

equivalent instrument at that point.

11/20/2015 Nattawoot Koowattanatianchai 69

Principles of option pricing

 Payoff values

 At expiration, a put option is worth either zero or

the difference between the exercise price and the underlying price, whichever is greater:

 pT = Max(0, X – ST)  PT = Max(0, X – ST)

 Arbitrage opportunities exist if these equalities do

not hold.

11/20/2015 Nattawoot Koowattanatianchai 70

Principles of option pricing

 Payoff values

 The value for Max(0, ST – X) calls or Max(0, X –

ST) for puts is called the option’s “intrinsic value”

• r “exercise value”.

 Intrinsic value is what the option is worth to exercise it

based on current conditions.

 Prior to expiration, an option will normally sell for more

than its intrinsic value.

 The difference between the market price of the option

and its intrinsic value is called its “time value” or “speculative value”. At expiration, the time value is zero.

11/20/2015 Nattawoot Koowattanatianchai 71

Principles of option pricing

 Example

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Op Option n Value ue Exam ample ple (X = 5 50) ST = 55 ST = 45 European call cT = Max(0, ST – X) 5 American call CT = Max(0, ST – X) 5 European put pT = Max(0, X – ST) 5 American put PT = Max(0, X – ST) 5

Principles of option pricing

 Example

11/20/2015 Nattawoot Koowattanatianchai 73

Principles of option pricing

 Example

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Principles of option pricing

 Boundary conditions

 The minimum value of any option is zero.

 c0 ≥ 0, C0 ≥ 0  p0 ≥ 0, P0 ≥ 0  No option can sell for less than zero, for in that case the

writer would have to pay the buyer.

 The maximum value of a call is the current value

• f the underlying.

 c0 ≤ S0, C0 ≤ S0  It would not make sense to pay more for the right to buy

the underlying than the value of the underlying itself.

11/20/2015 Nattawoot Koowattanatianchai 75

Principles of option pricing

 Boundary conditions

 The maximum value of a European put is the

present value of the exercise price. The maximum value of an American put is the exercise price.

 p0 ≤ X(1+r)T, P0 ≤ X  The best possible outcome for the put holder is that the

underlying goes to a value of zero. Then the put holder could sell a worthless asset for X. For an American put, the holder could sell it immediately and capture a value

• f X. For a European put, the holder would have to wait

until expiration; consequently, we must discount X from the expiration day to the present.

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Principles of option pricing

 Boundary conditions

 Example: Minimum and maximum values of

• ptions

11/20/2015 Nattawoot Koowattanatianchai 77

Option n Minimum um maximum um Exam ampl ple (S (S0=52 =52, X=50, 0, r=5%, T=1/2 /2 year ars) European call c0 ≥ 0 c0 ≤ S0 0 ≤ c0 ≤ 52 American call C0 ≥ 0 C0 ≤ S0 0 ≤ C0 ≤ 52 European put p0 ≥ 0 p0 ≤ X(1+r)T 0 ≤ p0 ≤ 48.80 [50/(1.05)0.5] American put P0 ≥ 0 P0 ≤ X 0 ≤ P0 ≤ 50

Principles of option pricing

 Boundary conditions

 The lower bound for American options

 For American options, which are exercisable

immediately, the lower bound of an American option price is its current intrinsic value.

 C0 ≥ Max(0, S0 – X)  P0 ≥ Max(0, X – S0)  If the option is in-the-money and is selling for less than its

intrinsic value, it can be bought and exercised to net an immediate risk-free profit.

11/20/2015 Nattawoot Koowattanatianchai 78

Principles of option pricing

 Boundary conditions

 The lower bound for European options

 Recall that European options can only be exercised at

• expiration. There is no way for market participants to

exercise an option selling for too little with respect to its intrinsic value. At this point, we will state lower bounds for European options first and prove them later.

 c0 ≥ Max[0, S0 – X/(1+r)T]  p0 ≥ Max[0, X/(1+r)T – S0]

11/20/2015 Nattawoot Koowattanatianchai 79

Principles of option pricing

 Boundary conditions

 The lower bound for European options

 Proof: The lower bound on a European call price  The following transactions must be worth more than zero

since they lead to a positive payoff regardless of what happens in the future.

• Buy a call
• Sell short the underlying
• Buy a zero-coupon bond with a face value of X that

matures on the option expiration day

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Principles of option pricing

 Boundary conditions

 The lower bound for European options

 Proof: The lower bound on a European call price

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Transac action

• n

Current nt value ue Value ue at expirati ation

• n

ST ≤ X ST > X Buy call c0 ST – X Sell short underlying

• S0
• ST
• ST

Buy bond X/(1+r)T X X Total c0 – S0 + X/(1+r)T X – ST ≥ 0

Principles of option pricing

 Boundary conditions

 The lower bound for European options

 Proof: The lower bound on a European put price  The following transactions must be worth more than zero

since they lead to a positive payoff regardless of what happens in the future.

• Buy a put
• Buy the underlying
• Issue a zero-coupon bond with a face value of X that

matures on the option expiration day

11/20/2015 Nattawoot Koowattanatianchai 82

Principles of option pricing

 Boundary conditions

 The lower bound for European options

 Proof: The lower bound on a European put price

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Transac action

• n

Current nt value ue Value ue at expirati ation

• n

ST ≤ X ST > X Buy put p0 X – ST Buy underlying S0 ST

• ST

Issue bond

• X/(1+r)T
• X
• X

Total p0 + S0 – X/(1+r)T ST – X ≥ 0

Principles of option pricing

 Boundary conditions

 Consolidations

 The lower bound for a European call is Max[0, S0 –

X/(1+r)T]. Except at expiration, this is greater than the lower bound for an American call, which is Max[0, S0 – X]. We could not, however, expect an American call to be worth less than a European call. Thus, the lower bound of the European call holds for American calls as well.

 c0 ≥ Max[0, S0 – X/(1+r)T]  C0 ≥ Max[0, S0 – X/(1+r)T]

11/20/2015 Nattawoot Koowattanatianchai 84

Principles of option pricing

 Boundary conditions

 Consolidations

 The lower bound for a European put is Max[0, X/(1+r)T –

S0]. Except at expiration, this is lower than the lower bound for an American put, which is Max[0, X – S0]. So, the American put lower bound is not changed to the European lower bound, the way we did for calls.

 p0 ≥ Max[0, X/(1+r)T – S0]  P0 ≥ Max[0, X – S0]

11/20/2015 Nattawoot Koowattanatianchai 85

Principles of option pricing

 Boundary conditions

 Example

 Consider call and put options expiring in 42 days, in

which the underlying is at 72 and the risk-free rate is 4.5%. The underlying makes no cash payments during the life of the options.

 A. Find the lower bounds for European calls and puts

with exercise prices of 70 and 75.

 70 call: Max(0,2.35) = 2.35  75 call: Max(0,-2.62) = 0  70 put: Max(0,-2.35) = 0  75 put: Max(0,2.62) = 2.62

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Principles of option pricing

 Boundary conditions

 Example

 B. Find the lower bounds for American calls and puts

with exercise prices of 70 and 75.

 70 call: Max(0,2.35) = 2.35  75 call: Max(0,-2.62) = 0  70 put: Max(0, 70 – 72) = 0  75 put: Max(0, 75 – 72) = 3

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Principles of option pricing

 The effect of a difference in exercise price

 Generally the higher the exercise price, the lower

the value of a call and the higher the value of a put.

 Proof: c0(X1) ≥ c0(X2) if X1 < X2

 The following transactions (known as a bull call spread)

lead to a positive payoff regardless of what happens in the future.

 Buy a European call with an exercise price of X1  Sell a European call with an exercise price of X2

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Principles of option pricing

 The effect of a difference in exercise price

 Proof: Proof: c0(X1) ≥ c0(X2) if X1 < X2

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Transac action

• n

Current nt value ue Value ue at expirati ation

• n

ST ≤ X1 X1< ST > X2 ST ≥ X2 Buy call (X =X1) c0(X1) ST – X1 ST – X1 Sell call (X =X2) c0(X2)

• (ST – X2)

Total c0(X1) - c0(X2) ST – X1 > 0 X2 – X1 > 0

Principles of option pricing

 The effect of a difference in exercise price

 Proof: p0(X2) ≥ p0(X1) if X1 < X2

 The following transactions (known as a bull put spread)

lead to a positive payoff regardless of what happens in the future.

 Buy a European put with an exercise price of X2  Sell a European put with an exercise price of X1

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Principles of option pricing

 The effect of a difference in exercise price

 Proof: Proof: c0(X1) ≥ c0(X2) if X1 < X2

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Transac action

• n

Current nt value ue Value ue at expirati ation

• n

ST ≤ X1 X1< ST > X2 ST ≥ X2 Buy put (X =X2) p0(X2) X2 – ST X2 – ST Sell put (X =X1) p0(X1)

• (X1 – ST)

Total p0(X2) – p0(X1) X2 – X1 > 0 X2 – ST > 0

Principles of option pricing

 The effect of a difference in exercise price

 Previous results also hold for American options.

 C0(X1) ≥ C0(X2) if X1 < X2  P0(X2) ≥ P0(X1) if X1 < X2

11/20/2015 Nattawoot Koowattanatianchai 92

Principles of option pricing

 The effect of a difference in time to expiration

 In general, a longer-term option has more time for

the underlying to make a favorable move. In addition, if the option is in-the-money by the end

• f a given period of time, it has a better chance of

moving even further in-the-money over a longer period of time. The additional time also gives it a better chance of moving out-of-the-money or further out-of-the-money, but the most that the

• ption holder can lose is the option premium.

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Principles of option pricing

 The effect of a difference in time to expiration

 Consider two European calls expiring at T1 and T2

respectively (T1 < T2 ).

 When the shorter-term call expires, the European call is

worth Max(0, ST1 – X). At that point in time, the longer- term European call is worth at least:

 Max[0, ST1 – X/(1+r)T1-T2], which is greater than Max(0, ST1 –

X).

 Thus, the longer-term European call is worth at least as

great as the shorter-term European call. These results still hold for American calls.

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Principles of option pricing

 The effect of a difference in time to expiration

 Consider two puts expiring at T1 and T2

respectively (T1 < T2 ).

 For puts, the longer term gives additional time for a

favorable move in the underlying to occur, but there is

• ne disadvantage to waiting the additional time. When a

put is exercised, the holder receives money. The lost interest on the money is a disadvantage of the additional

• time. Therefore, it is not always true that additional time

is beneficial to the holder of a European put. It is true, however, that the additional time is beneficial to the holder of an American put.

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Principles of option pricing

 The effect of a difference in time to expiration

 Summary

 c0(T1) ≥ c0(T2) if X1 < X2  C0(T1) ≥ C0(T2) if X1 < X2  p0(T2) can be either greater or less than p0(X1) if T1 < T2  P0(T2) ≥ P0(X1) if T1 < T2

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Put-call parity

 Fiduciary calls and protective puts

 Consider two option strategies referred to as “a

fiduciary call” and “a protective put

 Fiduciary call  Buy a European call with an exercise price of X  Buy a zero-coupon bond with a face value of X that

matures on the option expiration day

 Protective put  Buy a European put with an exercise price of X  Buy the underlying asset  Both strategies end up with the same value. To avoid

arbitrage, their value today must be the same.

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Put-call parity

 Fiduciary calls and protective puts

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Transac action

• n

Current nt value ue Value ue at expirati ation

• n

ST ≤ X ST > X Fiduciary call Buy call c0 ST – X Buy bond X/(1+r)T X X Total c0 + X/(1+r)T X ST Protective put Buy put p0 X – ST Buy underlying S0 ST ST Total p0 + S0 X ST

Put-call parity

 Call and synthetic call

 Synthetic call consists of the following

transactions that replicate the actual call.

 Buy a European put with the same underlying, exercise

price, and time to expiration to the European call

 Buy the underlying asset  Issue a zero-coupon bond with the face value of X that

matures on the option expiration day

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Put-call parity

 Call and synthetic call

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Transac action

• n

Current nt value ue Value ue at expirati ation

• n

ST ≤ X ST > X Call Buy call c0 ST – X Synthetic call Buy put p0 X – ST Buy underlying S0 ST ST Issue bond

• X/(1+r)T
• X
• X

Total p0 + S0 – X/(1+r)T ST – X

Put-call parity

 Put and synthetic put

 Synthetic put consists of the following transactions

that replicate the actual put.

 Buy a European call with the same underlying, exercise

price, and time to expiration to the European put

 Short sell the underlying asset  Buy a zero-coupon bond with the face value of X that

matures on the option expiration day

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Put-call parity

 Put and synthetic put

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Transac action

• n

Current nt value ue Value ue at expirati ation

• n

ST ≤ X ST > X Call Buy put p0 X – ST Synthetic call Buy call c0 ST – X Short underlying

• S0
• ST
• ST

Buy bond X/(1+r)T X X Total c0 – S0 + X/(1+r)T X – ST

Put-call parity

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Strat rategy gy Consis nsistin ing g of Wort rth Equat uates es to Strat rategy gy Consis nsistin ing g of Wort rth Fiduciary call Long call + long bond c0 + X/(1+r)T = Protective put Long put + long underlying p0 + S0 Long call Long call c0 = Synthetic call Long put + long underlying + short bond p0 + S0 – X/(1+r)T Long put Long put p0 = Synthetic put Long call + short underlying + long bond c0 + X/(1+r)T Long underlying Long underlying S0 = Synthetic underlying Long call + long bond + short put c0 + X/(1+r)T – p0 Long bond Long bond X/(1+r)T = Synthetic bond Long put + long underlying + short call p0 + S0 – c0

Put-call parity

 Example

 European put and call options with an exercise

price of 45 expire in 115 days. The underlying is priced at 48 and makes no cash payments during the life of the options. The risk-free rate is 4.5%. The put is selling for 3.75, and the call is selling for 8.00.

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Put-call parity

 Example

 A. Identify the mispricing by comparing the price

• f the actual call with the price of the synthetic

call.

 Using put-call parity, the following formula applies:  c0 = p0 + S0 – X/(1+r)T  The time to expiration is T = 115/365 = 0.3151.

Substituting values into the right-hand side:

 c0 = 3.75 + 48 – 45/(1.045)0.3151 = 7.37  Hence, the synthetic call is worth 7.37, but the actual

call is selling for 8.00 and is, therefore, overpriced.

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Put-call parity

 Example

 B. Based on your answer in Part A, demonstrate

how an arbitrage transaction is executed.

 Sell the call for 8.00 and buy the synthetic call for 7.37.

To buy the synthetic call, buy the put for 3.75, buy the underlying for 48.00, and issue a zero-coupon bond paying 45.00 at expiration. The bond will bring in 45/(1.045)0.3151 = 44.38 today. This transaction will bring in 8.00 – 7.37 = 0.63.

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Put-call parity

 Example

 B. Based on your answer in Part A, demonstrate

how an arbitrage transaction is executed.

 At expiration the following payoffs will occur  There will be no cash in or out at expiration.

11/20/2015 Nattawoot Koowattanatianchai 107

ST ≤ 45 ST > 4 45 Short call

• (ST – 45)

Long put 45 – ST Long underlying ST ST Issue bond

• 45
• 45

Total

American options

 American options can be exercised early.

Because early exercise is never mandatory, the right to exercise early may be worth something but could never hurt the option

• holder. Consequently:

 C0 ≥ c0  P0 ≥ p0

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American options

 Early exercise

 Suppose today, time 0, we are considering early

exercise an in-the-money American call.

 If we exercise, we pay X and receive an asset worth S0.

But we already determined that a European call is worth at least S0 – X/(1+r)T, which is more than S0 – X.

 Because the value we could obtain by selling this

call to someone else is more than the value we could obtain by exercising it, there is no reason to exercise the call early.

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American options

 Early exercise

 If the underlying makes a cash payment, there

may be reason to exercise the call early. If the underlying is a stock and pays a dividend, there may be sufficient reason to exercise just before the stock goes ex-dividend. By exercising, the

• ption holder throws away the time value but

captures the dividend.

 When the underlying makes no cash payments, C0 = c0  When the underlying makes cash payments during the

life of the option, C0 can be higher than c0

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American options

 Early exercise

 For puts, there is nearly always a possibility of

early exercise. Consider the most obvious case, an investor holding an American put on a bankrupt company. The stock is worth zero. It cannot go any lower. Thus, the put holder would exercise immediately.

 The American put is nearly always worth more than the

European put: P0 > p0

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The effect of cash flows

 Let PV(CF,0,T) be the present value of the

cash flows on the underlying over the life of the options. The lower bounds for European

• ptions can be restated as:

 c0 ≥ Max{0, [S0 – PV(CF,0,T)] – X/(1+r)T}  p0 ≥ Max{0, [X/(1+r)T – [S0 – PV(CF,0,T)] }

 Put call parity when the underlying makes

cash payments during the life of the options:

 c0 + X/(1+r)T = p0 + [S0 – PV(CF,0,T)]

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The effect of interest rates

 When interest rates are higher, call option

prices are higher.

 When investors buy call options instead of the

underlying, they are effectively buying an indirect leveraged position in the underlying. When interest are higher, buying the call instead of a direct leveraged position in the underlying is more

• attractive. Moreover, by using call options,

investors save more money by not paying for the underlying until a later date.

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The effect of interest rates

 When interest rates are higher, put option

prices are lower.

 For put options, higher interest rates are

• disadvantageous. When interest rates are higher,

investors lose more interest while waiting to sell the underlying when using puts. Thus, the

• pportunity cost of waiting is higher when interest

rates are higher.

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The effect of volatility

 Higher volatility increases call and put option

prices because it increases possible upside values and increases possible downside values of the underlying.

 The upside effect helps calls and does not hurt

puts.

 The downside effect does not hurt calls and helps

puts.

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Option price sensitivities

 Delta

 The sensitivity of the option price to a change in

the price of the underlying.

 Gamma

 A measure of how well the delta sensitivity

measure will approximate the option price’s response to a change in the price of the underlying.

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Option price sensitivities

 Rho

 The sensitivity of the option price to the risk-free

rate.

 Theta

 The rate at which the time value decays as the

• ption approaches expiration.

 Vega

 The sensitivity of the option price to volatility.

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4/6/2011 Natt Koowattanatianchai 118