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Topic 1Topic 1 Introduction To Derivatives Introduction To Derivatives

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This first lecture has four main goals:This first lecture has four main goals:1.1. Introduce you to the notion of risk and the role of Introduce you to the notion of risk and the role of

derivatives in managing risk.derivatives in managing risk. Discuss some of the general terms – such as short/long Discuss some of the general terms – such as short/long

positions, bid-ask spread – from finance that we need.positions, bid-ask spread – from finance that we need.

2.2. Introduce you to three major classes of derivative Introduce you to three major classes of derivative securities.securities. ForwardsForwards FuturesFutures OptionsOptions

3.3. Introduce you to the basic viewpoint needed to analyze Introduce you to the basic viewpoint needed to analyze these securities.these securities.

4.4. Introduce you to the major traders of these instruments.Introduce you to the major traders of these instruments.

BasicsBasics

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BasicsBasics Finance is the study of risk.Finance is the study of risk.

How to measure itHow to measure it How to reduce itHow to reduce it How to allocate itHow to allocate it

All finance problems ultimately boil down to three All finance problems ultimately boil down to three main questions:main questions: What are the cash flows, and when do they occur?What are the cash flows, and when do they occur? Who gets the cash flows?Who gets the cash flows? What is the appropriate discount rate for those cash What is the appropriate discount rate for those cash

flows?flows?

The difficulty, of course, is that normally none of The difficulty, of course, is that normally none of those questions have an easy answer.those questions have an easy answer.

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BasicsBasics As you know from other classes, we can generally classify As you know from other classes, we can generally classify

risk as being diversifiable or non-diversifiable:risk as being diversifiable or non-diversifiable: Diversifiable – risk that is specific to a specific investment – i.e. Diversifiable – risk that is specific to a specific investment – i.e.

the risk that a single company’s stock may go down (i.e. the risk that a single company’s stock may go down (i.e. Enron). This is frequently called Enron). This is frequently called idiosyncratic risk.idiosyncratic risk.

Non-diversifiable – risk that is common to all investing in Non-diversifiable – risk that is common to all investing in general and that cannot be reduced – i.e. the risk that the general and that cannot be reduced – i.e. the risk that the entire stock market (or bond market, or real estate market) will entire stock market (or bond market, or real estate market) will crash. This is frequently called crash. This is frequently called systematic risksystematic risk..

The market “pays” you for bearing non-diversifiable risk The market “pays” you for bearing non-diversifiable risk only – not for bearing diversifiable risk.only – not for bearing diversifiable risk. In general the more non-diversifiable risk that you bear, the In general the more non-diversifiable risk that you bear, the

greater the expected return to your investment(s).greater the expected return to your investment(s). Many investors fail to properly diversify, and as a result bear Many investors fail to properly diversify, and as a result bear

more risk than they have to in order to earn a given level of more risk than they have to in order to earn a given level of expected return.expected return.

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BasicsBasics In this sense, we can view the field of finance as In this sense, we can view the field of finance as

being about two issues:being about two issues: The elimination of diversifiable risk in portfolios;The elimination of diversifiable risk in portfolios; The The allocationallocation of systematic (non-diversifiable) risk to of systematic (non-diversifiable) risk to

those members of society that are most willing to bear those members of society that are most willing to bear it.it.

Indeed, it is really this second function – the Indeed, it is really this second function – the allocation of systematic risk – that drives rates of allocation of systematic risk – that drives rates of return.return. The expected rate of return is the “price” that the The expected rate of return is the “price” that the

market pays investors for bearing systematic risk.market pays investors for bearing systematic risk.

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BasicsBasics A A derivativederivative (or derivative security) is a financial (or derivative security) is a financial

instrument whose value depends upon the value instrument whose value depends upon the value of other, more basic, underlying variables.of other, more basic, underlying variables.

Some common examples include things such as Some common examples include things such as stock options, futures, and forwards. stock options, futures, and forwards.

It can also extend to something like a It can also extend to something like a reimbursement program for college credit. reimbursement program for college credit. Consider that if your firm reimburses 100% of Consider that if your firm reimburses 100% of costs for an “A”, 75% of costs for a “B”, 50% for costs for an “A”, 75% of costs for a “B”, 50% for a “C” and 0% for anything less. a “C” and 0% for anything less.

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Your “right” to claim this reimbursement, then is Your “right” to claim this reimbursement, then is tied to the grade you earn. The value of that tied to the grade you earn. The value of that reimbursement plan, therefore, is reimbursement plan, therefore, is derivedderived from from the grade you earn. the grade you earn.

We also say that the value is We also say that the value is contingentcontingent upon the upon the grade you earn. Thus, your claim for grade you earn. Thus, your claim for reimbursement is a “contingent” claim. reimbursement is a “contingent” claim.

The terms contingent claims and derivatives are The terms contingent claims and derivatives are used interchangeably.used interchangeably.

BasicsBasics

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BasicsBasics So why do we have derivatives and derivatives So why do we have derivatives and derivatives

markets?markets? Because they somehow allow investors to better control Because they somehow allow investors to better control

the level of risk that they bear.the level of risk that they bear. They can help eliminate idiosyncratic risk.They can help eliminate idiosyncratic risk. They can decrease or increase the level of systematic They can decrease or increase the level of systematic

risk.risk.

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A First ExampleA First Example There is a neat example from the bond-world of a There is a neat example from the bond-world of a

derivative that is used to move non-diversifiable derivative that is used to move non-diversifiable risk from one set of investors to another set that risk from one set of investors to another set that are, presumably, more willing to bear that risk.are, presumably, more willing to bear that risk.

Disney wanted to open a theme park in Tokyo, Disney wanted to open a theme park in Tokyo, but did not want to have the shareholders bear but did not want to have the shareholders bear the risk of an earthquake destroying the park.the risk of an earthquake destroying the park. They financed the park through the issuance of They financed the park through the issuance of

earthquake bonds.earthquake bonds. If an earthquake of at least 7.5 hit within 10 km of the If an earthquake of at least 7.5 hit within 10 km of the

park, the bonds did not have to be repaid, and there was park, the bonds did not have to be repaid, and there was a sliding scale for smaller quakes and for larger ones a sliding scale for smaller quakes and for larger ones that were located further away from the park.that were located further away from the park.

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A First ExampleA First Example Normally this could have been handled in the Normally this could have been handled in the

insurance (and re-insurance) markets, but there insurance (and re-insurance) markets, but there would have been transaction costs involved. By would have been transaction costs involved. By placing the risk directly upon the bondholders placing the risk directly upon the bondholders Disney was able to avoid those transactions costs. Disney was able to avoid those transactions costs. Presumably the bondholders of the Disney bonds are Presumably the bondholders of the Disney bonds are

basically the same investors that would have been holding basically the same investors that would have been holding the stock or bonds of the insurance/reinsurance companies.the stock or bonds of the insurance/reinsurance companies.

Although the risk of earthquake is not diversifiable to the Although the risk of earthquake is not diversifiable to the park, it could be to Disney shareholders, so this does beg park, it could be to Disney shareholders, so this does beg the question of why buy the insurance at all.the question of why buy the insurance at all.

This was not a “free” insurance. Disney paid This was not a “free” insurance. Disney paid LIBOR+310 on the bond. If the earthquake provision LIBOR+310 on the bond. If the earthquake provision was not it there, they would have paid a lower rate.was not it there, they would have paid a lower rate.

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A First ExampleA First Example This example illustrates an interesting notion – This example illustrates an interesting notion –

that insurance contracts (for property insurance) that insurance contracts (for property insurance) are really derivatives!are really derivatives!

They allow the owner of the asset to “sell” the They allow the owner of the asset to “sell” the insured asset to the insurer in the event of a insured asset to the insurer in the event of a disaster.disaster.

They are like put options (more on this later.)They are like put options (more on this later.)

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BasicsBasics Positions – In general if you are buying an asset – be Positions – In general if you are buying an asset – be

it a physical stock or bond, or the right to determine it a physical stock or bond, or the right to determine whether or not you will acquire the asset in the whether or not you will acquire the asset in the future (such as through an option or futures future (such as through an option or futures contract) you are said to be “LONG” the instrument.contract) you are said to be “LONG” the instrument.

If you are giving up the asset, or giving up the right If you are giving up the asset, or giving up the right to determine whether or not you will own the asset to determine whether or not you will own the asset in the future, you are said to be “SHORT” the in the future, you are said to be “SHORT” the instrument.instrument. In the stock and bond markets, if you “short” an asset, it In the stock and bond markets, if you “short” an asset, it

means that you borrow it, sell the asset, and then later buy means that you borrow it, sell the asset, and then later buy it back.it back.

In derivatives markets you generally do not have to borrow In derivatives markets you generally do not have to borrow the instrument – you can simply take a position (such as the instrument – you can simply take a position (such as writing an option) that will require you to give up the asset writing an option) that will require you to give up the asset or determination of ownership of the asset.or determination of ownership of the asset.

Usually in derivatives markets the “short” is just the Usually in derivatives markets the “short” is just the negative of the “long” positionnegative of the “long” position

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BasicsBasics Commissions – Virtually all transactions in the Commissions – Virtually all transactions in the

financial markets requires some form of commission financial markets requires some form of commission payment. payment. The size of the commission depends upon the relative The size of the commission depends upon the relative

position of the trader: retail traders pay the most, position of the trader: retail traders pay the most, institutional traders pay less, market makers pay the least institutional traders pay less, market makers pay the least (but still pay to the exchanges.)(but still pay to the exchanges.)

The larger the trade, the smaller the commission is in The larger the trade, the smaller the commission is in percentage terms.percentage terms.

Bid-Ask spread – Depending upon whether you are Bid-Ask spread – Depending upon whether you are buying or selling an instrument, you will get different buying or selling an instrument, you will get different prices. If you wish to sell, you will get a “BID” quote, prices. If you wish to sell, you will get a “BID” quote, and if you wish to buy you will get an “ASK” quote.and if you wish to buy you will get an “ASK” quote.

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BasicsBasics The difference between the bid and the ask can The difference between the bid and the ask can

vary depending upon whether you are a retail, vary depending upon whether you are a retail, institutional, or broker trader; it can also vary if institutional, or broker trader; it can also vary if you are placing very large trades.you are placing very large trades.

In general, however, the bid-ask spread is In general, however, the bid-ask spread is relatively constant for a given customer/position.relatively constant for a given customer/position.

The spread is roughly a constant percentage of The spread is roughly a constant percentage of the transaction, regardless of the scale – unlike the transaction, regardless of the scale – unlike the commission.the commission.

Especially in options trading, the bid-ask spread is Especially in options trading, the bid-ask spread is a much bigger transaction cost than the a much bigger transaction cost than the commission.commission.

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BasicsBasics Here are some example stock bid-ask spreads from Here are some example stock bid-ask spreads from

8/22/2006:8/22/2006: IBM: IBM: Bid – 78.77 Bid – 78.77 Ask – 78.79Ask – 78.79 0.025%0.025% ATT: ATT: Bid – 30.59Bid – 30.59 Ask – 30.60Ask – 30.60 0.033%0.033% Microsoft:Microsoft: Bid – 25.73 Bid – 25.73 Ask – 25.74Ask – 25.74 0.039%0.039%

Here are some example option bid-ask spreads (All Here are some example option bid-ask spreads (All with good volume)with good volume) IBM Oct 85 Call: IBM Oct 85 Call: Bid – 2.05Bid – 2.05 Ask – 2.20Ask – 2.20 7.3171%7.3171% ATT Oct 15 Call:ATT Oct 15 Call: Bid – 0.50Bid – 0.50 Ask –0.55Ask –0.55 10.000%10.000% MSFT Oct 27.5 : MSFT Oct 27.5 : Bid – 0.70Bid – 0.70 Ask –0.80.Ask –0.80. 14.285%14.285%

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BasicsBasics The point of the preceding slide is to demonstrate The point of the preceding slide is to demonstrate

that the bid-ask spread can be a huge factor in that the bid-ask spread can be a huge factor in determining the profitability of a trade. determining the profitability of a trade. Many of those option positions require at least a 10% Many of those option positions require at least a 10%

price movement before the trade is profitable.price movement before the trade is profitable. Many “trading strategies” that you see people Many “trading strategies” that you see people

propose (and that are frequently demonstrated propose (and that are frequently demonstrated using “real” data) are based upon using the using “real” data) are based upon using the average of the bid-ask spread. They usually lose average of the bid-ask spread. They usually lose their effectiveness when the bid-ask spread is their effectiveness when the bid-ask spread is considered.considered.

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BasicsBasics Market Efficiency – We normally talk about financial markets Market Efficiency – We normally talk about financial markets

as being efficient information processors.as being efficient information processors. Markets efficiently incorporate all publicly available information Markets efficiently incorporate all publicly available information

into financial asset prices.into financial asset prices. The mechanism through which this is done is by investors The mechanism through which this is done is by investors

buying/selling based upon their discovery and analysis of new buying/selling based upon their discovery and analysis of new information.information.

The limiting factor in this is the transaction costs associated with The limiting factor in this is the transaction costs associated with the market.the market.

For this reason, it is better to say that financial markets are For this reason, it is better to say that financial markets are efficient efficient to within transactions coststo within transactions costs. Some financial . Some financial economists say that financial markets are efficient to within the economists say that financial markets are efficient to within the bid-ask spread.bid-ask spread.

Now, to a large degree for this class we can ignore the bid-ask Now, to a large degree for this class we can ignore the bid-ask spread, but there are some points where it will be particularly spread, but there are some points where it will be particularly relevant, and we will consider it then.relevant, and we will consider it then.

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BasicsBasics Before we begin to examine specific contracts, Before we begin to examine specific contracts,

we need to consider two additional risks in the we need to consider two additional risks in the market:market: Credit risk – the risk that your trading partner might not Credit risk – the risk that your trading partner might not

honor their obligations.honor their obligations. Familiar risk to anybody that has traded on ebay!Familiar risk to anybody that has traded on ebay! Generally exchanges serve to mitigate this risk.Generally exchanges serve to mitigate this risk. Can also be mitigated by escrow accounts.Can also be mitigated by escrow accounts.

Margin requirements are a form of escrow account.Margin requirements are a form of escrow account.

Liquidity risk – the risk that when you need to buy or sell Liquidity risk – the risk that when you need to buy or sell an instrument you may not be able to find a an instrument you may not be able to find a counterparty.counterparty.

Can be very common for “outsiders” in commodities Can be very common for “outsiders” in commodities markets.markets.

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BasicsBasics So now we are going to begin examining the So now we are going to begin examining the

basic instruments of derivatives. In particular we basic instruments of derivatives. In particular we will look at (tonight):will look at (tonight): ForwardsForwards FuturesFutures OptionsOptions

The purpose of our discussion tonight is to simply The purpose of our discussion tonight is to simply provide a basic understanding of the structure of provide a basic understanding of the structure of the instruments and the basic reasons they might the instruments and the basic reasons they might exist. exist. We will have a more in-detail examination of their We will have a more in-detail examination of their

properties, and their pricing, in the weeks to come.properties, and their pricing, in the weeks to come.

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A A forward contractforward contract is an agreement between two is an agreement between two parties to buy or sell an asset at a certain future parties to buy or sell an asset at a certain future time for a certain future price.time for a certain future price. Forward contracts are normally not exchange traded.Forward contracts are normally not exchange traded. The party that agrees to buy the asset in the future is The party that agrees to buy the asset in the future is

said to have the said to have the longlong position. position. The party that agrees to sell the asset in the future is The party that agrees to sell the asset in the future is

said to have the said to have the shortshort position. position. The specified future date for the exchange is known as The specified future date for the exchange is known as

the delivery (the delivery (maturitymaturity) date.) date.

Forward ContractsForward Contracts

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The specified price for the sale is known as the The specified price for the sale is known as the deliverydelivery price, we will denote this as K. price, we will denote this as K. Note that K is set such that at initiation of the contract the Note that K is set such that at initiation of the contract the

value of the forward contract is 0. Thus, by design, no value of the forward contract is 0. Thus, by design, no cash changes hands at time 0. The mechanics of how to cash changes hands at time 0. The mechanics of how to do this we cover in later lectures.do this we cover in later lectures.

As time progresses the delivery price doesn’t change, As time progresses the delivery price doesn’t change, but the current spot (market) rate does. Thus, the but the current spot (market) rate does. Thus, the contract gains (or loses) value over time. contract gains (or loses) value over time. Consider the situation at the maturity date of the Consider the situation at the maturity date of the

contract. If the spot price is higher than the delivery contract. If the spot price is higher than the delivery price, the long party can buy at K and immediately sell at price, the long party can buy at K and immediately sell at the spot price Sthe spot price STT, making a profit of (S, making a profit of (STT-K), whereas the -K), whereas the short position could have sold the asset for Sshort position could have sold the asset for STT, but is , but is obligated to sell for K, earning a profit (negative) of (K-Sobligated to sell for K, earning a profit (negative) of (K-STT).).

Forward ContractsForward Contracts

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Example:Example: Let’s say that you entered into a forward contract to buy Let’s say that you entered into a forward contract to buy

wheat at $4.00/bushel, with delivery in December (thus wheat at $4.00/bushel, with delivery in December (thus K=$3.64.)K=$3.64.)

Let’s say that the delivery date was December 14 and that Let’s say that the delivery date was December 14 and that on December 14on December 14thth the market price of wheat is unlikely to be the market price of wheat is unlikely to be exactly $4.00/bushel, but that is the price at which you have exactly $4.00/bushel, but that is the price at which you have agreed (via the forward contract) to buy your wheat.agreed (via the forward contract) to buy your wheat.

If the market price is greater than $4.00/bushel, you are If the market price is greater than $4.00/bushel, you are pleased, because you are able to buy an asset for less than pleased, because you are able to buy an asset for less than its market price.its market price.

If, however, the market price is less than $4.00/bushel, you If, however, the market price is less than $4.00/bushel, you are not pleased because you are paying more than the are not pleased because you are paying more than the market price for the wheat.market price for the wheat.

Indeed, we can determine your net payoff to the trade by Indeed, we can determine your net payoff to the trade by applying the formula: payoff = Sapplying the formula: payoff = STT – K, since you gain an – K, since you gain an asset worth Sasset worth STT, but you have to pay $K for it. , but you have to pay $K for it.

We can graph the payoff function:We can graph the payoff function:

Forward ContractsForward Contracts

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Forward ContractsForward ContractsPayoff to Futures Position on Wheat

Where the Delivery Price (K) is $4.00/Bushel

-4

-3

-2

-1

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Wheat Market (Spot) Price, December 14

Pay

off

to

Fo

rwar

ds

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Example:Example: In this example you were the long party, but what about In this example you were the long party, but what about

the short party?the short party? They have agreed to sell wheat to you for $4.00/bushel They have agreed to sell wheat to you for $4.00/bushel

on December 14.on December 14. Their payoff is positive if the market price of wheat is Their payoff is positive if the market price of wheat is

less than $4.00/bushel – they force you to pay more for less than $4.00/bushel – they force you to pay more for the wheat than they could sell it for on the open market.the wheat than they could sell it for on the open market. Indeed, you could assume that what they do is buy it on Indeed, you could assume that what they do is buy it on

the open market and then immediately deliver it to you in the open market and then immediately deliver it to you in the forward contract.the forward contract.

Their payoff is negative, however, if the market price of Their payoff is negative, however, if the market price of wheat is greater than $4.00/bushel.wheat is greater than $4.00/bushel. They could have sold the wheat for more than $4.00/bushel They could have sold the wheat for more than $4.00/bushel

had they not agreed to sell it to you.had they not agreed to sell it to you. So their payoff function is the mirror image of your So their payoff function is the mirror image of your

payoff function:payoff function:

Forward ContractsForward Contracts

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Forward ContractsForward Contracts

Payoff to Short Futures Position on WheatWhere the Delivery Price (K) is $4.00/Bushel

-4

-3

-2

-1

0

1

2

3

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0 1 2 3 4 5 6 7 8

Wheat Market (Spot) Price, December 14

Pay

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Fo

rwar

ds

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Forward ContractsForward Contracts Clearly the short position is just the mirror image Clearly the short position is just the mirror image

of the long position, and, taken together the two of the long position, and, taken together the two positions cancel each other out:positions cancel each other out:

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Forward ContractsForward ContractsLong and Short Positions in a Forward Contract

For Wheat at $4.00/Bushel

-4

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-2

-1

0

1

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0 1 2 3 4 5 6 7 8

Wheat Price

Pay

off Long Position

Net Position

Short Position

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Futures ContractsFutures Contracts A futures contract is similar to a forward contract in A futures contract is similar to a forward contract in

that it is an agreement between two parties to buy that it is an agreement between two parties to buy or sell an asset at a certain time for a certain price. or sell an asset at a certain time for a certain price. Futures, however, are usually exchange traded and, Futures, however, are usually exchange traded and, to facilitate trading, are usually standardized to facilitate trading, are usually standardized contracts. This results in more institutional detail contracts. This results in more institutional detail than is the case with forwards.than is the case with forwards.

The long and short party usually do not deal with The long and short party usually do not deal with each other directly or even know each other for that each other directly or even know each other for that matter. The exchange acts as a clearinghouse. As matter. The exchange acts as a clearinghouse. As far as the two sides are concerned they are entering far as the two sides are concerned they are entering into contracts with the exchange. In fact, the into contracts with the exchange. In fact, the exchange guarantees performance of the contract exchange guarantees performance of the contract regardless of whether the other party fails.regardless of whether the other party fails.

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Futures ContractsFutures Contracts The largest futures exchanges are the Chicago The largest futures exchanges are the Chicago

Board of Trade (CBOT) and the Chicago Mercantile Board of Trade (CBOT) and the Chicago Mercantile Exchange (CME).Exchange (CME).

Futures are traded on a wide range of commodities Futures are traded on a wide range of commodities and financial assets.and financial assets.

Usually an exact delivery date is not specified, but Usually an exact delivery date is not specified, but rather a delivery range is specified. The short rather a delivery range is specified. The short position has the option to choose when delivery is position has the option to choose when delivery is made. This is done to accommodate physical made. This is done to accommodate physical delivery issues.delivery issues. Harvest dates vary from year to year, transportation Harvest dates vary from year to year, transportation

schedules change, etc.schedules change, etc.

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Futures ContractsFutures Contracts The exchange will usually place restrictions and The exchange will usually place restrictions and

conditions on futures. These include:conditions on futures. These include: Daily price (change) limits.Daily price (change) limits. For commodities, grade requirements.For commodities, grade requirements. Delivery method and place.Delivery method and place. How the contract is quoted.How the contract is quoted.

Note however, that the basic payoffs are the same Note however, that the basic payoffs are the same as for a forward contract.as for a forward contract.

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Options ContractsOptions Contracts Options on stocks were first traded in 1973. That Options on stocks were first traded in 1973. That

was the year the famous Black-Scholes formula was the year the famous Black-Scholes formula was published, along with Merton’s paper - a set of was published, along with Merton’s paper - a set of academic papers that literally started an industry.academic papers that literally started an industry.

Options exist on virtually anything. Tonight we are Options exist on virtually anything. Tonight we are going to focus on general options terminology for going to focus on general options terminology for stocks. We will get into other types of options later stocks. We will get into other types of options later in the class.in the class.

There are two basic types of options:There are two basic types of options: A A Call optionCall option is the right, but not the obligation, to buy the is the right, but not the obligation, to buy the

underlying asset by a certain date for a certain price.underlying asset by a certain date for a certain price. A A Put optionPut option is the right, but not the obligation, to sell the is the right, but not the obligation, to sell the

underlying asset by a certain date for a certain price.underlying asset by a certain date for a certain price. Note that unlike a forward or futures contract, the holder of Note that unlike a forward or futures contract, the holder of

the options contract does not have to do anything - they have the options contract does not have to do anything - they have the option to do it or not.the option to do it or not.

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Options ContractsOptions Contracts The date when the option expires is known as the The date when the option expires is known as the

exercise date, the expiration date, or the maturity exercise date, the expiration date, or the maturity date.date.

The price at which the asset can be purchased or The price at which the asset can be purchased or sold is known as the strike price.sold is known as the strike price.

If an option is said to be European, it means that If an option is said to be European, it means that the holder of the option can buy or sell (depending the holder of the option can buy or sell (depending on if it is a call or a put) only on the maturity date. on if it is a call or a put) only on the maturity date. If the option is said to be an American style If the option is said to be an American style option, the holder can exercise on any date up to option, the holder can exercise on any date up to and including the exercise date.and including the exercise date.

An options contract is always costly to enter as the An options contract is always costly to enter as the long party. The short party always is always paid long party. The short party always is always paid to enter into the contractto enter into the contract Looking at the payoff diagrams you can see why…Looking at the payoff diagrams you can see why…

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Options ContractsOptions Contracts Let’s say that you entered into a call option on Let’s say that you entered into a call option on

IBM stock:IBM stock: Today IBM is selling for roughly $78.80/share, so let’s say Today IBM is selling for roughly $78.80/share, so let’s say

you entered into a call option that would let you buy IBM you entered into a call option that would let you buy IBM stock in December at a price of $80/share.stock in December at a price of $80/share.

If in December the market price of IBM were greater than If in December the market price of IBM were greater than $80, you would exercise your option, and purchase the $80, you would exercise your option, and purchase the IBM share for $80.IBM share for $80.

If, in December IBM stock were selling for less than If, in December IBM stock were selling for less than $80/share, you could buy the stock for less by buying it $80/share, you could buy the stock for less by buying it in the open market, so you would not exercise your in the open market, so you would not exercise your option.option.

Thus your payoff to the option is $0 if the IBM stock is less than Thus your payoff to the option is $0 if the IBM stock is less than $80$80

It is (SIt is (STT-K) if IBM stock is worth more than $80-K) if IBM stock is worth more than $80 Thus, your payoff diagram is:Thus, your payoff diagram is:

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Options ContractsOptions ContractsLong Call on IBM

with Strike Price (K) = $80

-20

0

20

40

60

80

0 20 40 60 80 100 120 140 160

IBM Terminal Stock Price

Pa

yo

ff

K =

T

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Options ContractsOptions Contracts What if you had the short position?What if you had the short position? Well, after you enter into the contract, you have Well, after you enter into the contract, you have

grantedgranted the option to the long-party. the option to the long-party. If they want to exercise the option, you have to do so.If they want to exercise the option, you have to do so. Of course, they will only exercise the option when it is in Of course, they will only exercise the option when it is in

there best interest to do so – that is, when the strike there best interest to do so – that is, when the strike price is lower than the market price of the stock. price is lower than the market price of the stock.

So if the stock price is less than the strike price (SSo if the stock price is less than the strike price (STT<K), <K), then the long party will just buy the stock in the market, then the long party will just buy the stock in the market, and so the option will expire, and you will receive $0 at and so the option will expire, and you will receive $0 at maturity.maturity.

If the stock price is more than the strike price (SIf the stock price is more than the strike price (STT>K), >K), however, then the long party will exercise their option and however, then the long party will exercise their option and you will have to sell them an asset that is worth Syou will have to sell them an asset that is worth STT for $K. for $K.

We can thus write your payoff as: We can thus write your payoff as: payoff = min(0,Spayoff = min(0,STT-K), -K),

which has a graph that looks like:which has a graph that looks like:

3636

Options ContractsOptions ContractsShort Call Position on IBM Stock

with Strike Price (K) = $80

-85

-63.75

-42.5

-21.25

0

21.25

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Ending Stock Price

Pa

yo

ff t

o S

ho

rt P

os

itio

n

3737

Options ContractsOptions Contracts This is obviously the mirror image of the long This is obviously the mirror image of the long

position.position. Notice, however, that at maturity, the short Notice, however, that at maturity, the short

option position can NEVER have a positive payout option position can NEVER have a positive payout – the best that can happen is that they get $0.– the best that can happen is that they get $0. This is why the short option party always demands an This is why the short option party always demands an

up-front payment – it’s the only payment they are going up-front payment – it’s the only payment they are going to receive. This payment is called the option to receive. This payment is called the option premiumpremium or price.or price.

Once again, the two positions “net out” to zero: Once again, the two positions “net out” to zero:

3838

Options ContractsOptions ContractsLong and Short Call Options on IBM

with Strike Prices of $80

-100

-80

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-40

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Ending Stock Price

Pay

off

Long Call

Short Call

Net Position

3939

Options ContractsOptions Contracts Recall that a put option grants the long party the Recall that a put option grants the long party the

right to sell the underlying at price K.right to sell the underlying at price K. Returning to our IBM example, if K=80, the long Returning to our IBM example, if K=80, the long

party will only elect to exercise the option if the party will only elect to exercise the option if the price of the stock in the market is less than $80, price of the stock in the market is less than $80, otherwise they would just sell it in the market.otherwise they would just sell it in the market.

The payoff to the holder of the long put position, The payoff to the holder of the long put position, therefore is simplytherefore is simply

payoff = max(0, K-Spayoff = max(0, K-STT))

4040

Options ContractsOptions ContractsPayoff to Long Put Option on IBM

with Strike Price of $80

-10

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Pa

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4141

Options ContractsOptions Contracts The short position again has granted the option to The short position again has granted the option to

the long position. The short has to buy the stock the long position. The short has to buy the stock at price K, when the long party wants them to do at price K, when the long party wants them to do so. Of course the long party will only do this when so. Of course the long party will only do this when the stock price is less than the strike price.the stock price is less than the strike price.

Thus, the payoff function for the short put Thus, the payoff function for the short put position is:position is:

payoff = min(0, Spayoff = min(0, STT-K)-K)

And the payoff diagram looks like:And the payoff diagram looks like:

4242

Options ContractsOptions ContractsShort Put Option on IBMwith Strike Price of $80

-85

-63.75

-42.5

-21.25

0

0 20 40 60 80 100 120 140 160

Ending Stock Price

Pa

yo

ff

4343

Options ContractsOptions Contracts Since the short put party can never receive a Since the short put party can never receive a

positive payout at maturity, they demand a positive payout at maturity, they demand a payment up-front from the long party – that is, payment up-front from the long party – that is, they demand that the long party pay a they demand that the long party pay a premiumpremium to induce them to enter into the contract.to induce them to enter into the contract.

Once again, the short and long positions net out Once again, the short and long positions net out to zero: when one party wins, the other loses.to zero: when one party wins, the other loses.

4444

Options ContractsOptions ContractsLong and Short Put Options on IBM

with Strike Prices of $80

-100

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Ending Stock Price

Pa

yo

ff

Long Position

Short Position

Net Position

4545

Options ContractsOptions Contracts The standard options contract is for 100 units of The standard options contract is for 100 units of

the underlying. Thus if the option is selling for the underlying. Thus if the option is selling for $5, you would have to enter into a contract for $5, you would have to enter into a contract for 100 of the underlying stock, and thus the cost of 100 of the underlying stock, and thus the cost of entering would be $500.entering would be $500.

For a European call, the payoff to the option is:For a European call, the payoff to the option is: Max(0,SMax(0,STT-K)-K)

For a European put it is For a European put it is Max(0,K-SMax(0,K-STT))

The short positions are just the negative of The short positions are just the negative of these:these: Short call: -Max(0,SShort call: -Max(0,STT-K) = Min(0,K-S-K) = Min(0,K-STT))

Short put: -Max(0,K-SShort put: -Max(0,K-STT)) = Min(0,S = Min(0,STT-K)-K)

4646

Options ContractsOptions Contracts Traders frequently refer to an option as being “in Traders frequently refer to an option as being “in

the money”, “out of the money” or “at the money”. the money”, “out of the money” or “at the money”. An “in the money” option means one where the price of An “in the money” option means one where the price of

the underlying is such that if the option were exercised the underlying is such that if the option were exercised immediately, the option holder would receive a payout.immediately, the option holder would receive a payout.

For a call option this means that SFor a call option this means that Stt>K>K For a put option this means that SFor a put option this means that Stt<K<K

An “at the money” option means one where the strike and An “at the money” option means one where the strike and exercise prices are the same.exercise prices are the same.

An “out of the money” option means one where the price An “out of the money” option means one where the price of the underlying is such that if the option were exercised of the underlying is such that if the option were exercised immediately, the option holder would NOT receive a immediately, the option holder would NOT receive a payout.payout.

For a call option this means that SFor a call option this means that Stt<K<K For a put option this means that SFor a put option this means that Stt>K.>K.

4747

Options ContractsOptions ContractsLong Call on IBM

with Strike Price (K) = $80

-20

0

20

40

60

80

0 20 40 60 80 100 120 140 160

IBM Terminal Stock Price

Pa

yo

ff

K =

T

Out of the money In the money

At the money

4848

Options ContractsOptions Contracts One interesting notion is to look at the payoff One interesting notion is to look at the payoff

from just owning the stock – its value is simply from just owning the stock – its value is simply the value of the stock:the value of the stock:

4949

Options ContractsOptions Contracts

Payout Diagram for a Long Position in IBM Stock

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160

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0 20 40 60 80 100 120 140 160

Ending Stock Price

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5050

Options ContractsOptions Contracts What is interesting is if we compare the payout What is interesting is if we compare the payout

from a portfolio containing a short put and a long from a portfolio containing a short put and a long call with the payout from just owning the stock:call with the payout from just owning the stock:

5151

Options ContractsOptions ContractsPayout Diagram for a Long Position in IBM Stock

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Ending Stock Price

Pay

off

Long Call

Short Put

Stock

5252

Options ContractsOptions Contracts Notice how the payoff to the options portfolio has Notice how the payoff to the options portfolio has

the same shape and slope as the stock position – the same shape and slope as the stock position – just offset by some amount?just offset by some amount?

This is hinting at one of the most important This is hinting at one of the most important relationships in options theory – Put-Call parity.relationships in options theory – Put-Call parity.

It may be easier to see this if we examine the It may be easier to see this if we examine the aggregate position of the options portfolio:aggregate position of the options portfolio:

5353

Options ContractsOptions ContractsPayout Diagram for a Long Position in IBM Stock

-100

-50

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50

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Ending Stock Price

Pay

off

5454

Options ContractsOptions Contracts We will come back to put-call parity in a few weeks, We will come back to put-call parity in a few weeks,

but it is well worth keeping this diagram in mind.but it is well worth keeping this diagram in mind.

So who trades options contracts? Generally there So who trades options contracts? Generally there are three types of options traders:are three types of options traders: HedgersHedgers - these are firms that face a business risk. They - these are firms that face a business risk. They

wish to get rid of this uncertainty using a derivative. For wish to get rid of this uncertainty using a derivative. For example, an airline might use a derivatives contract to example, an airline might use a derivatives contract to hedge the risk that jet fuel prices might change. hedge the risk that jet fuel prices might change. 

Speculators Speculators - They want to take a bet (position) in the - They want to take a bet (position) in the market and simply want to be in place to capture market and simply want to be in place to capture expected up or down movements.expected up or down movements.

ArbitrageursArbitrageurs - They are looking for imperfections in the - They are looking for imperfections in the capital market.capital market.

5555

Financial EngineeringFinancial Engineering When we start examining the actual pricing of When we start examining the actual pricing of

derivatives (next week), one of the fundamental derivatives (next week), one of the fundamental ideas that we will use is the “law of one price”. ideas that we will use is the “law of one price”.

Basically this says that if two portfolios offer the Basically this says that if two portfolios offer the same cash flows in all potential states of the same cash flows in all potential states of the world, then the two portfolios must sell for the world, then the two portfolios must sell for the same price in the market – regardless of the same price in the market – regardless of the instruments contained in the portfolios.instruments contained in the portfolios. This is only true to “within transactions costs”, i.e. the This is only true to “within transactions costs”, i.e. the

bid-ask spread on each individual instrument.bid-ask spread on each individual instrument. Sometimes one portfolio will have such lower Sometimes one portfolio will have such lower

transactions costs that the law will only approximately transactions costs that the law will only approximately hold.hold.

5656

Financial EngineeringFinancial Engineering Financial engineering is the notion that you can use a Financial engineering is the notion that you can use a

combination of assets and financial derivatives to combination of assets and financial derivatives to construct cash flow streams that would otherwise be construct cash flow streams that would otherwise be difficult or impossible to obtain.difficult or impossible to obtain.

Financial engineering can be used to “break apart” a Financial engineering can be used to “break apart” a set of cash flows into component pieces that each set of cash flows into component pieces that each have different risks and that can be sold to different have different risks and that can be sold to different investors.investors.

Collateralized Bond Obligations do this for “junk” bonds.Collateralized Bond Obligations do this for “junk” bonds. Collateralized Mortgage Obligations do this for residential Collateralized Mortgage Obligations do this for residential

mortgages.mortgages.

Financial engineering can also be used to create cash Financial engineering can also be used to create cash flows streams that would otherwise be difficult to flows streams that would otherwise be difficult to obtain.obtain.

5757

Financial EngineeringFinancial Engineering The Schwab/First Union equity-linked CD is a good The Schwab/First Union equity-linked CD is a good

example of financial engineering.example of financial engineering. When it was issued (in 1999), the stock market was When it was issued (in 1999), the stock market was

(and had been) incredibly “hot” for several years.(and had been) incredibly “hot” for several years. Many investors wanted to be in the market, but did not Many investors wanted to be in the market, but did not

want to risk the market going down in value.want to risk the market going down in value. The equity-linked CD was designed to meet this The equity-linked CD was designed to meet this

need.need. As we will demonstrate, an investor could “roll their own” As we will demonstrate, an investor could “roll their own”

version of this, but in doing so would have incurred version of this, but in doing so would have incurred significant transaction costs.significant transaction costs.

Plus, many small investors (to whom this was targeted) Plus, many small investors (to whom this was targeted) probably could not get approval to trade options.probably could not get approval to trade options.

5858

Financial EngineeringFinancial Engineering The Contract:The Contract:

An investor buys the CD (Certificate of Deposit) today, and An investor buys the CD (Certificate of Deposit) today, and then earns 70% of the simple rate of return on S&P 500 index then earns 70% of the simple rate of return on S&P 500 index over the next 5.5 years.over the next 5.5 years.

If the S&P index ended up below the initial index level (so that If the S&P index ended up below the initial index level (so that the appreciation was negative), then the investor received the appreciation was negative), then the investor received their full initial investment back, but nothing else.their full initial investment back, but nothing else.

Thus, the payoff to the CD was simply:Thus, the payoff to the CD was simply:

So let’s say that you invested $10,000, and that in June of So let’s say that you invested $10,000, and that in June of 1999 the index was 1300 (so that you were, in essence, buying 1999 the index was 1300 (so that you were, in essence, buying $10,000/1,300 or 7.69 units of the index).$10,000/1,300 or 7.69 units of the index).

5.5

0

* 1 max 0, 1Maturity

IndexCD Investment

Index

5959

Financial EngineeringFinancial Engineering In 5.5 years your payoff will be based upon the In 5.5 years your payoff will be based upon the

index level. Potential index levels and payoffs index level. Potential index levels and payoffs include:include:

IndexIndex Simple Rate of ReturnSimple Rate of Return Cash ReceivedCash Received

10001000 - 23.07%- 23.07% $10,000$10,000

12001200 - 7.69%- 7.69% $10,000$10,000

13001300 0.00% 0.00% $10,000$10,000

14001400 7.69% 7.69% $10,538$10,538

15001500 15.38% 15.38% $11,076$11,076

20002000 53.85% 53.85% $13,769$13,769

(Note that on 12/30/2004 the S&P 500 was at 1211.92!)(Note that on 12/30/2004 the S&P 500 was at 1211.92!) The following chart demonstrates the payouts.The following chart demonstrates the payouts.

6060

Financial EngineeringFinancial EngineeringPayoff to Equity Linked Swap

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Pay

off

6161

Financial EngineeringFinancial Engineering Now, the first thing about that chart that you should Now, the first thing about that chart that you should

notice is that it looks an awful lot like the shape of a notice is that it looks an awful lot like the shape of a call option, although the slope of the upward-sloping call option, although the slope of the upward-sloping part is not as steep.part is not as steep.

This is our first indication that we may be able to This is our first indication that we may be able to decompose this into two simpler securities.decompose this into two simpler securities.

Indeed, one way of decomposing this security would Indeed, one way of decomposing this security would be to assume that we bought a bond that paid be to assume that we bought a bond that paid $10,000 at time 5.5, and that we bought 5.38 call $10,000 at time 5.5, and that we bought 5.38 call options with a strike of 1300 (70% of 10,000/1300.)options with a strike of 1300 (70% of 10,000/1300.)

The next graph demonstrates this position’s payoff.The next graph demonstrates this position’s payoff.

6262

Financial EngineeringFinancial Engineering

Bond Plus Call Payoff

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Index Value

Pay

off Option Payoff

Bond Payoff

Net

6363

Financial EngineeringFinancial Engineering This position is ALSO identical to a position consisting This position is ALSO identical to a position consisting

of:of: $10,000/1300 = 7.692 units of the index.$10,000/1300 = 7.692 units of the index. $10,000/1300 = 7.692 put options on the index (K=1300)$10,000/1300 = 7.692 put options on the index (K=1300) (-(1-.7)*$10,000/1300 = -2.30769) CALL options on the index.(-(1-.7)*$10,000/1300 = -2.30769) CALL options on the index.

The reason for the short call options is because the The reason for the short call options is because the CD only gives us 70% of the return on the index, so CD only gives us 70% of the return on the index, so we have to sell back some of that return via the call we have to sell back some of that return via the call option (note that we will earn a premium for this.)option (note that we will earn a premium for this.)

The following chart shows this:The following chart shows this:

6464

Financial EngineeringFinancial EngineeringLong Index, Long Put, Short Call

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Pay

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Index Payoff

Put Position

Call Position

Net

6565

Financial EngineeringFinancial Engineering Now, all three of these Now, all three of these shouldshould sell for the same price – but sell for the same price – but

there will be some differences because of transactions costs.there will be some differences because of transactions costs. Really, this is why the Schwab equity-linked CD can work: Really, this is why the Schwab equity-linked CD can work:

investors (retail investors) are willing to turn to the investors (retail investors) are willing to turn to the “prepackaged” asset to avoid transaction costs (and to avoid “prepackaged” asset to avoid transaction costs (and to avoid timing difficulties with unwinding their position.)timing difficulties with unwinding their position.)

Let’s just think of this as a bond and .7 long call options for a Let’s just think of this as a bond and .7 long call options for a moment.moment.

Clearly the call cannot be free, since the investor holds this Clearly the call cannot be free, since the investor holds this option they must pay something for it. How much do they option they must pay something for it. How much do they pay?pay? The interest that they could have earned on this money had The interest that they could have earned on this money had

they invested in a traditional CD.they invested in a traditional CD. At that time 5.5 year CDs were yielding 6%, so the investor At that time 5.5 year CDs were yielding 6%, so the investor

“gives up” $3,777 dollars in year 5.5 dollars. “gives up” $3,777 dollars in year 5.5 dollars.

5.5($10,000*1.06 ) 10,000 3,777

6666

Financial EngineeringFinancial Engineering The equity-linked CD is just one example of The equity-linked CD is just one example of

financial engineering – the notion that investors financial engineering – the notion that investors are really just purchasing potential future cash are really just purchasing potential future cash flows and that any two sets of identical potential flows and that any two sets of identical potential future cash flows must sell for the same price.future cash flows must sell for the same price.

This has led to a real revolution in finance, and This has led to a real revolution in finance, and we will discuss this idea throughout the semester.we will discuss this idea throughout the semester.

We will return to options pricing later in the We will return to options pricing later in the semester. Next, we turn our attention to the semester. Next, we turn our attention to the futures/forwards markets and pricing.futures/forwards markets and pricing.