ubs option fundamentals

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aUBS

Options:The Fundamentals

Options:TheFundamentals

IS

BN0-9701975-0-0


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Contents

Preface

A Simple Objective 1

Arbitrage 2

Forwards and Futures 9

Options 24

Options as Risk Management Tools:Break-Even Graphs 31

Options as Risk Management Tools:Combining Options with Underlying Positions 38

Option Valuation 44

Put/Call Parity 63

Option Trading 66

Beyond Option Value 76

Conclusion 82

Copyright 1999 UBS AG, Basel, SwitzerlandISBN 0-9701975-0-0


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Preface

The study of options and other derivative instruments can be an intimidatingexperience for a newcomer. Our objective in this book is to help introduce options in

a straightforward but rigorous way.

Most of the material in this work originated from the education program at UBS andits predecessor institutions - Swiss Bank Corporation and OConnor and Associates.In our training of traders, salespeople and other investment banking professionals, webelieve we have developed an intuitive, practical approach to derivatives.

The language we use is occasionally different from textbook language for good reason.Some textbooks talk about an option's "time value", combining several value

components in a way that leaves many people confused. When we talk about the sameideas, we distinguish between an option's basis value and its volatility value to eliminateconfusion. We dissect and examine the term "at-the-money", while most introductorydiscussions gloss over its possible interpretations.

We also debunk some option myths. One of the greatest of these myths is themisconception that option trading is a zero-sum market. We talk about the objectivesand viewpoints of option traders relative to the objectives and viewpoints of investorsand risk managers.

The perspectives, examples, and even the language we use have been very successfulin educating trading personnel. We hope the insights developed in our classrooms willmake options more understandable, accessible, and useful to a wider audience.

We wish to express our appreciation to the individuals who have contributed toproducing this book, principal among these are Glenn Satty, who contributed heavilyto the first edition of the book; Gary Gastineau, without whose persistent urging-onthe book would never have been written, and Tim Weithers who read the manuscriptand made many helpful suggestions. We are indebted to Springboard Design for theirsuperlative work on the production of the book. Inevitably, the author is responsiblefor any remaining errors.

Joe Troccolo, Managing DirectorUBS Financial Markets Education

Email: [email protected]


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A Simple Objective

In this book, we hope to demystify options and prepare readers to use these essentialtools to manage the market risks in portfolios of assets and liabilities. We want to

help the reader achieve a conceptual understanding of options and other derivativeinstruments, backed up by enough quantitative support to make the readercomfortable with these markets.

This publication is intended for anyone interested in the options markets: experiencedcorporate treasurers, fund managers, brokers, and traders, as well as new trainees.Even a complete understanding of the contents will not qualify you as a "rocketscientist" or a financial engineer, but you will be better equipped to discuss derivativeproducts intelligently and to evaluate the claims made about many financial markets

and instruments.

We begin our introduction to options with the concepts of arbitrage and expectedvalue. We introduce forwards and futures, define options, and show why options areuseful in risk management. We illustrate option payoffs using break-even graphs,examine some option strategies, and discuss the valuation of options. Finally, weexplore some popular option myths.

We offer this introduction as a prerequisite to more sophisticated option studies, notas the answer to all questions you might have about options and other derivative

instruments.


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Arbitrage

Before we can discuss options properly, we must understand the concept of arbitrage.Arbitrage is a powerful market force that helps to establish the value of many

financial instruments. When we talk about arbitrage, we distinguish betweendeterministic arbitrage and statistical arbitrage.

Deterministic Arbitrage

Deterministic arbitrage is the classic technique of simultaneously buying and sellingthe same or equivalent products at different prices to achieve a riskless profit. Forexample, what do we do if gold trades at $400 in New York and at $410 in London?The answer is quite simple: we buy low and we sell high. Specifically, we buy low for$400 in New York, and we sell high at $410 in London-and we make $10 risklessly.By "risklessly," we mean there is no risk of price movement. If we can execute thesetwo trades simultaneously, there is no gold price risk, and we can be confident of aprofit as long as other factors do not intervene.

One of the best ways to illustrate arbitrage opportunities is to use examples based onsimple raffles. Raffles define and clarify some of the ideas of profit and probabilitywe want to discuss.

Let's assume that a daily raffle sells exactly 100 tickets each morning. At the end ofthe day, 1 of these 100 tickets is chosen, and the holder of that ticket wins $1,000.What is the fair price or expected value of 1 ticket? To solve this problem, we dividethe $1,000 prize by 100 tickets for a fair price of $10.

If we pay $10 for the ticket and we do not win, we will not feel we were cheatedbecause we paid too much. We will attribute it to, say, the luck of the draw. Similarly,if we win, we will be ecstatic, but we will not think we bought a cheap ticket because,by the accident of probability, we won the prize.

When we divide the value of the prize by the number of tickets, we are able to derive thefair value or, in statistical terminology, the expected value or mean value of each ticket.

Buy low

sell high$10 risk-free profit

Buygoldfor$400inNewYork

Sellgoldat$410inLondon


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What do we do if tickets sell for $9 instead of $10? At $9, the ticket is cheap - socheap that we want to buy every ticket in the raffle. When the winning number is

drawn at the end of the day we stand to make a riskless profit of $100. If we hold allthe tickets, we win the $1,000 prize-having paid only $900. We can arbitrage adeterministic $100 profit - meaning we are certain to make that $100 - as long as theperson who created the raffle does not leave town after collecting our money!

It is important to understand that these cheap tickets will not be available at $9 very long.The raffle seller will lose money continuously at $9 a ticket and eventually will raise theprice of the tickets. What do we do if tickets sell for $11? Because $11 is too much topay for a ticket, we want to sell raffle tickets instead of buying them if we can. Wemight hold our own raffle selling tickets at $10.50. After selling 100 tickets, we willtake in $1,050. The prize we have to pay the raffle winner is $1,000, so we have $50left in our pockets. If any raffle ticket sells for less than fair value, we want to buy it.If it is overvalued, we want to sell it.

In reality, since raffles usually are held to raise money for non-profit organizations, itwould be difficult to start our own raffle for financial gain. In addition, rafflesgenerally do not compete against each other. People buy raffle tickets to support afavourite charity and are unconcerned if the ticket is overpriced or underpriced.However, if raffles did compete against each other, sellers would force the price of

overpriced raffles down, and buyers would force the price of underpriced raffles upuntil the price of raffle tickets stabilised at $10 and that price would be, in thestatistical sense, fair to everyone.

These raffle scenarios are similar to gold trading at $400 in New York and at $410in London. When there are many buyers of gold in New York, their demand forcesthe price up. When there are many sellers of gold in London, their supply forces theprice down. Nobody can say where the price will end up, but we presume it will fallbetween $400 and $410. Eventually, we know that the price will equilibrate to a fairvalue - the same price to buyers and sellers in all markets. The fair value of gold doesnot necessarily mean the price gold is "worth" - it means the efficient market valueor the price that eliminates arbitrage.

Expected Value

Buying a Whole Raffle

$9.00/ticket

x100 tickets$900.00 cost

$1,000.00 prize-$900.00 cost$100.00 profit

$10.50/ticketx100 tickets

$1,050.00 proceeds

$1,050.00 proceeds-$1,000.00 cost

$50.00 profit

Selling a Raffle


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Statistical Arbitrage

Locking in a sure profit by taking advantage of mispricings is deterministic arbitrage.While taking advantage of ticket mispricings is not within the spirit of a charity raffle,let's assume for purposes of illustration that raffle tickets sell for less than their fairvalue. Let's look at a different, but related, kind of arbitrage: statistical arbitrage.

What do we do if raffle tickets sell for $9 but we are permitted to buy only one ticketa day? Do we: (1) never buy a ticket, (2) buy a ticket once in a while, or (3) buy aticket every day? In this case, the ticket should be selling for $10 but only costs $9,so we have a $1 "edge," or expected profit per ticket. If we buy only one ticket a day,we are engaged in statistical arbitrage.

We do not have a certain profit if we cannot buy all the tickets in an underpricedraffle. We have to decide the best thing to do in an uncertain situation. In fact, thecorrect choice for someone who can accept the risk of a string of losses before he winsthe prize is not 1, 2, or even 3. The full answer is: buy one ticket every day until theuniverse ends. What do we expect to happen after buying a ticket every day until theend of time? We expect our fortune to grow without limit!

The key word in this scenario is expect. We are not certain to win this money, but we

do expect to win this money. We expect to earn an average of $1 a day on this raffle,because our edge, the expected value of our position, is $1. We expect to make thatedge on average after repeated trials. This is called a statistical arbitrage because it isnot certain or determined that we will make this profit - but we expect to make thisprofit over time.

What do we expect to happen after 100 days when we have spent $900 on raffletickets? We expect to win 1 time. Are we assured of winning 1 time? Absolutelynot - we could lose every time or we could win 2, 3, or even 99 times.

What do we expect to happen after 1,000 days? There will be 1,000 winners after1,000 days. With 1,000 winners and a 1 in 100 chance of winning each day, weexpect to win 10 times. We may not win 10 times - we may win only 8 times, or wemay win 12 times. In fact, there is about a 70% chance that we will win between 7and 13 times. If we play 1,000 times, we will be very surprised if we do not win atall - because the probability of not winning at all in 1,000 tries is extremely small, ifthe raffle is fair.

100 winners100 tickets per raffle

1 expected winning ticketvalue: $100

1,000 winners100 tickets per raffle

10 expected winning ticketsvalue: $1,000

1,000,000 winners100 tickets per raffle

10,000 expectedwinning tickets

value: $1,000,000


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The more times we play, the closer we should come to the number of times we expectto win and to the average of $1 a day we expect to make. If we play 1,000,000 times,

we expect to make close to $1,000,000 - our profit might fall a few hundred dollarsshort of $1,000,000 or rise a few hundred dollars beyond $1,000,000, but that is afairly small percentage variation compared to the percentage variation we might seeafter 100 days.

After 1,000,000 raffles, we expect to win about 10,000 times (1/100 of 1,000,000).Statisticians tell us we have about a 70% chance of winning between 9,900 and10,100 times. The larger number of raffles brings us closer to our expected averageof 1 win for every 100 times. The longer the period, the closer we expect to come tothe average payoff.

A different probability example illustrates additional important features of expectedvalue. The boards of two companies, A and B, are meeting to discuss a takeover. Ifthe takeover occurs, stock A will be worth $60.

If the takeover does not occur, the stock will be worth only $40. From our experience,we believe there is a 70% chance of a takeover. What is the fair value of stock A?

If there is a 70% chance that stock A will be worth $60, that leaves a 30% chance that

it will be worth $40. We multiply the probabilities by the values and add the result toget the expected value.

70% x $60 = $4230% x $40 = $12

$54 expected value

With these probabilities, the fair value of the stock is $54.

With the stock trading at $52, what do we do? We buy the stock, since it should tradeat $54, and we expect to make $2. Similarly, if the stock trades at $55, we sell it andexpect to make $1.

We determined the expected value of stock A - but did we determine our profit? Willwe make $2 if we buy the stock for $52? No. In fact, it will be impossible for us tomake $2. If we buy the stock for $52 and it trades at $60, we will make $8. If we buy

Stock A

With the stock at $52,we buy and expect to

make $2

70%

But the stock musttrade at $60 or at $40once the decision onthe takeover is made

If we do 1,000 trades

$8 x 700 - 12 x 300 =

$5,6000 - $3,6000 = $2,000

expected profit

Average Profit = $2/trade

Takeover No Takeover

Price $60 $40

Cost -$52 -$52

Profit/Loss = $8 = -$12

30% Fair value

$60

$40$54


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the stock at $52 and it trades at $40, we will lose $12. Expected value is notnecessarily the value of any individual trade.

If we buy stock like A in 1,000 takeovers at a price of $52, about 700 of those timeswe will make $8; and about 300 of those times, we will lose $12. After 1,000 times,we expect to net $2,000, an average or expected value of $2 a trade.

We have learned some interesting things about statistical arbitrage. First, we foundthat we must trade often. We cannot talk meaningfully about what happens at the endof one trade. We can begin to think about what happens at the end of 100 trades. Weare more comfortable thinking about what happens at the end of 1,000 trades, butwe prefer thinking about what happens after 1,000,000 trades. In our stock example,we can talk about the expected return after 1,000 such trades very easily, but wecannot know what happens after 1 trade even though we have determined that thefair value of the position is $54.

In addition to trading a large number of times, we must have correct probabilities toensure an accurate expected value. If our chances were 50/50 instead of 70/30, ourexpected value would become

50% x $60 = $30

50% x $40 = $20$50

making the stock we purchased at $52 expensive and causing us to lose money overtime. Unlike the simple raffle examples, where it should be possible to know theprobabilities, the statistical arbitrageur usually has to estimate probabilities.

There is another characteristic of statistical arbitrage that we have to think about. Totrade a number of times, we need to have "deep pockets." We need access to a largeamount of capital. If we are undercapitalized, we may lose all our money on the first

10 trades and not be able to trade anymore. If we cannot trade anymore, we cannotpossibly get our expected results. We do not want to run out of money before theprobabilities take over.

If there is a difference between market price and value, someone with deep pocketsand the opportunity to make a large number of transactions should be able toarbitrage the difference, absent other restrictions.


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Real World Applications

Insurance

One of the most important applications of statistical arbitrage is simple insurance.Insurance companies use statistical arbitrage to determine fair value by trying topredict how much they will pay out in claims. They divide the estimated amount theywill pay out by the number of insurance buyers and add an edge (profit) to determinethe premium they will charge each client. They expect to make the edge they addedon. With enough occurrences (customers) and with accurate estimates of theirprobable aggregate losses, they will make that edge.

Actually, managing the risk of an insurance operation is more complicated than thissimple example suggests, but statistical arbitrage is one of the most importantprinciples of insurance. Insurance companies employ a large number of analysts toensure accurate estimates of what they will need to pay out in claims. Similarly, in thefinancial industry, analysts are employed to project earnings, evaluate takeovers, andstudy other significant events, as well as to determine the value of financialinstruments, such as options.

Gambling

Another industry that bases its business on statistical arbitrage is casino gambling.

The odds (or edge) in a casino always favour the house. Roulette is a good exampleof a casino's use of statistical arbitrage. In roulette, there are 36 numbered spaces plus0 and double 0 for a total of 38 spaces. The fair odds are 37 to 1. If you win on agiven number, you should get your own chip back and 37 more. The house, underUnited States rules, pays only 35 to 1, keeping two chips out of 38 for itself.

Is it possible for the house to lose? Of course! Haven't you seen people sitting atroulette tables with piles and piles of money - at least in James Bond movies? WhenBond won all that money, did the house lose at that table on that night? Absolutely.Do you think the house loses at that table every night? If it lost every night, the house

would not operate that table. It is possible that the house loses at a particular tableon a particular night, but it is highly unlikely that the house loses in general. Thestatistical argument for the house is very convincing.


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An important difference between casinos on one hand and insurance companies andfinancial firms on the other is that casinos can calculate easily and accurately the

probabilities for their losses and then add an edge that almost certainly will protecttheir profits. The expected claims for insurance companies and the true values offinancial instruments are not as easily predicted or calculated. The fact that mostgambling probabilities are easily calculable makes simple gambling examples usefulin understanding the principles of probability.


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Forwards and Futures

There are three keys to understanding options. The first you already know about -arbitrage. The second is forwards or futures. Forwards and futures are financial

instruments that relate present and future prices, and they are very important inunderstanding what options are and how options are valued.

Distributions

Suppose gold is trading at $400 an ounce today. What price will gold be in one year?If we ask many different people this question, we will probably get many differentanswers. Some people will be very pessimistic, predicting a dramatic drop in goldprices. Others will be very optimistic, predicting a large rise in gold prices. A number

of people will fall in between the two extremes. After we accumulate all of theresponses, we will get a picture of where people think gold is going to trade for in ayear. The picture might look something like this:

The numbers at the top of each bar represent the percentage of people who predictthat the price of gold will fall in that price category in a year. Our first bar, centred at$350 an ounce, indicates that 3% of the people surveyed think gold will be around$350; about 7% of the people think it will be around $375; 12% predict a price near$400, and so forth. Many people are clustered in the middle--around $450.

25%

20%

15%

10%

5%

0%

Probability

Gold Price ($/ounce)

3%

7%

12%

16%

23%

13%11%

8%7%

350 375 400 425 450 475 500 525 550

Predictionsof the Priceof Gold inOne Year


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This picture is called a distribution. A distribution is characterised by its mean oraverage - its expected value. The expected value is the centre of the distribution.

When we add up and average the different responses from the people surveyed, wefind that, for this particular example, the mean is centred around $450. (The meancalculated from the actual responses is $453.50.) On average, the people in this groupfeel that the forward price of gold, that is the price of gold a year from now, shouldbe approximately $450.

A distribution also is characterised by its width or dispersion, which is sometimesexpressed as the standard deviation or the volatility. You can see in this distributionthat, even though it is centred around $450, there is a great deal of variation in theresponses. Some people estimate a price as low as $350, and some people estimate aprice as high as $550. The standard deviation measures the dispersion in thedistribution, and for our gold price survey distribution this measure of dispersion isaround $50.

The mean and the standard deviation are two important quantitative characteristics ofthe distribution. Many distributions we encounter in nature and in the financial marketshave a particular shape called a normal or bell-shaped distribution. Returns forcommodities, such as gold, or for Japanese yen or stocks or bonds have underlying

distributions that often are approximately normal. A normal distribution ischaracterised in part by its symmetry about the mean and by the fact that it is high inthe middle and low at the ends. We will assume for purposes of the illustration whichfollows that gold prices are approximately normally distributed.

The normal distribution has some very useful characteristics. In a normal distributionof gold price forecasts, we have 68% confidence that the future price will be withinone standard deviation of the mean. In our forecast distribution, the mean is about$450, and the standard deviation is about $50. Assuming this distribution is normal,we are 68% confident that the future price will fall somewhere between $400 and

$500 - $450 minus $50 and $450 plus $50. Adding the percentage responses in theprice range between $400 and $500 on our bar chart, we find that about 75% of ourforecasters feel the future price will be within one standard deviation of the mean.

We have 95% confidence the future price will be within two standard deviations ofthe mean in a normal distribution. Two standard deviations down from the mean is

Distribution

Mean

Normal Distribution

Standard deviation

Volatility

$450 + $50 = $50068% confidence

$450 - $50 = $40068% confidence

% ForecastResponses Price12% $400

16% $42523% $45013% $47511% $50075%


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about $350, and two standard deviations up from the mean is about $550; we haveabout 95% confidence that the forward price of gold will be somewhere between

$350 and $550. In our forecast distribution, all the responses are within that interval.

Finally, we are 99% confident that the future price will be within three standarddeviations of the mean in a normal distribution. Three standard deviations from themean is $150. We are 99% confident, or almost completely confident, that the futureprice of gold will be within a range of $300 to $600, that is, $150 down from themean and $150 up from the mean. These are very important properties of the normaldistribution.

The figure below illustrates the general shape of a normal distribution if responsescould be any price. It is almost identical to the shape of the bar graph distribution weexamined earlier. We used bars and large price intervals, but if we took smaller priceintervals, say from $350 to $355, $355 to $360, and so forth, the distribution wouldbecome more like a smooth curve. The high point in the middle of the normaldistribution curve is the mean, and the distribution tails off on both sides. Manyunderlying distributions look approximately like this one, and the assumption ofnormality is used as an input for many financial models.

11

Probability

Mean

normal

z value

A Normal Distribution

-4 -3 -2 -1 0 1 2 3 40

0.15

0.30

0.45

normal


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Market Participants

The expected future price of gold is important to anyone who uses the gold spotmarkets and the gold futures and forward markets:

A. InvestorsB. Long speculatorsC. Short speculatorsD. Long hedgersE. Short hedgersF. Arbitrageurs

It is useful to describe each of these market participants, discuss what they are doingin the market, and look at how their actions affect the spot and the future or forwardprice of an underlying commodity or security.

Let's look first at A, our investor. A thinks that gold is a good investment. She wantsto buy gold and hold it long term. When A buys gold, she tends to make the spot priceof gold go up because she is buying gold in the spot market. When the spot price ofgold rises, the future or forward price tends to rise as well.

B, our long speculator, is not a long-term investor, like A, but she thinks gold is goingup and would like to take a position that is going to be profitable if she is right. B can

accomplish this in a couple of different ways. First, B can buy spot gold. If gold goesup in a short period of time, she can sell that gold at a higher price and make a profit.

Alternatively, B can buy a future or forward contract on gold. B's purchase offorward gold tends to make the future or forward price go up; and, just as demandfor spot gold tends to raise future or forward prices, demand for forward gold tendsto raise spot prices. As we will see later, when B buys a future or a forward, she doesnot have to pay out much money. When A buys spot gold, say 100 ounces at $400,she has to pay $40,000, immediately.

Our short speculator, C, thinks that the gold price is going down. He is verypessimistic about the outlook for gold. Like A and B, C has two choices. He can sellgold, or he can sell a future or forward contract on gold.

If C happens to own gold, it will not be very difficult for him to sell it. Alternatively,if he is in a position to borrow gold relatively easily and sell it short, he might do that.


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But if he does not have direct access to gold, he probably will find it easier to sell afuture or forward contract on gold.

Short speculators like C affect the price of spot gold. The effect on the spot price isclear if the short speculators own gold and want to shift their investment tosomething else. If they are not trading in the spot market, the direct effect of their salewill be on the forward or future price, making it trade at a lower price.

D is someone who needs gold a year from now. D might be a dentist or a jeweller -someone who has all the gold he needs right now but who wants to be assured of aprice for gold a year from now. There is no advantage in buying the gold now andspending a large sum of money to store it. D probably will want to buy a future or

forward contract on gold, which will tend to push up the future or forward price.

E, our gold producer, is very busy digging gold out of the ground. He does not havemuch gold right now, but in a year he expects to have a large store of gold to sell. Hewants to make sure his hard work pays off and that he sells his gold at a good price.He cannot sell spot gold because he does not have it. However, he can sell a future orforward contract on gold now.

E is likely to affect the future or forward price, and since he is selling, his actions will

tend to make the future or forward trade at a lower price.

Our last market participant is the most important in some ways. F has no opinionabout gold's value and no opinion on the likely direction of gold prices, but he is veryaware of gold price relationships. Even if F did have a personal opinion about thevalue of gold or the direction of gold prices, his opinion would not affect his financialtransactions. F is an arbitrageur, and his actions in the marketplace tie the actions ofall other market participants together. F ensures the spot price of gold and the futureor forward price of gold are in their proper relationship. If the prices are notappropriate relative to one another, he will try to profit by buying in the cheap market

and selling in the expensive market. The actions of arbitrageurs are extremelyimportant in seeing that prices are kept in line.


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The Arbitrageurs Role

One of the reasons for emphasising the role of the arbitrageur in a discussion ofoptions is that arbitrageurs are an important factor in determining the spot/forwardprice relationship, and the first step in finding the value of an option is finding thevalue of the future or forward price.

If gold is trading at $400 today, at what price will someone agree today to buy or sellit one year in the future? We used a survey to forecast the price of gold in a year, butthere is a rational, deterministic relationship between spot price and the forward priceof gold that does not rely on an opinion survey or on anyones price forecast.

Suppose that the carry cost, that is, the cost of money or the cost of buying somethingand financing it for a year, is 10%. We will have to pay $400 to buy gold today.Holding the gold position will cost us 10% for a year, because if we had not spent the$400 on gold we could have left it in the bank and it would have continued to drawinterest at 10%. To own gold a year from now, it will cost us more than the $400 wepay now. In fact, it will cost us $440:

FUTURE OR FORWARD PRICEspot price of gold + cost of carry = future or forward price of gold

$400 + (10% x $400) = $440

If the present price of gold is $400, the future value of gold in one year is $440. Byfuture value, we do not necessarily mean the price gold will sell for in the future; it isthe value that future gold has to trade for today because of the cost of carrying goldfor a year.

Present value and spot value are the same thing. We have been talking about spotgold, but we can also talk about the spot value or current market value of a stock, abond, a currency, or some other underlying commodity or financial instrument.

Future value is the value of the future or the forward. There is a difference betweenfutures and forwards, but the difference is in how they are traded, not in howthey are valued. At this point we will treat futures and forwards as though theywere interchangeable. Later, we will discuss the practical differences between futuresand forwards.

Present (spot) value

Future value

Spot price+ Cost of carry

= Future or Forward price


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Basis

Basis is the difference between the forward value and the spot value. For manyproducts, the basis is defined as the cost minus the benefit of holding the underlying.

BASISbasis = cost of underlying - benefits of underlying

In our gold future value example, we have considered only the financial cost ofcarrying gold at 10%, because there are no economic or financial benefits fromowning gold. (It might give you a feeling of comfort to own gold, but we are notcounting that.) In some other products, there are benefits as well as costs.

For an equity, we consider the cost of carry, that is, the cost of the money to buy theequity, and we consider the possible benefit of receiving a cash dividend. Not allstocks pay dividends, but if they do, the dividend is a benefit.

In currency markets, the cost of carrying a foreign currency is the investor's domesticinterest rate. If a U.S. dollar-based investor wants Swiss francs (CHF), she has to giveup dollars to buy the CHF. When she gives up the dollars, she either takes them outof an interest-bearing account, which means she loses the interest income, or she

borrows the dollars from a bank and pays domestic interest. Either way she has acarry cost in the domestic currency, and that is the dollar cost. On the other hand,when she gets the CHF, she can invest them in a CHF account and earn interest atSwiss rates. The interest on the CHF account is a benefit of owningthe CHF.

Examining costs and benefits illustrates an important feature of the bond market.Often, traders or portfolio managers have to borrow money to carry bonds, Theyborrow money in the repo (repurchase agreement) market, posting the bonds ascollateralfor the loan. The repo rate is the term used for the interest rate charged on

such a loan. On the other hand, while they own the bond they are entitled to anycoupon interest that accumulates. For a bond, the repo rate is the carry cost of thebond, and the coupon is the benefit.

Stocks

Currencies

Bonds

Basis equalsCosts minus

Benefits


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The basis in each of these examples consists of the cost minus the benefit of owningthe underlying from the spot date to the forward date.

For gold, the basis is simply the cost of carrying the gold; For a stock, the basis is the cost of carrying the stock minus the dividends; For a currency, the basis is the cost of the interest on the domestic currency minus

the benefit of the interest earned on the foreign currency; For a bond, the basis is the cost of borrowing at the repo rate minus the benefit

of the coupon payment.

Spot Versus Forward Arbitrage

Applying the idea of basis to arbitrage examples will help clarify some spot andforward pricing relationships. Suppose that gold is trading for $400, and the forward,which we said before should be trading for $440, is trading for $450. It will cost us$400 plus 10% of $400 to buy gold and carry it for one year. That means we can buygold today by borrowing money from the bank, and in one year we have to repay thebank $440. With the gold forward trading today for $450, we can sell it and make aprofit. In one year, we do two more transactions:

Deliver gold for $450 an ounce to complete the forward contract

Repay bank -$440 an ounceNet difference $10 profit

With a spot price of $400 and a forward price of $450, we can buy gold todayand sell it forward, making a certain profit of $10 per ounce. This is a risklessprofit - an arbitrage profit - which we make by simultaneously buying and sellingequivalent instruments.

What are the equivalent instruments? It may seem as though we bought gold,but because we bought the gold by borrowing money and carrying it for a year, we

effectively bought the forward, paying $440 an ounce to own gold in one year.Simultaneously, we sold the actual forward, the one that is tradable, for $450.These are equivalent instruments, and they should be priced identically. Whenthe equivalent instruments do not trade at the same price, we can make anarbitrage profit.

16

Today

A Year from Today

Borrow $400 from the bank

Buy spot gold

Sell $450 forward gold

Deliver forward gold take in $450

Repay bank $400 +

($400 x 10%) = $440

Make a sure $10 profit


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Suppose that spot gold is trading at $400 and forward gold is trading at $420 - lowerthan the $440 forward price we calculated earlier. Forward gold at $420 sounds like

a bargain - let's buy the forward and simultaneously sell the spot to make a certainprofit of $20.

If we do not own gold, we must borrow it and sell it for $400, that is, sell spot goldshort. We can then invest the $400 from the sale of the gold. The bank will pay us10% interest, and we will have $440 in the bank a year later. Then, we will do twomore transactions to close out the arbitrage:

Withdraw money from the bank $440 an ounceAccept delivery of forward gold at -$420 an ounce

Net difference $20 profit

We have to return the gold we borrowed, and our profit may be reduced if we haveto pay a fee for borrowing the gold. In the first example, the gold forward wasexpensive compared to the $440 it would cost us to buy gold and carry it for a year.So we sold forward gold, bought spot gold, and carried it for a year. In the secondexample, the gold forward was cheap, so we bought forward gold, sold spot gold, andearned interest on a bank deposit.

In both cases, we arbitraged a profit with no price risk. In fact, we exchanged pricerisk for basis risk. It is important to emphasise that, in both cases, forwardtransactions were agreed on at the beginning of the period, and the actualtransactions were done at the end.

Index Arbitrage

Let's look at another example of arbitrage - one that many people have heard about,called index arbitrage. To understand how it works, we assume that a stock index is

trading at $200 or 200 index points, and the carry rate is 10%. The stocks in thisindex have a dividend flow of $8 or 8 index points over the next year.

We want to determine where the forward or future on the stock index should betrading. The carry cost on the index is 10%; 10% of $200 is $20, so it is going tocost $20 or 20 index points to carry the index for the next year. On the other hand,

$200 x 10% = $20

Spot+ Basis= Future


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we are entitled to an $8 or 8-index-point dividend flow over the next year, whichpartially offsets our carry cost.

$20 (Cost) - $8 (Benefit) = $12 (Basis)$200 (Spot) + $12 (Basis) = $212 (Future)The index future or forward should be trading at $212.

Suppose the one-year future is actually trading at $230. As in the first goldexample, the future is priced too high. Let's sell the future and buy appropriatequantities of all the stocks in the index - that is how we get to own the index.

To keep the numbers small, let's assume it will cost us $200 to buy the necessary

shares of each of the index stocks. It will cost us $12 (net of dividends) over thenext year to carry the index. At the end of one year, we will have $212 in netcosts, but we receive $230 for selling the future. Consequently, we will make anarbitrage profit of $18 or 18 index points.

Let's look at this example again with the forward or future price of the index toolow at $210. Suppose we have two long-term investors, A and B, who view themarket differently. A does not believe in futures. He is a long-term investor, andhe cares only about stocks. With the future trading too low at $210, A does

nothing - he simply continues to hold stocks.

B, on the other hand, decides to sell all of his stocks and buy the futuresimultaneously. B invests the $200 from the sale of the stock at 10%. Does thismean that B is not a long-term investor anymore? No. B is still a long-term investoreven though he sold his stock, because he immediately bought the equivalent of hisstock back by buying the future. B's new position illustrates that owning the futureis cheaper for B than owning the actual stock.

Let's assume the index is still at $200 at the expiration of the future in one year.

A never sold his stock. His stock was worth $200 at the beginning; it is stillworth $200. On the other hand, he received $8 worth of dividends, so from hispoint of view, he made 4% over the year.

How does B look at his position? B gave up the dividends but earned $20 ininterest on the proceeds of the stock that he sold. B is going to buy back his stock

With index at $200 at

expirationA

A keeps his stock He makes $8 in

dividends

B B sells his stock Invests $200 from the

sale at 10% interest

In a year, B withdraws$220

Pays out $210 forstock under theforward contract

B makes $10

Carrying the stocks inan index

Index arbitragecash flowsBuy Stock -$200Cost of basis -$12Sell future +$230Arbitrage profit $18


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at the $210 forward price he agreed to at the beginning of the transaction. Hisportfolio will consist of $200 in stock plus $10 in cash. A has $200 in stock plus

$8 in cash. By switching from stock to the underpriced future, B got a 5% returnover the year, 1% higher than As return.

What happens if the price of the index ends up at $210 at expiration? A is happy,because his stock is worth $210 and he got a 4% dividend flow, so he made 9%over the year. What about B? B's stock also is worth $210, but he still has that$10 in cash in his pocket, so B made10% - once again, 1% better than A.

Suppose something unfortunate happens: the index falls to $190 at the end of theyear. The stock positions of A and B are the same as before. A collects $8 in

dividends; B collects $20 in interest less the $10 from the spot/futures arbitrage($210 - $200), or a net of $10.

A started with $200 of stock, but now only has $190 worth of stock plus $8 individends, for a loss of $2 over the year-that is a -1% return. B has $190 worthof stock, but he has $10 in his pocket. Therefore, his total portfolio is still worth$200, and his return is zero. He may not like a 0% return, but it is better thanA's -1% return. In fact, no matter where the index ends up, B's return alwayswill be 1% better than A's.

Index arbitrage has gotten a bad name in the press - largely from people who do notunderstand the pricing relationships that lead to arbitrage. In our simplified example,we have tried to portray the reasons for this type of arbitrage realistically.

We see in our gold and index arbitrage examples that there is a rational relationshipbetween the price of the spot and the price of the forward. Regardless of what anyinvestor thinks the price of gold will be in a year, the actions of arbitrageurs - thepeople who monitor the relationships between the spot and the forward - will ensurethat the spot and the forward have the appropriate relationship to each other.

That does not mean the arbitrageurs determine the value of the spot or the value ofthe forward. While hedgers, speculators, and investors affect the spot price or theforward price, arbitrageurs ensure that the rational relationship between the spot andforward is preserved. We have emphasised this relationship because knowing how tovalue futures and forwards relative to spot is the first step toward knowing how tovalue options.

B is better off nomatter what happensto stock prices

5%

4%

3%

2%

1%

0%

A B


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The Difference Between Forwards and Futures

Although we have used the terms forwards and futures interchangeably, some of thedifferences between forwards and futures are significant.

Forward Contracts

A forward contract is an agreement between two parties made independently of anyorganised exchange market. The forward contract is a stand-alone, customisedcontract. It can be based on any amount of any good. It can be written for settlementat any time and at any price. It can be, as an extreme example, for delivery of 37,000

gallons of vodka in 52 days at $2 a gallon. The terms can be virtually anything solong as both parties agree to them.

One party agrees to sell the vodka at the contract price, and the other party agrees tobuy it at that price. Since the contract can be for any amount of any good, at any time,and at any price and since each party depends on the other party to meet contractualobligations, the forward contract cannot be traded freely with other counterparties.Forwards are not fungible; that is, one forward contract is not interchangeable withanother contract that has similar terms, and the present value of the forward contractis not easily converted to cash.

Each party has to be able to trust that the other party will uphold its side of theagreement, because the two counterparties are exposed to each other's ability andwillingness to perform on the contract. There is credit risk associated with a forwardcontract, which must be controlled. Often, the credit risk is controlled by banks,which act as intermediaries, assuring the performance of their clients. That is, if aclient fails to fulfil the agreement, the bank assumes the obligation. A very importantpoint to remember about forwards is that no money is exchanged by the parties untilthe actual exchange of goods or financial instruments at settlement. There is no

interim cash flow, unless there is a mark-to-market agreement between a party to thecontract and its bank.

ForwardsNo exchange marketCustomised contract

Nonfungible

Credit risk managedby banks


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Futures Contracts

A futures contract, in contrast to a forward, is an agreement between two partiesmade through their agents on an organised futures exchange. The parties who tradeon the exchange can represent themselves or they can represent customers. There arespecific rules and regulations that set the terms of the contract and the procedures fortrading. The contract is for a specific amount of a specific good to be delivered at aspecific time determined by the exchange, and the price discovery usually isdetermined by open outcry in a trading pit, that is, by people yelling and screamingat each other in a large room.

If you have been to a futures exchange, you may have seen just that - people yelling

and screaming and calling out numbers - yet the process is actually quite organised.This is the way prices are discovered and goods are exchanged in many efficientmarkets. Some markets now use computer systems which expose bids and offers to alarge number of potential traders at many locations, but the principle of bringing bidsand offers together is the same.

Several features of futures markets are worth noting.

First, a futures contract is always for a standardised amount of a specific good.We cannot set our own contract amount, say 37,000 bushels of soybeans. If thesoybean futures contract set by the exchange is 5000 bushels, we can trade anynumber of contracts, but each contract will be for 5000 bushels. The exchangealso specifies the quality of soybeans that can be delivered in fulfilment of thecontract (Grade A?) and exactly where the soybeans are to be physicallydelivered. Although we may not be interested in the details of the soybean futurescontract, every futures contract must be uniquely specified in order to be atradable instrument. Since all terms are determined in the futures contract, thecontracts can be traded freely with other counterparties. We can buy the contractfrom one person, sell the contract to another, and wash our hands of the

commitment. We can eliminate our obligation, because, with futures, the exchangetakes the other side of the contract.

The exchange is the ultimate counterparty for all futures trades, so the onlycredit risk is the creditworthiness of the exchange. Once a trade is completed, thetransaction is passed to the exchange clearing corporation. The futures contract

Organised futures exchange

Price discovery by open outcry

Standard contract specifiesamount and type of underlying

Exchange is counterparty

Margin

Futures


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buyers and sellers have agreements with the clearing corporation. Ultimately, if wetake delivery rather than offset our contract, the exchange will select someone

who is short the contract to make delivery. The party making delivery does nothave to be the party who sold us the contract originally. After the trade settles, theoriginal parties to the trade lose any direct tie to each other based on the trade.No credit intermediary is necessary, but performance bonds, sometimes calledmargin, must be deposited to ensure each party meets its obligations.

The exchange clearing corporation has to stand behind the creditworthiness ofits members, so it asks everyone for a deposit. Customers make a deposit withtheir broker, and the broker passes the deposit to the clearing corporation."Margin" is a misleading name for this deposit because people confuse this

performance bond margin with stock margin. While both types of margin protectcreditors, that is all they have in common. Stock margin is actually collateral fora loan. Futures margin is a good faith deposit, demonstrating ability andwillingness to meet contractual obligations.

How creditworthy is a clearing corporation? The Options Clearing Corporation,which is responsible for all exchange-traded securities options in the United States,has a AAA credit rating. Other clearing corporations would probably get similarratings if they sought them. There have been instances when futures investors havelost money due to credit problems after a trade has settled, but most of theseproblems have involved their brokers rather than the clearing corporation. Creditproblems in the U.S. futures markets are not a common experience.

Forwards Versus Futures

Let's look at an example that illustrates some of the differences between forwards andfutures. Hayashi agrees to buy 5,000 troy ounces of gold from Tanaka at $400 perounce exactly 57 days from today. Hayashi buys the forward, Tanaka sells the

forward - no money changes hands today. In 57 days, Hayashi pays Tanaka $2million and Tanaka gives Hayashi 5,000 troy ounces of gold regardless of what thespot price of gold is on that date. In other words, in 57 days, there is an exchange ofcash for gold.


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Now, let's look at a futures contract. On the floor of the New York COMEX, thecommodity exchange where gold is traded, Franz sells Heidi 50 gold futures

contracts. Each gold futures contract represents 100 troy ounces. So 50 futurescontracts cover 5,000 troy ounces of gold for near-term settlement at $400 per troyounce. The Hayashi/Tanaka forward agreement is basically the same as theHeidi/Franz futures agreement. No money changes hands today in either case, butFranz and Heidi must post margin (performance bonds) with the exchange.

Every day Franz's and Heidi's accounts will be adjusted for daily gold market movesuntil they close out their futures positions. If the price of gold goes up, a cashvariation margin payment is taken from Franz's (the seller's) account and an equalamount is deposited to Heidi's (the buyer's) account. If the price goes down the next

day, the cash transfer goes from the buyer's account to the seller's account. Thesevariation margin flows mark the position to the market each day.

What happens at settlement in 57 days, assuming that neither Franz nor Heidi hasoffset (traded out of) his/her position before that time? Franz delivers 5,000 troyounces of gold at the exchange's directions, and he is paid the spot rate. Note thatFranz does not receive the rate agreed on when the trade was made. He gets the spotrate in effect at delivery. Heidi pays the spot rate and receives 5,000 troy ounces. Thedaily flow of funds between the margin accounts over the life of the contracts makesup any difference between current market conditions and the original contract terms.

The net result of forward and futures transactions is identical, but the cash flowsduring the period from the original trade to expiration usually cause a difference ininterest earnings or interest costs. Statistically, rates are not expected to change to thesystematic advantage of the buyer or the seller.

At contract settlement

Franz delivers 5,000

oz. Gold at spot rate

Heidi pays the spotrate for 5,000 oz.

On the trade date

Franz sells Heidi

50 gold futurescontracts @ 400/oz

Franz and Heidi postmargin; account p/l

adjusted daily

A futures transaction

5,000 oz.x $400

$2,000,000

50 contracts

x 100 oz.5,000 oz.

Options


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Options

Introduction

Now we are ready to define options and compare graphical representations of spotand forward values with option values.

Lets start with a graph of the spot and forward values of an ounce of gold.

The green line represents the price of spot gold. When we buy an ounce of spot goldat $400, what happens to the value of our position as the value of gold changes? Ifthe spot gold price increases by $20, the value of the position increases by $20.Similarly, if the spot price of gold decreases, the value of the position decreases.

The blue line represents the one-year forward value for any given spot price of gold.If interest rates are 10%, the one-year forward value is $440 when gold is at $400.The vertical distance between the two lines is the basis. Notice that the basis increasesas the price of gold increases and decreases as the price of gold decreases. If gold is at$500, it costs $50 to carry the gold for a year. If gold is at $300, it costs $30 to carrythe gold for a year.

PositionValue

Spot Price400

Spot

Basis

Forward

ForwardValue

Spot Pricestrike

Forward

Option

Spot andForward

$400 + ($400 x 10%) = $440

$500 + ($500 x 10%) = $550

$300 + ($300 x 10%) = $330


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To the graphical representation of the forward price, we add the orange line. Theorange line tracks the forward in part, but it never drops below zero; we are protectedfrom any losses associated with the forward. This protection comes from holding acall option position. The orange line illustrates the forward value of the call. Weknow how to value a forward, and soon we will learn how to value the lossprotection associated with a call.

This protection feature is the third key to understanding options. We have examinedarbitrage and futures or forwards. When we understand this protection feature, wewill have a solid understanding of options.

Call Options

UBS defines a call option as a substitute for a long forward position with downsideprotection. The standard text book definition of a call, however, is the right but notthe obligation to buy a spot instrument or a commodity at a specific price at or beforea particular time in the future. While these two definitions are equivalent, the UBSdefinition emphasises the components of option value.

The right but not the obligation is equivalent to the protection in our definition.To buy is a long position. A spot instrument or a commodity at a specific priceat or before a particular time in the future is the specification of a transaction in thefuture, which corresponds to our term forward.

Options Versus Forwards

Lets look at an example that distinguishes options from forwards. Suppose we wantto buy a new car. We go to our favourite ZoomMobile* dealer and say How muchis this car? What is your best price? The dealer says, Ill give you my best price, but

you have to buy the car today. The best price is $50,000.

We do want to buy the car but we are not really sure that we couldnt get the car ata lower price from some other dealer. On the other hand, if we go away to look fora better price, we may lose out on the $50,000 price. That may be the best we coulddo after all. So we say to the dealer, "Heres $100. This is not a deposit on the car.

Call = right to buy

*Not a real brand!


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To keep the $100, just hold the price for one week. If I want to buy your car for$50,000, Ill come back within a week and do so. But whether I do or not, you keepthe $100." If the car dealer agrees, we have bought a "one week $50,000 call" on theZoomMobile.

Whats the next step? We visit other car dealers to try to find a better price for the samecar. If we can find a dealer who will sell us the car for less than $50,000, we will buy itfrom the dealer with the lowest such price. In this case the first dealer keeps our $100and we do not exercise i.e. use, our option. If, however, we cannot find another dealerwith a better price, we can return to the first dealer and buy the car. The car will cost usa total of $50,100 - $50,000 for the car and $100 for the option. Buying the car fromthe dealer for $50,000 is called exercising our option.

Suppose in fact we go back to the original dealer because we were not able to findanyone who would sell the car for less than $50,000. When we get there the dealersays, "That car you wanted is now priced at $55,000." We reply "Perfect! Now I canbuy the car from you for $50,000 and be very happy that I didnt have to pay thehigher price. Furthermore I can either keep the car at the bargain price of $50,000,or I can sell it to someone else for more than $50,000 and make a profit!"

We said we wanted to contrast the option with a forward. So suppose that we insteadbought a one week forward on the ZoomMobile. This means that we agree todaywith the dealer that we will come back in one week and buy the car for $50,000. (Wemight want to do this so that we would have one week to arrange financing for thecar.) A week later, we find that prices have risen to $55,000. This is no problem forus because we have "locked-in" the $50,000 price. Also we did not pay the $100 forthe option so we are really ahead compared to buying the option.

But what if we come back a week later and there is a big sign in the dealers windowsaying "All ZoomMobiles 10% off!" The price of the car is now $45,000 ($50,000minus the 10% discount of $5000). But we have agreed with the dealer that we will

buy the car for $50,000! We are probably angry with ourselves, or at least full ofregrets, that we did not spend the relatively small amount of $100 to buy the optioninstead of the forward. Why? Because we would now have the opportunity toparticipate in the lower price. We would not exercise our option. Instead we wouldlet it expire (go unused). We would have paid just $45,000 plus the $100 for theoption. Wouldnt we also regret having paid the $100 for the option? Maybe we

$50,000$45,000

ALL ZoomMobiles 10% OFF


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should have not done anything, i.e. just come back after one week and buy the car atwhatever price it is offered at that time? Thats fine if the price stays the same or goesdown, but if it goes up (as in the first example) we could end up paying a lot morethan we had planned. If we dont want to take the risk, we need the protection theoption gives us.

The difference between the option and the forward in this case is that the option letsus participate in an advantageous price move - we have the option to buy at a lowerprice. The option assures us that we will pay, at most, the agreed-upon price. Withthe forward, we will pay the original, lower, price if the price goes up. But if the pricegoes down, we will be obligated to pay the original, higher, price we agreed to whenwe bought the forward.

Option Terminology

Let's define some of the terminology that we will use throughout our discussion ofoptions. The spot instrument or commodity that is the subject of a forward, future,or option is called the underlying. In the case of the car, the ZoomMobile is theunderlying. The specific price at which we can buy the underlying, $50,000, is calledthe exercise or strike price. The date in the future that we agreed on for a possibletransaction is called the option's expiration date. In our ZoomMobile example, the

expiration date was one week from the day of our agreement.

If we choose to buy the car, we exercise our option. There are two common styles ofexercise - American and European. An American option allows you to exercise theoption at any time prior to expiration. The European option allows you to exercisethe option only on the expiration date. For example, if the car dealer says, "Comeback anytime during the week," we hold an American option, giving us theopportunity to exercise at any time. If he says, "I only work on Saturdays," then wehave a European option.

UnderlyingZoomMobile

Strike$50,000

Expirationone week

Exercise StyleAmerican/European


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Continental Etymology

Many investors wonder about the origin of the names for "European-style" options, which can be exercised only at expiration, and"American-style" options, which can be exercised at anytime duringtheir lives.

The genesis of these terms can be traced to an early option article byNobel laureate Paul Samuelson*. In this article, Samuelson calls awarrant that is exercisable only at the end of its life a European

warrant because of its similarity to call options traded in Europe. Hecontrasts these European instruments with the pre-exchange-tradedoptions market in the United States where put and call options couldbe exercised at the holders discretion.

Samuelson has revealed that his choice of the names European andAmerican for these exercise styles was not based solely on hisobservations of the exercise patterns in European and Americanmarkets. As an inside joke, he chose the name American for the more

complex option because he had grown tired of Europeans telling himthat options and warrants were too complex for unsophisticatedAmericans to understand. The importance of Samuelsons paper,which anticipates many features of the later Black-Scholes optionformula, extends well beyond its contribution to optionsnomenclature.

* Paul Samuelson, "Rational Theory of Warrant Pricing," IndustrialManagement Review 6 no. 2 (Spring 1965); also in P.H. Cootner,editor, The Random Character of Stock Market Prices, (Cambridge:

M.I.T. Press, 1964), 506-532.


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Put Options

So far we have been discussing a call - a long forward with downside protection. Butthere is another type - a put option. UBS defines a put as a substitute for a shortforward position with upside protection. If we are short a forward, and the price goesup, we will lose money. A put protects us against a loss on the way up. What is a shortposition? A short position is one that increases in value as the market price declines.

There are two types of short positions. Agreeing today to sell gold one year fromtoday means we have a short forward position. If we expect to have gold to sell oneyear from now, because we are gold producers, we have a covered short position. Ifwe are selling in anticipation of a lower price without holding gold, our position is

uncovered. It is possible to take a short position in the spot market, too. In order toshort spot, we need to borrow the underlying to make delivery on our sale. We willhave to buy the underlying back later and return it.

We would like to cover our short position at a cheaper price. Remember, "buy low,sell high." In the case of a short position, first we sell high, then we buy back low,still measuring our profit by the price difference as we would in the more familiarlong position. The danger in a short position is that we may sell high and have to buyback higher. If prices go up, we want to be protected against potential loss.

Buying a put option is an alternative to taking an outright short forward position. Theclassic book definition of a put is the right but not the obligation to sell a spot instrumentor a commodity at a specific price at or before a particular time in the future. "The rightbut not the obligation" represents the "protection" of our definition. "To sell"corresponds to "short;" "a spot instrument or a commodity at a specific price at orbefore a particular time in the future" means the specifics of a contract in the future,which corresponds to our use of the term "forward."

A jeweller gives her customer a "money-back" guarantee on a diamond ring. The

customer buys the ring for $10,000 and has the right, but not the obligation (doesthis sound familiar?) to return the ring to the jeweller and get the $10,000 purchaseprice back. The dealer has given the customer a put option. Perhaps the ring is apresent and the person the customer wants to give it to doesnt want it (sad story).Well at least our, possibly heart-broken, customer wont have a monetary loss becausehe can return the ring for the full $10,000. If the price has fallen in the meantime to$9,000, he has avoided a $1,000 loss.


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On the other hand, if the price has risen to $11,000, he could now sell the ring tosomeone else, possibly the jeweller, or maybe some other hopeful suitor, for a profit.Note once again the protection and the opportunity the option has given to the ownerof the option.

To contrast a put option with a forward lets consider a different example. Youremployer has told you that you are going to be transferred in 3 months to anotheroffice and this will require you to move house. You find a buyer for your house andagree that you can live there for another three months but at the end of the 3 months,you will sell the house for $1,000,000 (you are an important executive!). Threemonths from now it turns out that house prices in your neighbourhood haveincreased dramatically and your house is now worth $1,100,000. Unfortunately you

will not be able to participate in this higher price because you have agreed to sell itfor less.

Suppose instead that your employer wants to make you feel better about moving andsays, " I know you are unhappy about moving so I will agree to buy your house fromyou for $1,000,000 3 months from now, if you havent already sold it." You can nowadvertise your house at a higher price. If someone offers you $1,100,000 you can sellthe house to them rather than your employer you are allowing your option to expire and have a profit of $100,000 compared to the $1,000,000 price you wereguaranteed. What if instead, after putting your house on the market, the best bid youreceived was $950,000? Then you can exercise i.e. use, your option and make youremployer buy the house for $1,000,000, thereby avoiding a $50,000 loss. Youremployer gave you a put option.

In summary, a call is a substitute for a long forward with downside protection, whichassures a maximum price and enables the holder to pay less if the price declines. Aput is a substitute for a short forward with upside protection, which locks in aminimum price and enables the holder to sell for more if the price rises.

Put = right to sell

Puts and insurance

Options as Risk Management Tools:


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p gBreak-Even GraphsWhen we buy properly-priced forwards, no money changes hands. However, it isdifferent with options. When we buy options, we have to pay out money. Why do wepay out money? This is a fairly complicated question, so we are going to look at onesimple aspect of it in this section - namely, what we can expect at expiration when webuy an option. We are going to look at options in terms of risk management, usingbreak-even graphs for illustration.

Buying a Call

First, let's look at a call break-even graph where we have purchased a $100 strike callfor $6 - meaning we have paid $6 for the right to buy the underlying at the strike price

of $100.

In the figure, we, as the call buyer, will break-even when the underlying reaches $106,because if we exercise our option and buy the underlying for $100 and then sell it forthe current market price of $106, we make $6. However, we paid $6 for the calloption, so we just break even on the whole transaction.

At the underlying price of $100, we lose all the money we paid for the call option.We paid $6, but the underlying ended up at $100, so we do not want to exercise theoption. We can exercise, but, since the underlying is trading for $100 anyway, it doesnot matter whether we exercise the option to buy the underlying for $100 or simplybuy it for $100 in the market. At any underlying price below $100, of course, we do

10

0

-6

Profit/Loss

Price of Underlying

80 100 106 120-10

Break-evenof a LongCall

We pay out $6 for a$100 strike call

$100 strike+$6 premium

$106 break-even

Underlying at strike-$106 effective cost+$100 value

-$6 loss

Underlying at $107-$106 effective cost+$107 value

$1 profit


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not exercise the option - we do not want to exercise the right to buy something for$100 that is worth less than $100. However, above the underlying price of $100, theoption has value. Above the underlying price of $106, the option has a positive netpay off. In fact, the option position profits dollar for dollar above $106. For example,if the underlying is at $107 and we pay $106 total for something worth $107, wemake a $1 profit.

If, at expiration, the price of the underlying is at $101, we will exercise our option,even though we will show a net loss of $5. A $5 loss - while not desirable - is moretolerable than a $6 loss.

Buy option ($6)

Exercise option ($100)Sell spot $101Net profit (loss) ($5)

At an underlying price of $110, we make a $4 profit overall:

Buy option ($6)Exercise option ($100)Sell spot $110Net profit $4

Leverage and Risk

Options have been much maligned since their appearance in ancient times, and, as aresult, option myths occasionally overshadow the helpful function options can servein the market place. One option myth is that high leverage means the same thing ashigh risk.

When we talk about leverage, we mean "an investment or operating position subject to

a multiplied effect on profit or position value from a small change in sales quantity orprice." Options traditionally have been associated with high leverage. When we buy the$6 call to get the right to buy stock at $100, we are highly leveraged because we arepaying only $6 for the right to buy something selling for $100. Is it also true, however,that we have a high-risk position?

Option MythThe myth that highleverage means high risk


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Suppose we want to buy a stock for $100, but we buy a call with a strike price of$100 for $6, instead - and buy a $100 T bill* priced at $98 today for a total outlayof 104. In a year, if the price of the stock is above $100, we can exercise our optionand buy the stock with the proceeds from the T bill. The call provides leverage - onecall is covering an entire $100 share of stock - but we do not have high risk. In fact,we have used the leverage of the call to lower our risk. We can understand thereduction in risk by comparing this strategy to simply buying the stock.

What happens if we buy the stock outright? We could lose much more than the optionpremium. If we buy the stock at $100 and it ends up at $90, we will lose $10. Bybuying the call for $6, and the T bill for $98 we end up with enough money in ouraccount to buy the stock at $90 and have some $10 cash left over.

Why, then, do people say that buying options is risky? Because options can be usedin risky ways. For instance, we could be greedy and use the $98 to buy junk bondsor to gamble at the black-jack tables - instead of buying the T bill - thereby increasingour risk. Or we could buy 16 calls with our $100 for a total of $96 invested in calls.If the price of the underlying stock does not exceed $100 at expiration, we will notwant to exercise our options, they will expire worthless, and we will lose the entire$96 - even though the stock may be only a dollar or two below $100. By spendingnearly all of our money on calls, we can use the leverage of the calls to increase ourrisk. It is possible to use calls to take on risk, but it is an investment choice, not an

unavoidable outcome.

Selling a Call

We have been talking about buying a call option. Now, let's sell a call option, usinga short call position break-even graph to illustrate the results. At expiration, with theprice of the underlying stock above $100, the option buyer exercises the call option,and we have to sell her the stock for $100.

For example, if the stock is trading at $103 at expiration, the call buyer will certainlyexercise her right to buy the stock at $100 - and we will be obligated to sell her the stockfor $100. If we do not already own the stock, we have to go to the market, buy the stockfor $103, then turn around and sell it to the call buyer for $100 - for a $3 loss on thetransaction. Of course, we are compensated for the loss by taking in a $6 premium for theoption; so at $103, our net profit is $3.

Stock price at expiration:Above 100 -Exercise optionBelow 100 -Buy stock

in market

High Leverage and High Risk

16 number of callsx$6 price of calls$96 amount invested

in calls

* A T bill is sold at a discount to its face value. The differential 100-98 is interest earned on the bill.

We take in $6 for a $100strike call

+$6 sell call+$100 stock called

-$103 buy stock$3

High Leverage and Low Risk$100 stock price

(6) call price

(98) T bill pricepay 104 today


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We receive $6 for undertaking the potential obligation to sell the underlying at thestrike price. We break even when the underlying reaches $106 and lose dollar fordollar as the stock rises above $106. If the underlying is above $106 at expiration, wehave not received enough of a fee to cover the risk we took on in this instance. If theunderlying is lower than $106 at expiration, we make a profit.

Buying and Selling a Put

Let's look at the break-even graph of a put option. Let's buy a $100 strike price putfor $6. What have we bought? We have bought the right to sell the underlying for$100, and we have paid $6 for this right.

We will break even when the underlying reaches $94. Why $94? We will exercise ouroption only if the underlying is below $100 at expiration, because we do not want toexercise our option to sell the underlying at $100 if we can sell it at a higher price inthe market. As the underlying falls below the strike price of $100, we, as the ownerof the long put, begin to benefit from the price decline. Since we have paid $6 for this

option, the underlying will have to drop at least $6 below the exercise price or strikeprice of the option for us to break even.

As the underlying continues to fall below $94, we continue to profit dollar for dollaras illustrated in the break-even graph of a long put.

Underlying at break-even+$6 sell call

+$100 stock called-$106 buy stock

0

10

0

6

Profit/Loss

Price of Underlying

80 100 106 120-10

Break-even of aShort Call

Buying a put

We pay out $6 for a $100strike put

Underlying at break-even-$6 buy put

+$100 sell stock-$94 buy stock

0


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Again, if the underlying is at $100 or higher at expiration, we lose all the money wepaid for the put. However, below $100, we start to recoup our option premium and,at even lower prices, we earn a profit. Even if the underlying is $98 at expiration, forexample, we will exercise our $100 put, recapturing $2, and netting a loss of $4 onthe transaction:

Buy option ($6)Exercise option $100Buy spot ($98)

Net profit (loss) ($4)

Now, let's become a put seller. We may be obligated to buy the underlying at $100 ifthe put buyer chooses to exercise his option - but the put buyer will not exerciseunless the stock price at expiration is below $100.

Obviously, if the underlying is trading for less than $100 in the market, we do notwant to buy it for $100; but if we sell a $100 put option, we have the obligation todo just that. However, we take in a $6 fee or premium for undertaking this potential

obligation. Because we receive $6 for the potential obligation, we break even whenthe underlying hits $94, and we lose dollar for dollar below $94. The graph for theshort put position, shown on the next page, is the mirror image of the longput position.

Selling a put

We take in $6 for a $100strike put

10

0

6

-10

Profit/Loss

Price of Underlying80 100 12494

strike

Break-evenof a Long

Put

Underlying at break-even+$6 sell put

-$100 buy stock+$94 sell stock0


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Above $100, we get to keep the entire $6 premium, but below $100, we lose thepremium dollar for dollar until the underlying reaches $94, the break-even point.Below $94, we lose, period.

This position is the subject of another option myth - that selling puts uncovered or"naked" is risky. To sell puts naked means to sell a put on an underlying instrumentwithout having a short position in the underlying. Presumably, naked put sellersbelieve the price of the stock is not going to fall. If the price goes down, the naked putseller can lose a great deal of money. If the naked put seller simply is "betting" that

the underlying will not go down, then selling puts can be a very risky undertakingindeed.

Selling naked puts received a great deal of negative publicity following the crash ofmany stock markets around the world in October 1987. Many investors lost a lot ofmoney on this strategy. Some of these investors were unaware that they could lose morethan they had "invested." Neither the investors nor, in many cases, their advisorsunderstood options well enough to use them safely. Our hope is to improveunderstanding of options and option strategies.

As with many myths, reality is far more mundane than the popular fable. Suppose,for example, we want to buy a stock for $100. If the stock is trading at $100, we canbuy it in the market. However, if the stock is trading a bit above $100, we mightdecide we do not want to buy it at that price. What can we do? One way we can tryto implement a $100 stock purchase is to call up a broker and place a $100 limit

10

0

6

Profit/Loss

Price of Underlying84 100 12494

strike

-10

Break-evenof a Short

Put

Limit order to buy

Option MythThe myth that sellingnaked puts is risky


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order, meaning an order to buy the stock at $100. The broker holds our order and, ifthe stock trades down to $100, he buys the stock for us.

Alternatively, we can sell a $100 strike put for $6 and place $100 in the bank to earninterest. If the underlying is below $100 at expiration, the put buyer will exercise, and wewill have to buy the stock for $100. But there are two advantages in taking this position:

1) if we deposit our money in the bank, we earn interest, which will help offsetany loss on a stock purchase, and

2) we collect a $6 fee for giving someone else the right to sell us the stockfor $100.

Effectively, we buy the stock for $94 rather than $100. At expiration, the stock couldbe trading considerably below $100; it could be trading for $95 or even for $90. Ifowning the stock at $100 is our goal and we place a limit order instead of selling aput, we definitely will buy that stock for $100 if the price drops. We may be better offpursuing the naked put seller's strategy, because we may be able to buy the stock at alower price. And if the stock does not trade below $100, we have $6 plus interest onthe purchase price!

Of course, there is a disadvantage to the short "naked" put position. Suppose the stock

starts to trade down, and suddenly we decide we do not want to own the stock at $100anymore. That is, the stock looks appealing at $102 - and even at $100; but as the stockbegins to trade lower and lower, we decide to back out. If we place a limit order withour broker, we can call the broker and quickly cancel the order. If we are quick enough,we do not have to buy the stock.

If we sell the put we have to go into the market and buy that put back in order to getout of our position. Buying the put back might be expensive. If the put is worth $6when the stock is trading higher, it is likely to be worth more than $6 at a lower

underlying price.

Is it risky to sell naked puts? It can be risky if the investor simply takes a position onwhich way the stock is going to trade. But if the investor sells the put because he likesthe stock at a certain strike price and invests the purchase price of the stock to earninterest, selling naked puts can be a low risk strategy. Again, it is very important toevaluate option characteristics with respect to one's objective.

Buy $100 stock or

Sell $100 put for $6;deposit $100 in the bank

Options as Risk Management Tools:C bi i O ti ith U d l i P iti


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Combining Options with Underlying Positions

Long Synthetic Call

Suppose we own an underlying security that is trading at $100, and we buy a $95

strike put for $3. Since a long put gains dollar for dollar below the strike, it negatesor offsets the potential loss from a long underlying position. As a put buyer, we areprotected from a loss below $95 at a cost of $3. The combination of the longunderlying position and the long put creates a new position which, as we can see inthe figure below, looks like a call. We call this position a long synthetic call.

Let's examine the long synthetic call position more closely. We are long theunderlying. What is our risk? Our risk is that the price of the underlying may decline.When we buy a stock, a bond, or a currency, we incur price risk (among other typesof risk). We can hedge price risk by buying something that will protect us if the priceof the underlying moves against us.

We own the underlying, and we are worried that its price will go down. We canprotect ourselves by buying a put.

Remember, buying a forward plus downside protection is like buying a call. When webuy the underlying, we are long the forward. If we then buy the protection providedby the put, we are effectively buying a call. This combination of a long forwardposition and a long put creates a long synthetic call. Let's look at the maximum lossand possible gain on the long synthetic call break-even graph.

10

0

5

Profit/Loss

Price of Underlying90 100 105 110

-5

95

Long Synthetic Call

Long Underlying

Long Put

-10

Break-even of aLong SyntheticCall

We are long an underlying

at $100

We pay out $3 for $95strike put

Buying protection


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In the figure, we are long the underlying at $100 (represented by the blue line). If theunderlying price subsequently goes up, we will gain dollar for dollar on that position.If the underlying price goes down, we will lose dollar for dollar. To protect ourselves

against the price decline, we have bought a put (represented by the green line).

Notice that the protection the put gives us does not begin until we have alreadysuffered a $5 loss on the underlying position.

We paid $3 for the long put, so if the underlying price ends up at $95, we have a $5loss on our underlying plus the $3 cost of the put premium for a total loss of $8 onthe position. This is the maximum amount we can lose on the position, however, sinceat any price below $95, we gain dollar for dollar on our long put.

If the underlying price ends up at $100 at expiration, the $95 put is not worthanything, and we have neither gained nor lost on our underlying position - we are stillout the $3 for the put premium, though. The underlying must rise to $103 in orderfor us to break even on the overall position. At $103, we have a gain on ourunderlying position that exactly offsets the loss of the premium we pay for the put.

When we combine the underlying and the put positions, we create a long syntheticcall - the orange line. This line is plotted by simply adding the gains and subtractingthe losses of the long underlying and long put positions.

Why is this combination position called a long synthetic call? Let's take it apart. First,the orange line in the graph looks like the graph of a long call position. We havelimited loss on the downside and unlimited gain on the upside. Also, it looks like weare paying $8 for the protection because at $95 we have a maximum loss of $8. Then,as the price goes up from $95, we gain dollar for dollar, until we get to $103, wherewe break even.

The position is called synthetic because we are not actually buying a call - not directly.

We do not go into the market and buy a call; we construct it ourselves withinstruments that have the same risk profile but different cash flows. We buy theunderlying, and we buy protection.

Remember, a put is like a short forward plus upside protection; a call is like a longforward plus downside protection. In our figure, it looks as though we have a

Worst case-$100 buy at spot+$95 sell at strike

-$5 underlying loss-$3 put premium-$8 net loss

Underlying unchanged-$100 original cost+$100 sell at spot

0 underlyingprofit/loss

-$3 put premium-$3 net loss

Protective put


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protected long forward. Since we constructed the strategy from a long underlyingposition and a long put, we call it a synthetic call instead of a real call. The put optionin this case is called a protective put because it protects the underlying position from

a price decline.

Covered Write - Short Synthetic Put

Let's look at another strategy, a position called a buy-write or a covered write becausewe sell or write a call, which is collateralised by a long underlying position. Let'sassume we are long the underlying at $100 (represented by the blue line). Again, as theunderlying goes down, we lose, and as the underlying goes up, we gain. We sell a $105strike call for $3 (represented by the green line). Below $105 at expiration, we keepthe premium from the call, but at any underlying price above $105, we lose dollar fordollar on the call as the underlying moves up, offsetting the gain in our spot position.

When we put these two positions together, we create a synthetic position that lookslike the graph of a short put (the orange line). The return pattern is the same patternwe saw when we examined the short put position. We have a maximum gain of $8 (a$5 gain on the price of the underlying and a $3 gain from selling the option) at anunderlying price of $105. Below $105, we lose dollar for dollar, breaking even on theposition at $97. Below $97, we lose outright.

If

ubs option fundamentals

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