Paul Wilmott on Quantitative Finance

prolog to the second edition

This book is a greatly updated and expanded version of the first edition. The content continues to reflect my own interests and prejudices, based on my skills, such as they are. In the period between the first and second editions, the financial markets have expanded, the tools available to the modeler have expanded, and my girth has expanded. On a personal basis I have spent as much time being a practitioner in a hedge fund as being an independent researcher. Much of the new material therefore represents both my desire as a scientist to build the best, most accurate models, and my need as a practitioner to have models that are fast and robust and simple to understand. As I said, this book is a very personal account of my areas of expertise. Since the subject of quant finance has been galloping apace of late, I advise that you supplement this book with the specialized books that I recommend throughout, and in particular those in the quant library at the end.

I would like to re-thank those people I mentioned in the prolog to the first edition: Arefin Huq, Asli Oztukel, Bafkam Bim, Buddy Holly, Chris McCoy, Colin Atkinson, Daniel Bruno, Dave Thomson, David Bakstein, David Epstein, David Herring, David Wilson, Edna Hepburn-Ruston, Einar Holstad, Eli Lilly, Elisabeth Keck, Elsa Cortina, Eric Cartman, Fouad Khennach, Glen Matlock, Henrik Rassmussen, Hyungsok Ahn, Ingrid Blauer, Jean Laidlaw, Jeff Dewynne, John Lydon, John Ockendon, Karen Mason, Keesup Choe, Malcolm McLaren, Mauricio Bouabci, Patricia Sadro, Paul Cook, Peter Jäckel, Philip Hua, Philipp Schönbucher, Phoebus Theologites, Quentin Crisp, Rich Haber, Richard Arkell, Richard Sherry, Sam Ehrlichman, Sandra Maler, Sara Statman, Simon Gould, Simon Ritchie, Stephen Jefferics, Steve Jones, Truman Capote, Varqa Khadem, and Veronika Guggenbichler.

I would also like to thank the following people. My partners in various projects: Paul and Jonathan Shaw at 7city, unequaled in their dedication to training and their imagination for new projects. Also Riaz Ahmad and Seb Lleo who have helped make the Certificate in Quantitative Finance so successful, and for taking some of the pressure off me; Everyone involved in the magazine, especially Aaron Brown, Alan Lewis, Bill Ziemba, Caitlin Cornish, Dan Tudball, Ed Lound, Ed Thorp, Elie Ayache, Espen Gaarder Haug, Graham Russel, Henriette Präst, Jenny McCall, Kent Osband, Liam Larkin, Mike Staunton, Paula Soutinho and Rudi Bogni. I am particularly fortunate and grateful that John Wiley & Sons have been so supportive in what must sometimes seem to them rather wacky schemes; Thanks to Ron Henley, the best hedge fund partner a quant could wish for, "It’s just a jump to the left. And then a step to the right.’ And to John Morris of Fulcrum, interesting times: and to Nassim Nicholas Taleb for interesting chats.

Thanks to, John, Grace, Sel and Stephen, for instilling in me their values: values which have invariably served me well. And to Oscar and Zachary who kept me sane throughout many a series of unfortunate events!

Finally, thanks to my number one fan, Andrea Estrella, from her number one fan, me.

ABOUT THE AUTHOR

Paul Wilmott’s professional career spans almost every aspect of mathematics and finance, in both academia and in the real world. He has lectured at all levels, founded a magazine, the leading website for the quant community, and a quant certificate program. He has managed money as a partner in a very successful hedge fund. He lives in London, is married, and has two sons. His only remaining dream is to get some sleep.

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PART ONE

mathematical and financial foundations; basic theory of derivatives; risk and return

The first part of the book contains the fundamentals of derivatives theory and practice. We look at both equity and fixed income instruments. I introduce the important concepts of hedging and no arbitrage, on which most sophisticated finance theory is based. We draw some insight from ideas first seen in gambling, and we develop those into an analysis of risk and return.

The assumptions, key concepts and results in Part One make up what is loosely known as the ‘Black–Scholes world,’ named for Fischer Black and Myron Scholes who, together with Robert Merton, first conceived them. Their original work was published in 1973, after some resistance (the famous equation was first written down in 1969). In October 1997 Myron Scholes and Robert Merton were awarded the Nobel Prize for Economics for their work, Fischer Black having died in August 1995. The New York Times of Wednesday. 15th October 1997 wrote: ‘Two North American scholars won the Nobel Memorial Prize in Economic Science yesterday for work that enables investors to price accurately their bets on the future, a breakthrough that has helped power the explosive growth in financial markets since the 1970’s and plays a profound role in the economics of everyday life.’¹

Part One is self contained, requiring little knowledge of finance or any more than elementary calculus.

Chapter 1: Products and Markets An overview of the workings of the financial markets and their products. A chapter such as this is obligatory. However, my readers will fall into one of two groups. Either they will know everything in this chapter and much, much more besides. Or they will know little, in which case what I write will not be enough.

Chapter 2: Derivatives An introduction to options, options markets, market conventions. Definitions of the common terms, simple no arbitrage, put-call parity and elementary trading strategies.

Chapter 3: The Random Behavior of Assets An examination of data for various financial quantities, leading to a model for the random behavior of prices. Almost all of sophisticated finance theory assumes that prices are random, the question is how to model that randomness.

Chapter 4: Elementary Stochastic Calculus We’ll need a little bit of theory for manipulating our random variables. I keep the requirements down to the bare minimum. The key concept is Itô’s lemma which I will try to introduce in as accessible a manner as possible.

Chapter 5: The Black–Scholes Model I present the classical model for the fair value of options on stocks, currencies and commodities. This is the chapter in which I describe delta hedging and no arbitrage and show how they lead to a unique price for an option. This is the foundation for most quantitative finance theory and I will be building on this foundation for much, but by no means all, of the book.

Chapter 6: Partial Differential Equations Partial differential equations play an important role in most physical applied mathematics. They also play a role in finance. Most of my readers trained in the physical sciences, engineering and applied mathematics will be comfortable with the idea that a partial differential equation is almost the same as ‘the answer,’ the two being separated by at most some computer code. If you are not sure of this connection I hope that you will persevere with the book. This requires some faith on your part; you may have to read the book through twice: I have necessarily had to relegate the numerics, the real ‘answer,’ to the last few chapters.

Chapter 7: The Black–Scholes Formulae and the ‘Greeks’ From the Black–Scholes partial differential equation we can find formulae for the prices of some options. Derivatives of option prices with respect to variables or parameters are important for hedging. I wall explain some of the most important such derivatives and how they are used.

Chapter 8: Simple Generalizations of the Black–Scholes World Some of the assumptions of the Black–Scholes world can be dropped or stretched with ease. I will describe several of these. Later chapters are devoted to more extensive generalizations.

Chapter 9: Early Exercise and American Options Early exercise is of particular importance financially. It is also of great mathematical interest. I will explain both of these aspects.

Chapter 10: Probability Density Functions and First-exit Times The random nature of financial quantities means that we cannot say with certainty what the future holds in store. For that reason we need to be able to describe that future in a probabilistic sense.

Chapter 11: Multi-asset Options Another conceptually simple generalization of the basic Black–Scholes world is to options on more than one underlying asset. Theoretically simple, this extension has its own particular problems in practice.

Chapter 12: How to Delta Hedge Not everyone believes in no arbitrage, the absence of free lunches. In this chapter we see how to profit if you have a better forecast for future volatility than the market.

Chapter 13: Fixed-income Products and Analysis: Yield, Duration and Convexity This chapter is an introduction to the simpler techniques and analyses commonly used in the market. In particular I explain the concepts of yield, duration and convexity. In this and the next chapter I assume that interest rates are known, deterministic quantities.

Chapter 14: Swaps Interest-rate swaps are very common and very liquid. I explain the basics and relate the pricing of swaps to the pricing of fixed-rate bonds.

Chapter 15: The Binomial Model One of the reasons that option theory has been so successful is that the ideas can be explained and implemented very easily with no complicated mathematics. The binomial model is the simplest way to explain the basic ideas behind option theory using only basic arithmetic. That’s a good thing, right? Yes, but only if you bear in mind that the model is for demonstration purposes only, it is not the real thing. As a model of the financial world it is too simplistic, as a concept for pricing it lacks the elegance that makes other methods preferable, and as a numerical scheme it is prehistoric. Use once and then throw away, that’s my recommendation.

Chapter 16: How Accurate is the Normal Approximation? One of the major assumptions of finance theory is that returns are Normally distributed. In this chapter we take a look at why we make this assumption, and how good it really is.

Chapter 17: Investment Lessons from Blackjack and Gambling We draw insights and inspiration from the not-unrelated world of gambling to help us in the treatment of risk, return, and money/risk management.

Chapter 18: Portfolio Management If you are willing to accept some risk how should you invest? I explain the classical ideas of Modern Portfolio Theory and the Capital Asset Pricing Model

Chapter 19: Value at Risk How risky is your portfolio? How much might you conceivably lose if there is an adverse market move? These are the topics of this chapter.

Chapter 20: Forecasting the Markets? Although almost all sophisticated finance theory assumes that assets move randomly, many traders rely on technical indicators to predict the future direction of assets. These indicators may be simple geometrical constructs of the asset price path or quite complex algorithms. The hypothesis is that information about short-term future asset price movements are contained within the past history of prices. All traders use technical indicators at some time. In this chapter I describe some of the more common techniques.

Chapter 21: A Trading Game Many readers of this book will never have traded anything more sophisticated than baseball cards. To get them into the swing of the subject from a practical point of view I include some suggestions on how to organize your own trading game based on the buying and selling of derivatives. I had a lot of help with this chapter from David Epstein who has been running such games for several years.

¹ We’ll be hearing more about these two in Chapter 44 on ‘Derivatives **** Ups.’

CHAPTER 1

products and markets

In this Chapter …

the time value of money

an introduction to equities, commodities, currencies and indices

fixed and floating interest rates

futures and forwards

no-arbitrage, one of the main building blocks of finance theory

1.1 INTRODUCTION

This first chapter is a very gentle introduction to the subject of finance, and is mainly just a collection of definitions and specifications concerning the financial markets in general. There is little technical material here, and the one technical issue, the ‘time value of money,’ is extremely simple. I will give the first example of ‘no arbitrage.’ This is important, being one part of the foundation of derivatives theory. Whether you read this chapter thoroughly or just skim it will depend on your background; mathematicians new to finance may want to spend more time on it than practitioners, say.

1.2 THE TIME VALUE OF MONEY

The simplest concept in finance is that of the time value of money; $1 today is worth more than $1 in a year’s time. This is because of all the things we can do with $1 over the next year. At the very least, we can put it under the mattress and take it out in one year. But instead of putting it under the mattress we could invest it in a gold mine, or a new company. If those are too risky, then lend the money to someone who is willing to take the risks and will give you back the dollar with a little bit extra, the interest. That is what banks do, they borrow your money and invest it in various risky ways, but by spreading their risk over many investments they reduce their overall risk. And by borrowing money from many people they can invest in ways that the average individual cannot. The banks compete for your money by offering high interest rates. Free markets and the ability to change banks quickly and cheaply ensure that interest rates are fairly consistent from one bank to another.

I am going to denote interest rates by r. Although rates vary with time I am going to assume for the moment that they are constant. We can talk about several types of interest. First of all there is simple and compound interest. Simple interest is when the interest you receive is based only on the amount you invest initially, whereas compound interest is when you also get interest on your interest. Compound interest is the main case of relevance. And compound interest comes in two forms, discretely compounded and continuously compounded. Let me illustrate how they each work.

Suppose I invest $1 in a bank at a discrete interest rate of r paid once per annum. At the end of one year my bank account will contain

If the interest rate is 10% I will have one dollar and ten cents. After two years I will have

or one dollar and twenty-one cents. After n years I will have (1 + r)n dollars. That is an example of discrete compounding.

Now suppose I receive m interest payments at a rate of r/m per annum. After one year I will have

(1.1) equation

(I have dropped the $ sign, taking it as read from now on.)

I am going to imagine that these interest payments come at increasingly frequent intervals, but at an increasingly smaller interest rate: I am going to take the limit m → ∞. This will lead to a rate of interest that is paid continuously. Expression (1.1) becomes

This is a simple application of Taylor series when r/m is small. And that is how much money I will have in the bank after one year if the interest is continuously compounded. Similarly, after a time t I will have an amount

(1.2) equation

in the bank. Almost everything in this book assumes that interest is compounded continuously.

Another way of deriving the result (1.2) is via a differential equation. Suppose I have an amount M(t) in the bank at time t, how much does this increase in value from one day to the next? If I look at my bank account at time t and then again a short while later, time t + dt, the amount will have increased by

where the right-hand side comes from a Taylor series expansion of M(t + dt). But I also know that the interest I receive must be proportional to the amount I have, M, the interest rate, r, and the time step, dt. Thus

Dividing by dt gives the ordinary differential equation

the solution of which is

If the initial amount at t = 0 was $1 then I get (1.2) again.

This equation relates the value of the money I have now to the value in the future. Conversely, if I know I will get one dollar at time T in the future, its value at an earlier time t is simply

I can relate cashflows in the future to their present value by multiplying by this factor. As an example, suppose that r is 5% i.e. r = 0.05, then the present value of $1,000,000 to be received in two years is

The present value is clearly less than the future value.

Interest rates are a very important factor determining the present value of future cashflows. For the moment I will only talk about one interest rate, and that will be constant. In later chapters I will generalize.

Important Aside

What mathematics have we seen so far? To get to (1.2) all we needed to know about are the two functions e (or exp) and log, and Taylor series. Believe it or not, you can appreciate almost all finance theory by knowing these three things together with ‘expectations.’ I’m going to build up to the basic Black–Scholes and derivatives theory assuming that you know all four of these. Don’t worry if you don’t know about these things yet, take a look at Appendix A where I review these requisites and show how to interpret finance theory and practice in terms of the most elementary mathematics.

Just because you can understand derivatives theory in terms of basic math doesn’t mean that you should. I hope that there’s enough in the book to please the Ph.D.s¹ as well.

1.3 EQUITIES

The most basic of financial instruments is the equity, stock or share. This is the ownership of a small piece of a company. If you have a bright idea for a new product or service then you could raise capital to realize this idea by selling off future profits in the form of a stake in your new company. The investors may be friends, your Aunt Joan, a bank, or a venture capitalist. The investor in the company gives you some cash, and in return you give him a contract stating how much of the company he owns. The shareholders who own the company between them then have some say in the running of the business, and technically the directors of the company are meant to act in the best interests of the shareholders. Once your business is up and running, you could raise further capital for expansion by issuing new shares.

This is how small businesses begin. Once the small business has become a large business, your Aunt Joan may not have enough money hidden under the mattress to invest in the next expansion. At this point shares in the company may be sold to a wider audience or even the general public. The investors in the business may have no link with the founders. The final point in the growth of the company is with the quotation of shares on a regulated stock exchange so that shares can be bought and sold freely, and capital can be raised efficiently and at the lowest cost.

Figures 1.1 and 1.2 show screens from Bloomberg giving details of Microsoft stock, including price, high and low, names of key personnel, weighting in various indices (see below) etc. There is much, much more info available on Bloomberg for this and all other stocks. We’ll be seeing many Bloomberg screens throughout this book.

Figure 1.1 Details of Microsoft stock.

Source: Bloomberg L.P.

Figure 1.2 Details of Microsoft stock continued.

Source: Bloomberg L.P.

In Figure 1.3 I show an excerpt from The Wall Street Journal Europe of 14th April 2005. This shows a small selection of the many stocks traded on the New York Stock Exchange. The listed information includes highs and lows for the day as well as the change since the previous day’s close.

Figure 1.3 The Wall Street Journal Europe of 14th April 2005. Reproduced by permission of Dow Jones & Company, Inc.

The behavior of the quoted prices of stocks is far from being predictable. In Figure 1.4 I show the Dow Jones Industrial Average over the period January 1950 to March 2004. In Figure 1.5 is a time series of the Glaxo–Wellcome share price, as produced by Bloomberg.

Figure 1.4 A time series of the Dow Jones Industrial Average from January 1950 to March 2004.

Figure 1.5 Glaxo–Wellcome share price (volume below).

Source: Bloomberg L.P.

If we could predict the behavior of stock prices in the future then we could become very rich. Although many people have claimed to be able to predict prices with varying degrees of accuracy, no one has yet made a completely convincing case. In this book I am going to take the point of view that prices have a large element of randomness. This does not mean that we cannot model stock prices, but it does mean that the modeling must be done in a probabilistic sense. No doubt the reality of the situation lies somewhere between complete predictability and perfect randomness, not least because there have been many cases of market manipulation where large trades have moved stock prices in a direction that was favorable to the person doing the moving.

To whet your appetite for the mathematical modeling later, I want to show you a simple way to simulate a random walk that looks something like a stock price. One of the simplest random processes is the tossing of a coin. I am going to use ideas related to coin tossing as a model for the behavior of a stock price. As a simple experiment start with the number 100 which you should think of as the price of your stock, and toss a coin. If you throw a head multiply the number by 1.01, if you throw a tail multiply by 0.99. After one toss your number will be either 99 or 101. Toss again. If you get a head multiply your new number by 1.01 or by 0.99 if you throw a tail. You will now have either 1.01² × 100, 1.01 × 0.99 × 100 = 0.99 × 1.01 × 100 or 0.99² × 100. Continue this process and plot your value on a graph each time you throw the coin. Results of one particular experiment are shown in Figure 1.6. Instead of physically tossing a coin, the series used in this plot was generated on a spreadsheet like that in Figure 1.7. This uses the Excel spreadsheet function RAND ( ) to generate a uniformly distributed random number between 0 and 1. If this number is greater than one half it counts as a ‘head’ otherwise a ‘tail.’

Figure 1.6 A simulation of an asset price path?

Figure 1.7 Simple spreadsheet to simulate the coin-tossing experiment.

1.3.1 Dividends

The owner of the stock theoretically owns a piece of the company. This ownership can only be turned into cash if he owns so many of the stock that he can take over the company and keep all the profits for himself. This is unrealistic for most of us. To the average investor the value in holding the stock comes from the dividends and any growth in the stock’s value. Dividends are lump sum payments, paid out every quarter or every six months, to the holder of the stock.

The amount of the dividend varies from year to year depending on the profitability of the company. As a general rule companies like to try to keep the level of dividends about the same each time. The amount of the dividend is decided by the board of directors of the company and is usually set a month or so before the dividend is actually paid.

When the stock is bought it either comes with its entitlement to the next dividend (cum) or not (ex). There is a date at around the time of the dividend payment when the stock goes from cum to ex. The original holder of the stock gets the dividend but the person who buys it obviously does not. All things being equal a stock that is cum dividend is better than one that is ex dividend. Thus at the time that the dividend is paid and the stock goes ex dividend there will be a drop in the value of the stock. The size of this drop in stock value offsets the disadvantage of not getting the dividend.

This jump in stock price is in practice more complex than I have just made out. Often capital gains due to the rise in a stock price are taxed differently from a dividend, which is often treated as income. Some people can make a lot of risk-free money by exploiting tax ‘inconsistencies.’

I discuss dividends in depth in Chapter 8 and again in Chapter 64.

1.3.2 Stock Splits

Stock prices in the US are usually of the order of magnitude of $100. In the UK they are typically around £1. There is no real reason for the popularity of the number of digits, after all, if I buy a stock I want to know what percentage growth I will get, the absolute level of the stock is irrelevant to me, it just determines whether I have to buy tens or thousands of the stock to invest a given amount. Nevertheless there is some psychological element to the stock size. Every now and then a company will announce a stock split (see Figure 1.8). For example, the company with a stock price of $900 announces a three-for-one stock split. This simply means that instead of holding one stock valued at $900, I hold three valued at $300 each.²

Figure 1.8 Stock split info for Microsoft.

Source: Bloomberg L.P.

1.4 COMMODITIES

Commodities are usually raw products such as precious metals, oil, food products etc. The prices of these products are unpredictable but often show seasonal effects. Scarcity of the product results in higher prices. Commodities are usually traded by people who have no need of the raw material. For example they may just be speculating on the direction of gold without wanting to stockpile it or make jewelry. Most trading is done on the futures market, making deals to buy or sell the commodity at some time in the future. The deal is then closed out before the commodity is due to be delivered. Futures contracts are discussed below.

Figure 1.9 shows a time series of the price of pulp, used in paper manufacture.

Figure 1.9 Pulp price.

Source: Bloomberg L.P.

1.5 CURRENCIES

Another financial quantity we shall discuss is the exchange rate, the rate at which one currency can be exchanged for another. This is the world of foreign exchange, or Forex or FX for short. Some currencies are pegged to one another, and others are allowed to float freely. Whatever the exchange rates from one currency to another, there must be consistency throughout. If it is possible to exchange dollars for pounds and then the pounds for yen, this implies a relationship between the dollar/pound, pound/yen and dollar/yen exchange rates. If this relationship moves out of line it is possible to make arbitrage profits by exploiting the mispricing.

Figure 1.10 is an excerpt from The Wall Street Journal Europe of 14th April 2005. At the top of this excerpt is a matrix of exchange rates. A similar matrix is shown in Figure 1.11 from Bloomberg.

Figure 1.10 The Wall Street Journal Europe of 14th April 2005, currency exchange rates. Reproduced by permission of Dow Jones & Company, Inc.

Figure 1.11 Key cross currency rates.

Source: Bloomberg L.P.

Although the fluctuation in exchange rates is unpredictable, there is a link between exchange rates and the interest rates in the two countries. If the interest rate on dollars is raised while the interest rate on pounds sterling stays fixed we would expect to see sterling depreciating against the dollar for a while. Central banks can use interest rates as a tool for manipulating exchange rates, but only to a degree.

At the start of 1999 Euroland currencies were fixed at the rates shown in Figure 1.12.

Figure 1.12 Euro fixing rates.

Source: Bloomberg L.P.

1.6 INDICES

For measuring how the stock market/economy is doing as a whole, there have been developed the stock market indices. A typical index is made up from the weighted sum of a selection or basket of representative stocks. The selection may be designed to represent the whole market, such as the Standard & Poor’s 500 (S&P500) in the US or the Financial Times Stock Exchange index (FTSE100) in the UK, or a very special part of a market. In Figure 1.4 we saw the DJIA, representing major US stocks. In Figure 1.13 is shown JP Morgan’s Emerging Market Bond Index. The EMBI+ is an index of emerging market debt instruments, including external-currency-denominated Brady bonds, Eurobonds and US dollar local markets instruments. The main components of the index are the three major Latin American countries, Argentina, Brazil and Mexico. Bulgaria, Morocco, Nigeria, the Philippines, Poland, Russia and South Africa are also represented.

Figure 1.13 JP Morgan’s EMBI Plus.

Figure 1.14 shows a time series of the MAE All Bond Index which includes Peso and US dollar denominated bonds sold by the Argentine Government.

Figure 1.14 A time series of the MAE All Bond Index.

Source: Bloomberg L.P.

1.7 FIXED-INCOME SECURITIES

In lending money to a bank you may get to choose for how long you tie your money up and what kind of interest rate you receive. If you decide on a fixed-term deposit the bank will offer to lock in a fixed rate of interest for the period of the deposit, a month, six months, a year, say. The rate of interest will not necessarily be the same for each period, and generally the longer the time that the money is tied up the higher the rate of interest, although this is not always the case. Often, if you want to have immediate access to your money then you will be exposed to interest rates that will change from time to time, as interest rates are not constant.

These two types of interest payments, fixed and floating, are seen in many financial instruments. Coupon-bearing bonds pay out a known amount every six months or year etc. This is the coupon and would often be a fixed rate of interest. At the end of your fixed term you get a final coupon and the return of the principal, the amount on which the interest was calculated. Interest rate swaps are an exchange of a fixed rate of interest for a floating rate of interest. Governments and companies issue bonds as a form of borrowing. The less creditworthy the issuer, the higher the interest that they will have to pay out. Bonds are actively traded, with prices that continually fluctuate.

Fixed-income modeling and products are the subject of Chapters 13 and 14 and the whole of Part Three.

1.8 INFLATION-PROOF BONDS

A recent addition to the list of bonds issued by the US government is the index-linked bond. These have been around in the UK since 1981, and have provided a very successful way of ensuring that income is not eroded by inflation.

In the UK inflation is measured by the Retail Price Index or RPI. This index is a measure of year-on-year inflation, using a ‘basket’ of goods and services including mortgage interest payments. The index is published monthly. The coupons and principal of the index-linked bonds are related to the level of the RPI. Roughly speaking, the amounts of the coupon and principal are scaled with the increase in the RPI over the period from the issue of the bond to the time of the payment. There is one slight complication in that the actual RPI level used in these calculations is set back eight months. Thus the base measurement is eight months before issue and the scaling of any coupon is with respect to the increase in the RPI from this base measurement to the level of the RPI eight months before the coupon is paid. One of the reasons for this complexity is that the initial estimate of the RPI is usually corrected at a later date.

Figure 1.15 shows the UK gilts prices published in The Financial Times of 14th April 2005. The index-linked bonds are on the right.

Figure 1.15 UK gilts prices from The Financial Times of 14th April 2005. Reproduced by permission of The Financial Times.

In the US the inflation index is the Consumer Price Index (CPI). A time series of this index is shown in Figure 1.16.

Figure 1.16 The CPI index.

Source: Bloomberg L.P.

The dynamics of the relationship between inflation and short-term interest rates is particularly interesting. Clearly the level of interest rates will affect the rate of inflation directly through mortgage repayments, but also interest rates are often used by central banks as a tool for keeping inflation down.

We look at the modeling of inflation in Chapter 71.

1.9 FORWARDS AND FUTURES

A forward contract is an agreement where one party promises to buy an asset from another party at some specified time in the future and at some specified price. No money changes hands until the delivery date or maturity of the contract. The terms of the contract make it an obligation to buy the asset at the delivery date, there is no choice in the matter. The asset could be a stock, a commodity or a currency.

A futures contract is very similar to a forward contract. Futures contracts are usually traded through an exchange, which standardizes the terms of the contracts. The profit or loss from the futures position is calculated every day and the change in this value is paid from one party to the other. Thus with futures contracts there is a gradual payment of funds from initiation until maturity.

Forwards and futures have two main uses, in speculation and in hedging. If you believe that the market will rise you can benefit from this by entering into a forward or futures contract. If your market view is right then a lot of money will change hands (at maturity or every day) in your favor. That is speculation and is very risky. Hedging is the opposite, it is avoidance of risk. For example, if you are expecting to get paid in yen in six months’ time, but you live in America and your expenses are all in dollars, then you could enter into a futures contract to lock in a guaranteed exchange rate for the amount of your yen income. Once this exchange rate is locked in you are no longer exposed to fluctuations in the dollar/yen exchange rate. But then you won’t benefit if the yen appreciates.

1.9.1 A First Example of No Arbitrage

Futures and forwards provide us with our first example of the no-arbitrage principle.

Consider a forward contract that obliges us to hand over an amount $F at time T to receive the underlying asset. Today’s date is t and the price of the asset is currently $S(t), this is the spot price, the amount for which we could get immediate delivery of the asset. When we get to maturity we will hand over the amount $F and receive the asset, then worth $S(T). How much profit we make cannot be known until we know the value $S(T), and we can’t know this until time T. From now on I am going to drop the ‘$’ sign from in front of monetary amounts.

We know all of F, S(t), t and T. But is there any relationship between them? You might think not, since the forward contract entitles us to receive an amount S(T) – F at expiry and this is unknown. However, by entering into a special portfolio of trades now we can eliminate all randomness in the future. This is done as follows.

Enter into the forward contract. This costs us nothing up front but exposes us to the uncertainty in the value of the asset at maturity. Simultaneously sell the asset. It is called going short when you sell something you don’t own. This is possible in many markets, but with some timing restrictions. We now have an amount S(t) in cash due to the sale of the asset, a forward contract, and a short asset position. But our net position is zero. Put the cash in the bank, to receive interest.

When we get to maturity we hand over the amount F and receive the asset, this cancels our short asset position regardless of the value of S(T). At maturity we are left with a guaranteed – F in cash as well as the bank account. The word ‘guaranteed’ is important because it emphasizes that it is independent of the value of the asset. The bank account contains the initial investment of an amount S(t) with added interest, which has a value at maturity of

Our net position at maturity is therefore

Since we began with a portfolio worth zero and we end up with a predictable amount, that predictable amount should also be zero. We can conclude that

(1.3) equation

This is the relationship between the spot price and the forward price. It is a linear relationship, the forward price is proportional to the spot price.

The cashflows in this special hedged portfolio are shown in Table 1.1.

Table 1.1 Cashflows in a hedged portfolio of asset and forward.

Figure 1.17 shows a path taken by the spot asset price and its forward price. As long as interest rates are constant, these two are related by (1.3).

Figure 1.17 A time series of a spot asset price and its forward price.

If this relationship is violated then there will be an arbitrage opportunity. To see what is meant by this, imagine that F is less than S(t)er(T–t). To exploit this and make a riskless arbitrage profit, enter into the deals as explained above. At maturity you will have S(t)er(T–t) in the bank, a short asset and a long forward. The asset position cancels when you hand over the amount F, leaving you with a profit of S(t)er(T–t) – F. If F is greater than that given by (1.3) then you enter into the opposite positions, going short the forward. Again you make a riskless profit. The standard economic argument then says that investors will act quickly to exploit the opportunity, and in the process prices will adjust to eliminate it.

1.10 SUMMARY

The above descriptions of financial markets are enough for this introductory chapter. Perhaps the most important point to take away with you is the idea of no arbitrage. In the example here, relating spot prices to futures prices, we saw how we could set up a very simple portfolio which completely eliminated any dependence on the future value of the stock. When we come to value derivatives, in the way we just valued a future, we will see that the same principle can be applied albeit in a far more sophisticated way.

FURTHER READING

For general financial news visit www.bloomberg.com and www.reuters.com. CNN has online financial news at www.cnnfn.com. There are also online editions of The Wall Street Journal, www.wsj.com. The Financial Times, www.ft.com and Futures and Options World, www.fow.com.

For more information about futures see the Chicago Board of Trade website www.cbot.com.

Many, many financial links can be found at Wahoo!, www.io.com/~gibbonsb/wahoo.html.

In the main, we’ll be assuming that markets are random. For insight about alternative hypotheses see Schwager (1990, 1992).

See Brooks (1967) for how the raising of capital for a business might work in practice.

Cox, Ingersoll & Ross (1981) discuss the relationship between forward and future prices.

¹ And Nobel laureates.

² In the UK this would be called a two-for-one split.

CHAPTER 2

derivatives

In this Chapter …

the definitions of basic derivative instruments

option jargon

how to draw payoff diagrams

no arbitrage and put-call parity

simple option strategies

2.1 INTRODUCTION

The previous chapter dealt with some of the basics of financial markets. I didn’t go into any detail, just giving the barest outline and setting the scene for this chapter. Here I introduce the theme that is central to the book, the subject of options, a.k.a. derivatives or contingent claims. This chapter is nontechnical, being a description of some of the most common option contracts, and explanation of the market-standard jargon. It is in later chapters that I start to get technical.

Options have been around for many years, but it was only on 26th April 1973 that they were first traded on an exchange. It was then that The Chicago Board Options Exchange (CBOE) first created standardized, listed options. Initially there were just calls on 16 stocks. Puts weren’t even introduced until 1977. In the US options are traded on CBOE. the American Stock Exchange, the Pacific Exchange and the Philadelphia Stock Exchange. Worldwide, there are over 50 exchanges on which options are traded.

2.2 OPTIONS

If you are reading the book in a linear fashion, from start to finish, then the last topics you read about will have been futures and forwards. The holder of future or forward contracts is obliged to trade at the maturity of the contract. Unless the position is closed before maturity the holder must take possession of the commodity, currency or whatever is the subject of the contract, regardless of whether the asset has risen or fallen. Wouldn’t it be nice if we only had to take possession of the asset if it had risen in value?

The simplest option gives the holder the right to trade in the future at a previously agreed price but takes away the obligation. So if the stock falls, we don’t have to buy it after all.

As an example, consider the following call option on Microsoft stock. It gives the holder the right to buy one of Microsoft stock for an amount $25 in one month’s time. Today’s stock price is $24.5. The amount ‘25’ which we can pay for the stock is called the exercise price or strike price. The date on which we must exercise our option, if we decide to, is called the expiry or expiration date. The stock on which the option is based is known as the underlying asset.

Let’s consider what may happen over the next month, up until expiry. Suppose that nothing happens, that the stock price remains at $24.5. What do we do at expiry? We could exercise the option, handing over $25 to receive the stock. Would that be sensible? No, because the stock is only worth $24.5, either we wouldn’t exercise the option or if we really wanted the stock we would buy it in the stock market for the $24.5. But what if the stock price rises to $29? Then we’d be laughing, we would exercise the option, paying $25 for a stock that’s worth $29, a profit of $4.

We would exercise the option at expiry if the stock is above the strike and not if it is below. If we use S to mean the stock price and E the strike then at expiry the option is worth

This function of the underlying asset is called the payoff function. The ‘max’ function represents the optionality.

Why would we buy such an option? Clearly, if you own a call option you want the stock to rise as much as possible. The higher the stock price the greater will be your profit. I will discuss this below, but our decision whether to buy it will depend on how much it costs; the option is valuable, there is no downside to it unlike a future. In our example the option was valued at $1.875. Where did this number come from? The valuation of options is one of the subjects of this book, and I’ll be showing you how to find this value later on.

What if you believe that the stock is going to fall, is there a contract that you can buy to benefit from the fall in a stock price?

The holder of a put option wants the stock price to fall so that he can sell the asset for more than it is worth. The payoff function for a put option is

Now the option is only exercised if the stock falls below the strike price.

Figure 2.1 is an excerpt from The Wall Street Journal Europe of 14th April 2005 showing options on various stocks. The table lists closing prices of the underlying stocks and the last traded prices of the options on the stocks. To understand how to read this let us examine the prices of options on Apple. Go to ‘AppleC’ in the list, there are several instances. The closing price on 13th April 2005 was $41.35, (the LAST column, second from the right). Calls and puts are quoted here with strikes of $37.50, $40, …, $47.50, $50, others may exist but are not included in the newspaper. The expiries mentioned are April, May and July. Part of the information included here is the volume of the transactions in each series, we won’t worry about that but some people use option volume as a trading indicator. From the data, we can see that the April calls with a strike of $40 were worth $2.40. The puts with same strike and expiry were worth $1.20. The April calls with a strike of $42.50 were worth $1.20 and the puts with same strike and expiry were worth $2.45. Note that the higher the strike, the lower the value of the calls but the higher the value of the puts. This makes sense when you remember that the call allows you to buy the underlying for the strike, so that the lower the strike price the more this right is worth to you. The opposite is true for a put since it allows you to sell the underlying for the strike price.

Figure 2.1 The Wall Street Journal Europe of 14th April 2005, Stock Options. Reproduced by permission of Dow Jones & Company, Inc.

There are more strikes and expiries available for options on indices, so let’s now look at the Index Options section of The Wall Street Journal Europe 5th January 2000, this is shown in Figure 2.2.

Figure 2.2 The Wall Street Journal Europe of 5th January 2000, Index Options. Reproduced by permission of Dow Jones & Company, Inc.

In Figure 2.3 are the quoted prices of the March and June DJIA calls against the strike price. Also plotted is the payoff function if the underlying were to finish at its current value at expiry, the closing price of the DJIA was 10997.93 on the day the option prices were quoted.

Figure 2.3 Option prices versus strike, March and June series of DJIA.

This plot reinforces the fact that the higher the strike the lower the value of a call option. It also appears that the longer the time to maturity the higher the value of the call. Is it obvious that this should be so? As the time to expiry decreases what would we see happen? As there is less and less time for the underlying to move, so the option value must converge to the payoff function.

One of the most interesting features of calls and puts is that they have a non-linear dependence on the underlying asset. This contrasts with futures which have a linear dependence on the underlying. This non-linearity is very important in the pricing of options since the randomness in the underlying asset and the curvature of the option value with respect to the asset are intimately related.

Calls and puts are the two simplest forms of option. For this reason they are often referred to as vanilla because of the ubiquity of that flavor. There are many, many more kinds of options, some of which will be described and examined later on. Other terms used to describe contracts with some dependence on a more fundamental asset are derivatives or contingent claims.

Figure 2.4 shows the prices of call options on Glaxo–Wellcome for a variety of strikes as of January. All these options are expiring in October. The table shows many other quantities that we will be seeing later on.

Figure 2.4 Prices for Glaxo–Wellcome calls expiring in October.

Source: Bloomberg L.P.

2.3 DEFINITION OF COMMON TERMS

The subjects of mathematical finance and derivatives theory are filled with jargon. The jargon comes from both the mathematical world and the financial world. Generally speaking the jargon from finance is aimed at simplifying communication, and to put everyone on the same footing.¹ Here are a few loose definitions to be going on with, some you have already seen and there will be many more throughout the book.

Premium: The amount paid for the contract initially. How to find this value is the subject of much of this book.

Underlying (asset): The financial instrument on which the option value depends. Stocks, commodities, currencies and indices are going to be denoted by S. The option payoff is defined as some function of the underlying asset at expiry.

Strike (price) or exercise price: The amount for which the underlying can be bought (call) or sold (put). This will be denoted by E. This definition only really applies to the simple calls and puts. We will see more complicated contracts in later chapters and the definition of strike or exercise price will be extended.

Expiration (date) or expiry (date): Date on which the option can be exercised or date on which the option ceases to exist or give the holder any rights. This will be denoted by T.

Intrinsic value: The payoff that would be received if the underlying is at its current level when the option expires.

Time value: Any value that the option has above its intrinsic value. The uncertainty surrounding the future value of the underlying asset means that the option value is generally different from the intrinsic value.

In the money: An option with positive intrinsic value. A call option when the asset price is above the strike, a put option when the asset price is below the strike.

Out of the money: An option with no intrinsic value, only time value. A call option when the asset price is below the strike, a put option when the asset price is above the strike.

At the money: A call or put with a strike that is close to the current asset level.

Long position: A positive amount of a quantity, or a positive exposure to a quantity.

Short position: A negative amount of a quantity, or a negative exposure to a quantity. Many assets can be sold short, with some constraints on the length of time before they must be bought back.

2.4 PAYOFF DIAGRAMS

The understanding of options is helped by the visual interpretation of an option’s value at expiry. We can plot the value of an option at expiry as a function of the underlying in what is known as a payoff diagram. At expiry the option is worth a known amount. In the case of a call option the contract is worth max (S – E. 0). This function is the bold line in Figure 2.5.

Figure 2.5 Payoff diagram for a call option.

Figure 2.6 shows Bloomberg’s standard option valuation screen and Figure 2.7 shows the value against the underlying and the payoff.

Figure 2.6 Bloomberg option valuation screen, call.

Source: Bloomberg L.P.

Figure 2.7 Bloomberg scenario analysis, call.

Source: Bloomberg L.P.

The payoff for a put option is max(E – S, 0), this is the bold line plotted in Figure 2.8. Figure 2.9 shows Bloomberg’s option valuation screen and Figure 2.10 shows the value against the underlying and the payoff.

Figure 2.8 Payoff diagram for a put option.

Figure 2.9 Bloomberg option valuation screen, put.

Source: Bloomberg L.P.

Figure 2.10 Bloomberg scenario analysis, put.

Source: Bloomberg L.P.

These payoff diagrams are useful since they simplify the analysis of complex strategies involving more than one option.

Make a note of the thin lines in all of these figures. The meaning of these will be explained very shortly.

2.4.1 Other Representations of Value

The payoff diagrams shown above only tell you about what happens at expiry, how much money your option contract is worth at that time. It makes no allowance for how much premium you had to pay for the option. To adjust for the original cost of the option, sometimes one plots a diagram such as that shown in Figure 2.11. In this profit diagram for a call option I have subtracted from the payoff the premium originally paid for the call option. This figure is helpful because it shows how far into the money the asset must be at expiry before the option becomes profitable. The asset value marked S* is the point which divides profit from loss; if the asset at expiry is above this value then the contract has made a profit, if below the contract has made a loss.

Figure 2.11 Profit diagram for a call option.

As it stands, this profit diagram takes no account of the time value of money. The premium is paid up front but the payoff, if any, is only received at expiry. To be consistent one should either discount the payoff by multiplying by e−r(T–t) to value everything at the present, or multiply the premium by er(T–t) to value all cashflows at expiry.

Figure 2.12 shows Bloomberg’s call option profit diagram. Note that the profit today is zero; if we buy the option and immediately sell it we make neither a profit nor a loss (this is subject to issues of transaction costs).

Figure 2.12 Profit diagram for a call.

Source: Bloomberg L.P.

2.5 WRITING OPTIONS

I have talked above about the rights of the purchaser of the option. But for every option that is sold, someone somewhere must be liable if the option is exercised. If I hold a call option entitling me to buy a stock some time in the future, who do I buy this stock from? Ultimately, the stock must be delivered by the person who wrote the option. The writer of an option is the person who promises to deliver the underlying asset, if the option is a call, or buy it, if the option is a put. The writer is the person who receives the premium.

In practice, most simple option contracts are handled through an exchange so that the purchaser of an option does not know who the writer is. The holder of the option can even sell the option on to someone else via the exchange to close his position. However, regardless of who holds the option, or who has handled it. the writer is the person who has the obligation to deliver or buy the underlying.

The asymmetry between owning and writing options is now clear. The purchaser of the option hands over a premium in return for special rights, and an uncertain outcome. The writer receives a guaranteed payment up front, but then has obligations in the future.

2.6 MARGIN

Writing options is very risky. The downside of buying an option is just the initial premium, the upside may be unlimited. The upside of writing an option is limited, but the downside could be huge. For this reason, to cover the risk of default in the event of an unfavorable outcome, the clearing houses that register and settle options insist on the deposit of a margin by the writers of options. Clearing houses act as counterparty to each transaction.

Margin comes in