Published on: **Mar 4, 2016**

- 1. Final degree project Pricing options: From binomial trees to Black-Scholes formula Carles P´erez Guallar June 2015 carlesperezguallar@gmail.com Degree in Mathematics Supervised by: Dr. Luis Ortiz Gracia Financial Mathematics and Risk Control Group, Centre de Recerca Matem`atica.
- 2. CONTENTS Contents 1 Introduction 2 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Background of futures and options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Derivatives and arbitrage pricing 6 2.1 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3 Risk-free interest rate and the value of the money . . . . . . . . . . . . . . . . . . 8 2.1.4 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 An introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Arbitrage and the risk-neutral world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 The fundamental theorem of arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 The concept of risk-neutral probabilities . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Put–call parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Binomial option pricing model 20 3.1 Building up the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1.1 One-step binomial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1.2 Multi-period market and binomial trees . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.3 Call option pricing under the binomial model . . . . . . . . . . . . . . . . . . . . . 24 3.1.4 Properties of the multi-period binomial market . . . . . . . . . . . . . . . . . . . . 28 3.2 Model calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.1 The Cox, Ross and Rubinstein model (CRR) . . . . . . . . . . . . . . . . . . . . . 34 3.3 The passage to the limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 The Black-Scholes formula 38 4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 Background of Black-Scholes formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Connection with the reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3.1 Risk-free interest rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3.2 Historical and implied volatilities and the Greek letters . . . . . . . . . . . . . . . 41 4.3.3 Volatility smile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 Conclusions 48 6 Appendices 49 6.1 Appendix A: numerical and graphical examples . . . . . . . . . . . . . . . . . . . . . . . . 49 6.2 Appendix B: further mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.3 Appendix C: simulation codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7 References 63 Carles P´erez Guallar 1
- 3. 1 INTRODUCTION 1 Introduction 1.1 Preface Some people visualize maths as a theoretical ﬁeld, far from reality and, in many cases, useless. When I hear such opinions, I do my best to change their minds. However, I have to say, in their defence, that during my degree almost all subjects seemed to have no relation with the real word, despite the fact that I know they have application in a wide range of situations. Because of that, I wanted to discuss an applied topic in my ﬁnal project, to highlight that maths may be a powerful tool in real problems. Moreover, I was interested in how mathematical models can be employed to make proﬁts, specially in how maths play a cen- tral role in the complex ﬁnancial markets of nowadays. Thus, when I heard that one of the formulas used in these markets appears always in classiﬁcations such list of equations that changed the world 1 or, furthermore, gives rise to a piece of news with the headline The maths formula linked to the ﬁnancial crash in the BBC, I did not hesitate about the fact that I had to understand what this formula expressed and where it came from. As we will see, behind this formula there is a mathematical model, arising from some hypotheses about the market. I ﬁnd particularly interesting and ingenious how these hy- potheses are translated into the language of mathematics, and then how eﬃcacious turns out to be this translation. The statements obtained from these hypotheses allow to obtain important properties, announced in form of lemmas and theorems. Speciﬁcally, there is a theorem which is diﬃcult to prove: the fundamental arbitrage theorem. Most of books and other sources that I consulted skipped its proof, and those which prove it, usually did it after the introduction of dozens of lemmas that pave the way. Hence I found diﬃcult to demonstrate this theorem directly, and I spent many hours before eventually get a proof, with some steps done by myself. On the other hand, I also spent much time learning about the ﬁnancial aspect of this work, that is, learning what is a ﬁnancial option and how they are traded. Consequently, I did an eﬀort to explain as properly as I could the ﬁnancial basis, writing many pages on this topic. However, I must admit that if it had been in the other way round, if I had been expert in ﬁnance but I would have had to learn maths, it would have been much more diﬃcult to reach the goal. I believe that, when a problem in the real world which implies maths is considered, mathematicians have already done the most diﬃcult part, whatever is the ﬁeld of application. Finally, after showing the implications of all the hypotheses and reaching the formu- las that the model allows to deduce, I include a chapter in which the relation between the theoretical framework and the real world is assessed. As far as I know, one of the most common problems that mathematicians have when they start to work in real problems is the comprehension of the fact that models never ﬁt perfectly in real situations. Or that assumptions never are perfectly met. 1 For instance, the book called In Pursuit of the Unknown: 17 Equations That Changed the World, by the famous mathematician and popular-science writer, Ian Stewart. Among these 17 equations one can ﬁnd the Pythagoras theorem, the mass–energy equivalence deduced by Einstein and also the Black-Scholes formula ([22]) Carles P´erez Guallar 2
- 4. 1 INTRODUCTION 1.2 Abstract This work begins by discussing the historical background of the origin of options, followed by an introduction which assess the current situation, that motivates the work. Next, there is a glossary that introduces the ﬁnancial basis, followed by an introductory example. After these sections, the work can be itemized in four parts: 1. In the ﬁrst one, the Black-Scholes model with its assumptions is presented. These assumptions are formalized in mathematical language, which allows us to ﬁnd impli- cations of these assumptions, among them the Fundamental Theorem of Arbitrage. Next, the concept of risk neutral probabilities is explained and we demonstrate the put-call parity formula, which relates the price of a put and call option through a simple formula. From this last formula stems the fact that, in following sections, we deal only with call options. 2. In the second one, the binomial market model is deﬁned based on the concepts and hypotheses of the Black-Scholes model. After that, we deduce how to compute the price of a call option under the binomial model, by means of a hedging strategy. 3. In the third one, we introduce the calibration of the model, that is, we discuss about the parameters involved in the binomial market and in the formula for the call valuation and we study where they come from. We also present the Cox-Ross- Rubinstein model, which is a particular case of the binomial model. The expressions obtained in the model calibration permit to take the limit as the number of steps in a binomial model tends to inﬁnity. By this means we obtain the Black-Scholes formula. 4. In the last one, we discuss how to estimate, from real data, the parameters that appears in Black-Scholes formula, and the connection of this formula with the real world. We also present the so-called volatility smile which are patterns that arise in the real world and which contradicts what Black-Scholes predicts. We show, with real data, what is a volatility smile. Finally, and before the conclusion, we present some graphical representations, both in 2- dimensional and 3-dimensional. These ﬁgures support and help a better understanding of what we have studied. We also carry out simulations of a N-step binomial model. Then, in an appendix, all codes are included. My aim has been to write each chapter in such a way that someone who has never heard about options can understand all the addressed issues. For this reason, this work has more text than it is common in mathematics, increasing its extension beyond what is expected from a ﬁnal degree project. Carles P´erez Guallar 3
- 5. 1 INTRODUCTION 1.3 Background of futures and options We are going to start with a basic example, very similar to that which can be found in [24], to understand in an intuitive way the background and concepts in which this work is based. Suppose that in six months we plan to go to England and for this reason we have saved 400 euros that, at some point before the trip, we should exchange for pounds. We consult the change and see that a euro corresponds to 0.73 pounds. But we know that, a month ago, this exchange rate was 0.71, and two months ago, 0.74. What do we have to do? Exchange our euros now? Wait with the expectation that the pound will drop against the euro? But the opposite may happen. These situations, dominated by the uncertainty, appear constantly in the ﬁnance ﬁeld. The problem is how to evaluate and quantify this uncertainty. For instance, we have said that the exchange of euros to pounds in the last three months were, respectively, 0.74, 0.71 and 0.73. However, these could have been 0.20, 1.5, 0.73. In what case seems to be more risk of losses (or proﬁts) depending on when we choose to make the exchange? It seems that in the second. As we will see, this uncertainty is modeled in terms of random variables and the risk is associated with the dispersion of values, this is, with the variance. Moreover, problems like the one explained here, caused the invention of futures and options. When hearing about ﬁnancial options or future contracts, many people suppose that these are sophisticated ﬁnancial instruments invented in Wall Street, for their own spec- ulative purposes. But, in fact, this class of contracts did not appear in Wall Street and other ﬁnancial markets until the 70’s, and their origin dates back much further in time. As it is well known, the harvest of some cereals take place in a speciﬁc time of year. Back in the Middle Age, traders were interested in make an agreement, in advance, be- fore the harvest, about the amount and the price at which they were going to buy the cereals. Thus, the farmer and the trader come to terms regarding the price, the ﬁrst to sell and the second to purchase, an speciﬁc amount of cereals in the future. This kind of contracts are known as future contracts. Notice that they do not require any initial payment, it is just an agreement between both parts. In contrast, early in the XVII century, tulips were a valued product in Netherlands and its market was expanding rapidly in other parts of Europe. To hedge their business against bad crops, which could trigger an escalation of prices, the traders of tulips made the proposal to the farmers to cut a deal in advance on the price of tulips, with the partic- ularity that, by the time the harvest was ready, traders could choose whether to buy for the accorded price or not. Farmers signed such contracts, provided that traders paid a certain amount of money in exchange, since the ﬁnal decision was in their hands. This class of contracts, which give the right but not the obligation to buy at a given price, are called options. The tulips trade continued growing, as so did the options associated to it. The growth was so big that a parallel market, where only options were traded, appeared. There, such contracts were bought and sold, for only speculative purposes. Carles P´erez Guallar 4
- 6. 1 INTRODUCTION 1.4 Motivation Some people argue that the source of the current ﬁnancial crisis lies, partly, on maths. In 2008, after a ﬁnancial rescue package was approved in USA, an eminent American profes- sor of ﬁnance was interviewed. Speaking about the relation between the ﬁnancial crisis and the complex ﬁnancial products that have been created since the 70’s he aﬃrmed that: ”These complex ﬁnancial instruments were actually designed by mathematicians and physi- cists, who used algorithms and computer models...” Obviously they turned out to be wrong because you can’t model human behaviour with math..” [4] Whether these claims were justiﬁed or not, what is an objective fact is that maths have played a central role in the ﬁnancial world during the last 40 years. And in turn, the ﬁnancial world has played an important role during the current crisis. One of the most important changes in ﬁnancial markets during the last 40 years has been the rise of derivative contracts. These ﬁnancial products are neither money nor goods. Sometimes it is said that these products are bets on bets or investments in in- vestments. It is estimated that the present size of the derivative market is more than 100 times what it was in 1970. To get an idea of its importance, one can consider a data comparison: according to BIS 2 , derivative products accounted, at the end of 2013, for 700 billion dollars 3 . On the other hand, the global Gross Domestic Product (GDP) accounted for 74.31 billion dollars. The derivative market was ten times the global GDP!. 4 And, in this market, mathematical models are fundamental. In the following pages we will present the most successful of these models. After explaining and formalizing the market hypotheses that we will employ, one of the most simple option pricing model will be introduced: the binomial model. This model stems from modelling the ups and downs of prices as the result of head or tail when tossing a coin. This mathematical model, as we will see, allows a passage to the limit resulting in the Black-Scholes formula. This method of obtaining this famous expression is diﬀerent from the original one, which was presented by these two economist, Black and Scholes, one of them PhD in maths, in a paper published in 1973. Their paper changed the course of the ﬁnancial world, and it is impossible to explain the rise of ﬁnancial options without quoting this mathematical expression. 2 Bank for International Settlements 3 1 billon=1012 4 This comparison, although it helps to give an idea and is common in the media, it is not absolutely precise. This stems from the fact that there are several ways to compute the amount of money that derivatives account for. For instance, another way to compute this quantity estimates that, to meet the payment of all derivative contracts, it would be necessary 25 billions dollars, which remains being the frightening quantity of one third of the global GDP. Carles P´erez Guallar 5
- 7. 2 DERIVATIVES AND ARBITRAGE PRICING 2 Derivatives and arbitrage pricing 2.1 Glossary 2.1.1 Basic concepts The goal of this section is to introduce concisely and clearly the ﬁnancial concepts that we present in this work. A ﬁnancial market is a physical or virtual space in which the exchange of ﬁnancial products take place and where their prices are set. It is there, in a ﬁnancial market, where both assets and derivative contracts are traded. An asset is any possession with an economic value or which is expected to produce proﬁts in the future. An asset can be tangible, for instance a building, or intangible, for instance a share. Some assets, such as some kinds of shares of a certain company, give periodically an amount of money to their owners. This amount, which is related with the proﬁts of that company, is called dividend. For sake of simplicity, we will omit the existence of divi- dends. A portfolio is a collection of assets, such as bonds or derivatives, owned by a certain entity. Big corporations have huge portfolios with plenty of assets. For instance, it is common that some shares are used in order to speculate whereas derivatives are used with hedging objectives, to reduce the risk. Later on we will be able to understand how derivatives may be used in this regard. When an individual invests in assets, two positions can be taken: long position and short position. The long position is the most plain: it consists in buying the asset with the expectation that its price will increase. On the other hand, roughly speaking, a short position consists in borrow and sell the asset, in order to repurchase in the future expecting a drop in its value. Without going into details, a short position can be under- stood as borrowing something that one does not possess and selling it with the purpose to repurchase it in the future and return it to the original owner. These two types of positions will be fundamental in this work and we will assume that both positions can be taken freely: at any moment, for any asset and in any amount. 2.1.2 Derivatives A derivative is a contract whose price depends on (derives) the price of a certain asset. This asset is called the underlying asset, and can be, for instance, the gold, the value of a given badge, a stock market index, etc. There are several classes of derivatives but we will introduce only those related with this work. A future contract or forward contract is a contract between two parts which sets Carles P´erez Guallar 6
- 8. 2 DERIVATIVES AND ARBITRAGE PRICING the price for which one of the parts will sell while the other will buy a given asset at a certain future point. The transaction will take place whether the price of the asset rises or falls. This kind of contract is called future when it is traded in an organized mar- ket, with preﬁxed rules, and forward when the contract is made without any speciﬁc rules. Related with these contracts, we ﬁnd options, which are the derivatives discussed in this work. An option is a contract which gives to its owner the right, but not the obliga- tion, to buy or sell a certain amount of assets at an agreed price, called strike price, on or before a given future date. When the option’s owner decides to carry out the transaction, it is said that the option has been exercised. Obviously, the buyer (owner) of the option pays a price, a premium, to the seller for having such rights. This is the price we aim to ﬁgure out in this work. Notice that this price must depend on the asset over which the option conveys rights, that is, on the underlying asset. There are two basic classes of options. An option conveying to its owner the right to buy at a pre-agreed price is called a call option or simply a call. That which conveys to its owner the right of sell at a pre-agreed price is called a put option or simply a put. Both of them are commonly traded and, as we will see later, under certain hypotheses a relationship between the price of a call and a put with the same underlying asset can be established. From this relationship stems the fact that, in theoretical approaches, often only call options are discussed. There is another important classiﬁcation in two major classes of options. European options are those that only can be exercised at a certain and unique future time T, while American options are those that can be exercised at any time between the agreement of the contract and a given future time T. In both cases T is known as expiration time or maturity. The so-called option’s payoﬀ is the value of the option at maturity. For instance, if St denotes the price of the underlying asset of an European call option with a strike price K, the payoﬀ of this option at maturity T is given by max{ST − K, 0}. The explanation is simple: if at time T one has that ST > K, then the option will be exercised and the owner will be buying cheap with a proﬁt equal to ST − K. Conversely, supposing one has that ST < K, then the option will not be exercised (since otherwise the owner would be buying expensive). We must keep in mind this easy expression, as we will use it many times. As aforementioned, in forwards, futures and also in options there is one who sells a right and one who buys a right. The one who sells the right (sells the option) is said to be taking a short position, whereas the one who purchases the right (who buys the option) is said to be taking a long position. A counterintuitive fact is that it is possible to take a short position (sell) in a call option (right to buy) over a speciﬁc asset without possessing it. This practice is known as naked short selling. This technique consist in selling the call option without possessing the Carles P´erez Guallar 7
- 9. 2 DERIVATIVES AND ARBITRAGE PRICING asset and, if the Call’s owner decides to exercise the option, buy the asset in order to resell it at the pre-agreed strike price. 2.1.3 Risk-free interest rate and the value of the money The risk-free interest rate, denoted by r or rf , is a theoretical concept which stems from the assumption that an investment without any risk exists. Such investment oﬀers a se- cure return in a given period of time, without any possibility of losses. In practice it is considered, for example, the rate of return of Treasury bonds from countries with strong economies such as Germany or the United States, where the risk of default is considered virtually null. Because it is assumed that, at any time, an investor can make riskless proﬁts at the rate given by the risk-free interest rate, it follows that any investment with risk requires a greater premium than that given by the risk-free interest rate. We can understand this through the concept of ’risk premium’ which is frequent in news. For instance, the Spanish risk premium measures the diﬀerence between the interest that is paid on Spanish bonds over German bonds. Such risk premium is positive, since German economy is stronger than Spanish economy, the interest rate oﬀered by Spanish bonus have to be higher than the interest rate oﬀered by German bonus. Another way to interpret the risk-free interest rate is through the interest that a de- posit account in a bank oﬀers. In a certain way this bank account can be considered a safe investment. In any case, r will be a model parameter that one will have to estimate. Continuous interest formula: Assume that there is a safe investment. The interests of such investment can be paid in one temporal unit, for instance, each year, with interest rate r. Hence if we have invested C in the safe investment, after a year we will have C(1 + r). Suppose that the interests are paid every six months. Then, after a year, we will have C(1 + r 2 )(1 + r 2 ). Generally, if during a year the interests are paid in n instalments, we will have C(1 + r n )n . And, ﬁnally, if the interests are paid continuously, one obtains: lim n→∞ C(1 + r n )n = Cer (2.1) Along the present work, we will assume, unless stated otherwise, that the interests are paid continuously. A euro today is worth more than a euro tomorrow. Without taking into con- sideration concepts such as inﬂation, deﬂation or currency devaluation, we will discuss Carles P´erez Guallar 8
- 10. 2 DERIVATIVES AND ARBITRAGE PRICING why this sentence is true. Suppose we can choose between 100 euros today or 100 euros after a time T. Our answer shall be bold: we want the money today. If we have it today we can invest this money, for instance, in a safe Treasury bond and after a year our 100 euros will become 100er . And notice there is no doubt that 100er > 100. Therefore, from this inequality arises the sentence above about the time value of money. Symmetrically, using the same argument, it can be justiﬁed that the 100 euros we would receive after a time T at the current time have a value equal to 100e−r . This last inter- pretation will be taken into account continuously along this work. 2.1.4 Arbitrage In ﬁnance, an arbitrage is a practice that consist in making a proﬁt from the price diﬀer- ence of a certain asset between two markets. Those who engage in arbitrage are known as arbitrageurs and their beneﬁts stem from buying at a certain price and afterwards selling it more expensive. When the word arbitrage is used in a theoretical or academic context, an arbitrage is a transaction which does not need a net investment at a current time but, nevertheless, it oﬀers proﬁts at a certain future time. We will see that, with the assumptions that we will use in this work, the above deﬁ- nition of arbitrage can be translated as the possibility of getting insured proﬁts, without risk, with a rate of return larger than that given by the risk-free interest rate. In real markets arbitrage exists but, as a rule, the price diﬀerences are small and only big investors can make proﬁts from these diﬀerences. Moreover, the duration of an arbitrage is always short, since the market tends to an equilibrium (consider the law of supply and demand). 5 Therefore, it seems suitable that one of the hypothesis that is usually formulated in the quantitative ﬁnance ﬁeld, and specially when pricing derivatives, is the absence of arbitrage. This is commonly summarized with the expression: there is no thing such a free lunch. 5 Besides, given the advancement in technology it has become extremely diﬃcult to make proﬁt from mispricing in the market. Any ineﬃcient pricing setups are detected quickly and the opportunity is often eliminated in a matter of seconds. Carles P´erez Guallar 9
- 11. 2 DERIVATIVES AND ARBITRAGE PRICING 2.2 An introductory example The arbitrage In this example we will leave aside abstract mathematics and we will attempt to give a general idea about what we will discuss in this work. The purpose of this example is only introductory and its absolute comprehension would probably require some previous knowledge that will be subsequently introduced. Let Λ be a portfolio consisting of λ1 shares whose price is 20 for each unit. Since the monetary unit is not important, we will omit it in this example, and also along this work. Suppose that the price of such shares, after time T, can take two values: 18 or 23. Also suppose that we have λ2 bonds, each of them with a price given by 5 1 + r , with an interest rate r, which is paid at maturity T (these bonds are riskless, i.e. r is the risk-free interest rate). The initial value of our portfolio is: Λ(0) = 20λ1 + 5 1 + r λ2 As it has been said, after time T, there are two possible values for our portfolio, two possible scenarios, one in which the share’s price has risen (up), and one in which the share’s price has dropped (down): Λu(T) = 23λ1 + (1 + r) 5 1+r λ2 Λ(0) 55 )) Λd(T) = 18λ1 + (1 + r) 5 1+r λ2 Assume we have two speciﬁc amounts of each asset, for instance λ1 = −2 and λ2 = 10. The negative value λ1 = −2 is not contradictory: it means that a short position is taken in two shares. That is to say, we sell it with the compromise of repurchasing it in a certain future time. Assume, as well, that the rate of return for the bonds is r = 0.25, which is very high but it will be useful to assess what is an arbitrage strategy. With this numbers, the initial value of our portfolio turns out to be: Λ(0) = 20 · (−2) + 5 1 + 0.25 · 10 = 0 That is, its total price is zero. On the other hand, the two possible values of our portfolio at time T will be: Λu(T) = 23 · (−2) + 5 · 10 = 4 Λd(T) = 18 · (−2) + 5 · 10 = 14 Carles P´erez Guallar 10
- 12. 2 DERIVATIVES AND ARBITRAGE PRICING In any case we make proﬁts! Something whose value was zero at the beginning, has a positive price at time T with absolutely certainty (either 4 or 14). This kind of situations are what cannot occur in a market with no arbitrage. In fact, in this example the arbi- trage stems from assuming a large interest rate r. We can study, for this speciﬁc example, when the arbitrage (that is, proﬁts without risk and without net investment) appears. This happens if and only if for a given r there exist λ1 and λ2 such that: Λ(0) = Λ(0) = 20λ1 + 5 1 + r λ2 ≤ 0 Λu(T) = 23λ1 + 5λ2 ≥ 0 with at least one strict inequality (= 0) Λd(T) = 18λ1 + 5λ2 ≥ 0 Notice that the ﬁrst equality assures that there is no net investment, whereas the second and the third assure that in both scenarios there are no losses. The condition ’at least one strict inequality =’ assures that, at least in one case, there are proﬁts. It is easy to study the system above for r and show that the equations have no solu- tion in λ1 and λ2 when r ∈ (−0.1, 0.15). Since it does not make sense to have a negative interest rate, one obtains that there is no arbitrage if and only if r ∈ [0, 0.15). Later on, a more formal deﬁnition of arbitrage will be given, and a theorem which estab- lishes that, in absence of arbitrage (in this example, for r ∈ (−0.1, 0.15)) there exist two positive numbers p1 and p2 such that: 20(1 + r) = 18p1 + 23p2 1 = p1 + p2 The ﬁrst equation equals the price of the shares at time 0 with a linear combination of the two possible prices at time T, adding a factor (1 + r), given by the risk-free interest rate, that always has to appear when we relate the value of money at present time with its value at a future time. The second equation imposes that the sum of coeﬃcients of this linear combination is one, so as it can be seen as a probability distribution. Solving the system, one obtains: p1 = 3 5 − 4r p2 = 2 5 − r It is straightforward to see that 0 ≤ p1 ≤ 1 and 0 ≤ p2 ≤ 1 if and only if r ∈ [0, 0.15), which is exactly the same condition we have already obtained when we imposed the absence of arbritrage (which means that p1, p2 can be interpreted as a probability distribution if and only if there is no arbitrage). We regard, by now in an arbitrary way, p1 and p2 as the probabilities to be in one or in the other scenario (which follows that p1 is the probability to be in a down scenario Carles P´erez Guallar 11
- 13. 2 DERIVATIVES AND ARBITRAGE PRICING whereas p2 is the probability to be in aup scenario). If we take, for instance, r = 0.05, we obtain: p1 = 2 5 p2 = 3 5 (2.2) We will use these values below. Continuing the example: pricing a call option Continuing with the example above, we will see how to price a call European option, whose price we denote by X(t), over one share of the above example. We denote by S(t) the price of one share, and we assume a risk-free interest rate r = 0.05. Let us suppose that the call oﬀers a strike price K = 22 with expiration at T (which means that, at time T, the owner of the call has the right, but not the obligation, to buy one share for 22). Remember that, at time 0, the share price is 20 and at time T its price can be either 18 or 23. Then, the payoﬀ of the Call option at time T is given by: payoﬀ = max{S(T) − K, 0} (2.3) I.e, if S(T) = 23 the right of buy will be exercised, paying 22 for something that cost 23, and the owner of the call option will obtain a payoﬀ of 1. Conversely, if it costs 18, the option will not be exercised and the payoﬀ will be zero. In form of diagram, it can be represented as: (1, 23)t=T (X, 20)t=0 77 '' (0, 18)t=T The problem is to compute the value of X at time 0, where X denotes the price of the call. This graphic disposition, where the position of the call price at t = 0, X, is then, at time T, occupied, in each scenario, by its payoﬀ, arises from the natural interpretation that the call value at the expiration time T is its payoﬀ. To ﬁnd the fair price X, we will carry out two methods: Hedging: this method is based on building a portfolio, say Λ, with no risk. To achieve this goal, we purchase λ > 0 shares (we take a long position) and we sell a call option over one share (we are using a naked short selling, agreeing to sell one share that we do not possess). Notice that the long position is betting for a rise of price whereas the sell of a call option is betting for its drop. The purpose of this technique (the hedging) is to eliminate the risk, as we will see. The initial investment or the initial value of our portfolio is: Λ(0) = 20 · λ − 1 · X Carles P´erez Guallar 12
- 14. 2 DERIVATIVES AND ARBITRAGE PRICING At time T, the portfolio takes two possible values since there are two possible scenarios: • Up: the price of shares at time T rises to 23. Then, the other part decides to exercise its right to buy for 22. Therefore, we should buy one share for 23 and sell it for 22, loosing 1. Thus, in this scenario the value of our portfolio is: Λu(T) = 23λ − 1 • Down: the price of shares at time T drops to 18. Then, the other part does not exercise its right to buy and the value of our portfolio is: Λd(T) = 18λ The portfolio is risk-free if we choose λ such that the value of what we possess at time T is equivalent (but not exactly equal because of the existence of the risk-free interest rate or, equivalently, due to the time value of money) to what we invested at time t = 0. This equivalence has to be met in both scenarios. To make this to happen, ﬁrst of all the value of Λ in both scenarios at time T must be equal: Λu(T) = Λd(T) ⇔ 23λ − 1 = 18λ ⇔ λ = 1/5. With this value of λ, the portfolio costs the same at time T in both scenarios: Λ(T) = 3.6. Hence, we could think that the price at time t = 0 should be 3.6. However, this is wrong since, as it has been said, the time value of money has to be taken into account. This value evolves according to the risk-free interest rate, which we assumed to be r = 0.05 and paid at time T. This leads to the fact that the price of the portfolio at time t = 0 has to be: Λ(0)(1 + r) = Λ(T) ⇔ Λ(0) = Λ(T) 1 1 + r = 3.6 · 1 1.05 = 3.4289 Equalling the expression above with which we already have for Λ(0), and using λ = 1/5, we have an equation with X as unknown, which allows us to obtain: Λ(0) = 20 · λ − 1 · X = 3.4289 ⇔ X = 20 · 1 5 − 3.4289 = 0.573 We eventually obtain the call price at time 0. This price is usually known as the fair price or the arbitrage-free price of the option. Let us see why. Keeping all the assumptions, if the call price were higher than that we have obtained, we would have (1+r)Λ(0) < Λ(T). This would allow us to make proﬁts without net inversion. Indeed, we could borrow money, at interest rate r, to the amount of Λ(0) with maturity T (as we will discuss later, we suppose this operation is always possible). Therefore, at time 0 we have Λ(0) in cash, and we can use this money to build the portfolio Λ. The net investment is then zero. At time T we have to return the borrowed money, so we have to pay Λ(0)(1 + r) but, at the same time, the value of our portfolio is Λ(T) > (1 + r)Λ(0), making net proﬁts without risk and without investment. It follows that there is arbitrage, Carles P´erez Guallar 13
- 15. 2 DERIVATIVES AND ARBITRAGE PRICING and this is a contradiction under the hypotheses we will assume. Notice that, in the above discussion, to ﬁnd secure gains with a return higher than that given by risk-free interest rate, we could simply argue that we buy Λ at time 0 and we wait until time T. However, we did an equivalent approach, yet more cumbersome, to obtain gains without net investment. We use this approach because it is the way in which we will formulate the idea of arbitrage. In contrast, if the call price were lower than the one we have already obtained, we would have (1 + r)Λ(0) > Λ(T). If that were met, we could take a naked short position (sell without owning) Λ(0) and then, at time T, repurchase the portfolio, now at price Λ(T). By doing this one makes proﬁts which overcome those given by the risk-free interest rate. Or, in other words, one makes proﬁts after discounting the time value of money. But this is not so obvious. To ﬁgure this out, one can be smart and proceed similarly but with a zero initial investment. At time 0, we take a naked short position in Λ(0), obtaining Λ(0) in cash. We use this cash to buy Λ(0) bonds. At time T the bonds mature and we have (1 + r)Λ(0) in cash. Also at this time, we are required to repurchase Λ(T), and we do, but since Λ(T) < (1 + r)Λ(0) we make net proﬁts. Note, and the understanding of this is essential, that we call net proﬁts when this proﬁts overcome the increasing value of money over the time, i.e., proﬁts after discounting the time-value of money. Since this increase is given, by assumption, by the risk-free inter- est rate, we can refer to this net proﬁts also as proﬁts that overcome those given by the risk-free interest rate. By using probabilities: We will see, in a diﬀerent way and by now somewhat heuristic, how to obtain the call fair price. As it has been already seen, the option’s payoﬀ is either 0 in case of drop of share prices or 1 in case of a raise of price. We will interpret the values of p1 and p2 we already computed (refejintr) as the respective probabilities of each scenario. Then, the expected proﬁts will be given by 0p1 + 1p2 . On the other hand, it is consistent to assume that the fair price for the call option at time t = 0 has to be equal to the expected proﬁts that it will oﬀer, that is, the expected payoﬀ, taking into account the risk-free interest rate: X = 1 1 + r E[X(T)] = 1 1 + r (0p1 + 1p2) = 3/5 1 − 0.05 = 0.573 Which, indeed, matches the result above. This last computation is commonly termed a discounted expected value of the future payoﬀ, where discounted refers to the fact that one takes into account the risk-free interest rate. It has to be highlighted the vast diﬀerence in proﬁts and also in losses when money is invested through options. One of the most important uses of derivative products is to speculate. Nonetheless, perhaps, the main purpose of derivative products, surpassing the speculation, is the use of those products as an insurance against losses, as we have seen in the example in which we have priced the Call option by means of the hedging method. Carles P´erez Guallar 14
- 16. 2 DERIVATIVES AND ARBITRAGE PRICING 2.3 Arbitrage and the risk-neutral world In order to formalize the ideas given in the example, we need to introduce the hypothesis that we will use. The present work is based on the market model known as Black-Scholes model. The hypotheses of this model are the following ones: • The market has, at least, assets with a certain risk, which we call shares, and assets without risk, which we call bonds. Assumptions on the assets: • The rate of return of the risk-free asset, the bonds, is constant and is the so-called risk-free interest rate. • There are no dividends. Assumptions on the market: • There are no arbitrage opportunities. • It is possible to borrow and lend money in any amount, also in fractions, at an interest given by the risk-free interest rate. • It is possible to buy or sell assets in any amount, also in fractional amounts (and even including the naked short selling positions). • All transactions are free from taxes or costs (this hypothesis is known as frictionless market). 2.3.1 The fundamental theorem of arbitrage In this section we will deﬁne mathematically the concepts and assumptions, and we will prove the theorem which gives name to this section. From now on, we will assume that the risk-free interest rate is paid continuously (and therefore we have to use the factor erT ). Deﬁnition 1. Consider a set of M diﬀerent assets: A1, A2, . . . , AM with which one can trade freely. Assume that one of them, say A1, is riskless and satisﬁes that, at time t = T, its price does not depend on the market scenario (thus it represents the bond or the risk- free asset). Let Sj 0 be the price of one unit of the j asset at t = 0. Assume, without loss of generality, that S1 0 = 1 (this assumption can be understood as ﬁxing the monetary unit). Suppose that, at time T, the market can be in a ﬁnite number N of possible scenarios ω1, ω2, . . . , ωN . Let deﬁne Ω = {ω1, ω2, . . . , ωN } (this set of scenarios summarizes the uncertainty in the market). Carles P´erez Guallar 15
- 17. 2 DERIVATIVES AND ARBITRAGE PRICING Let us denote by Sj T the price of the asset j at time T. Notice that, since A1 is riskless, in all scenarios one has S1 T = erT , whereas S2 T , . . . , SM T are functions that depend on the scenarios ωi, i.e., the price of the j asset at time t in the i scenario is Sj T (ωi). This type of construction is called a single period market. The expression one can trade freely refers to the possibility of taking long positions and short positions in any asset and in any amount, as we assumed in the hypotheses. It is important to remark now that, assuming it is possible to deﬁne a probability function π over Ω, the Sj T functions can be seen as random variables depending on the diﬀerent market scenarios ωi. Deﬁnition 2. A portfolio is a real valued vector such that: Λ = (Λ1, Λ2, . . . , ΛM ) ∈ RM The entry Λj represents the number of units of the asset Aj that one owns. A negative value, Λj < 0, indicates that one has taken a short position in |Λj| units of the asset Aj. The portfolio value at time t = 0 is given by: V0(Λ) = M j=1 ΛjSj 0 (2.4) And the value of the portfolio Λ at time t = T in the market scenario ωi is: VT (Λ, ωi) = M j=1 ΛjSj T (ωi) (2.5) Deﬁnition 3. An arbitrage is a portfolio which makes proﬁts without net initial invest- ment. Or, in other words, makes money out of nothing. It is deﬁned as a portfolio Λ which satisﬁes one of these two following properties: V0(Λ) ≤ 0 and VT (Λ,ωi) > 0 ∀i = 1, . . . , N or V0(Λ) < 0 and VT (Λ,ωi) ≥ 0 ∀i = 1, . . . , N (2.6) Deﬁnition 4. Let π be a probability distribution deﬁned over the set Ω of the possible market scenarios. It is said that π is a risk-neutral measure or an equilibrium measure if for any asset Aj, the price of Aj at time t = 0, which is denoted by Sj 0, can be computed as: Sj 0 = e−rT N i=1 π(ωi)Sj T (ωi) (2.7) Carles P´erez Guallar 16
- 18. 2 DERIVATIVES AND ARBITRAGE PRICING Note that this expression can be understood as an expected value under the probability function π, properly discounted by the risk-free interest rate. Theorem 1. (Fundamental theorem of arbitrage) In a single period market M a risk- neutral measure exists if and only if there is no arbitrage. Proof. First suppose that a risk-neutral measure π exists. Let Λ be a portfolio. Consider the equation (2.7), and multiplying both sides by Λj, one obtains: Sj 0Λj = e−rT N i=1 πi(ωi)Sj T (ωi)Λj (2.8) Summing the equation above from j = 1 to M, we have: M j=1 Sj 0Λj = e−rT N i=1 πi(ωi) M j=1 Sj T (ωi)Λj (2.9) By using equations (2.4) and (2.5), the above equality is written as: V0(Λ) = e−rT N i=1 πi(ωi)VT (Λ, ωi) (2.10) We now can see that, from this expression, one can deduce that an arbitrage is impossible. Suppose that VT (Λ, ωi) > 0 for all scenario ωi. Then, by the equality (2.10), V0(Λ) > 0. On the other hand, if V0(Λ) < 0 then it is impossible, again by (2.10), to meet the con- dition VT (Λ, ωi) ≥ 0,∀i. Hence there is no arbitrage. The other implication is considerably more complicated to prove. Due to its extension, the proof can be found in details in the appendix 6.2. Deﬁnition 5. An arbitrage-free market is said to be a complete market if and only if a unique equilibrium measure exists. As outlined below, this uniqueness is important because it will mean that one can get a single valuation of the option. It is possible to give a mathematical characterization of a complete market, depending on the existence or not of a certain kind of portfolio. This characterization can be found in [17]. We will not use this characterization, and we will show that the market is complete in our particular model when it is required. 2.3.2 The concept of risk-neutral probabilities Let us make a small digression here which will be useful for the understanding of the relation of this work (or our world) to the real world. It is coherent to assume that the price of an asset depends basically on its risk. Also, it is coherent to assume that investors may be prone to avoid that risk and that, if any risk is accepted, more possible proﬁts will be demanded. In other words, investors demand more proﬁt for bearing more uncertainty. Carles P´erez Guallar 17
- 19. 2 DERIVATIVES AND ARBITRAGE PRICING Logically, when one has to price an asset, it has to be taken into account the associated risk. In general, though, the risk quantiﬁcation has subjective factors, depending on the investor, and it is very diﬃcult to obtain an accurate measure. In an arbitrage-free complete market there is an alternative to evaluate the risk of an asset, and this alternative has not relation at all with subjectivity. This alternative stems from the existence of a set of probabilities such that, somehow, consider all the infor- mation about the possibles outcomes in the future, that is, it takes into account all the possible scenarios. These probabilities are those given by the equilibrium measure π, that exists always in absence of arbitrage, and which associates a probability to each scenario. By using π, one can compute the fair price of an asset as the expected value of its price in the future, discounted by the risk-free interest rate (recall the expression (2.7)). In absence of uniqueness for the equilibrium measure, this computation can be done for each equi- librium measure, thus more than one price can be obtained for each asset. Therefore, the use of risk-neutral probabilities requires the market to be complete. It is common to confuse the constructed probability distribution with the real-world probabilities. They are diﬀerent since, in the real world investors can be risk seekers or, conversely, more conservatives, whereas in a risk-neutral world there are no such cat- egories. Let us see this through an example: Suppose we can choose between 50 euros or bet 0 or 100 on heads or tails. In both cases the expected value is 50 euros and in a risk-neutral world both options are equiv- alent. For a risk-neutral investor both options are exactly the same and these investors cannot distinguish between them. However, it is clear that in the real world these choices are essentially diﬀerent. A risk seeker investor would choose to toss the coin whereas a conservative investor would choose the 50 euros. Although the method of risk-neutral probabilities may seem artiﬁcial, it gives a useful framework to apply mathematics when evaluating some ﬁnancial products and currently is widely used in derivative pricing. The following pages will rely, as we will see, on the existence of such probabilities. Besides, the risk-neutral probabilities are closely linked to the Black-Scholes model. Actually, the main factor of success of the Black-Scholes model is that it was the ﬁrst option pricing model without any dependency on subjective quantities. 2.4 Put–call parity In this section it is justiﬁed the reason why from now on we only deal with Call options. To do this, we will prove a simple formula which relates the price of a European call option with the price of a European put option. In the formal deﬁnition of portfolio that we have Carles P´erez Guallar 18
- 20. 2 DERIVATIVES AND ARBITRAGE PRICING already made, we did not regard options as assets. However, without loss of generality, we will consider portfolios including options as assets. The reason is that an option can be seen as an asset with a certain price at time t = 0 and other at t = T (depending on its payoﬀ). Lemma 1. Let P be the price of a European put option, whose strike price is K and whose expiration time is T, over some underlying asset. Let C be the price of a European call option whose strike price is also K and whose expiration time is also T, over the same underlying asset. Then, for each t ∈ [0, T], the following formula holds: P = C − S0 + Ke−rt (2.11) Where S0 is the price of the underlying asset at time t = 0 and r is the risk-free interest rate. Proof. Consider, under the hypotheses given by the lemma, a portfolio Λ1 composed of: Λ1 = Long position in (we buy) a call option C Short position in (we borrow and sell) a put option P One has, denoting by Vt(Λ1), that the value of the portfolio at time t is: V0(Λ1) = P − C Vt(Λ1) = max{S(t) − K, 0} − max{K − S(t), 0} = S(t) − K Where we use the payoﬀ formulas of a put and a call options, and where, for the put, we subtract its value due to our short position in it. Now consider the portfolio Λ2 composed of: Λ2 = Long position in (we buy) one unit of the underlying asset. Short position in (we borrow) Ke−rt bonds One has, denoting by Vt(Λ2), that the value of the portfolio at time t is: V0(Λ2) = Ke−rt − S0 Vt(Λ2) = S(t) − Kert e−rt = S(t) − K Notice that Vt(Λ1) = Vt(Λ2). We will argue that in the case that two portfolios cost the same at time t, then they cost the same at time 0. Suppose, to reach a contradiction, that V0(Λ1) < V0(Λ2). Otherwise, the proof is the same. Then, one can take a long position in Λ1 and a short position in Λ2, making net proﬁts at time 0. And then, at time t, since Vt(Λ1) = Vt(Λ2), no payment has to be made. This clearly violates the absence of arbitrage and proves that V0(Λ1) = V0(Λ2), which directly gives the equality we had to prove. 6 6 Notice that, if one wants to be formal with the non-arbitrage deﬁnition we have made, one has to use bonds in the argument we did, in order to obtain a zero investment at t = 0 and net proﬁts at time t = T Carles P´erez Guallar 19
- 21. 3 BINOMIAL OPTION PRICING MODEL 3 Binomial option pricing model 3.1 Building up the model In this chapter we will use the market hypothesis and properties from last section to introduce the binomial pricing model. This model arises, essentially, from supposing a market which evolves in a discrete time and such that the price of the underlying asset evolves with it. That is, given an option, one divides its expiration time in N discrete periods or steps. Between two intermediate steps, from step n to step n + 1, a ﬁnite set of outcomes (scenarios) is considered, generating a lattice or tree whose nodes are scenarios. A probability is assigned to every change of scenario (to each branch). Then, the outcomes and probabilities are considered backwards through the tree until one ﬁnds a fair value for the option at time 0. 3.1.1 One-step binomial model Let M be a single period market which evolves from time t = 0 to time t = T, in which only two assets are traded: the bond B an a risky asset, A, for instance some kind of shares. Let S0 be the price of one unit of A at time 0. Suppose that, at time T, two scenarios are possible: one in which the price of A is larger than it was at time t = 0 and other in which the price of A is less than it was at time t = 0. The ﬁrst is denoted by u (from up) whereas the second one is denoted by d (from down). Following this notation, one can choose two numbers u and d such that 0 < d < 1 < u and through them one can express the two possible scenarios as: ST (u) = S0 · u > S0 S0 = S(0) 55 )) ST (d) = S0 · d < S0 This is the so-called one-step binomial model, which has already appeared in the example discussed in the ﬁrst section. Notice that both u and d have two meanings. On the one hand, they are numbers, as it has been stated. On the other hand, they denote the respective scenarios. This will never be confusing since the context always will make clear what they are expressing. Theorem 2. In a one-step binomial model M, the following condition: d < erT < u 7 (3.1) 7 Notice that this means that the increase or decrease in the price of the risky asset has to overcome the rate of change (the discount) given by the risk-free interest rate, or equivalently, the change in the value of money over the time. Carles P´erez Guallar 20
- 22. 3 BINOMIAL OPTION PRICING MODEL is equivalent to the property that M is an arbitrage-free and complete market (i.e. there is one and only one risk-neutral measure π). More precisely, if one deﬁnes: p := erT − d u − d (3.2) one has that the risk-neutral measure is given by π(u) = p and π(d) = (1 − p) Proof. The proof is a simple computation. It has to be seen that there is one and only one equilibrium measure, that is, a function π whose values range in [0, 1], depending on the scenarios u and d, which satisﬁes: S0 = e−rT π(u)S0u + π(d)S0d and π(u) + π(d) = 1 (3.3) This system of linear equations is straightforward to solve, obtaining: π(u) = erT − d u − d = p π(d) = 1 − erT − d u − d = 1 − p (3.4) And π(u), π(d) ∈ [0, 1] if and only if 0 ≤ π(u) ≤ 1, if and only if 0 ≤ erT − d ≤ u − d, if and only if d < erT < u. This ﬁnishes the proof. Note that what this result shows is that the probabilities of a rise or a fall in the asset price (the A price) are not subjective: under the assumption of no arbitrage and the completeness in a one-step binomial market, these probabilities are predetermined (after assuming known values for u, d and r). The values p and 1 − p are the risk-neutral prob- abilities, about which we have already discussed. In a later chapter we will see how to choose the values of u, d and r. Summarizing, we have seen that in the one-step binomial market, choosing properly the values of u, d and r, the arbitrage-free and completeness properties are met. It can be easily seen that an example of an arbitrage-free market but not complete is a trinomial model (i.e. with three possible scenarios). In this case, choosing a third value such that d < s < u, with d and u meeting the conditions given by the last theorem, one has a market with inﬁnite equilibrium measures. This is straightforward to see: following the same steps that in the proof above, the obtained system has three equations and two unknowns, so it has inﬁnite solutions, each of them giving a speciﬁc equilibrium measure (this example can be found in [17]). 3.1.2 Multi-period market and binomial trees We have deﬁned a single period market as the market which evolves over time in one single step, from t = 0 to t = T. Let us generalize this idea to N steps. We will suppose that the transactions and changes of price can take place in a discrete-time in the instants given by: 0 = t0 < t1 < · · · < tN = T Carles P´erez Guallar 21
- 23. 3 BINOMIAL OPTION PRICING MODEL Where, in the transition between ti and ti+1, one has the deﬁnition of a single period market. This structure is the so-called multi-period market. From now on we will deal with multi-period markets, with N temporal steps that, for simplicity, we will consider such that: ∆t = tn − tn−1 = T N n = 1, . . . , N We will keep the assumption of two assets, A and B, respectively the risky (e.g. shares) and the riskless asset (e.g. bonds). Thus the portfolio is given by a pair of values (A, B), where A and B denote the respective units of shares and bonds. Meanwhile, S(tn) = Sn will denote the price of each unit of A at time tn, with j = 0, . . . , N. As before, the price of one bond is set as being equal to 1 and consequently its price at time tn will be ertn which is equal to ern∆t . Our aim is to generalize the one-step binomial model to N steps or periods. I.e., the asset A starts with a certain initial price at time t = 0. Then, at time t1, this price will increase or decrease. At time t2, the prices that one had at t1 will again increase or decrease, and so on, with all the increments and decrements given, respectively, by the same factor. This generates a set of branches known as binomial tree. Let us formalize this model. We assume that the price of the asset follows an stochastic dynamics such that, between two time instants tn−1 and tn, the asset price can increase or decrease in accordance with: Sn = Sn−1 · zn n = 1, . . . , N (3.5) Where z1, . . . , zN are independent and identically distributed Bernoulli random variables. Hence each zN satisﬁes: zn = u with probability p d with probability (1 − p) Where, as it has been stated, 0 < d < 1 < u. Then, considering a single time increment: Sn = uSn−1 with probability p dSn−1 with probability (1 − p) By this, all the possible market scenarios are determined in each tn. Under these assump- tions, {S(ti) : i = 0, . . . , N} may be seen as a time series or a stochastic process, with which a set of trajectories can be associated. A trajectory for N = 3 might be: (S0, uS0, udS0, u2 dS0) or equivalently, summarized as (u, d, u) This trajectory also can be expressed as the vector (u, ud, u2 d). Regardless of the chosen representation, the trajectory is determined by the values taken by the random variables zi, i = 1, . . . , 3. Let Ω be the set of all possible trajectories: Ωn = {(e1, e2, . . . , eN ) : ek = u o ek = d, j = 1, . . . , N} Carles P´erez Guallar 22
- 24. 3 BINOMIAL OPTION PRICING MODEL Then Ω has 2N elements. Returning to the example N = 3, notice the fact that the set of all trajectories can be graphically represented as: S = S o S = u S S=dSo S= uuSo S=udSo S=ddSo S=uuuSo S=uudSo S=uddSo S=dddSo o Figure 3.1: Binomial tree for N = 3. The binomial model introduced here, such that if the underlying asset moves up and then down (u, d), the price will be the same as if it had moved down and then up (d, u), is the so-called recombinant tree model — since the two paths merge or recombine-. Later on we will see how to compute the option price by means of binomial trees. The recombinant property reduces the number of tree nodes, and thus accelerates the computation of the option price. Moreover, as we will see, this property also allows to compute that price not only by using a recursive method, but also via a closed formula, avoiding the necessity to build the tree. Characterization of the scenarios: observe that, in a binomial tree of N steps, each node can be characterized in a unique way by two numbers, the ﬁrst one is n, with 0 ≤ n ≤ N, which indicates the time step or the height in the tree, the second one is j, with 0 ≤ j ≤ n, which indicates the number of increases u that have taken place (and therefore also sets the number of decreases d, which is n − j). Then, each node, char- acterized by (n, j) can be seen as an scenario ωn,j. This construction, and especially its graphical representation, is called a binomial tree. Also note that a N-step binomial tree will have N + 1 ﬁnal nodes, which are charac- terized in a unique way by j = 0, 1, . . . , N. Sometimes one is interested only in these ﬁnal nodes, as if it were a single period market with N+1 scenarios given by ωj, j = 0, 1, . . . , N. Subsequently this interpretation will be used. Now, we will see a lemma that provides a mathematical basis for the binomial trees in terms of a discrete-time and homogeneous Markov chain (see the last page of Appendix A (6.2) for the deﬁnition of this type of random process). Lemma 2. The stochastic process S = {St : t ∈ {t0, . . . , tN }} given by a binomial model of N steps is a discrete-time and homogeneous Markov chain. Carles P´erez Guallar 23
- 25. 3 BINOMIAL OPTION PRICING MODEL Proof. It turns out to be evident that the future state relies only upon on the present state, having only two possibilities: an increase u or a decrease d, which in no way dependent on past states. The proof only formalizes this idea. The state space E is the set of states, Sj tn = S0uj dn−j , with 0 ≤ n ≤ N and 0 ≤ j ≤ n, and each of them can be expressed, as we have already stated, by the pairs (n, j). Thus consider m + 1 states such that: P{St0 = (n0, j0), . . . , Stm = (nm, jm)} > 0 (3.6) We have to demonstrate that the Markov property is met: P{Stm+1 = (nm+1, jm+1) | St0 = (n0, j0), . . . , Sm = (nm, jm)} = (3.7) P{Stm+1 = (nm+1, jm+1) | Stm = (nm, jm)} (3.8) First of all, notice that (3.6) holds if and only if n0 = 0, n1 = 1, . . . , nm = m, since other- wise it would be a jump between two non-consecutive heights of the tree, which happens with 0 probability. For the same reason, if nm+1 = m + 1, then the equality (3.7) is met in form of 0 = 0. Besides, unless the trajectory (n0, j0), . . . , (nm, jm) leads to one of the two possible nodes previous to the node Stm+1 = (nm+1, jm+1), which are the nodes such that jm = jm+1 − 1 or jm = jm+1, the equality (3.7) is met again in form of 0 = 0. Hence for the sequence (n0, j0), . . . , (nm, jm), the only free parameter is jm, which can take two values: jm = jm+1 − 1 or jm = jm+1. Therefore, the future state depends only on the present state. That is: P{Stm+1 = (m + 1, jm+1) | St0 = (1, j0), . . . , Sm = (m, jm)} = P{Stm+1 = (m + 1, jm+1) | Stm = (m, jm)} = = u if jm+1 = jm + 1 d if jm+1 = jm And, since these probabilities do not depend on the value of m, one has that the homo- geneous property is met. The Markov property for the binomial trees, although is not used anymore in this work, helps to understand the binomial tree structure to those who are familiar with Markov chains, and gives the possibility to apply the several properties and tools of analysis of Markov chains to the binomial trees. Last but not least, this characterization also emphasizes the fact that a binomial tree can be interpreted as a particular case of random walk, a fact that is clearly intuitive. 3.1.3 Call option pricing under the binomial model Consider again a one-step binomial market. Our goal is to ﬁnd the fair price of a Euro- pean call option at time t = 0 over one unit of the asset A, with a given strike price K. The fair price is the one that does not violate the arbitrage-free hypothesis. Carles P´erez Guallar 24
- 26. 3 BINOMIAL OPTION PRICING MODEL The method is the same that has been employed in the introductory example: setting up a portfolio by using a hedge strategy under the arbitrage-free hypothesis to obtain the so-called fair price of a call option. Consequently we start by taking a long position in a certain amount of the assets A and B and selling a call option (short position) over one unit of A with the purpose to get a risk-free portfolio. Let X0 be the price of the call option at time t = 0. So the initial value of our portfolio is: V0(Λ) = AS0 + B − X0 (3.9) Let Xu and Xd be the Call payoﬀs in the respective scenarios, i.e., the losses we will have due to the obligation to cover the Call option. At time T two scenarios are possible: VT (Λ)u = AuS0 + erT B − Xu (3.10) VT (Λ)d = AdS0 + erT B − Xd (3.11) Aiming to eliminate the risk of our portfolio we equal the above equations, i.e., we impose VT (Λ)u = VT (Λ)d , as in the introductory example, and we get: ˆA = Xu − Xd S0(u − d) (3.12) Now, using this value ˆA, the portfolio is free of risk, and due to the assumption of no arbitrage, it has to be met that V0(Λ) = e−rT VT (Λ)u , where we regard the time-dependence in money’s value. Imposing this equality and using the value ˆA, one easily obtains: X0 = Xu 1 − de−rT u − d + Xd ue−rT − 1 u − d (3.13) Recalling the expression of p, (3.2), given by theorem 2, the expression above can be rewritten as: X0 = e−rT (pXu + (1 − p)Xd) (3.14) Note that p is the probability of an up movement, i.e. the probability of a payoﬀ equal to Xu. Similarly, (1 − p) is the probability of a down movement, i.e. the probability of a payoﬀ equal to Xd. So that the following equality is met: X0 = e−rT E[XT ] (3.15) Where we understand XT as the random variable whose values are the possible values of the call option at maturity, interpreting the payoﬀ at time T as the value of the call option at time T. Thus, we have obtained the fair price of the Call option X0 as its expected proﬁts, with a factor e−rT that represents the rate of return given by the risk-free interest rate or, equivalently, the time value of money. The goal now is to generalize this result, and to carry out this one must switch the Carles P´erez Guallar 25
- 27. 3 BINOMIAL OPTION PRICING MODEL single period market by a multi-period market. Assume a multi-period market which has, again, the assets A and B. Assume that this market evolves under a binomial model in N steps, 0 = t0 < t1 < · · · < tN = T, all of them with length ∆t, generating a binomial tree. Keep in mind that each node or scenario can be uniquely expressed as: S(ωn,j) = S0uj dn−j (3.16) By the formula shown above (3.14), and noting that one can consider each scenario as a node such that at the next step divides into two nodes (being a one-step tree), one has: Xj n = e−r∆t (pXj+1 n+1 + (1 − p)Xj n+1) = e−r∆t E[Xn+1|Sn = Sj n] (3.17) for 1 ≤ n ≤ N − 1 0 ≤ j ≤ n. Where the notation E[Xn|Sn = Sj n] refers to the fact that the expectation has to be com- puted conditioned on the starting node. Given all the tree and taking into account the last expression, we have obtained a re- cursive method to price the call option, which stems from going backwards on the tree. In our next step it will be shown that from this recursive method one can obtain a closed formula which gives the fair price of the call option at time t = 0. Proposition 1. The fair price of an European Call option can be obtained from the ﬁnal nodes of a N-step binomial tree as: X0 = e−r∆tN N j=0 N j pj (1 − p)N−j Xj N (3.18) Where, by deﬁnition, ∆tN = T The interpretation of this formula is simple: it is again an expected value corrected by the risk-free interest rate. In the summation there are exactly one term for each ﬁnal node or, equivalently, for each ﬁnal scenario (characterized by each j), in which the Call payoﬀ is Xj N . By using basic combinatorics, one gets that the probability of being in the node j is given by: N j pj (1 − p)N−j Where pj (1 − p)N−j is the probability of a trajectory that leads to that node and where N k are the total number of trajectories that lead to that scenario. X0 = e−rT E[XN ] (3.19) Carles P´erez Guallar 26
- 28. 3 BINOMIAL OPTION PRICING MODEL Proof. The proof is by induction on N. For N = 1 we have already seen (equation (3.14)) that: X0 = er∆t (pXu + (1 − p)Xd ) (3.20) Assume that the formula holds up to N − 1, and let us show the N case. In a N-step binomial tree, the N − 1 nodes satisfy, by using the case N = 1, the following equality: Xj N−1 = e−r∆t pXj+1 N + (1 − p)Xj N (3.21) By using the induction hypothesis on N − 1, one has, for a N-step binomial tree, that: X0 = e−r∆t(N−1) N−1 j=0 N − 1 j pj (1 − p)N−j−1 e−r∆t [pXj+1 N + (1 − p)Xj N ] (3.22) Which can be rewritten as: X0 = e−r∆tN I N−1 j=0 N − 1 j pj+1 (1 − p)N−j−1 Xj+1 N + II (1 − p) N−1 j=0 N − 1 j pj (1 − p)N−j−1 Xj N (3.23) Thus the factor e−r∆tN has been obtained. Now we have to simplify and rearrange the sum I + II to ﬁnish the proof. By changing the sum index, I can be written as: I = (1 − p) N j=1 N − 1 j − 1 pj (1 − p)N−j−1 Xj N (3.24) This last expression allows us to work with I + II together (by adding the summands). We are going to study the coeﬃcient of Xj N in I + II. If we see that they are equal to N j pj (1 − p)N−j for j = 0, . . . , N, we will ﬁnish the proof. For j = 0, one only has to look at II. The summand is: (1 − p) N − 1 0 p0 (1 − p)N−1 X0 N = (1 − p)N X0 N = N j pj (1 − p)N−j Xj N |j=N (3.25) For 0 < j < N both I and II have to be taken into account, and the resulting summand is: [ N − 1 j − 1 + N − 1 j ]pj (1 − p)N−j Xj N (3.26) Notice that the value between brackets ([ ]) can be rewritten as: N − 1 j − 1 + N − 1 j = j N N j + N − j N N j = N j (3.27) Which gives the coeﬃcient we had to obtain for Xj N with 0 < j < N. It only remains the case of N. We only have to look at I, and the term is: N − 1 N − 1 pN XN N = pN XN N = N j pj (1 − p)N−j Xj N |j=0 (3.28) Carles P´erez Guallar 27
- 29. 3 BINOMIAL OPTION PRICING MODEL We can further analyze the obtained formula from the fact we have a set of factors Xj N which represent the payoﬀ. Now we will see how this formula is written for a European call option, with strike price K and with expiration time T, and whose payoﬀ is given by: Xj N = max{Sj N − K, 0} Since Sj N is an increasing sequence in j, in general a j0 such that if j ≥ j0 then Xj N = Sj N − K will exist. Thus, the value of the option at time zero is: X0 = e−rT N j=j0 N j pj (1 − p)N−j (Sj N − K) (3.29) Besides, recalling that Sj N = S0uj dN−j , j0 is the minimum integer such that: S0uj dN−j ≥ K, Which directly leads to: j0 = min{j ≥ ln K S0dN ln u d : j ∈ {0, 1, . . . , N}} (3.30) Finally we obtain the following closed formula to price a European call option: X0 = e−rT S0 N j=j0 N j pj (1 − p)N−j uj dN−j − e−rT K N j=j0 N j pj (1 − p)N−j (3.31) with j0 given by (3.30). 3.1.4 Properties of the multi-period binomial market As it has been stated before, if we take into account only the instants t = 0 and T in a multi-period binomial market, then the resulting market is a single period market with N + 1 scenarios (one for each ﬁnal node). Each scenario can be denoted by ωN,j, j = 0, . . . , N. Let M be that market. We will brieﬂy present the properties of M. First of all, notice that, in M, the closed formula to price a call option (3.18) stills having sense since it depends only on the ﬁnal scenarios (or time T). Besides, more properties can be shown. Proposition 2. In a N-step binomial market the following formula holds: S0 = e−rT N j=0 N j pj (1 − p)N−j Sj N (3.32) Carles P´erez Guallar 28
- 30. 3 BINOMIAL OPTION PRICING MODEL Proof. The similarity with the formula for X0 (3.18) is clear, which we have already proved by induction using the case N = 1. In the case of the formula presented here we also have proved the case N = 1 in theorem 2 and the induction steps to prove the formula of this proposition are exactly the same than those we used to prove (3.18). In fact, the formula to price the call option is a particular case of the formula introduced here, since as it has been outlined an option may be thought as an asset in a portfolio. Again, this formula depends only on 0 and T. Recalling the deﬁnition of an equilibrium measure, it is natural to deﬁne: π(ωN,j) = N j pj (1 − p)N−j And one has: S0 = e−rT N j=0 π(ωN,j)Sj N (3.33) It is clear that π is a probability function since in particular it follows the binomial distribution. Hence π is an equilibrium measure in M. From that fact follows that M is an arbitrage-free market (remember that we are considering only two assets, A and B, for A the last preposition has proved that π satisﬁes the condition of being an equilibrium measure whereas for B it is straightforward to prove). Now the natural question is whether this equilibrium measure is unique or not, i.e. if M is a complete market. Proposition 3. The market M is arbitrage-free and complete. Proof. We have already seen that an equilibrium measure π exists, so it is an arbitrage- free market. We have to show that the equilibrium measure is unique. For N = 1 we have demonstrated it in theorem 2. To prove the case N, the steps are identical to those that one has to use to prove (3.33), which in turn are the same we have used to prove (3.18). If one revises carefully this proof, one can realize that from it arises the uniqueness of π. As a matter of fact, the case N = 1 is a Bernoulli distribution, and if one repeats N independent Bernoulli trials, one obtains a binomial distribution with N trials, which means that no other probability function is possible. The fact that the so-called fair price for the call options exists comes from the arbitrage- free property, whereas the uniqueness of the fair price stems from the completeness of the binomial market. Finally, it has to be highlighted that the binomial model is also a powerful tool to evaluate American options (those which can be exercised at any time between t = 0 and T) since at the intermediate nodes one knows the payoﬀ for diﬀerent t ∈ [0, T] and the right time to exercise the option can be computed. Carles P´erez Guallar 29
- 31. 3 BINOMIAL OPTION PRICING MODEL 3.2 Model calibration The model calibration lies in determining the parameters of the model from observation of the real world. In the binomial model, these parameters are u, d and the risk-free interest rate r, since the value of p is determined by this three parameters. For now, we will assume that r is observable in the market and afterwards we will go deeper into that assumption. In this section we will introduce new concepts and new variables that will bring us tools to calibrate the model, i.e., to estimate through real data the values of u and d and, consequently, of p. Recall that the binomial model is based on the equality Sn = Sn−1zn, where zn is a random variable with Bernoulli distribution with probability p. Thus Sn can be inter- preted as a random variable. We will assume, for simplicity and as usual in the ﬁeld of ﬁnance, that the time t is measured in years. Deﬁnition 6. The logarithmic rate of return Y of an asset A whose price is S is deﬁned as the random variable that satisﬁes: ST = S0eY T (3.34) Where S0 is the price at time t = 0 and ST is the price at time tN = T. Notice that, for the risk-free asset B, one has Y = r, from which stems that the random variable Y is, in some way, a measure of the risk of the asset. Isolating Y : Y = 1 T ln( SN S0 ) (3.35) Since SN S0 = SN SN−1...S1 SN−1...S1S0 and using the logarithm properties, Y can be written as: Y = 1 T N n=1 ln( Sn Sn−1 ) = 1 T N j=1 ln(zn) (3.36) Where in the last equality we use that Sn Sn−1 = zn. Noting that: z = u with probability p d with probability (1 − p) We are able to compute the expected value and the variance of Y : µy = E[Y ] = E[ 1 T N j=1 ln(zn)] = 1 T N j=1 E[ln(zi)] = 1 ∆t (p ln(u) + (1 − p) ln(d)) (3.37) Carles P´erez Guallar 30
- 32. 3 BINOMIAL OPTION PRICING MODEL Where in the last equality we use that zi are i.i.d. random variables and that N T = 1 ∆t . The variance is given by: σ2 y = V ar[Y ] = 1 T2 NV ar[z0] (3.38) Where: V ar[z0] = E[z2 0] − E[z0]2 = (p ln(u)2 + (1 − p) ln(d)2 ) + (p ln(u) + (1 − p) ln(d))2 = = p(1 − p)[ln(u)2 + ln(d)2 − 2 ln(u) ln(d)] = p(1 − p)[ln(u) − ln(d)]2 = p(1 − p) ln2 ( u d ) (3.39) Thus ﬁnally: σ2 y = 1 T∆t ln2 ( u d )p(1 − p) (3.40) Both computed parameters are important. On one hand, the expected value µy is known as annual rate of return, average rate of return or, simply, rate of return. From its deﬁnition it can be understood why it is known by this name: as it has been said Y represents the analogous of the risk-free interest rate though for a risky asset. This risk causes a variability which is the reason why it is expressed through a random variable and one has to compute the expected value. On the other hand, one has the volatility, denoted by σ, which is deﬁned as the standard deviation of the annualized logarithmic rate of return or log returns. I.e., σy with T = 1, i.e.: σ = 1 ∆t ln2 ( u d )p(1 − p) (3.41) For now, we will assume that the volatility is ﬁxed and in a later section we will discuss how this parameter is estimated. Our target is to ﬁnd a way to determine u and d. We deﬁne the auxiliary parameters: u := ue−r∆t d := de−r∆t (3.42) Then, recalling the expression for p in the binomial model (3.2), p = er∆t − d u − d , one obtains: u p + (1 − p)d = 1 (3.43) Further, we deﬁne ξ as the value which satisﬁes the following equality: u d = u d = e2ξ √ ∆t (3.44) Notice that ξ > 0 since u/d > 1 and, hence, the exponent is positive. Also notice that, isolating: ξ = 1 4 ln2 ( u d ) 1 √ ∆t (3.45) Carles P´erez Guallar 31
- 33. 3 BINOMIAL OPTION PRICING MODEL And then the volatility (3.41) can be expressed as: σ2 = 4p(1 − p)ξ2 (3.46) Which lead us to the equation: p(1 − p) = σ2 4ξ2 (3.47) It is easy to see (deriving) that the function p(1−p), with 0 < p < 1, has its maximum at p = 1/2 whose value is 1/4, so the last equation (3.47) has sense only for ξ ≥ σ. Moreover, its solutions are: p = 1 2 (1 ± 1 − σ2 ξ2 ) (3.48) Besides, isolating from (3.43), we obtain: p = 1 − d u − d (3.49) From the equation above, (3.49), one has u = 1 − (1 − p)d p , and from (3.44), d = u e−2ξ √ ∆t . Combining both expressions it holds that: u = 1 p + (1 − p)e−2ξ √ ∆t (3.50) And: d = 1 pe2ξ √ ∆t + (1 − p) (3.51) Finally, using (3.42) and multiplying the numerator and the denominator by eξ √ ∆t for u and by e−ξ √ ∆t for d we get the expressions for u and d: u = eξ √ ∆t+r∆t peξ √ ∆t + (1 − p)e−ξ √ ∆t d = e−ξ √ ∆t+r∆t peξ √ ∆t + (1 − p)e−ξ √ ∆t (3.52) And these are the possible values for u and d under the no arbitrage hypothesis. Observe that, given a value for σ and for √ ∆t, one has yet dependence on ξ (or equiva- lently on p, since ξ and p are related by the equation (3.48)). Namely, the above expres- sions for u and d have one degree of freedom. In the literature, there are some common choices for this degree of freedom allowing to obtain values for u and d. The two most common are: • Taking ξ = σ one has, by (3.47), p = 1/2, and the corresponding values of u and d given by expressions (3.52). Recall that we have assumed that σ is known. • Assuming u · d = 1. We will see, in the following section, the importance of this choice. Carles P´erez Guallar 32
- 34. 3 BINOMIAL OPTION PRICING MODEL We have found out that in a N-step binomial market with speciﬁc values for the model parameters, one can compute the fair price or arbitrage-free price of a call option. It is now natural to ask what happens when the number of steps N increases. That is, what happens when the time length between steps becomes smaller, when ∆t → 0. We wonder whether the model is stable or not, i.e., if when N increases the value of the call option tends to a speciﬁc number. The following ﬁgure allows to foresee the answer: Convergence of the binomial model N Callprice 5.80 5.85 5.90 5.95 6.00 6.05 6.10 20 40 60 80 100 Figure 3.2: Valuation of a call option (X0) under the binomial model with parameters T = 10, S = 10, K = 11, r = 0.02, σ = 0.5, for N from 5 to 100, with the assumption that u · d = 1 In the following chapter we will deeply study this convergence, but before reaching it, we will make a quick note on the model known as the Cox, Ross and Rubinstein model. This model is the most common one in literature and has great historical signiﬁcance for the present work. Carles P´erez Guallar 33
- 35. 3 BINOMIAL OPTION PRICING MODEL 3.2.1 The Cox, Ross and Rubinstein model (CRR) The model based on binomial trees for option pricing was ﬁrst proposed in 1979 by the economists and engineers John Carrington Cox, Stephen Ross and Mark Edward Rubinstein. 8 In that article, the authors carried out the passage to the limit and, by doing this, they obtained the Black-Scholes formula, in a diﬀerent manner from that used by Black and Scholes back in 1973. Furthermore, before carrying out the passage to the limit, Cos, Ross and Rubinstein assumed that u·d = 1 and then they took u = eσ √ ∆t and d = e−σ √ ∆t . Due to the importance of this article, we will present, in the way they did, where this values comes from. Thus we will see how by taking u · d = 1, one can deduce the values u and d as a function of σ used in the CRR model. Let us assume u · d = 1, and taking into account equations (3.37) and (3.41) we have: σ2 ∆t = 4p(1 − p) ln(u)2 2p = 1 + µy ln(u) (3.53) Solving the system: u = eσ √ ∆t √ 1+∆t( µy σ )2 d = e−σ √ ∆t √ 1+∆t( µy σ )2 (3.54) Expanding the exponent to ﬁrst order: σ √ ∆t 1 + ∆t( µy σ )2 = σ √ ∆t + O( √ ∆t) (3.55) Therefore, for ∆t small, one has: : u = eσ √ ∆t d = e−σ √ ∆t (3.56) With this choice, they took the limit, getting the Black-Scholes formula. A similar ap- proach will be introduced in the subsequent pages. However, in our case we will not assume the condition u · d = 1. The explanation is that, in the limit, all choices become the same, provided they satisfy the equations presented in this chapter. The proof of this property is implicit in the following chapter, in which we will obtain the Black-Scholes formula without assuming any speciﬁc relationship between u and d, beyond those given by the set of equations already presented. 8 In their paper, Option pricing: a simpliﬁed approach, [5]. The notation of that article is very similar to that used in all the current literature as well as in this work. Carles P´erez Guallar 34
- 36. 3 BINOMIAL OPTION PRICING MODEL 3.3 The passage to the limit Suppose ﬁxed values u and d. The expressions deduced in last section will be used to achieve the passage to the limit, i.e. to compute the limit when the number of steps N tends to inﬁnity, N → ∞, or equivalently, ∆t → 0. Lemma 3. The random variable Y has asymptotically normal distribution given by: Y ∼ N(r − σ2 2 , σ2 T ) (3.57) Proof. Recall the expression of Y , (3.36). We can rewrite it as: Y = 1 N N i=1 N T ln(zi) = 1 N N i=1 1 ∆t ln(zi) (3.58) Notice that 1 ∆t ln(zi) are i.i.d. random variables, thus we can apply the Central Limit Theorem in the last expression, since our goal is to ﬁnd the behaviour when N → ∞. The ﬁrst we need is the value of E[ 1 ∆t ln(zi)]. We have already seen that: E[( 1 ∆t ln(zi))] = 1 ∆t (p ln( u d ) + ln(d)) (3.59) Now, we substitute in the expression above u and d and u/d for some of the expressions we have deduced, ((3.52),(3.44)). For convenience, we consider each summand separately: p ln( u d ) = p2ξ √ ∆t using (3.44) ln(d) = −ξ √ ∆t + r∆t− ln(peξ √ ∆t + (1 − p)e−ξ √ ∆t ) I using (3.52) (3.60) Let us consider only the last term of the above expression, which we called I. Taking common factor and by means of the logarithm properties, one obtains: I = ln(peξ √ ∆t + (1 − p)e−ξ √ ∆t ) = ln(e−ξ √ ∆t ) + ln(1 + p(e2ξ √ ∆t − 1)) (3.61) Notice that, if ∆t → 0 then p(e2ξ √ ∆t − 1) → 0, and taking into account the Taylor series ln(1 + x) = x − x2 2 + .., we have: I = −ξ √ ∆t + p(e2ξ √ ∆t − 1) − 1 2 p2 (e2ξ √ ∆t − 1)2 + O(( √ ∆t)3 ) The notation O indicates, as usual, the behaviour in the inﬁnity ∆t → 0 of the terms that have not been written. Moreover, in this notation, we have replaced (e √ ∆t − 1)3 Carles P´erez Guallar 35
- 37. 3 BINOMIAL OPTION PRICING MODEL by( √ ∆t)3 . This has been made by convenience and it is easy to justify by applying the Hˆopital rule to show that these two expressions are equivalent inﬁnitesimals, i.e.: lim ∆t→∞ e √ ∆t − 1 √ ∆t = 1 Now, we use the Taylor series of ex up to second order in (3.61), to get: I = −ξ √ ∆t + p(2ξ √ ∆t − 4ξ2 ∆t) − p2 (2ξ2 ∆t) + O(( √ ∆t)3 ) Now, replacing this last expression of I in the expression of ln(d) (3.60), and then using the obtained expression of ln(d) in (3.59), to ﬁnally take into account the expression of p ln(u/d) given by (3.60), one gets: µy = 1 ∆t (p2ξ √ ∆t+ξ √ ∆t−r∆t−ξ √ ∆t+p(2ξ √ ∆t−4ξ2 ∆t)+p2 (2ξ2 ∆T)+O( √ ∆t)3 ) = = r + 2p2 ξ2 − 2pξ2 + O( √ ∆t) = r − 2ξ2 p(1 − p) + O( √ ∆t) And considering (3.46): E[( 1 ∆t ln(zi))] = r − σ2 2 + O( √ ∆t) (3.62) The variance of Y , σY , has already been computed, (3.40)) , and from the deﬁnition of the volatility σ (3.41) one has σ2 y = σ2 T . Hence by the Central Limit Theorem, Y is asymptotically normal with parameters: Mean given by E[( 1 ∆t ln(zi))] = r − σ2 2 Variance given by σ2 T As we aimed to prove. Lemma 4. The random variable ST follows, in the limit ∆t → 0, a log-normal distribution such that: ST = S0eσ √ TZ+(r−σ2 2 )T Z ∼ N(0, 1) (3.63) Proof. Recall the ﬁrst deﬁnition of Y , (3.34), which was ST = S0eyT . Now, since asymp- totically Y ∼ N(r − σ2 2 , σ2 T ), we can write: Y = σ √ T Z + (r − σ2 2 ) Z ∼ N(0, 1) (3.64) And substituting it in the expression of ST we obtain the statement of the lemma. Carles P´erez Guallar 36
- 38. 3 BINOMIAL OPTION PRICING MODEL Corollary 1. More generally, for any t ∈ (0, T] it holds that: St = S0eσ √ tZ+(r−σ2 2 )t Z ∼ N(0, 1) (3.65) Proof. It suﬃces noticing that all the above analysis can be carried out for any t ∈ (0, T]. One simply has to take the interval [0, t] and divide it into N + 1 equal intervals ∆t = t N . Then one takes the limit in the same way we have done and the same expressions, depending on t instead of T, are obtained. We will apply these asymptotic results to the valuation of a European call option over an asset whose price is S. Let f be the function deﬁned as f(SN ) = max{SN −K, 0}. Then the payoﬀ of a European call option can be written as XN = f(SN ). Now, our goal is to ﬁnd an expression for the price of the call at t = 0, denoted by X, in the limit. Theorem 3. The price X of an European Call option satisﬁes: lim ∆t→0 X = e−rT 1 √ 2π ∞ −∞ f(S0ezσ √ T+(r−σ2 2 )T )e −z2 2 dz (3.66) Proof. We have shown that, under the binomial model, the option price can be computed as: X = e−rT E[f(SN )] And, by Lemma 4, in the limit, one gets: X = e−rT E[f(S0eσ √ Tz+(r−σ2 2 )T )] z ∼ N(0, 1) Now we have to regard the property that, given a random variable Z with density hz(z) and given a function g, then the expected value of g(Z) can be computed as: E[g(z)] = ∞ −∞ g(z)hz(z)dz (3.67) Actually, this equality requires g to be a measurable function. In particular, we will take g as the payoﬀ f(SN ), which is a monotonically increasing function. This property is enough to ensure that a function is measurable. 9 . Indeed, applying (3.67) to the function of z given by g(z) = f(S0eσ √ Tz+(r−σ2 2 )T ) and taking into account that the density of a standard normal is h(z) = 1 √ 2π e −z2 2 , one obtains: lim ∆t→0 X = lim ∆t→0 e−rT E[f(SN )] = e−rT 1 √ 2π ∞ −∞ f(S0ezσ √ T+(r−σ2 2 )T )e −z2 2 dz (3.68) 9 If one has a background in measure theory, demonstrate that a monotonically increasing function is measurable does not require more than half a page. Carles P´erez Guallar 37
- 39. 4 THE BLACK-SCHOLES FORMULA 4 The Black-Scholes formula 4.1 Derivation We will use Theorem 3 to deduce the Black-Scholes formula. Suppose the risk-free interest rate is r, the strike price is K, the volatility is σ and the expiration time is T. We know that f(SN ) = max{SN − K, 0}. Therefore, using this equality in the expression shown by the last Theorem one gets: X = e−rT 1 √ 2π ∞ −∞ max{S0ezσ √ T+(r−σ2 2 )T − K, 0}e −z2 2 dz (4.1) Notice that the integrand is zero from −∞ up to a certain value of z. For convenience we will call this value −d2. It can be directly calculated: S0e(−d2)σ √ T+(r−σ2 2 )T = K ⇔ −d2 = 1 σ √ T (ln( K S0 ) − (r − σ2 2 )T) (4.2) Using −d2, we can write: X = e−rT 1 √ 2π ∞ −d2 S0 I ezσ √ T+(r−σ2 2 )T−z2 2 dz I − e−rT K √ 2π ∞ −d2 e −z2 2 dz II . (4.3) Now we will simplify this integral. For the ﬁrst summand, (I), we consider the change of variable s = z − σ √ T. That is, z = s + σ √ T. Then, the exponent of the term I is: sσ √ T + σ2 T + r − σ2 T 2 − s2 2 − sσ √ T − σ2 T 2 = rT − s2 2 (4.4) And since ds = dz, one obtains: X = 1 √ 2π S0 ∞ −d1 e −s2 2 ds − e−rT K 1 √ 2π ∞ −d2 e −z2 2 dz (4.5) Where, due to the change of variable, −d1 = −d2 − σ √ T. Note that both integrands are density functions of the standard normal distribution. Therefore, both integrals can be written in terms of the standard normal cumulative distribution function, which we will denote by N. It is well known that, due to the symmetry of the standard normal density, it holds that N(a) = 1 − N(−a). This is: 1 − N(−a) = 1 √ 2π ∞ −a e −x2 2 dx = 1 √ 2π a −∞ e −x2 2 dx = N(a) (4.6) And by means of this property, from (4.5) we get: X = 1 √ 2π S0 d1 −∞ e −s2 2 ds − e−rT K 1 √ 2π d2 −∞ e −z2 2 dz (4.7) Carles P´erez Guallar 38
- 40. 4 THE BLACK-SCHOLES FORMULA Where: d1 = 1 σ √ T [ln( S0 K ) + (r + σ2 2 )T] d2 = 1 σ √ T [ln( S0 K ) + (r − σ2 2 )T] = d1 − σ √ T (4.8) And one can write the obtained expression of X in a compact form in terms of N: X = N(d1)S0 − N(d2)Ke−rT (4.9) And the equivalent expressions (4.7) and (4.9) are known as Black-Scholes formula. Finally it has to be pointed out that, because of the put-call parity relationship (Section 2.11), this formula may be used for pricing a put option. From the put-call parity formula: P = Ke−rT (1 − N(d2)) − S0(1 − N(d1)) (4.10) And by the properties of N: P = Ke−rT N(−d2) − S0N(−d1) (4.11) It is not simple to carry out an exact interpretation of why Black-Scholes formula has this form (4.9) or where each factor comes from. Nevertheless, an intuitive approach can be attempted. First, notice that the option price X depends on S0, which is the current price of the asset, minus the exercise price K, this last one discounted by the rate given by r since K represents a future value. It is coherent that, roughly speaking, the option price depends on the diﬀerence between the real price of what we will get (S) and the price we will pay for it (K). However, notice that both S and e−rT K are multiplied by N, which means that, in some way, are weighted by the normal cumulative distribution function, evaluated respectively in d1 and d2. The interpretation of these parameters is even more complex. What can be easily seen is that both d1 and d2 increase when the ratio S0/K increases. Thus, the larger the diﬀerence between the current price and the strike price, larger the weights in the formula. In other words, insofar as this ratio increases, then the probability to exercise the option is higher, and, in some way, the inﬂuence of each term is also higher. Finally, roughly speaking, d1 ∝ σ whereas d2 ∝ −σ, and then it seems that X increases in σ. We will prove, later on, that X is strictly increasing in σ. 4.2 Background of Black-Scholes formula As it has been said in the introduction, the model known as Black-Scholes model (some- times also known as Black-Scholes-Merton model) appeared for the ﬁrst time in 1973 in a paper by the Americans Fischer Black and Myron Scholes, entitled The Pricing of Options and Corporate Liabilities ([1]). Not much later, the American economist Robert C. Merton published another article in which he enlarged and gave more supports to this Carles P´erez Guallar 39
- 41. 4 THE BLACK-SCHOLES FORMULA model. In 1999, Scholes and Merton were awarded the Nobel Price in economy for this work (Black had passed away two years before). In their paper, Black and Scholes used this model to deduce the expression now known as Black-Scholes equation. This equation is a partial diﬀerential equation that does not appear in this work. Then, they imposed as boundary conditions the market hypotheses, for instance the absence of arbitrage, as it has been done here, along with a hedging strategy. By doing this, they obtained a speciﬁc equation. This equation turned out to have the same form than the equation that describes the distribution of heat in a given region over the time (the so-called heat equation, which is a parabolic partial diﬀeren- tial equation). This equation had been widely studied and solved in physics. Black and Scholes had therefore no problem to solve that equation and reached an expression, now known as Black-Scholes formula, which is the formula we have obtained here. Moreover, the market hypotheses presented in their article were the same than those we have used here, with only one exception. Black and Scholes added the hypothesis that the price of the assets follows, in the words of the original article, a random walk in continuous time. In fact, it is quite intuitive that this behaviour has appeared here, since as it has been pointed out, an N-step binomial model might be interpreted as a random walk. Hence, in the passage to the limit, we implicitly considered the so-called random walk in continuous time. The Black-Scholes formula allowed (and allows) to price an European option by means of a closed mathematical formula, thus giving a scientiﬁc framework, free of subjectivity. This caused a revolution in the usage of these ﬁnancial products. However, it might be thought that, using this expression, everyone can know exactly the price of an option. Then, how does one make proﬁts? This question will be answered in the following chap- ter, and it is from the explanation we will give where most of the incorrect and abusive uses of this equation comes from. This alleged misuse and abuse of this equation has been directly related to the ﬁnan- cial crisis 10 . It is said that this equation changed the way in which transactions took place. Before Black-Scholes, options did not represent a huge amount of money and, besides, transactions were made based on common sense and experience. However, af- ter the success of that formula, transactions began to be carried out based on whether a computer’s answer was ’yes or no’, most of times without assessing any of the model hypotheses. Lastly, remark that, as already noted in the original paper, by relaxing some hypotheses (for instance, supposing σ to be a function of t, assuming the existence of dividends...) many variations of Black-Scholes formula can be deduced. Despite the fact that in the real world most of investors use some of these variations, we will not get into details about these variations. 10 Among many other articles, BBB, April 7, 2012, Black-Scholes: The maths formula linked to the ﬁnancial crash. The Guardian, February The mathematical equation that caused the banks to crash Carles P´erez Guallar 40
- 42. 4 THE BLACK-SCHOLES FORMULA 4.3 Connection with the reality In all classes of mathematical models it is very important to assess how close the model hypotheses are to the real world. Skipping this evaluation is one of the most common mistakes. In the same way, the parameters play also a central role and their estimation could lead to winning or losing money, even though the model is always the same. Notice that Black-Scholes formula depends on some known and ﬁxed parameters (which are directly observable): the strike price K, the expiration time T and the initial price of the underlying asset S. However, it depends also on two other parameters which one has to estimate since they are not observables: these are the risk-free interest rate r and the volatility σ. 4.3.1 Risk-free interest rate Despite the fact that the risk-free interest rate is a theoretical concept which seems artiﬁ- cial, its estimation usually is as simple as taking a reference value. This value can be, for instance, the interest rate of treasury bonds from a solid economy (in practice, Germany or the United States, among others). Therefore, this parameter is commonly considered as observable. 4.3.2 Historical and implied volatilities and the Greek letters Recalling that volatility is deﬁned as the standard deviation of the logarithmic rate of return of a given asset, one can deduce that it is a measure of the risk of an asset or, equivalently, a measure of the variations in the price of an asset. As we will see, its estimation is more complex than the estimation of the risk-free interest rate. The assessment of some diﬀerent methods to estimate the volatility, as well as the advantages and disadvantages of each of them, would be enough to write several works like this. Hence, we will just brieﬂy explain that there are two major types of volatility, which turn out to be essentially diﬀerent: the historical volatility or statistic volatility and the implied volatility. Yet they yield diﬀerent results, commonly one computes and uses both volatilities. Historical volatility The historical volatility is a measure of the variations in the price of an asset, computed from data of past prices. It measures how the ﬂuctuations of prices have been during a given past period. Roughly speaking, it is usually obtained by measuring the variation of price of an asset during consecutive days, then computing the standard deviation of the logarithm of the rate of price variation, and ﬁnally expressing the result as an annual percent. We will see it next. Also it has to be pointed out that short-term investors use few measures: 10 or 20 days for example while long-term investors use more measures: 60, 180 or 360 days for example. When using time series models to ﬁt or predict the volatility, the number and periodicity of those measures is an important factor. Carles P´erez Guallar 41