Introduction to Options Pricing Theory

Transcription

1 Introuction to Options Pricing Theory Simone Calogero Chalmers University of Technology

2 Preface This text presents a self-containe introuction to the binomial moel an the Black-Scholes moel in options pricing theory. It is the main literature for the course Options an Mathematics at Chalmers, which provies the stuents with a first ruimentary knowlege in mathematical finance (in particular, without using stochastic calculus). The pre-requisites to follow this text are the stanar basic courses in mathematics, such as calculus an linear algebra. No previous knowlege on probability theory an finance are require. Each chapter is complemente with a number of exercises an Matlab coes. The exercises marke with the symbol (?) aim to critical thinking an o not necessarily have a well-efine unique solution. The solution of the exercises marke with the symbol ( ) an the answer to those marke with the symbol ( ) can be foun in appenixes B an C at the en of the text. Further exercises are foun in Appenix D. Remark: The Matlab coes presente in this text are not optimize. Moreover the powerful vectorization tools of Matlab are not employe, in orer to make the coes easily aaptable to other computer softwares an languages. The task to improve the coes presente in this text is left to the intereste reaer. 1

3 Contents 1 Warm-up Basic financial concepts Qualitative properties of option prices Binomial markets The binomial stock price Binomial markets Arbitrage portfolio Computation of the binomial stock price with Matlab European erivatives The binomial price of European erivatives Example: A stanar European erivative Example: A non-stanar European erivative Heging portfolio Computation of the binomial price of stanar European erivatives with Matlab American erivatives The binomial price of American erivatives Optimal exercise time of American put options Example of American put option Heging portfolio processes of American erivatives Computation of the fair price of American erivatives with Matlab Introuction to Probability Theory Finite Probability Spaces Ranom Variables Expectation an Variance Inepenence an Correlation Conitional expectation Stochastic processes. Martingales Applications to the binomial moel

4 5.4.1 General iscrete options pricing moels Quantitative vs funamental analysis of a stock Computing probabilities with Matlab Infinite Probability Spaces Joint istribution. Inepenence Central limit theorem Brownian motion Black-Scholes options pricing theory Black-Scholes markets Black-Scholes price of stanar European erivatives Black-Scholes price of European call an put options The Black-Scholes price of other stanar European erivatives Binary call option Butterfly options strategy Chooser option Implie volatility Stanar European erivatives on a ivien-paying stock Optimal exercise time of American calls on ivien-paying stocks A The Markowitz portfolio theory 129 B Solutions to selecte exercises 134 C Answer to selecte exercises 155 D Aitional exercises 156 3

5 Chapter 1 Warm-up The purpose of this chapter is threefol: (1) introuce a few basic financial concepts, (2) formulate an iscuss the main assumptions behin the stanar theory of options pricing, (3) erive some funamental qualitative properties of option prices. 1.1 Basic financial concepts For a more etaile iscussion on the concepts introuce in this section, see [2]. Financial assets The term asset may be use to ientify any economic resource capable of proucing value an which, uner specific legal terms, can be bought an sol (i.e., converte into cash). Assets may be tangible (e.g., lans, builings, commoities, etc.) or intangible (e.g., patents, copyrights, stocks, etc.). Assets are also ivie into real assets, i.e, assets whose value is erive by an intrinsic property (e.g., tangible assets), an financial assets, such as stocks, options, bons, etc., whose value is instea erive from a contractual claim on the income generate by another (possibly real) asset. For example, upon holing shares of the Volvo stock (a financial asset), we can make a profit from the prouction an sale of cars even if we o not own an auto plant (a real asset), since the performance of the Volvo company is reflecte in the value of our stock. As we consier only financial assets in these notes, the terms asset an financial asset will be henceforth use interchangeably. The price of a financial asset is agree by the buyer an the seller of the asset as a result of some kin of negotiation. More precisely, the ask price is the minimum price at which the seller is willing to sell the asset, while the bi price is the maximum price that the buyer is willing to pay for the asset. When the ifference between these two values, calle bi-ask sprea, becomes zero, the exchange of the asset takes place at the corresponing price. A generic financial asset will be enote by U an its price at time t by Π U (t). Prices are generally positive, although some financial assets (e.g., forwar contracts) have zero price. 4

6 The asset price refers to the price of a share of the asset, where share stans for the minimum amount of an asset which can be trae, an is typically measure in some unit of currency (e.g., ollars). In these notes all prices are given in a fixe currency, which is however left unspecifie. Markets Financial assets can be trae in official markets or in over the counter (OTC) markets. In the former case all traes are subject to a common legislation, while in the latter the exchange conitions are agree upon by the iniviual traers. Example of OTC markets are the currency markets (Forex) an the bon markets, while stock markets, option markets an futures markets are all examples of official markets. Buyers an sellers of assets in a market will be calle investors or agents. Long an short position Besies the usual operations of buying an selling the asset, we nee to consier an aitional common type of transaction, which is calle short-selling. Short-selling an asset (typically a stock) is the practice of selling the asset without actually owning it. Concretely, an investor is short-selling N shares of an asset if the investor borrows the shares from a thir party an then sell them immeiately on the market. The reason for short-selling an asset is the expectation that the price of the asset will ecrease in the future. More precisely, assume that N shares of an asset U are short-sol at time t = 0 for the price Π U (0) an let T > 0 be the time at which the shares must be returne to their original owner. If Π U (T ) < Π U (0), then upon re-purchasing the N shares at time T, an returning them to the lener, the short-seller will make the profit N(Π U (0) Π U (T )). An investor is sai to have a long position on an asset if the investor owns the asset an will therefore profit from an increase of the price of the asset. Conversely, the investor is sai to have a short position on the asset if the investor will profit from a ecrease of its value, as it happens for instance when the investor is short-selling the asset. Finally we remark that any transaction in the market is subject to transaction costs (e.g., broker s commissions an lening fees for short-selling) an transaction elays (traing in real markets is not instantaneous). Stocks an iviens The capital stock of a company is the part of the company equity capital that is mae publicly available for traing. Stocks are trae in official markets (stock markets). For instance, over 300 company stocks are trae in the Stockholm exchange market. A stock market inex is a weighte average of the value of a collection of stocks trae in one or more markets. For example, S&P500 (Stanar an Poor 500) measures the average value of 500 stocks trae at the New York stock exchange (NYSE) an NASDAQ markets. The price per share at time t > 0 of a generic stock will be enote by S(t). 5

7 A stock may occasionally pay a ivien to its shareholers. This means that a fraction of the stock price is eposite to the bank account of the shareholers. The ay at which the ivien is pai, as well as its amount in percentage of the opening stock price at this ay, are known in avance. After the ivien has been pai, the price of the stock iminishes of exactly the amount pai by the ivien. Portfolio position an portfolio process Consier an agent that invests on N assets U 1,..., U N uring the time interval [0, T ]. Assume that the agent traes on a 1 shares of the asset U 1, a 2 shares of the asset U 2,..., a N shares of the asset U N. Here a i Z, where a i < 0 means that the investor has a short position in the asset U i, while a i > 0 means that the investor has a long position in the asset U i (the reason for this interpretation will become soon clear). The vector A = (a 1, a 2,..., a N ) Z N is calle a portfolio position, or simply a portfolio. The value of the portfolio at time t is given by N V A (t) = a i Π U i (t), t [0, T ], (1.1) i=1 where Π U i (t) enotes the price of the asset U i at time t. The value of the portfolio measures the wealth of the investor: the higher is V (t), the richer is the investor at time t. Now we see that when the price of the asset U i increases, the value of the portfolio increases if a i > 0 an ecreases if a i < 0, which explains why a i > 0 correspons to a long position on the asset U i an a i < 0 to a short position. We also remark that portfolios can be ae by using the linear structure on Z N, namely if A, B Z N, A = (a 1,..., a N ), B = (b 1,..., b N ) are two portfolios an α, β Z, then C = αa + βb is the portfolio C = (αa 1 + βb 1,..., αa N + βb N ). In the efinition of portfolio position an portfolio value given above, the investor keeps the same number of shares of each asset uring the whole time interval [0, T ]. Suppose now that the investor changes the position on the assets at some times 0 = t 0 < t 1 < t 2 < < t M = T ; for simplicity we assume that at each time t 1,... t M the change in the portfolio position occurs instantaneously. Let A 0 enote the initial (at time t = t 0 = 0) portfolio position of the investor an A j enote the portfolio position of the investor in the interval of time (t j 1, t j ], j = 1,..., M. As positions hol for one instance of time only are clearly meaningless, we may assume that A 0 = A 1, i.e., A 1 is the portfolio position in the close interval [0, t 1 ]. The vector (A 1,..., A M ) is calle a portfolio process. If we enote by a ij the number of shares of the asset i in the portfolio A j, then we see that a portfolio process is in fact equivalent to the N M matrix A = (a ij ), i = 1,..., N, j = 1,..., M. The value V (t) of the portfolio process at time t is given by the value of the corresponing portfolio position at time t as efine by (1.1). Hence for t (t j 1, t j ] an j = 1,..., M the value of the portfolio process is given by N V (t) = V Aj (t) = a ij Π U i (t). 6 i=1

8 The initial value V (0) = V A0 (0) = V A1 (0) of the portfolio, when it is positive, is calle the initial wealth of the investor. A portfolio process is sai to be self-financing if no cash is ever withrawn or infuse in the portfolio. Let us give an example. Suppose that at time t 0 = 0 the investor is short 400 shares on the asset U 1, long 200 shares on the asset U 2 an long 100 shares on the asset U 3. This correspons to the portfolio whose value is A 0 = ( 400, 200, 100), V A0 = 400 Π U 1 (0) Π U 2 (0) Π U 3 (0). If this value is positive, the investor nees an initial wealth to set up this portfolio position: the income eriving from short selling the asset U 1 oes not suffice to open the esire long position on the other two assets. As mentione before, we may assume that the investor keeps the same position in the interval (0, 1], i.e., A 1 = A 0. The value of the portfolio at time t = 1 is V A1 (1) = 400 Π U 1 (1) Π U 2 (1) Π U 3 (1). Now suppose that at time t = 1 the investor buys 500 shares of U 1, sells x shares of U 2, an sells all the shares of U 3. Then in the interval (1, 2] the investor has a new portfolio which is given by A 2 = (100, 200 x, 0), whose limit value at time t = 1 is V A2 (1) = 100 Π U 1 (1) + (200 x) Π U 2 (1). The ifference between the value of the two portfolios immeiately after an immeiately before the transaction is then V A2 (1) V A1 (1) = 100 Π U 1 (1) + (200 x) Π U 2 (1) ( 400 Π U 1 (1) Π U 2 (1) Π U 3 (1)) = 500 Π U 1 (1) x Π U 2 (1) 100 Π U 3 (1). If this ifference is positive, then the new portfolio cannot be create from the ol one without extra cash. Conversely, if this ifference is negative, then the new portfolio is less valuable than the ol one, the ifference being equivalent to cash withrawn from the portfolio. Hence for self-financing portfolio processes we must have V A2 (1) V A1 (1) = 0, i.e., x = 500ΠU 1 (1) 100Π U 3 (1) Π U 2 (1). Of course, x will be an integer only in exceptional cases, which means that perfect selffinancing strategies in real markets are almost impossible. 7

9 The return of a self-financing portfolio process in the interval [0, T ] is given by R(T ) = V (T ) V (0), (1.2) where V (t) enotes the value of the portfolio at time t. If the return is positive, the investor makes a profit in the interval [0, T ], if it is negative the investor incurs in a loss. When V (0) > 0 we may also compute the relative return of the portfolio, which is given by R (T ) = V (T ) V (0). (1.3) V (0) Finally we remark that investment returns are commonly annualize by iviing the return R(T ) by the time T expresse in fraction of years (e.g., T = 1 week = 1/52 years). Historical volatility The historical volatility of an asset measures the amplitue of the time fluctuations of the asset price, thereby giving information on its level of uncertainty. It is compute as the stanar eviation of the log-returns of the asset base on historical ata. More precisely, let [t 0, t] be some interval of time in the past, with t enoting possibly the present time, an let T = t t 0 > 0 be the length of this interval. Let us ivie [t 0, t] into n equally long perios, say t 0 < t 1 < t 2 <... t n = t, t i t i 1 = h, for all i = 1,... n. The set of points {t 0, t 1,... t n } is calle a partition of the interval [t 0, t]. Assume for instance that the asset is a stock. The log-return of the stock price in the interval [t i 1, t i ] is given by 1 ( ) S(ti ) R i = log S(t i ) log S(t i 1) = log, i = 1,... n. (1.4) S(t i 1 ) The (correcte) sample variance of the log-returns is then where (t) = 1 n 1 R = 1 n n i=1 n (R i R) 2, i=1 R i = 1 ( ) S(t) n log S(t 0 ) (1.5) is the sample mean of log-returns. To obtain the T-historical variance of the asset we ivie (t) by h measure in fraction of years, that is σ T 2 (t) = 1 1 h n 1 n (R i R) 2 (T -historical variance). (1.6) i=1 1 Throughout these notes, log x stans for the natural logarithm of x > 0 (which is also frequently enote by ln x in the literature). 8

10 The square root of the T historical variance is the T-historical volatility: σ T (t) = 1 1 n (R i h n 1 R) 2 (T -historical volatility). (1.7) i=1 Note carefully that the historical volatility epens on the partition being use to compute it. Suppose for example that t t 0 = T = 20 ays, which is quite common in the applications, an let t 1,... t 20 be the market closing times at these ays. Let h = 1 ay = 1/365 years. Then σ 20 (t) = n (R i 19 R) 2 is calle the 20-ays historical volatility. We remark that h = 1/252 is also commonly use as normalization factor, since there are 252 traing ays in one year. As a way of example, Figure 1.1 shows the 20-ays volatility of four stocks in the Stockholm exchange market from January 1 st, 2014 until May 2 n, 2014 (88 traing ays). These ata have been obtaine with MATHEMATICA by running the following comman on May 3 r, 2014: FinancialData["ticker", "Volatility20Day", {2014, 1, 1}] Upon running this comman, the software connects to Yahoo Finance an collects the 20- ays volatility ata for the stock ientifie by the ticker symbol ticker, starting from the ate {2014, 1, 1} (year, month, ay) until the present ay. Note that in a few cases the historical volatility remains approximately constant within perios of about 20 ays. Financial erivatives. Options A financial erivative (or erivative security) is an asset whose value epens on the performance of one (or more) other asset(s), which is calle the unerlying asset. There exist various types of financial erivatives, the most common being options, futures, forwars an swaps. In this section we iscuss option erivatives on a single asset (typically a stock). A call option is a contract between two parties: the buyer, or owner, of the call an the seller, or writer, of the call. The contract gives the owner the right, but not the obligation, to buy the unerlying asset for a given price, which is fixe at the time when the contract is stipulate, an which is calle strike price of the call. If the buyer can exercise this right only at some given time T in the future then the call option is calle European, while if the option can be exercise at any time earlier than or equal to T, then the option is calle American. The time T is calle maturity time, or expiration ate of the call. The writer of the call is oblige to sell the asset to the buyer if the latter ecies to exercise the option. If the option to buy in the efinition of a call is replace by the option to sell, then the option is calle a put option. 9 i=1

11 Getinge 10 Electrolux (a) ELUXY-B.ST (b) GETIB.ST Seb Scania (c) SCV-B.ST () SEB-C.ST Figure 1.1: 20-ays volatility of 4 stocks in the Stockholm exchange market on May 2 n, The caption in each graph shows the ticker of the stock. In exchange for the option, the buyer must pay a premium to the seller (options are not free). Suppose that the option is a European option with strike price K an maturity time T. Assume that the unerlying is a stock with price S(t) at time t T an let Π 0 be the premium pai by the buyer to the seller. In which case is it then convenient for the buyer to exercise the option at maturity? Let us efine the pay-off of the European call as Y = (S(T ) K) + := max(0, S(T ) K) (call), i.e., Y > 0 if the stock price at the expiration ate is greater than the strike price of the call, while Y = 0 otherwise; similarly, we efine the pay-off of the European put by Y = (K S(T )) + (put). Clearly, the buyer shoul exercise the option if an only if Y > 0, as in this case it is more convenient to buy (for the call), resp. sell (for the put), the stock at the strike price rather than at the market price. The return for the owner of the option is given by N(Y Π 0 ), 10

12 SHtL 600 call in the money K 200 call out of the money 100 T 0 t Figure 1.2: The call option with strike K = 200 an maturity T is in the money in the upper region an out of the money in the lower region. The put option with the same parameters is in the money in the lower region an out of the money in the upper region. where N is the number of option contracts in the buyer portfolio2, hence the buyer makes a profit only if Y > Π0. One of the main problems in options pricing theory is to set a reasonable fair value for the price Π0 of options (an other erivatives). Let us introuce some further terminology. The European call (resp. put) with strike K is sai to be in the money at time t if S(t) > K (resp. S(t) < K). The call (resp. put) is sai to be out of the money if S(t) < K (resp. S(t) > K). If S(t) = K, the (call or put) option is sai to be at the money at time t. The meaning of this terminology is self-explanatory, see Figure 1.2. The pay-off of American calls exercise at time t is Y (t) = (S(t) K)+, while for American puts we have Y (t) = (K S(t))+. The quantity Y (t) is also calle intrinsic value of the American option. In particular, the intrinsic value of an out-of-the-money American option is zero. Option markets Option markets are relatively new compare to stock markets. The first one has been establishe in Chicago in 1974 (the Chicago Boar Options Exchange, CBOE). In an option market anyone (after a proper authorization) can be the buyer or the seller of an option. Market options are available on ifferent assets (stocks, ebts, inexes, etc.) an for if2 Options are typically sol in multiples of 100 shares, hence the minimum amount of options that one can buy is 100, which cover 100 shares of the unerlying asset. 11

13 ferent strikes an maturities, which can vary between one week an several months. Most commonly, market options are of American style. Clearly, the eeper in the money is the option, the higher will be the price of the option in the market, while the price of an option eeply out of the money is usually quite low (but never zero!). It is also clear that the buyer of the option is the party holing the long position on the option, since the buyer owns the option an thus hopes for an increase of its value, while the writer is the holer of the short position. One reason why investors buy call options is to protect a short position on the unerlying asset. In fact, suppose that an investor is short-selling 100 shares of a stock at time t = 0 for the price S(0) an let t 0 > 0 be the time at which the shares must be returne to the lener. At time t = 0 the investor buys 100 shares of an American call option on the stock with strike K S(0) an maturity later than t 0. If at time t 0 the price of the stock is no lower than S(0), the investor will exercise the call an thus obtain 100 shares of the stock for the price K S(0). So oing the investor will be able to return the shares to the lener with minimal losses. At the same fashion, investors buy put options to protect a long position on the unerlying asset 3. Exercise 1.1 (?). Can you think of a reason why investors sell options? Of course, speculation is also an important factor in option markets. However the stanar theory of options pricing is firmly base on the interpretation of options as erivative securities an oes not take speculation into account. European, American an Asian erivatives European call an put options are examples of more general contracts calle European erivatives. Given a function g : (0, ) R, the stanar European erivative with pay-off Y = g(s(t )) an maturity time T > 0 is a contract that pays to its owner the amount Y at time T > 0. Here S(T ) is the price of the unerlying stock at time T, while g is the pay-off function of the erivative (e.g., g(x) = (x K) + for European call options, while g(x) = (K x) + for European put options). Hence, the pay-off of stanar European erivatives epens only on the price of the stock at maturity an not on the earlier history of the stock price. An example of stanar European erivative which is actually trae in the market (other than call an put options) is the igital option. Denote by H(x) the Heavisie function, { 1, for x > 0 H(x) = 0, for x 0, an let K, L > 0 be constants expresse in units of some currency (e.g., ollars). The stanar European erivative with pay-off function g(x) = LH(x K) is calle cash-settle igital call option; this erivative pays the amount L if S(T ) > K, an nothing otherwise. The physically-settle igital call option has the pay-off function g(x) = xh(x K), which means that at maturity the buyer receives either the stock (when S(T ) > K), or 3 A short-selling strategy that is not covere by a suitable security is sai to be nake. 12

14 K 5 10 premium SHT L SHT L SHT L K 10 premium -2 (a) Call option (b) Put option premium L 10 4 premium 2 K SHT L 5 K (c) Digital option (cash-settle) () Digital option (physically-settle) Figure 1.3: Pay-off function (continuous line) an return (ashe line) of some stanar European erivatives. nothing. Digital options are also calle binary options. Figure 1.3 shows the pay-off function of call, put an igital call options with strike K = 10. Drawing the pay-off function of a erivative helps to get a first insight onto its properties. Exercise 1.2. Given K > 0, consier the stanar European erivative with maturity T an pay-off function g(x) = (x K + K)+ 2(x K)+ + (x K K)+. Draw the graph of g an erive the range of S(T ) for which the erivative expires in the money. If the pay-off epens on the history of the stock price uring the interval [0, T ], an not just on S(T ), the European erivative is sai to be non-stanar. An example of non-stanar European R T erivative is the so-calle Asian call option, the pay-off of which 1 is given by Y = ( T 0 S(t) t K)+. The value at time t of the European erivative with pay-off Y an expiration ate T will be enote by ΠY (t) (we o not inclue the expiration ate in our notation). The term European refers to the fact that the contract cannot be exercise before time T. For a stanar American erivative the buyer can exercise the contract at any 13

15 time t (0, T ] an so oing the buyer will receive the amount Y (t) = g(s(t)), where g is the pay-off function of the American erivative. Non-stanar American erivatives can be efine similarly to the European ones, but with the further option of earlier exercise. Exercise 1.3. Look for the efinition of the following options: Bermua option, Compoun option, Lookback option, Barrier option, Chooser option. Classify them as American/European, stanar/non-stanar an write own their pay-off function. Money market A money market is a (OTC) market consisting of risk-free assets, i.e., assets whose value is always increasing in time. Like options, assets in the money market have finite maturity, which varies between one ay an one year 4. Examples of risk-free assets in the money market are commercial papers an repurchase agreements (repo). In contrast to stock an option markets, money markets are typically accessible only by financial institutions an not by private investors. The price of a generic risk-free asset in the money market at time t will be enote by B(t); the fact that the asset is risk-free means that B(t 2 ) > B(t 1 ), for all t 2 > t 1, the ifference B(t 2 ) B(t 1 ) being etermine by the interest rate of the asset in the interval [t 1, t 2 ]. We say that a risk-free asset has instantaneous interest rate r(t) > 0 in the interval [t 1, t 2 ] if ( t ) B(t) = B(t 1 ) exp r(s) s, for t 1 t t 2. (1.8) t 1 We remark that there exists other ways to efine the interest rate of a risk-free asset; in particular, the interest rate may be compoune iscretely in time, rather then continuously as in (1.8). Inasmuch as in these notes we use only (1.8) to compute the value of assets in the money market, we shall refer to r(t) simply as the interest rate of the risk-free asset, i.e., the wor instantaneous will be omitte for brevity. Moreover we shall always assume that the interest rate is a constant r, so that (1.8) simplifies to B(t) = B(t 1 ) exp(r(t t 1 )), for t 1 t t 2. (1.9) The interest rate is measure in yearly percentage. For example, if one share of a risk-free asset has initial value B(0) = 10 at time t = 0 an interest rate 10% per year, then after 1 month=1/12 years its value is B(1/12) = 10 exp(0.1/12) To see how the money market works in practice, suppose that an investor buys a riskfree asset at time t = 0 which expires at time T > 0. The seller will receive the quantity B 0 = B(0). As part of the agreement, the seller promises to re-purchase the risk-free asset at time T for B(T ) > B 0. Hence buying an asset in the money market is equivalent to len money to the seller, while selling an asset in the money market is equivalent to borrow money from the buyer. To this regar we remark that the esignation of assets in the money market as risk-free presupposes that the party issuing the asset will be able to repay the ebt to 4 Risk-free assets with maturity longer than one year are calle bons an are trae in the bon market. 14

16 the buyer, i.e., the possibility of bankruptcy is isregare. This assumption is reasonable only when the risk-free asset is issue by a soli financial institution. The seller of the risk-free asset is the party holing the short position on the asset, while the buyer hols the long position, although strictly speaking the long/short position refers to the interest rate (since the value of risk-free assets cannot ecrease). Note that so far we have introuce three strategies that investors can unertake to obtain cash: short-selling an asset, writing an option or borrowing from the money market. Frictionless markets As all mathematical moels, also those in options pricing theory are base on a number of assumptions. Some of these assumptions are introuce only with the purpose of simplifying the analysis of the moels an often correspon to facts that o not occur in reality. Among these simplifying assumptions we impose that 1. There is no bi/ask sprea 5 2. There are no transaction costs an traes occur instantaneously 3. An investor can trae any fraction of shares 4. No lack of liquiity: there is no limit to the amount of cash that can be borrowe from the money market We have seen in the previous sections that real markets o not satisfy exactly these assumptions, although in some case they o it with reasonable approximation. For instance, if the investor is an agent working for a large financial institution, then the above assumptions reflect reality quite well. However they work very baly for private investors. We summarize the valiity of these assumptions by saying that the market has no friction. The iea is that when the above assumptions hol, traing procees smoothly without resistance. In a frictionless market we may efine the portfolio process of an agent who is investing on N assets uring the time interval [0, T ] as a function A : [0, T ] R N, A(t) = (a 1 (t),..., a N (t)), i.e., by assumptions 2 an 3, the number of shares a i (t) of each single asset at time t is now allowe to be any real number an to change at any arbitrary time in the interval [0, T ]. Portfolio processes can be ae using the linear structure in R N, namely if B = (b 1 (t),..., b N (t)), then A + B is the portfolio A + B = (a 1 (t) + b 1 (t),..., a N (t) + b N (t)). The value at time t of the portfolio process A is V A (t) = N a(t)π U i (t), i=1 5 In particular, any offer to buy/sell an asset is matche by an offer to sell/buy the asset. 15

17 an clearly V A (t) + V B (t) = V A+B (t). Moreover it is clear that, thanks to assumption 3, perfect self-financial portfolio processes in frictionless markets always exist. A further simplifying assumption that we make in the rest of these notes is the following: 5. All risk-free assets in the the money market have the same constant interest rate r Although r is always positive in the applications, we shall sometimes allow r = 0 or even a negative interest rate. We shall refer to r as the interest rate of the money market. In the applications it is customary to choose the value of r to be an interbank offere rate, such as LIBOR, or EURIBOR, etc., that is the average interest rate at which banks in a given geographical zone len money to one another. 1.2 Qualitative properties of option prices The purpose of this section is to erive some qualitative properties of option prices using only basic principles, without invoking any specific mathematical moel for the market ynamics. The following notation will be use. S(t) enotes the price at time t > 0 of a given stock, C(t, S(t), K, T ) enotes the price at time t [0, T ] of the European call option on the stock with strike K > 0 an maturity T > 0. The price of the European put option with the same parameters will be enote by P (t, S(t), K, T ); finally Ĉ(t, S(t), K, T ) an P (t, S(t), K, T ) enote the values of the corresponing American call an put option. No-ummy investor principle Probably the most self-evient of all financial principles is the following, which we call the no-ummy investor principle 6 : Investors prefer more to less an o not unertake traing strategies which result in a sure loss. This principle has a number of straightforwar consequences. For example, an investor will never exercise an option which is out of the money, while an option that expires in the money is always exercise 7. Moreover the price of stocks an options (prior to expire) is always positive. Exercise 1.4 (?). Use the no-ummy investor principle to justify the following properties. (i) The price of a financial erivative tens to its the pay-off as maturity is approache. 6 More commonly (an respectfully) known as rational investor principle. 7 Provie of course the owner of the option can affor to exercise. For instance, the buyer of a call option may not have the cash require to buy the unerlying when the call expires in the money. 16

18 In particular, for European call/put options, C(t, S(t), K, T ) (S(T ) K) +, P (t, S(t), K, T ) (K S(T )) +, as t T an similarly for American options; (ii) An American erivative is at least as valuable as its European counterpart. In particular, for call/put options, Ĉ(t, S(t), K, T ) C(t, S(t), K, T ), P (t, S(t), K, T ) P (t, S(t), K, T ) (iii) The price of an American erivative is always larger or equal to its intrinsic value. In particular, for American call/put options, Ĉ(t, S(t), K, T ) (S(t) K) +, P (t, S(t), K, T ) (K S(t))+. Any reasonable mathematical moel for the price of options must be consistent with the properties (i)-(iii) in the previous exercise. In the rest of this section they are assume to hol without any further comment. Arbitrage-free principle An arbitrage opportunity is an investment strategy that requires no initial wealth an which ensures a positive profit without taking any risk. For example, suppose that at time t = 0 an investor sells one share the American call option with strike K an maturity T 1 an buys one share of the American call on the same stock with the same strike but with maturity T 2 > T 1. Suppose that the price of the latter option is lower than the price of the former, i.e., Ĉ 2 := Ĉ(0, S(0), K, T 2) < Ĉ(0, S(0), K, T 1) := Ĉ1. The investor will then have the cash Ĉ1 Ĉ2 available to buy shares of a risk-free asset in the money market. This portfolio is clearly riskless: if the buyer of the option with maturity T 1 ecies to exercise at some time t T 1, the investor can pay-off the buyer by exercising his/her own option. Hence this investment is an example of arbitrage opportunity: it requires no initial wealth, it entails no risk an it ensures a positive profit. However, why shoul the investor be able to fin someone willing to pay more for an option that expires earlier? This woul be of course a ummy investment for the buyer. Due to the complexity of moern markets, arbitrage opportunities o actually exist, but only for a very short time, as they are quickly exploite an trae away by investors. The previous iscussion leas us to assume the valiity of the so-calle arbitrage-free principle: Asset prices in a market are such that no arbitrage opportunities can be foun. Asset prices that are consistent with this principle are sai to be arbitrage free or fair. 17

19 Dominance principle The arbitrage-free principle can be use to erive a number of qualitative properties of option prices, which are not as obvious as (i)-(iii) in Exercise 1.4. To this purpose we nee first to express the arbitrage-free principle in a more quantitative form. There are several ways to o this, e.g. by requiring the absence of arbitrage portfolios in the market (see next chapter), or by imposing the so-calle ominance principle: Dominance Principle: Suppose that t < T is the present time an consier a portfolio which oes not contain ivien-paying assets or short positions on American erivatives. If the value of the portfolio is non-negative at time T, i.e., V (T ) 0, then V (t) 0. The fact that the ominance principle must hol as a consequence of the arbitrage-free principle is clear. In fact, if V (t) < 0, then the investor nees no initial wealth to open the portfolio, while on the other han the portfolio return is positive, since V (T ) V (t) V (t) > 0. Hence the given portfolio ensures a positive profit without taking any risk, which violates the arbitrage-free principle. Let us comment further on the formulation of the ominance principle. First of all, the requirement that the portfolio oes not contain short positions on American erivatives is necessary, otherwise there is no guarantee that the portfolio exists up to time T (the buyer may exercise the erivative prior to T ). The reason to require that the assets pay no ivien is the following. Suppose that a stock pays a ivien of 2 % at time T. Just before that the investor open a short position on the stock an invest 99% of the income on a risk-free asset. Hence the value of this portfolio is negative, but it becomes instantaneously positive when the ivien is pai 8. The following simple theorem will be use for our applications of the ominance principle. Theorem 1.1. Assume that the ominance principle hols an let A, B be two portfolios which o not contain ivien pay assets or American erivatives. Then, for t < T : (a) If V A (T ) = 0, then V A (t) = 0; (b) If V A (T ) = V B (T ), then V A (t) = V B (t); (c) If V A (T ) V B (T ), then V A (t) V B (t). Proof. (a) The ominance principle implies V A (t) 0. As the portfolio A oes not contain American erivatives, then the portfolio A oes not contain short positions on American erivatives, hence the ominance principle applies to the portfolio A. We obtain V A (t) = V A (t) 0, or V A (t) 0. Hence V A (t) = 0. (b) follows by part (a) an the relation V A B (t) = V A (t) V B (t). The proof of (c) is similar. The next theorem collects a number of properties that must be satisfie by option prices 8 In practice this is not a feasible strategy as the profit is extremely small an highly surpasse by transaction costs. Moreover certain markets require to own the stock for a sufficiently long perio of time in orer to be entitle to the next ivien. 18

20 as a consequence of the ominance principle. We enote these properties by (iv)-(vii) in orer to continue the list (i)-(iii) given in Exercise 1.4. Theorem 1.2. Assume that the ominance principle hols an let r be the interest rate of the money market. Then, for all t < T, (iv) The put-call parity hols S(t) C(t, S(t), K, T ) = Ke r(t t) P (t, S(t), K, T ). (1.10) (v) If r 0, then C(t, S(t), K, T ) (S(t) K) + ; the strict inequality C(t, S(t), K, T ) > (S(t) K) + hols when r > 0. (vi) If r 0, the map T C(t, S(t), K, T ) is non-ecreasing. (vii) The maps K C(t, S(t), K, T ) an K P (t, S(t), K, T ) are convex 9. Proof. (iv) Consier a constant portfolio A which is long one share of the stock an one share of the put option, an is short one share of the call an K/B(T ) shares of the risk-free asset. The value of this portfolio at maturity is V A (T ) = S(T ) + (K S(T )) + (S(T ) K) + Hence, by (a) of Theorem 1.1, V A (t) = 0, for t < T, that is which is the claim. K B(T ) = 0. B(T ) S(t) + P (t, S(t), K, T ) C(t, S(t), K, T ) Ke r(t t) = 0, (v) We can assume S(t) K, otherwise the claim is obvious (the price of a call cannot be negative). By the put-call parity, using that P (t, S(t), K, T ) 0, C(t, S(t), K, T ) = S(t) Ke r(t t) + P (t, S(t), K, T ) S(t) Ke r(t t) ; the right han sie equals S(t) K for r = 0 an is strictly greater than this quantity for r > 0. As S(t) K = (S(t) K) + for S(t) K, the claim follows. (vi) Consier a portfolio A which is long one call with maturity T 2 an strike K, an short one call with maturity T 1 an strike K, where T 2 > T 1 t. By the claim (v) we have C(T 1, S(T 1 ), K, T 2 ) (S(T 1 ) K) + = C(T 1, S(T 1 ), K, T 1 ), i.e., V A (T 1 ) 0, for t < T 1. Hence V A (t) 0, i.e., C(t, S(t), K, T 2 ) C(t, S(t), K, T 1 ), which is the claim. 9 Recall that a real-value function f on an interval I is convex if f(θx + (1 θ)y) θf(x) + (1 θ)f(y), for all x, y I an θ (0, 1). 19

21 (vii) We prove the statement for call options, the argument for put options being the same. Let K 0, K 1 > 0 an 0 < θ < 1 be given. Consier a portfolio A which is short one share of a call with strike θk 1 + (1 θ)k 0 an maturity T, long θ shares of a call with strike K 1 an maturity T, long (1 θ) shares of a call with strike K 0 an maturity T. The value of this portfolio at maturity is V A (T ) = (S(T ) (θk 1 + (1 θ)k 0 )) + + θ(s(t ) K 1 ) + + (1 θ)(s(t ) K 0 ) +. The convexity of the function f(x) = (S(T ) x) + gives V A (T ) 0 an so V A (t) 0 by the ominance principle. The latter inequality is C(t, S(t), θk 1 + (1 θ)k 0, T ) θc(t, S(t), K 1, T ) + (1 θ)c(t, S(t), K 0, T ), which is the claim for call options. Exercise 1.5. Consier the following table of European options prices at time t = 0: CALL Maturity Strike Price 1 month month month month PUT Maturity Strike Price 1 month month month month Assume that the money market has interest rate r = 0 an that the price of the unerlying asset at time t = 0 is S(0) = 100. Explain why these prices are incompatible with the ominance principle. Fin a constant portfolio position which violates the ominance principle. HINT: Look for violations of the properties (iv) (vii). Exercise 1.6. Assume that the ominance principle hols an prove the following. (viii) If K 0 K 1, then C(t, S(t), K 0, T ) C(t, S(t), K 1, T ), i.e., the price of European call options is non-increasing with the strike price. Similarly the price of put options is non-ecreasing with the strike price. (ix) C(t, S(t), K, T ) S(t) an P (t, S(t), K, T ) Ke r(t t). Exercise 1.7 ( ). Assume that the ominance principle hols. Consier the European erivative U with maturity time T an pay-off Y given by Y = min[(s(t ) K 1 ) +, (K 2 S(T )) + ], where K 2 > K 1 an (x) + = max(0, x). Fin a constant portfolio consisting of European calls an puts expiring at time T which replicates the value of U (i.e., whose value at any time t < T equals the value of U). 20

22 Exercise 1.8. Suppose K > 0. A butterfly sprea on call options pays the amount max(0, S(T ) K + K) 2 max(0, S(T ) K) + max(0, S(T ) K K) at the maturity T. Show that the value of this option is non-negative at any point of time. Exercise 1.9 ( ). The price of a contract at time t is N units of currency an it pays at the maturity ate T > t the amount N + αn(s(t ) K) +. Show that if C(t, S(t), K, T ) > 0 an N 0. α = t) 1 e r(t C(t, S(t), K, T ) Optimal exercise time of American options Consier now a no-ummy investor owning an American put option. investor exercise the option? At any time t < T we have, by (iii), (1.11) When shoul the either P (t, S(t), K, T ) > (K S(t))+ or P (t, S(t), K, T ) = (K S(t))+. Exercising the American put at a time t when the strict inequality P (t, S(t), K, T ) > (K S(t)) + hols is a ummy ecision, because the resulting pay-off is lower than the value of the erivative 10. On the other han, if the equality P (t, S(t), K, T ) = (K S(t)) + hols at time t, then the optimal strategy for the investor is to exercise the American put, as in this case the pay-off equals the value of the erivative, i.e., the investor takes full avantage of the American put. This leas us to introuce the following efinition. Definition 1.1. A time t < T is calle an optimal exercise time for the American put with value P (t, S(t), K, T ) if P (t, S(t), K, T ) = (K S(t)) +. A similar efinition can be justifie for American call options, i.e., the optimal exercise time of the American call is a time t at which Ĉ(t, S(t), K, T ) = (S(t) K) +. However, assuming that the ominance principle hols (an that the money market has positive interest rate), we have Ĉ(t, S(t), K, T ) C(t, S(t), K, T ) > (S(t) K) +, for t < T, see (v) in Theorem 1.1. It follows that, in an arbitrage-free market, it is never optimal to exercise American call options prior to maturity when the unerlying stock pays no ivien. As oppose to this, it will be shown in Chapter 6 that when the unerlying stock pays a ivien prior to the expiration ate of the American call, it is optimal to exercise the American call just before the ivien is pai, provie the price of the stock is sufficiently high, see Theorem 6.9. As in the absence of iviens the optimal strategy is to hol the American call until maturity, the ominance principle leas us to the following, last property on the fair price of options: 10 Of course the buyer might want to close the position on the American put for other reasons. In this case however it is more profitable to sell the option rather than exercising it. 21

23 (x) When the unerlying stock pays no ivien (an the money market has positive interest rate), the fair price of European calls an American call with equal parameters are the same, i.e., Ĉ(t, S(t), K, T ) = C(t, S(t), K, T ). Final remarks: The properties (i)-(x) are quite well represente in real markets, thereby giving inirect support to the arbitrage-free principle. Note also that these properties epen only on the valiity of the arbitrage-free principle (in the form of the ominance principle) an not on the specific market ynamics. In the following chapters we shall give an alternative proof of (some of) these properties by using explicit moels for the price of stocks an options in the market. Exercise 1.10 (Comparison with market ata). Call an put options on Nasaq 100 are of European style an so they can be use to test the properties erive in this section. The market price of call an put options on Nasaq 100, for ifferent strikes an maturities, can be foun at the homepage http: // www. marketwatch. com/ investing/ inex/ nx/ options. Compile a table of prices for options nearly at the money, for instance using the first 10 options in an out of the money (if an option appears with zero price it means that it has not yet been trae; you can just skip it). Use these ata to plot (with Matlab for instance) the price of call an put options in terms of the strike price an the time of maturity. Are the properties (vi), (viii), (viii) verifie? Next use the put-call parity ientity to compute the value of the interest rate r for each pair of call-put with the same strike an maturity. What can you conclue? Do the ata support the put-call parity? 22

24 Chapter 2 Binomial markets In this an the following two chapters we present a time-iscrete moel for the fair price of options first propose in [4] an which is known uner the name of binomial options pricing moel. The moel is very popular among practitioners ue to its implementation simplicity. The present chapter eals with the ynamics of the unerlying asset, which we assume to be a stock. The following two chapters are concerne with European an American erivatives on the stock. The time-continuum analogue of the binomial moel is the Black-Scholes moel, which will be stuie in Chapter The binomial stock price The binomial asset price is a moel for the evolution in time of the price of financial assets. It is often applie to stocks, hence we enote by S(t) the price of the asset at time t. We are intereste in monitoring the stock price in some finite time interval [0, T ], where T > 0 coul be for instance the expiration ate of an option on the stock. The price of the stock at time t = 0 is enote by S(0) or S 0 an is assume to be known. The binomial stock price can only change at some given pre-efine times 0 = t 0 < t 1 < t 2 < < t N = T ; moreover the price at time t i+1 epens only on the price at time t i an the result of tossing a coin. Precisely, letting u, R, u >, an p (0, 1), we assume { S(ti 1 )e S(t i ) = u, with probability p, S(t i 1 )e, with probability 1 p, for all i = 1,..., N. Here we may interpret p as the probability to get a hea in a coin toss (p = 1/2 for a fair coin). We restrict to the stanar binomial moel, which assumes that the parameters u,, p are time-inepenent an that the stock pays no ivien in the interval [0, T ]. In the applications one typically chooses u > 0 an < 0 (e.g., = u is quite common), hence u stans for up, since S(t i ) = S(t i 1 )e u > S(t i ), while stans for own, for S(t i ) = S(t i 1 )e < S(t i ). In the first case we say that the stock price goes up at time t i, in the secon case that it goes own at time t i. 23

25 Next we introuce a number of assumptions which simplify the analysis of the moel without compromising its generality 1. Firstly we assume that the times t 0, t 1,... t N are equiistant, that is t i t i 1 = h > 0, for all i = 1,..., n. In the applications the value of h must be chosen much smaller than T. Without loss of generality we can pick h = 1, an so t 1 = 1, t 2 = 2,..., t N = T = N, with N >> 1. For instance, if N = 67 (the number of traing ays in a perio of 3 months), then h = 1 ay an S(t), for t {1,... N}, may refer to the closing price of the stock at each ay. It is convenient to enote I = {1,..., N}. Hence, from now on, we assume that the binomial stock price is etermine by the rule S(0) = S 0 an { S(t 1)e u, with probability p S(t) =, t I. (2.1) S(t 1)e, with probability 1 p Remark 2.1 (Notation). The notation use in the present notes is the same as in [3], although it is slightly ifferent from the one use in the stanar literature on the binomial moel, see e.g., [6]. In fact the binomial stock price is more commonly written as { S(t 1)u, with probability p S(t) = S(t 1), with probability 1 p, with 0 < < u. All the results in the present text can be translate into the stanar notation by the substitutions e u u, e. In our notation the log-returns of the stock take a slightly simpler form, which is useful when passing to the time-continuum limit (see Section 6.1). Each possible sequence (S(1),..., S(N)) of the future stock prices etermine by the binomial moel is calle a path of the stock price. Clearly, there exists 2 N possible paths of the stock price in a N-perio moel. Letting 2 {u, } N = {x = (x 1, x 2,..., x N ) R N : x t = u or x t =, t I} be the space of all possible N-sequences of ups an owns, we obtain a unique path of the stock price (S(1),..., S(N)) for each x {u, } N. For instance, for N = 3 an x = (u, u, u) the corresponing stock price path is given by S 0 S(1) = S 0 e u S(2) = S(1)e u = S 0 e 2u S(3) = S(2)e u = S 0 e 3u, 1 We come back to the general moel in Section 2.4, where it is implemente with Matlab. 2 Note carefully that in the set {u, } N, the letters u, mean up an own an shoul not be confuse with the numerical parameters u, in the binomial stock price (2.1). 24