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73323274-13-BSM-model - T he Black-ScholesMerton Model In...

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Unformatted text preview: T he Black-ScholesMerton Model In the early 1970s, Fischer Black, Myron Scholes, and Robert Merton made a major breakthrough in the pricing of stock options. I This involved the development of what has become known as the Black-Scholes (or Black-Scholes-Merton) model. The model has had a huge influence on the way that traders price and hedge options. It has also been pivotal to the growth and success of financial engineering in the last 20 years. In 1997, the importance of the model was recognized when Robert Merton and Myron Scholes were awarded the Nobel prize for economics. Sadly, Fischer Black died in 1995, otherwise he too would undoubtedly have been one of the recipients of this prize. This chapter shows how the Black-Scholes model for valuing European call and put options on a non-dividend-paying stock is derived. It explains how volatility can be either estimated from historical data or implied from option prices using the model. It shows how the risk-neutral valuation argument introduced in Chapter 11 can be used. It also shows how the Black-Scholes model can be extended to deal with European call and put options on dividend-paying stocks and presents some results on the pricing of American call options on dividend-paying stocks. 13.1 LOGNORMAL PROPERTY OF STOCK PRICES The model of stock price behavior used by Black, Scholes, and Merton is the model we developed in Chapter 12. It assumes that percentage changes in the stock price in a short period of time are normally distributed. We define fJ..: Expected return on stock per year 0-: Volatility of the stock price per year The mean of the percentage change in the stock price in time .6.t is J.L.6.t and the 1 See F. Black and M. Scholes, "The Pricing of Options and Corporate Liabilities," JOllmal oj Political Economy, 81 (May/June 1973): 637-59; R.C. Merton, "Theory of Rational Option Pricing," Bell Journal oj Economics and Managemelll Science, 4 (Spring 1973): 141-83. 281 282 CHAPTER 13 standard deviation of this percentage change is O'.;;s:i, so that (13.1) where /::;.S is the change in the stock price S in time distribution with mean 111 and standard deviation s. As shown in Section 12.6, the model implies that InST -lnSo /::;.t, and ¢(111, s) denotes a normal ~ ¢[ (tL - ~2)T, o'vT ] From this, it follows that ~: ~ ¢[ (tL - ~2) T, o'vT ] (13.2) ~ ¢[In So + (tL - ~2) T, o'vT ] (13.3) In and In ST where ST is the stock price at a future time T and So is the stock price at time O. Equation (13.3) shows that In ST is normally distributed, so that ST has a lognormal O'2 /2)T and the standard deviation distribution. The mean of In ST is In So + is o'vT. (tL - Example 13.1 Consider a stock with an initial price of $40, an expected return of 16% per annum, and a volatility of 20% per annum. Fr()m equation (13.3), the probability distribution of the stock price ST in 6 months' time is given by In ST ~ ¢[ln40 + (0.16 - 0.2 2 /2) x 0.5, 0.2.JQ.5] InST ~ ¢(3.759, 0.141) There is a 95% probability that a normally distributed variable has a value within 1.96 standard deviations of its mean. Hence, with 95% confidence, 3.759 - 1.96 x 0.141 < In ST < 3.759 + 1.96 x 0.141 This can be written e3.759-1.96xO.141 < ST < e3.759+1.96xO.141 or 32.55 < ST < 56.56 Thus, there is a 95% probability that the stock price in 6 months will lie between 32.55 and 56.56. A variable that has a lognormal distribution can take any value between zero and infinity. Figure 13.1 illustrates the shape of a lognormal distribution. Unlike the normal distribution, it is skewed so that the mean, median, and mode are all different. From equation (13.3) and the properties of the lognormal distribution, it can be shown that the expected value E(ST) of ST is given by E(ST) = SoellT (13.4) 283 T he Black--Scholes-Nlerton Nlodel Figure 13.1 Lognormal distribution. o This fits in with the definition of J-t as the expected rate of return. The variance var(ST) of ST, can be shown to be given by 2 (13.5) Example 13.2 Consider a stock where the current price is $20, the expected return is 20% per annum, and the volatility is 40% per annum. The expected stock price, E(ST), and the variance of the stock price, var(ST), in 1 year, are given by E(ST) = 20eO. 2xl = 24.43 and var(ST) = 400e2xO.2XI(e°.41xl - 1) = 103.54 The standard deviation of the stock price in 1 year is .)103.54, or 10.18. 13.2 THE DISTRIBUTION OF THE RATE OF RETURN The lognormal property of stock prices can be used to provide information on the probability distribution of the continuously compounded rate of return earned on a stock between times 0 and T. If we define the continuously compounded rate of return per annum realized between times 0 and T as x, it follows that ST = Soe xT so that 1 ST x=-lnT So (13.6) From equation (13.2), it follows that (13.7) See Technical Note 2 on the author's website for a proof of the results in equations (13.4) and (13.5). For a more extensive discussion of the properties of the lognormal distribution, see J. Aitchison and J. A. C. Brown, The Lognormal Distribution. Cambridge University Press, 1966. 2 284 CHAPTER Thus, the continuously compounded rate of return per annum is normally distrit)ut(~d with mean fL - (i /2 and standard deviation a /.JT. As T increases, deviation of x declines. To understand the reason for this, consider two cases: T = and T = 20. We are more certain about the average return per year over 20 years we are about the return in anyone year. Example 13.3 Consider a stock with an expected return of 17% per annum and a volatility 20% per annum. The probability distribution for the average rate of return tinuously compounded) realized over 3 years is normal, with mean 0.17 - o·t = 0.15 or 15% per annum, and standard deviation ~= 0.1155 or 11.55% per annum. Because there is a 95% chance that a normally dlsltntlUt(~d variable will lie within 1.96 standard deviations of its mean, we can be confident that the average return realized over 3 years will be between and +37.6% per annum. 13.3 THE EXPECTED RETURN The expected return, fL, required by investors from a stock depends on the riskiness the stock. The higher the risk, the higher the expected return. It also depends on level of interest rates in the economy. The higher the level of interest rates, the the expected return required on any given stock. Fortunately, we do not have to corlcer'n ourselves with the determinants of fL in any detail. It turns out that the value of a option, when expressed in terms of the value of the underlying stock, does not on fL at all. Nevertheless, there is one aspect of the expected return from a stock frequently causes confusion. Equation (13.1) shows that fL!::lt is the expected percentage change in the stock in a very short period of time, !::It. It is natural to assume from this that fL is the eXI)ected continuously compounded return on the stock. However, this is not the case. continuously compounded return, x, actually realized over a period of time of length is given by equation (13.6) as x = .!.In ST T So and, as indicated in equation (13.7), the expected value E(x) of x is fL - a 2 /2. The reason why the expected continuously compounded return is different from fL subtle, but important. Suppose we consider a very large number of very short periods time of length !::It. Define Sj as the stock price at the end of the ith interval and !::lSj Sj+1 - Sj. Under the assumptions we are making for stock price behavior, the ave:ra!~e of the returns on the stock in each interval is close to fL. In other words, fL !::It is close the arithmetic mean of the !::lS;/ Sj. However, the expected return over the whole 285 T he Black-Scholes-J.1/[erton Nlodel Business Snapshot 13.1 covered by the data, expressed with a compounding interval of i3.t, is close to fL - a 2 /2, not fL. 3 Business Snapshot 13.1 provides a numerical example concerning the mutual fund industry to illustrate the point being made here. For a mathematical explanation of what is going on, we start with equation (13.4): Taking logarithms, we get In[E(ST)] = In(So) + fLT It is now tempting to set In[E(ST)] = E[ln(ST)]' so that E[ln(ST)] -In(So) = fLT, or E[ln(Sri So)] = fLT, which leads to E(R) = fL. However, we cannot do this because In The arguments in this section show that the term "expected return" is ambiguous. It can refer either to JL or to JL - 02/2. Unless otherwise stated, it will be used to refer to fL throughout this book. J CHAPTER 13 286 is a nonlinear function. In fact, In[E(ST)] > E[ln(ST)]' so that E[ln(SrlSo)] < J1-T, which leads to E(x) < J1-. (As pointed out above, E(x) = J1- - (J2/2.) 13.4 VOLATILITY The volatility (J of a stock is a measure of our uncertainty about the returns provided by the stock. Stocks typically have a volatility between 15% and 60%. From equation (13.7), the volatility of a stock price can be defined as the standard deviation of the return provided by the stock in 1 year when the return is 'expressed using continuous compounding. When T is small, equation (13.1) shows that (J-JT is approximately equal to the standard deviation of the percentage change in the stock price in time T. Suppose that (J = 0.3, or 30%, per annum and the current stock price is $50. The standard deviation of the percentage change in the stock price in 1 week is approximately 30 x f£ = 4.16% A one-standard-deviation move in the stock price in 1 week is therefore 50 x 0.0416, or $2.08. Equation (13.1) shows that our uncertainty about a future stock price, as measured by its standard deviation, increases-at least approximately-with the square root of how far ahead we are looking. For example, the standard de;viation of the stock price in 4 weeks is approximately twice the standard deviation in 1 week. Estimating Volatility from Historical Data To estimate the volatility of a stock price empirically, the stock price is usually observed at fixed intervals of time (e.g., every day, week, or month). . Define: n + 1: Number of observations Si: Stock price at end of ith interval, with i = 0, 1, ... , n r: Length of time interval in years and let Ui = In (.!i-) Si-l for i = 1, 2, ... , n. The usual estimate, s, of the standard deviation of the S = 1 --1 n- L:1I. I (Ui - 1= Ui is given by _? u)- or S= where it is the mean of the lL:lI? u":"- -- n- 1 Ui. i=l 1 1 n(n _ 1) (L:1I)2 u· i=[ 1 287 T he Black-Scholes-.lVIertoll .lVIodel From equation (13.2), the standar~ deviation of the Ui is a,Jr. The variable s is therefore an estimate of a,Jr. It follows that a itself can be estim~ted as 8, where ,f'" ~ s a=- ,Jr The standard error of this estimate can be shown to be approximately 8/.,fiii. Choosing an appropriate value for 12 is not easy. More data generally lead to more accuracy, but a does change over time and data that are too old may not be relevant for predicting the future volatility. A compromise that seems to work reasonably well is to use closing prices from daily data over the most recent 90 to 180 days. An often-used rule of thumb is to set n equal to the number of days to which the volatility is to be applied. Thus, if the volatility estimate is to be used to value a 2-year option, daily data for the last 2 years are used. More sophisticated approaches to estimating volatility involving GARCH models are discussed in Chapter 19. Example 13.4 Table 13.1 shows a possible sequence of stock prices during 21 consecutive trading days., In this case, I>i = 0.09531 and I>f = 0.00326 Table 13.1 Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Computation of volatility. Closing stock price (dollars) Price relative S;/Si-l 20.00 20.10 19.90 20.00 20.50 20.25 20.90 20.90 20.90 20.75 20.75 21.00 21.10 20.90 20.90 21.25 21.40 21.40 21.25 21.75 22.00 1.00500 0.99005 1.00503 1.02500 0.98780 1.03210 1.00000 1.00000 0.99282 1.00000 1.01205 1.00476 0.99052 1.00000 1.01675 1.00706 1.00000 0.99299 1.02353 1.01149 Daily return = In(S;/ Si-d Ui 0.00499 0.01000 0.00501 0.02469 -0.01227 0.03159 0.00000 0.00000 -0.00720 0.00000 0.01198 0.00475 -0.00952 0.00000 0.01661 0.00703 0.00000 -0.00703 0.02326 0.01143 288 CHAPTER 13 and the estimate of the standard deviation of the daily return is 0.00326 19 or 1.216%. Assuming that there are 252 trading days per year, r = 1/252 and the data give an estimate for the volatility per annum of 0.01216J252 =0.193, or 19.3%. The standard error of this estimate is 0.193 = 0.031 ..j2 x 20 or 3.1 % per annum. The foregoing analysis assumes that the stock pays no dividends, but it can be adapted to. accommodate dividend-paying stocks. The return, lei> during a time interval that includes an ex-dividend day is given by lei S·+D =In-'-Si_1 where D is the amount of the dividend. The return in other time intervals is still lei = 1n SSi i-I . However, as tax factors playa part in determining returns around an ex-dividend date, it is probably best to discard altogether data for intervals that include an ex-dividend date. Trading Days vs. Calendar Days An important issue is whether time should be measured in calendar days or trading days when volatility parameters are being estimated and used. As shown in Business Snapshot 13.2, research shows that volatility is much higher when the exchange is open for trading than when it is closed. As a result, practitioners tend to ignore days when the exchange is closed when estimating volatility from historical data and when calculating the life of an option. The volatility per annum is calculated from the volatility per trading day using the formula Volatility per annum Volatility = per trading day x Number of trading days per annum This is what we did in Example 13.4 when calculating volatility from the data in Table 13.1. The number of trading days in a year is usually assumed to be 252 for stocks. The life of an option is also usually measured using trading days rather than calendar days. It is calculated as T years, where T Number of trading = - - - - - - - - - ' - - - ' -days-until 'option- maturity '-- --- =-- ----"252 T he Black-ScJlOles-J.l!lertoll J.l!lodel 289 Business Snapshot 13.2 It is natural to assume that the volatility of a stock is caused by new information reaching the market. This new information causes people to revise their.opinions about the value of the stock. The price of the stock changes and vol 'ty results. This view of what causes volatility is not supported by researcli: With 1years of daily stock price data,. researchers can calculate: 1. The variance of stock price returns between the close 0 tradin", on one day and the close of trading on the next day. when there. are no intervening nontrading days 2. The variance of the stock price returns between t and the close of trading on Monday The second variance is the variance of returns over a 3-day perio e fi sa variance over a I-day period. We might reasonably expect the second variance to be three times as great as the first variance. Fama (1965), F 0), a ch and Roll (1986) show that this is not the case. These thre studie te the second variance to be, respectively, 22%, 19%, and 1 her than the first variance. . At this stage you might be tempted to argue that these resul more news reaching the market when the market is open for trad' Roll (1984) does not support tIus explanation. Roll looked at t juice futures. By far the most important news for orange juice fu s prices is news about the weather and news about the weather is equally likely to arrive at any time. When Roll did a similar analysis to that just described for stocks, he found that the second (Friday-to-Monday) variance is only 1.54 times the first variance. The only reasonable conclusion from all "this is that volatility is to a la '" ' tent caused by trading itself. (Traders usually have no difficulty accepting this conclusion!) 13.5 CONCEPTS UNDERLYING THE BLACK-SCHOLES-MERTON DIFFERENTIAL EQUATION The Black-Scholes-Merton differential equation is an equation that must be satisfied by the price of any derivative dependent on a non-dividend-paying stock. The equation is derived in the next section. Here we consider the nature of the arguments we will use. The arguments are similar to the no-arbitrage arguments we used to value stock options in Chapter 11 for the situation where stock price movements are binomial. They involve setting up a riskless portfolio consisting of a position in the derivative and a position in the stock. In the absence of arbitrage opportunities, the return from the portfolio must be the risk-free interest rate, r. This leads to the Black-Scholes-Merton differential equation. The reason a riskless portfolio can be set up is that the stock price and the derivative price are both affected by the same underlying source of uncertainty: stock price movements. In any short period of time, the price of the derivative is perfectly correlated with the price of the underlying stock. When an appropriate portfolio of the stock and the derivative is established, the gain or loss from the stock position 290 CHABTER 13 Figure 13.2 Relationship between call price and stock price. Current stock price is So. Call price Stock price So always offsets the gain or loss from the derivative position so that the overall value of the portfolio at the end of the short period of time is known with certainty/ Suppose, for example, that at a particular point in time the relationship between a small change f:.S in the stock price and the resultant small change f:.c in the price of a European call option is given by f:.c = O.4f:.S This means that the slope of the line representing the relationship between c and Sis 0.4, as indicated in Figure 13.2. The riskless portfolio would consist of: 1. A long position in 0.4 shares 2. A short position in one call option There is one important difference between the Black-Scholes-Merton analysis and our analysis using a binomial model in Chapter 11. In Black-Scholes-Merton, the position in the stock and the derivative is riskless for only a very short period of time. (Theoretically, it remains riskless only for an instantaneously short period of time.) To remain riskless, it must be adjusted, or rebalanced, frequently.4 For example, the relationship between f:.c and t::.S in our example might change from t::.c = 0.4 t::.S today to t::.c = 0.5 t::.S in 2 weeks. This would mean that, in order to maintain the riskless position, an extra 0.1 share would have to be purchased for each call option sold. It is nevertheless true that the return from the riskless portfolio in any very short period of time must be the risk-free interest rate. This is the key element in the Black-Scholes analysis and leads to their pricing formulas. Assumptions The assumptions we use to derive the Black-Scholes-Merton differential equation are as follows: 1. The stock price follows the process developed in Chapter 12 with f.L and (J constant. 2. The short selling of securities with full use of proceeds is permitted. 4 We discuss the rebalancing of portfolios in more detail in Chapter 15. 291 The Black-S choles-lVIertOll lv[odel 3. 4. 5. 6. 7. There are no transactions costs or taxes. All securities are perfectly divisible. There are no dividends during the life of the derivative. There are no riskless arbitrage opportunities. Security trading is continuous. The risk-free rate of interest, r, is constant and the same for all maturities. As we discuss in later c4apters, some of these assumptions can be relaxed. For example, a and r can be known functions of t. We can even allow interest rates to be stochastic . provided that the stock price distribution at maturity of the option is stil110gnormal. 13.6 DERIVATION OF THE BLACK-SCHOLES-MERTON DIFFERENTIAL EQUATION The stock price process we are assuming is the one we developed in Section 12.3: dS = JLS dt + as dz (13.8) Suppose that f is the price of a call option or other derivative contingent on S. The variable f must be some function of Sand t. Hence, from equation (12.14), (13.9) The discrete versions of equations (13.8) and (13.9) are !:is = JLS !:it + as !:iz (13.10) and (13.11) where !:is and !:if are the changes in f and S in a small time interval !:it. Recall from the discussion of Ito's lemma in Section 12.5 that the Wiener processes underlying f and S are the same. In other words, the !:iz (= E.jl;i) in equations (13.10) and (13.11) are the same. It follows that, by choosing a portfolio of the stock and the derivative, the Wiener process can be eliminated. The appropriate portfolio is -1: derivative +a fI as: shares The holder of this portfolio is short one derivative and long an amount shares. Define IT as the value of the portfolio. By definition af IT = -f+-S as aflas of (13.12) 292 CHAPTER 13 The change 6.TI in the value of the portfolio in the time interval 6.t is given by af (13.13) 6.TI = -6.f +-6.S as Substituting equations (13.10) and (13.11) into equation (13.13) yields 11) 2 af Iaf 6.TI= ( -----O'-S- 6.t at 2 as 2 (13.14) Because this equation does not involve 6.z, the portfolio must be riskless during time 6.t. The assumptions listed in the preceding section imply that the portfolio must instantaneously earn the same rate of return as other short-term risk-free securities. If it earned more than this return, arbitrageurs could make a riskless profit by borrowing money to buy the portfolio; if it earned less, they could make a riskless profit by shorting the portfolio and buying risk-free securities. It follows that (13.15) 6.TI = rTI 6.t where t is the risk-free interest rate. Substituting from equations (13.12) and (13.14) into (13.15), we obtain 1a f - -f S (a 28s 2 2) 6.t=r (aas ) 6.t· at 2 f -f+ - - 0 ' S 2 so thal (13.16) Equation (13.16) is the Black-Scholes-Merton differential equation. It has many solutions, corresponding to all the different derivatives that can be defined with S as the underlying variable. The particular derivative that is obtained when the equation is solved depends on the boundary conditions that are used. These specify the values of the derivative at the boundaries of possible values of Sand t. In the case of a European call option, the key boundary condition is f = max(S - K, 0) when t= T In the case of a European put option, it is f = max(K - S, 0) when t = T One point that should be emphasized about the portfolio used in the derivation of equation (13.16) is that it is not permanently riskless. It is riskless only for an infinitesimally short period of time. As Sand t change, 8f/aS also changes. To keep the portfolio riskless, it is therefore necessary to frequently change the relative proportions of the derivative and the stock in the portfolio. Example 13.5 A forward contract on a non-dividend-paying stock is a derivative dependent on the stock. As such, it should satisfy equation (13.16). From equation (5.5), we .know that the value of the forward contract, f, at a general time t is given in terms T he Blqck-S(:holes-Ivlertoll Ivlodel 293 of the stock price S at this time by f =. S - Ke-r(T-I) where K is the delivery price. This means that of -at = _rKe-r(T-tl, of 1 as =, 02 f -0 as 2 - When these are substituted into the left-hand side of equation (13.16), we obtain _rKe-r(T-t) + rS This equals rf, showing that equation (13.16) is indeed satisfied. The Prices of Tradeable Derivatives Any function f(S, t) that is a solution of the differential equation (13.16) is the theoretical price of a derivative that could be traded. If a derivative with that price existed, it would not create any arbitrage opportunities. Conversely, if a function f(S, t) does not satisfy the differential equation (13.16), it cannot be the price of a derivative without creating arbitrage opportunities for traders. To illustrate this point, consider first the function eS • This does not satisfy the differential equation (13.16). It is therefore not a candidate for being the price of a derivative dependent on the stock price. If an instrument whose price was always e S existed, there would be an arbitrage opportunity. As a second example, consider the function e(ci -2r)(T-I) S This does satisfy the differential equation, and so is, in theory, the price of a tradeable security. (It is the price of a derivative that pays off 1/ ST at time T.) For other examples of tradeable derivatives, see Problems 13.11, 13.12, 13.23, and 13.28. 13.7 RISK-NEUTRAL VALUATION We introduced risk-neutral valuation in connection with the binomial model in Chapter 11. It is without doubt the single most important tool for the analysis of derivatives. It arises from one key property of the Black-Scholes-Merton differential equation (13.16). This property is that the equation does not involve any variables that are affected by the risk preferences of investors. The variables that do appear in the equation are the current stock price, time, stock price volatility, and the risk-free rate of interest. All are independent of risk preferences. The Black-Scholes-Merton differential equation would not be independent of risk preferences if it involved the expected return, fL, on the stock. This is because the value of f.L does depend on risk preferences. The higher the level of risk aversion by investors, the higher fL will be for any given stock. It is fortunate that fL happens to drop out in the derivation of the differential equation. Because the Black-Scholes-Merton differential equation is independent of risk preferences, an ingenious argument can be used. If risk preferences do not enter the 294 CHAPTER 13 equation, they cannot affect its solution. Any set of risk preferences can, therefore, be used when evaluating f. In particular, the very simple assumption that all investors are risk neutral can be made. In a world where investors are risk neutral, the expected return on all investment assets is the risk-free rate of interest, r. The reason is that risk-neutral investors do not require a premium to induce them to take risks. It is also true that the present value of any cash flow in a risk-neutral world can be obtained by discounting its expected value at the risk-free rate. The assumption that the world is risk neutral does, therefore, considerably simplify the analysis of derivatives. Consider a derivative that provides a payoff at one particular time. It can be valued using risk-neutral valuation by using the following procedure: 1. Assume that the expected return from the underlying asset is the risk-free interest rate, r (i.e., assume JL = r). _ 2. Calculate the expected payoff from the derivative. 3. Discount the expected payoff at the risk-free interest rate. It is important to appreciate that risk-neutral valuation (or the assumption that all investors are risk neutral) is merely an artificial device for obtaining solutions to the Black-Scholes differential equation. The solutions that are obtained are valid in all worlds, not just those where investors are risk neutral. When we move from a riskneutral world to a risk-averse world, two things happen. The expected growth rate in the stock price changes and the discount rate that must be used for any payoffs from the deri~ative changes. It happens that these two changes always offset each other exactly. Application to Forward Contracts on a Stock We valued forward contracts on a non-dividend-paying stock in Section 5.7. In Example 13.5, we verified that the pricing formula satisfies the Black-Scholes differential equation. In this section we derive the pricing formula from risk-neutral valuation. We make the assumption that interest rates are constant and equal to r. This is somewhat more restrictive than the assumption in Chapter 5. Consider a long forward contract that matures at time T with delivery price, K. As indicated in Figure 1.2, the value of the contract at maturity is ST-K where ST is the stock price at time T. From the risk-neutral valuation argument, the value of the forward contract at time 0 is its expected value at time T in a risk-neutral world discounted at the risk-free rate of interest. Denoting the value of the forward contract at time zero by f, this means that f = e- rT E(ST - K) where E denotes the expected value in a risk-neutral world. Since K is a constant, this equation becomes (13.17) The -expected return JL on the stock becomes r in a risk-neutral world. Hence, from T he Black-Scholes-l1Ifertoll 1Vlodel 295 equation (13.4), we have ~ E(ST) = Soe rT (13.18) Sub?tituting equation (13.18) into equation (13.17) gives f= So - Ke- rT (13.19) This is in agreement with equation (5.5). 13.8 BLACK-SCHOLES PRICING FORMULAS The Black-Scholes formulas for the prices at time 0 of a European call option on a non-dividend-paying stock and a European put option on a non-dividend-paying stock are (13.20) and (13.21) where _In(SolK)+(r+O'212)T d1 O'-IT 2 _ In (Sol K) + (r - O' /2)T _ d ;;;; d7 ;;;; I-O'yT O'yT The function N(x) is the cumulative probability distribution function for a standardized normal distribution. In other words, it is the probability that a variable with a standard normal distribution, ¢(O, 1), will be less than x. It is illustrated in Figure 13.3. The remaining variables should be familiar. The variables c and p are the European call and European put price, So is the stock price at time zero, K is the strike price, r is the Figure 13.3 Shaded area represents N(x). 296 CHAPTER 13 continuously compounded risk-free rate, a is the stock price volatility, and T is the time to maturity of the option. One way of deriving the Black-Scholes formulas is by solving the differential equation (13.16) subject to the boundary condition mentioned in Section 13.6. 5 Another approach is to use risk-neutral valuation. Consider a European call option. The expected value of the option at maturity in a risk-neutral world is E[max(ST - K, 0)] where, as before, E denotes the expected value in a risk-neutral world. From the riskneutral valuation argument, the European call option price c is this expected value discounted at the risk-free rate of interest, that is, c = e- rT E[max(ST - K, 0)] (13.22) The appendix at the end of this chapter shows that this equation leads to the result in equation ( 1 3 . 2 0 ) . " To provide an interpretation of the terms in equation (13.20), we note tqat it can be written . (13.23) The expression N(d2 ) is the probability that the option will be exercised in a risk-neutral world, so that KN(d2 ) is the strike price times the probability that the strike price will be paid. The expression SoN(d\)erT is the expected value of a variable that is equal to ST if ST > K and to zero otherwise in a risk-neutral world. Since the European price equals the American price when there are no dividends (see Section 9.5), equation (13.20) also gives the value of an American call option on a nondividend-paying stock. Unfortunately, no exact analytic formula for the value of an American put option on a non-dividend-paying stock has been produced. Numerical procedures for calculating American put values are discussed in Chapter 17. When the Black-Scholes formula is used in practice the interest rate r is set equal to the zero-coupon risk-free interest rate for a maturity T. As we show in later chapters, this is theoretically correct when r is a known function of time. It is also theoretically correct when the interest rate is stochastic provided that the stock price at time T is lognormal and the volatility parameter is chosen appropriately. As mentioned earlier, time is normally measured as the number of trading days left in the life of the option divided by the number of trading days in I year. Properties of the Black-Scholes Formulas We now show that the Black-Scholes formulas have the right general properties by considering what happens when some of the parameters take extreme values. When the stock price, So, becomes very large, a call option is almost certain to be exercised. It then becomes very similar to a forward contract with delivery price K. 5 The differential equation gives the call and put prices at a general time t. For example, the call price that satisfies the differential equation is c = SN(d l ) - Ke-r(T-r)N(d2), where d l and d2 ;, = In(S/ K) + (r + a2 /2)(T u.;r:::t t) dl - u-lT - t. See Problem 13.17 to prove that the differential equation is satisfied. T he Black-Scholes-}YIertoll lVIodel 297 From equation (5.5), we expect the call price to be This is, in fact, the call price given by equation (13.20) because, when So becomes very large, both d l and d2 become very large, and N(d l ) and N(d2) are both close to 1.0. When the stock price becomes very large, the price of a European put option, p, approaches zero. This is consistent with equation (13.21) because N( -d l ) and N( -d2) are both close to zero .. Consider next what happens when the volatility a approaches zero. Because the stock is virtually riskless, its price will grow at rate r to SoerT at time T and the payoff from a call option is max(Soe rT - K, 0) Discounting at rate r, the value of the call today is e-rT max(SoerT - K, 0) = max(So - Ke- rT , 0) To. show that this is consistent with equation (13.20), consider first the case where So > Ke- rT • This implies that In (Sol K) + rT > O. As a tends to zero, d l and d2 tend to +00, so that N(d l ) arid N(d2 ) tend to 1.0 and equation (13.20) becomes c = So - Ke- rT When So < Ke- rT , it follows that In(Sol K) + rT < O. As a tends to zero, d l and d2 tend to -00, so that N(d() and N(d2 ) tend to zero and equation (13.20) gives a call price of zero. The call price is therefore always max(So - Ke- rT , 0) as a tends to zero. Similarly, it can be shown that the put price is always max(Ke-rT - So, 0) as a tends to zero. 13.9 CUMULATIVE NORMAL DISTRIBUTION FUNCTION The only problem in implementing equations (13.20) and (13.21) is in calculating the cumulative normal distribution function, N(x). Tables for N(x) are provided at the end of this book. The NORMSDIST function calculates N(x) in Excel. A polynomial approximation that gives six-decimal-place accuracy is 6 N(x) = !1-NI(X)(a(k+a2k2+a3k3+a4k4+askS) whenx~O 1-N(-x) . whenx<O where 1 1 +yx k=--, a( = 0.319381530, a3 6 = 1.781477937, a4 y=0.2316419 a2 = -0.356563782 = -1.821255978, as = 1.330274429 See M. Abramowitz and 1. Stegun, Handbook of Mathematical Functions. New York: Dover Publications, 1972. CHAPTER 13 2 98 and Example 13.6 The stock price 6 months from the expiration of an option is $42, the exercise price of the option is $40, the risk-free interest rate is 10% per annum, and the volatility is 20% per annum. This means that So = 42, K = 40, r = 0.1, a = 0.2, T=0.5, d 1 = In(42/40) + (0.1 + 0.22 /2) x 0.5 = 0.7693 0.2..)[5 d = In(42/40) 7 + (0.1 - -. 2 0.2 /2) x 0.5 = 0.6278 0.2..)[5 and Ke- rT = 40e-O.05 = 38.049 Hence, if the option is a European call, its value c is given by c = 42N(0.7693) - 38.049N(0.6278) If the option is a European put, its value p is given by p = 38.049N(-0.6:?78) - 42N(-0.7693) Using the polynomial approximation, N(0.7693) = 0.7791, N( -0.7693) = 0.2209 = 0.7349, N(-0.6278) = 0.2651 N(0.6278) so that c = 4.76, p = 0.81 Ignoring the time value of money, the stock price has to rise by $2.76 for the purchaser of the call to break even. Similarly, the stock price has to fall by $2.81 for the purchaser of the put to break even. 13.10 WARRANTS AND EXECUTIVE STOCK OPTIONS The exercise of a regular call option on a company has no effect on the number of the company's shares outstanding. If the writer of the option does not own the company's shares, he or she must buy them in the market in the usual way and then sell them to the option holder for the strike price. As explained in Chapter 8, warrants and executive stock options are different from regular call options in that exercise leads to the company issuing more shares and then selling them to the option holder for the strike price. As the strike price is less than the market price, this dilutes the interest of the existing shareholders. How should potential dilution affect the way we value outstanding warrants and executive stock options? The answer is that it should not! Assuming markets are 299 T he Black-Scholes-l\!Iertoll l\!Iodel Business Snapshot 13.3 WarraI~ts, Executive 'Stock Options, and Dilu Consider a company with 100,000 shares each worth $50. It su an announcement that it is granting 100,000 stock optionst strike price of $50 and a vesting period of 3 years. If the marke the shareholders from the employee stock options in the form more highly motivated managers, the stock price will declin iatel announcement of the employee stock options. If the stock price declines dilution cost to the current shareholders is $5. per share or $500,000 ota!. Suppose that the company does well during the vesting period so the vesting period the share price is $100. Suppose further that a exercised at this point. The payoff to the employees is $50 per opti to argue that there will be further dilution in that 100,000 share share are now merged with 100,000 shares for which only $50 is al share price reduces to $75 and (b) the payoff to the option hoI option. However, this argument is flawed. The exercise of t by the market and already reflected in the share pri exercised is $50. This example illustrates the general point that w inlpact of dilution from executive stock options or wa price as soon as they are announced and does not need again when the options are valued. efficient the stock price will reflect potential dilution from all outstanding warrants and executive stock options. This is explained in Business Snapshot 13.3. 7 Consider next the situation a company is in when it is contemplating a new issue of warrants (or executive stock options). We suppose that the company is interested in calculating the cost of the issue assuming that there are no compensating benefits. We assume that the company has N shares worth So each and the number of new options contemplated is M, with each option giving the holder the right to buy one share for K. The value of the company today is NSo. This value does not change as a result of the warrant issue. Suppose that without the warrant issue the share price will be ST at the warrant's maturity. This means that (with or without the warrant issue) the total value of the equity and the warrants at time Twill NST • If the warrants are exercised, there is a cash inflow from the strike price increasing this to NS T + MK. This value is distributed among N + M shares, so that the share price immediately after exercise becomes NST+MK N+M Therefore the payoff to an option holder if the option is exercised is NST+MK -K N+M Analysts sometimes assume that the.sum of the values of the warrants and the equity (rather than just the value of the equity) is lognormal. The result is a Black-Scholes type of equation for the value of the warrant in terms of the value of the warrant. See Technical Note 3 on the author's website for an explanation of this model. 7 300 CHAPTER 13 or N --(ST-K) N+M This shows that the value of each option is the value of N N+M regular call options on the company's stock. Therefore the total cost of the options is M times this. , Example 13.7 A company with 1 million shares worth $40 each is considering issuing 200,000 warrants each giving holder the right to buy one share with a strike price of $60 in 5 years. It wants to know the cost of this. The interest rate is 3% per annum, and the volatility is 30% per annum. The company pays no dividends. From equation (13.20), the value of a 5-year European call option on the stock is $7.04. In this case, N = 1,000,000 and M = 200,000, so that the value of each warrant is 1,000,000 1,000,000 + 200,000 x 7.04 = 5.87 or $5.87. The total cost of the warrant issue is 200,000 x 5.87 = $1.17 million. Assuming the market perceives no benefits from the warrant issue, we expect the . stock price to decline by $1.17 to $38:83. 13.11- IMPLIED VOLATILITIES The one parameter in the Black-Scholes pncmg formulas that cannot be directly observed is the volatility of the stock price. In Section 13.4, we discussed how this can be estimated from a history of the stock price. In practice, traders usually work with what are known as implied volatilities. These are the volatilities implied by option prices observed in the market. To illustrate how implied volatilities are calculated, suppose that the value of a European call option on a non-dividend-paying stock is 1.875 when So = 21, K = 20, r = 0.1, and T = 0.25. The implied volatility is the value of u that, when substituted into equation (13.20), gives c = 1.875. Unfortunately, it is not possible to invert equation (13.20) so that u is expressed as a function of So, K, r, T, and c. However, an iterative search procedure can be used to find the implied u. For example, we can start by trying u = 0.20. This gives a value of c equal to 1.76, which is too low. Because c is an increasing function of u, a higher value of u is required. We can next try a value of 0.30 for u. This gives a value of c equal to 2.10, which is too high and means that u must lie between 0.20 and 0.30. Next, a value of 0.25 can be tried for u. This also proves to be too high, showing that u lies between 0.20 and 0.25. Proceeding in this way, we can halve the range for u at each iteration and the correct value of a can be calculated to any required accuracy.s In this example, the implied volatility is 0.235, or 23.5%, per This method -is presented for illustration. Other more powerful methods, such as the Newton-Raphson method, are often used in practice (see footnote 5 of Chapter 4). DerivaGem can be used to calculate implied volatilities_ 8 T he B lack-Scholes-IV!ertOll lV!odel 301 annum. A similar procedure can be used in conjunction with binomial trees to find implied volatilities for American options. Implied volatilities are used to monitor the market's opinion ab»ut the volatility of a particular stock. Traders like to calculate implied volatilities from actively traded options on a certain asset and interpolate between them to calculate the appropriate volatility for pricing a less actively traded option on the same stock. We explain this procedure in Chapter 16. It is important to note that the prices of deep-in-the-money and deep-out-of-the-money options are relatively insensitive· to volatility. Implied volatilities calculated from these options, therefore, tend to be unreliable. 13.12 DIVIDENDS Up to now, we have assumed that the stock upon which the option is written pays no dividends. In this section, we modify the Black-Scholes model to take account of dividends. We assume that the amount and timing of the dividends during the life of an option can be predicted with certainty. For short-life options this is not an unreasonable assumption. For long-life options it is usual to assume that the dividend yield rather the cash dividend payments are known. Options can then be valued as will be described in the next chapter. The date on which the dividend -is paid should be assumed to be the ex-dividend date. On this date the stock price declines by the amount of the dividend. 9 European Options European options can be analyzed by assuming that the stock price is the sum of two components: a riskless component that corresponds to the known dividends during the life of the option and a risky component. The riskless component, at any given time, is the present value of all the dividends during the life of the option discounted from the ex-dividend dates to the present at the risk-free rate. By the time the option matures, the dividends will have been paid and the riskless component will no longer exist. The Black-Scholes formula is therefore correct if So is equal to the risky component of the stock price and (j is the volatility of the process followed by the risky component. IO Operationally, this means that the Black-Scholes formula can be used provided that the stock price is reduced by the present value of all the dividends during the life of the option, the discounting being done from the ex-dividend dates at the risk-free rate. A dividend is counted as being during the life of the option only if its ex-dividend date occurs during the life of the option. For tax reasons the stock price may go down by somewhat less than the cash amount of the dividend. To take account of this phenomenon, we need to interpret the word 'dividend' in the context of option pricing as the reduction in the stock price on the ex-dividend date caused by the dividend. Thus, if a dividend of SI per share is anticipated and the share price normally goes down by 80% of the dividend on the ex-dividend date, the dividend should be assumed to be SO.80 for the purposes of the analysis. 9 10 In theory, this is not quite the same as the volatility of the stochastic process followed by the whole stock price. The volatility of the risky component is approximately equal to the volatility of the whole stock price multiplied by So/(So - D), where D is the present value of the dividends. However, an adjustment is only necessary when volatilities are estimated using historical data. An implied volatility is calculated after the present value of dividends have been subtracted from the stock price and is the volatility of the risky component. 302 CHAPTER 13 Example 13.8 Consider a European call option on a stock when there are ex-dividend dates in two months and five months. The dividend on each ex-dividend date is expected to be $0.50. The current share price is $40, the exercise price is $40, the stock price volatility is 30% per annum, the risk-free rate of interest is 9% per annum, and the time to maturity is six months. The present value of the dividends is 0.5e-O.1667xO.09 + 0.5e- 0.4167xO.09 = 0.9741 The bption price can therefore be calculated from the Black-Scholes· formula, with So = 40 - 0.9741 = 39.0259, K = 40, r = 0.09, (J" = 0.3, and T = 0.5: d = In(39.0259/40) I d? - + (0.09 + 0.3 2 /2) x 0.5 = 0.2017 0.3.JQ.5 = In(39.0259/40) + (0.09 - 2 0.3 /2) x 0.5 = -0.0104 0.3.JQ.5 Using the polynomial approximation in Section 13.9 gives us N(d l ) = 0.5800, N(d2 ) = 0.4959 and, from equation (13.20), the call price is 39.0259 x 0.5800 - 40e-O.09xO.5 x 0.4959 = 3.67 or $3.67. American Options Consider next American call options. In Section 9.5, we showed that in the absence of dividends American options should never be exercised early. An extension to the argument shows that when there are dividends, it is optimal to exercise only at a time immediately before the stock goes ex-dividend. We assume that n ex-dividend dates are anticipated and that they are at times tlo t2,"" tn, with tI < t2 < ... < tn' The dividends corresponding to these times will be denoted by Db D 2 , .•. , DID respectively. We start by considering the possibility of early exercise just prior to the final ex-dividend date (i.e., at time t n ). If the option is exercised at time tn' the investor receives sCtn) - K where Set) denotes the stock price at time t. If the option is not exercised, the stock price drops to sCtn) - Dn. As shown by equation (9.5), the value of the option is then greater than It follows that, if that is, (13.24) The Black-ScllOles-1YIerton lVlodel 303 it cannot be optimal to exercise at time tIl' On the other hand, if (13.25) for any reasonable assumption about the stochastic process followed by the stock price, it can be shown that it is always optimal to exercise at time til for a sufficiently high value of set,,). The inequality in (13.25) will tend to be satisfied when the final exdividend date is fairly close to the maturity of the option (i.e., T - tIl is small) and the dividend is large. Consider next time t,,-I> the penultimate ex-dividend date. If the option is exercised immediately prior to time t,,-I> the investor receives S(t,,_I) - K. If the option is not exercised at time t,,-I> the stock price drops to S(t,,_I) - D,,_I and the earliest subsequent time at which exercise could take place is tIl' Hence, from equation (9.5), a lower bound to the option price if it is not exercised at time t,,_1 is It follows that if or it is not optimal to exercise immediately prior to time t,,-I. Similarly, for any i < n, if (13.26) it is not optimal to exercise immediately prior to time t i . The inequality in (13.26) is approximately equivalent to Di :;:;;; Kr(t i+ 1 - t i ) Assuming that K is fairly close to the current stock price, the dividend yield on the stock has to be either close to or above the risk-free rate of interest for this inequality not to be satisfied. This is often not the case. We can conclude from this analysis that, in many circumstances, the most likely time for the early exercise of an American call is immediately before the final ex-dividend date, til" Furthermore, if inequality (13.26) holds for i = 1,2, ... ,n - 1 and inequality (13.24) holds, we can be certain that early exercise is never optimal. Black's Approximation Black suggests an approximate procedure for taking account of early exercise in call options. II This involves calculating, as described earlier in this section, the prices of European options that mature at times T and til' and then setting the American price equal to the greater of the two. This approximation seems to work well in most cases. 12 See F. Black, "Fact and Fantasy in the Use of Options," Financial Analysts JOl/1'1lal, 31 (July/August 1975): 36-41, 61-72. II 12 For an exact formula, suggested by Roll, Geske, and Whaley, for valuing calls when there is only one exdividend date, see Technical Note 4 on the author's website. This involves the cumulative bivariate normal distribution function. A procedure for calculating this function is given in Technical Note 5 also on the author's website. 3 04 CHAPTER 13 Example 13.9 Consider the situation in Example 13.8, but suppose that the option is American rather than European. In this case Dl = D2 = 0.5, So = 40, K = 40, r = 0.09, tl = 2/12, and t2 =5/12. Since K[l - e- r (r1- t \l] = 40(1 - e-O.09xO.25) = 0.89 is greater than 0.5, it follows (see inequality (13.26)) that the option should never be exercised immediately before the first ex-dividend date. In addition, since K[l - e- r (T-t 1l ] = 40(1 _ [O.09xO.0833) = 0.30 is less than 0.5, it follows (see inequality (13.25)) that, when it is sufficiently deep in the money, the option should be exercised immediately before the second exdividend date. We now use Black's approximation to value the option. The present value of the first dividend is 0.5e-O.1667xO.09 = 0.4926 so that the value of the option, on the assumption that it expires just before the final ex-dividend date, can be calculated using the Black-Scholes formula with So = 40 - 0.4926 = 39.5074, K = 40, r = 0.09, II = 0.30, and T = 0.4167. It is $3.52. Black's approximation involves taking the greater of this and the value ,of the option when it can only be exercised at the end of 6 months. From Example 13.8, we know that the latter is $3.67. Black's approximation, therefore, gives the value of the American call as $3.67. The value of the option given by DerivaGem using "Binomial American" with 500 time steps is $3.72. There are two reasons for differences between the Binomial Model (BM) and Black's approximation (BA). The first concerns the timing of the early exercise decision; the second concerns the way volatility is applied. The timing of the early exercise decision tends to make BM greater than BA. In BA, the assumption is that the holder has to decide today whether the option will be exercised after 5 months or after 6 months; BM allows the decision on early exercise at the 5-month point to depend on the stock price. The way in which volatility is applied tends to make BA greater than BM. In BA, when we assume exercise takes place after 5 months, the volatility is applied to the stock price less the present value of the first dividend; when we assume exercise takes place after 6 months, the volatility is applied to the stock price less the present value of both dividends. SUMMARY We started this chapter by examining the properties of the process for stock prices introduced in Chapter 12. The process implies that the price of a stock at some future time, given its price today, is lognormal. It also implies that the continuously compounded return from the stock in a period of time is normally distributed. Our uncertainty about future stock prices increases as we look further ahead. The standard deviation of the logarithm of the stock price is proportional to the square root of how far afiead we are looking. T he Black~Scholes-lVlertonlVlodel 305 To estimate the volatility a of a stock price empirically, the stock price is observed at fixed intervals of time (e.g., every day', every week, or every month). For each time period, the natural logarithm of the ratio of the stock price at the end of the time period to the stock price at the beginning of the time period is calculat€d. The volatility is estimated as the standard deviation of these numbers divided by the square root of the length of the time period in years. Usually, days when the exchanges are closed are ignored in measuring time for the purposes of volatility calcula~ions. The differential equation for the price of any derivative dependent on a stock can be obtained by creating a riskless position in the option and the stock. Because the derivative and the stock price both depend on the same underlying source of uncertainty, this can always be done. The position that is created remains riskless for only a very short period of time. However, the return on a riskless position must always be the risk-free interest rate if there are to be no arbitrage opportunities. The expected return on the stock does not enter into the Black-Scholes differential equation. This leads to a useful result known as risk-neutral valuation. This result states that when valuing a derivative dependent on a stock price, we can assume that the world is risk neutral. This means that we can assume that the expected return from the stock is the risk-free interest rate, and then discount expected payoffs at the risk-free interest rate. The Black-Scholes equations for European call and put options can be derived by either solving their differential equation or by using risk-peutral valuation. An implied volatility is the volatility that, when used in conjunction with the BlackScholes option pricing formula, gives the market price of the option. Traders monitor implied volatilities and commonly use the implied volatilities from actively traded options to estimate the appropriate volatility to use to price a less actively traded option on the same asset. Empirical results show that the volatility of a stock is much higher when the exchange is open than when it is closed. This suggests that, to some extent, trading itself causes stock price volatility. The Black-Scholes results can be extended to cover European call and put options on dividend-paying stocks. The procedure is to use the Black-Scholes formula with the stock price reduced by' the present value of the dividends anticipated during the life of the option, and the volatility equal to the volatility of the stock price net of the present value of these dividends. In theory, American call options are liable to be exercised early, immediately before any ex-dividend date. In practice, only the final ex-dividend date usually needs to be considered. Fischer Black has suggested an approximation. This involves setting the American call option price equal to the greater of two European call option prices. The first European call option expires at the same time as the American call option; the second expires immediately prior to the final ex-dividend date. FURTHER READING On the Distribution of Stock Price Changes B1attberg, R., and N. Gonedes, "A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices," Journal of Business, 47 (April 1974): 244-80. Fama, E. F., "The Behavior of Stock Market Prices," Journal of Business, 38 (January 1965): 34-105. 306 CHAPTER 13 Kon, S. J., "Models of Stock Returns-A Comparison," Journal of Finance, 39 (March 1984): 147-65. Richardson, M., and T. Smith, "A Test for Multivariate Normality in Stock Returns," Journal of Business, 66 (1993): 295-32l. On the Black-Scholes Analysis Black, F. "Fact and Fantasy in the Use of Options and Corporate Liabilities," Financial Analysts Journal, 31 (July/August 1975): 36-41, 61-72. Black, F. "How We Came Up with the Option Pricing Formula," Journal.of Portfolio Management, 15, 2 (1989): 4-8. Black, F., and M. Scholes, "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, 81 (May/June 1973): 637-59. Merton, R. c., "Theory of Rational Option Pricing;" Bell Journal of Economics and Management Science, 4 (Spring 1973): 141-83. On Risk-Neutral Valuation . Cox, J.C., and S.A. Ross, "The Valuation of Options for Alternative Stochastic Processes," Journal of Financial Economics, 3 (1976): 145-66. Smith, C. W., "Option Pricing: A Review," Journal of Financial Economics, 3 (1976): 3-54. On the Causes of Volatility Fama, E. F. "The Behavior of Stock Market Prices." Journal of Business, 38 (January 1965): 34-105. French, K. R. "Stock Returns and the Weekend Effect." Journal of Financial Economics. 8 (March 1980): 55-69. . French, K. R., and R. RoIl "Stock Return Variances: The Arrival of Information and the Reaction of Traders." Journal of Financial Economics, 17 (September 1986): 5-26. RoIl R. "Orange Juice and Weather," American Economic Review, 74, 5 (December 1984): 861-80. Questions and Problems (Answers in Solutions Manual) 13.1. What does the Black-Scholes stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about the continuously compounded rate of return on the stock during the year? 13.2. The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day? 13.3. Explain the principle of risk-neutral valuation. 13.4. Calculate the price of a 3-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum. 13.5. What difference does it make to your calculations in Problem 13.4 if a dividend of $1.50 is expected in 2 months? . 13.6. What is implied volatility? How can it be calculated? 13.7. A stock price is currently $40. Assume that the expected return from the stock is 15% and that its volatility is 25%. What is the probability distribution for the rate of return (with continuolls compounding) earned ovei'a 2-year period? T he Black-ScllOles-1VIertoll 1VIodel 307 13.8. A stock price follows geometric Brownjan motion with an expected return of 16% and a . volatility of 35%. The current price is $38. (a) What is the probability that a European call option on the stock with an exercise 'price of $40 and a maturity date in 6 months will be exercised? (b) What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised? 13.9. Using the notation in this chapter, prove that a 95% confidence interval for between ST is 13.10. A portfolio manager announces that the average of the returns realized in each year of the last 10 years is 20% per annum. In what respect is this statement misleading? 13.11. Assume that a non-dividend-paying stock has an expected return of JL and a volatility of (J. An innovative financial institution has just announced that it will trade a security that pays off a dollar amount equal to In ST at time T, where ST denotes the value of the stock price at time T. (a) Use risk-neutral valuation to calculate the price of the security at time t in terms of the stock price, S, at time t. (b) Confirm that your price satisfies the differential equation (13.16). 13.12. Consider a derivative that pays off Sf} at time T, where ST is the stock price at that time. When the stock price follows geometric Brownian motion, it can be shown that its price at time t (t ~ T) has the form h(t, T)S" where S is the stock price at time t and h is a function only of t and T. (a) By substituting into the Black-Scholes-Merton partial differential equation, derive an ordinary differential equation satisfied by h(t, T). (b) What is the boundary condition for the differential equation for h(t, T)? (c) Show that h(t, T) = 13.13. 13.14. 13.15. 13.16. 13.17. e[0.5,,211 (1I-1)+r(1l-1)](T -c) where r is the risk-free interest rate and (J is the stock price volatility. What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is 3 months? What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is 6 months? Consider an American call option on a stock. The stock price is $70, the time to maturity is 8 months, the risk-free rate of interest is 10% per annum, the exercise price is $65, and the volatility is 32%. A dividend of $1 is expected after 3 months and again after 6 months. Show that it can never be optimal to exercise the option on either of the two dividend dates. Use DerivaGem to calculate the price of the option. A call option on a non-dividend-paying stock has a market price of $2t. The stock price is $15, the exercise price is $13, the time to maturity is 3 months, -and the risk-free interest rate is 5% per annum. What is the implied volatility? With the notation used in this chapter: (a) What is N'(x)? 308 CHAPTER 13 (b) Show that SN'(dd = Ke-r(T-t)N'(d2 ), where S is the stock price at time d 1 = In(S/ K) + (r + O' t and 2 /2)(T - t) o'~ 2 d, = In(S/ K) + (r - O' /2)(T - t) O'-vlT - t (c) Calculate adl/aS and ad2 /aS. (d) Show that when c = SN(d l ) - Ke-r(T-t)N(d2) , it follows that aaC = _rKe-r(T-t) N(d2) - SN'(d 1) ~ t 2 T-t where c is the price of a call option on a non-dividend-paying stock. (e) Show that ac/as = N(d l ). (f] Show that c satisfies the Black-Scholes differential equation. (g) Show that c satisfies the boundary condition for a European call option, i.e., that c = max(S - K, 0) as t ------7 T. . 13.18. Show.that the Black-Scholes formulas for call and put options satisfy put-'--Call parity. 13.19. A stock price is currently $50 and the risk-free interest rate is 5%. Use the DerivaGem software to translate the following table of European call options on the stock into a table of implied volatilities, assuming no dividends. Are the option prices <:;onsistent with the assumptions underlying Black-Scholes? - Maturity (months) Strike price ($) 3 6 12 45 50 55 7.0 3.7 1.6 8.3 5.2 2.9 10.5 7.5 5.1 13.20. Explain carefully why Black's approach to evaluating an American call option on a dividend-paying stock may give an approximate answer even when only one dividend is anticipated. Does the answer given by Black's approach understate or overstate the true option value? Explain your answer. 13.21. Consider an American call option on a stock. The stock price is $50, the time to maturity is 15 months, the risk-free rate of interest is 8% per annum, the exercise price is $55, and the volatility is 25%. Dividends of $1.50 are expected in 4 months and 10 months. Show that it can never be optimal to exercise the option on either of the two dividend dates. Calculate the price of the option. 13.22. Show that the probability that a European call option will be exercised in a risk-neutral world is, with the notation introduced in this chapter, N(d2). What is an expression for the value of a derivative that pays off $100 if the price of a stock at time T is greater than K? 1 13.23. Show that S-2r/a could be the price of a traded security. 13.24. A corp.pany has an issue of executive stock options outstanding. Should dilution be taken into account when the options are valued? Explain your answer. T he Black-Scholes-111ertoll 1110del 309 13.25. A company's stock price is $50 and 1.0 million shares are outstanding. The company is considering giving its employees 3 million at-the-money 5-year . call options. Option exercises will be handled by issuing more shares. The stock pric{ volatility is 25%, the 5-year risk-free rate is 5% and the company does not pay dividends. Estimate the cost to the company of the employee stock option issue. Assignment Questions 13.26. A stock price is currently $50. Assume that the expected return from the stock is 18% and its volatility is 30%. What is the probability distribution for the stock price in 2 years? Calculate the mean and standard deviation of the distribution. Determine the 95% confidence interval. 13.27. Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows: 30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0, 32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2 13.28. 13.29. 13.30. 13.31. Estimate the stock price volatility. What is the standard error of your estimate? A financial institution plans to offer a security that pays off a dollar amount equal to slat time T. (a) Use risk-neutral valuation to calculate the price of the security at time t in terms of the stock price S at time t. (Hint: The expected value of sl- can be calculated from the mean and variance of ST given in Section 13.1.) (b) Confirm that your price satisfies the differential equation (13.16). Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5%, the volatility is 25% per annum, and the time to maturity is 4 months. (a) What is the price of the option if it is a European call? (b) What is the price ~f the option if it is an American call? (c) What is the price of the option if it is a European put? (d) Verify that put--eall parity holds. Assume that the stock in Problem 13.29 is due to go ex-dividend in I,} months. The expected dividend is 50 cents. (a) What is the price of the option if it is a European call? (b) What is the price of the option if it is a European put? (c) If the option is an American call, are there any circumstances under which it will be exercised early? Consider an American call option when the stock price is $18, the exercise price is $20, the time to maturity is 6 months, the volatility is 30% per annum, and the risk-free interest rate is 10% per annum. Two equal dividends are expected during the life of the option with ex-dividend dates at the end of 2 months and 5 months. Assume the dividends are 40 cents. Use Black's approximation and the DerivaGem software to value the option. How high can the dividends be without the American option being worth more than the corresponding European option? 310 CHAPTER 13 APPENDIX PROOF OF THE BLACK-SCHOLES-MERTON FORMULA We will prove the Black-Scholes result by first proving another key result that will also be useful in future chapters. Key Result If V is lo~ormally distributed and the standard deviation of In V is w, then (13A.l) E[max(V - K, 0)] = E(V)N(d]) - KN(d2 ) where d] = In[E(V)/ K] + w 2 /2 W 2 In[E(V)/ K] - w /2 d? = -.-::.---:--:..:---:..----'-- W and 1j: denotes the expected value. Proof of Key Result Define g(V) as the probability density function of V. It follows that E[max(V - K, 0)] = J: (13A.2) (V - K)g(V) dV The variable In V is normally distributed with standard deviation w. From the properties of the lognormal distribution, the mean of In V is m, where]3 m = In[E(V)] - w2 /2 (13A.3) Define a new variable In V-m Q=--w (13AA) This variable is normally distributed with a mean of zero and a standard deviation of 1.0. Denote the density function for Q by !z(Q) so that 1 _Q2 j ? !z(O) = - e - - rr;= yL.1L Using equation (BAA) to convert the expression on the right-hand side of equation (l3A.2) from an integral over V to an integral over Q, we get E[max(V - K, 0)] = Joo (e Qw +1II - K)lz(Q)dQ (InK-III)jw . or E[max(V - K, 0)] . 13 = Joo eQw+lII lz(Q)dQ - KJoo (lnK-III)jw Fo~ a proof of this, see Technical Note 2 OIi'lhe author's website. (lnK-III)jw lz(Q)dQ (13A.5) 311 T he Black-Scholes-lYlerton lYlodel Now 2 = em+w /2 eHQ-w)2j/2 .j2ii = em+w2 / 21z(Q _ w) This means that equation (13A.5) becomes E[max(V - K, 0)] , = em+w-/2 Joo h(Q - w)dQ - K Joo (In K-m)/w h(Q)dQ (13A.6) (In K-m)/w If we define N(x) as the probability that a variable with a mean of zero and a standard deviation of 1.0 is less than x, the first integral in equation (13A.6) is 1- N[(1nK - m)jw - w] or N[(-lnK+m)jw+w] Substituting for m from equation (l3A.3) leads to Similarly the second integral in equation (13A.6) is N(d2 ). Equation (13A.6), therefore, becomes Substituting for m from equation (13A.3) gives the key result. The Black-Scholes-Merton Result We now consider a call option on a non-dividend-paying stock maturing at time T. The strike price is K, the risk-free rate is r, the current stock price is So, and the volatility is a. As shown in equation (13.22), the call price c is given by c = e-rT E[max(ST - K, 0)] (13A.7) where ST is the stock price at time T and E denotes the expectation in a risk-neutral world. Under the stochastic process assumed by Black-Scholes, ST is lognormal. Also, from equations (13.3) and (13.4), E(ST) = Soe rT and the standard deviation of In ST is a-/T. From the key result just proved, equation (l3A.7) implies c = e-rT[Soe rT N(d[) - KN(d2 )] or 312 CHAPTER 13 where d _In[E(ST)/ K] . d? = - In(So/ K) + (r + O' /2)T O'.JT O'.JT I- and 2 + O'2T /2 h? In[E(ST)/ K] - O'-T /2 O'.JT = In(So/ K) + (r - ? 0'- /2)T --:...-"-'----=,----'---- O'.JT ...
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This note was uploaded on 01/30/2012 for the course MATH 174 taught by Professor Donblasius during the Spring '11 term at UCLA.

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