12 -0 Finance 457 The Black–Scholes Model 12 Chapter Twelve Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -1 Finance 457 Chapter Outline 12. 1 Log-Normal Property of Stock Prices 12. 2 The distribution of the Rate of Return 12. 3 The Expected Return 12. 4 Volatility 12. 5 Concepts Underlying the Black-Scholes-Merton Differential Equation 12. 6 Derivation of the Black-Scholes-Merton Differential Equation 12. 7 Risk Neutral Valuation 12. 8 Black-Scholes Pricing Formulae Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -2 Finance 457 Chapter Outline (continued) 12. 9 Cumulative Normal Distribution Function 12. 10 Warrants Issued by a company on its own Stock 12. 11 Implied Volatilities 12. 12 The Causes of Volatility 12. 13 Dividends 12. 14 Summary Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -3 Finance 457 Prospectus: • In the early 1970 s, Fischer Black, Myron Scholes and Robert Merton made a major breakthrough in the pricing of options. • In 1997, the importance of this work was recognized with the Nobel Prize. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -4 Finance 457 12. 1 Log-Normal Property of Stock Prices • This is fully developed in chapter 11. • Assume that percentage changes in stock price in a short period of time are normally distributed. • Let: m: expected return on the stock : volatility on the stock • The mean of the percentage change in time dt is mdt • The standard deviation of this percentage change is Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -5 Finance 457 12. 1 Log-Normal Property of Stock Prices • The percentage changes in stock price in a short period of time are normally distributed: • Where d. S is the change in stock price in time dt, and f(m, s) denotes a normal distribution with mean m and standard deviation s. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -6 Finance 457 12. 1 Log-Normal Property of Stock Prices • As shown in section 11. 7, the model implies that • From this it follows that • and Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -7 Finance 457 12. 1 Log-Normal Property of Stock Prices The above equation shows that ln ST is normally distributed. This means that ST has a lognormal distribution. A variable with this distribution can take any value between zero and infinity. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -8 Finance 457 Properties of the Log-Normal Distribution A variable with this distribution can take any value between zero and infinity. 0 Unlike a normal distributions, it is skewed so that the mean, median, and mode are all different. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -9 Finance 457 12. 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 time zero and T. • Define the continuously compounded rate of return per annum realized between times zero and T as h. • It follows that so that Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -10 Finance 457 12. 2 The Distribution of the Rate of Return It follows from: that Thus, the continuously compounded rate of return per annum in normally distributed with mean Mc. Graw-Hill/Irwin and standard deviation Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -11 Finance 457 12. 3 The Expected Return • The expected return, m, required by investors from a stock depends on the riskiness of the stock. • The higher the risk, the higher m, ceteris paribus. • m also depends on interest rates in the economy • We could spend a lot of time on the determinants of m, but it turns out that the value of a stock option, when expressed in terms of the value of the underlying stock, does not depend on m at all. • There is however, one aspect of m that frequently causes confusion and is worth explaining. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -12 Finance 457 A subtle but important difference • Shows that mdt is the expected percentage change in the stock price in a very short period of time d t. • This means that m is the expected return in a very short period of time dt. • It is tempting to assume that m is also the continuously compounded return on the stock over a relatively long period of time. • However, this is not the case. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -13 Finance 457 A subtle but important difference • The continuously compounded return on the stock over T years is: Equation (12. 7): Shows that the expected value of this is The distinction between m and is subtle but important. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -14 Finance 457 A subtle but important difference Start with Taking logarithms: Since ln is a nonlinear function, So we cannot say In fact, we have: So the expected return over the whole period T, expressed with compounding dt, is close to Not m Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -15 Finance 457 A subtle but important difference • The above shows that a simple term like expected return is ambiguous. • It can refer to or m • For example, if your portfolio has had the following returns over the last five years: 30%; 20%; 10%; – 20%; – 40%; • What is the expected return? • Unless otherwise stated m will be expected return. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -16 Finance 457 12. 4 Volatility • The volatility of a stock, s, is a measure of our uncertainty about the returns. • Stocks typically have a volatility between 20% and 50% • From • The volatility of a stock price can be defined as the standard deviation of the return provided by the stock in one year when the return is expressed using continuous compounding. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -17 Finance 457 The Volatility • The volatility of an asset is the standard deviation of the continuously compounded rate of return in 1 year • As an approximation it is the standard deviation of the percentage change in the asset price in 1 year Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -18 Finance 457 Estimating Volatility from Historical Data (page 239 -41) 1. Take observations S 0, S 1, . . . , Sn at intervals of t years 2. Calculate the continuously compounded return in each interval as: 3. Calculate the standard deviation, s , of the ui´s 4. The historical volatility estimate is: Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -19 Finance 457 12. 5 Concepts Underlying the Black. Scholes-Merton Differential Equation • The arguments are similar to the no-arbitrage arguments we used to value stock options using binomial valuation in Chapter 10. • Set up a riskless portfolio consisting of a position in the derivative and a position in the stock. • In the absence of profitable arbitrage, the portfolio must earn the risk-free rate, r. • This leads to the Black–Scholes–Merton differential equation. • An important difference is the length of time that the portfolio remains riskless. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -20 Finance 457 Assumptions 1. The stock price follows a lognormal process with m and constant. 2. Short selling with full use of proceeds permitted. 3. No transactions costs or taxes. 4. No dividends during the life of the derivative. 5. No riskless arbitrage opportunities. 6. Security trading is continuous. 7. The risk-free rate, r, is constant and the same for all maturities. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -21 Finance 457 12. 6 Derivation of the Black-Scholes. Merton Differential Equation • The stock price process we are using: d. S = m. Sdt + Sdz • Let f be the price of a call option or other derivative contingent upon S. The variable f must be some function of S and t. From Itô’s lemma Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -22 Finance 457 12. 6 Derivation of the Black-Scholes. Merton Differential Equation • The appropriate portfolio is: short one derivative and long shares • Define P as the value of the portfolio. • By definition, The change, d. P, in the value of the portfolio in time dt Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -23 Finance 457 12. 6 Derivation of the Black-Scholes. Merton Differential Equation Substituting d. S = m. Sdt + Sdz and yields Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -24 Finance 457 12. 6 Derivation of the Black-Scholes. Merton Differential Equation Because d. P does not involve dz, the portfolio must be riskless during time dt. The no-arbitrage condition is therefore: d. P = r. Pdt Substituting from above yields: Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -25 Finance 457 12. 6 Derivation of the Black-Scholes. Merton Differential Equation It’s a short step to: This is the Black–Scholes–Merton differential equation. It has many solutions, corresponding to the different derivatives that can be defined with S as the underlying variable. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -26 Finance 457 12. 6 Derivation of the Black-Scholes. Merton Differential Equation The particular solution that is obtained when the equation is solved depends on the boundary conditions that are used. In the case of a European call, the key boundary condition is: f = max(S – K, 0) when t = T Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -27 Finance 457 Example • Consider a forward contract on a non-dividend paying stock. • From chapter 3 we have Clearly this satisfies the Black–Scholes–Merton differential equation: Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -28 Finance 457 12. 7 Risk Neutral Valuation • Without a doubt, the single most important tool for the analysis of derivatives. • Note that the Black–Scholes–Merton differential equation does not involve any variable that is affected by the risk preferences of investors. • The only variables are S 0, T, s, and r. • So any set of risk preferences can be used when evaluating f. Let’s use risk neutrality. • Now we can calculate the value of any derivative by discounting its expected payoff at the risk-free rate. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -29 Finance 457 Risk Neutral Valuation of Forwards • Consider a long forward contract that matures at time T with delivery price K. • The payoff at maturity is ST – K • The value of the forward contract is the expected value at time T in a risk-neutral world discounted at the risk-free rate. f = e–r. T Ê(ST – K) • Since K is constant, f = e–r. T [Ê(ST) – K] • In a risk-neutral world, m becomes r so Ê(ST) = S 0 er. T • We have f = S 0 – K e–r. T which is the no-arbitrage result we have from chapter 3. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -30 Finance 457 12. 8 Black–Scholes Pricing Formulae The Black-Scholes formulae for the price of a European call and a put written on a non-dividend paying stock are: N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -31 Finance 457 A Black–Scholes Example Find the value of a six-month call option on the Microsoft with an exercise price of $150 The current value of a share of Microsoft is $160 The interest rate available in the U. S. is r = 5%. The option maturity is 6 months (half of a year). The volatility of the underlying asset is 30% per annum. Before we start, note that the intrinsic value of the option is $10 —our answer must be at least that amount. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -32 Finance 457 A Black–Scholes Example Let’s try our hand at using the model. If you have a calculator handy, follow along. First calculate d 1 and d 2 Then, Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -33 Finance 457 A Black–Scholes Example N(d 1) = N(0. 52815) = 0. 7013 N(d 2) = N(0. 31602) = 0. 62401 Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -34 Finance 457 Another Black–Scholes Example Assume S = $50, K = $45, T = 6 months, r = 10%, and = 28%, calculate the value of a call and a put. From a standard normal probability table, look up N(d 1) = 0. 812 and N(d 2) = 0. 754 (or use Excel’s “normsdist” function) Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -35 Finance 457 12. 8 Black–Scholes Pricing Formulae To provide intuition, rewrite the Black-Scholes call formula as: N(d 2) is the probability that the option will be exercised in a risk-neutral world, so KN(d 2) is the expected value of the cost of exercise. S 0 N(d 1)er. T is the expected value of a variable that equals ST if ST > K and is zero otherwise in a riskneutral world. The present value at the risk-free rate is the value of a call Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -36 Properties of the Black–Scholes Formulae Finance 457 Consider what happens when ST becomes large. The option is almost certain to finish in-the-money, so the call becomes like a forward contract. From chapter 3 we have f = S 0 – K e–r. T When S 0 becomes large, d 1 and d 2 become large, so N(d 2) and N(d 1) become close to 1 The Black-Scholes call price reduces to the futures price. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -37 Properties of the Black–Scholes Formulae Finance 457 Consider what happens when s approaches zero. Because the stock is riskless, ST = S 0 er. T At expiry, the payoff from the call will be max(S 0 er. T – K, 0) If we discount at r c = e–r. Tmax(S 0 er. T – K, 0) = max(S 0 – K e–r. T, 0) Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -38 Properties of the Black–Scholes Formulae Finance 457 If S 0 > K e–r. T When s approaches zero, d 1 and d 2 tend to + , so N(d 2) and N(d 1) become close to 1. The Black–Scholes call price is then: S 0– K e–r. T If S 0 < K e–r. T When s approaches zero, d 1 and d 2 tend to – , so N(d 2) and N(d 1) become close to zero. The Black–Scholes call price is then 0 So, the Black–Scholes value of a call when s approaches zero c = max(S 0 – K e–r. T, 0) Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -39 Finance 457 12. 9 Cumulative Normal Distribution Function • NORMSDIST in Excel rocks. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -40 Finance 457 12. 10 Warrants Issued by a company on its own Stock • • There is a dilution effect. We can use the Black-Scholes formula for the value of a call if: 1. The stock price S 0 is replaced by S 0 + (M/N)W 2. The volatility is the volatility of the equity (I. e. the volatility of the shares plus the warrants, not just the shares). 3. The formula is multiplied by Ng/(N + Mg) Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -41 Finance 457 12. 11 Implied Volatilities • These are the volatilities that are implied by the observed prices of options in the market. • It is not possible to solve • For • In practice, use goal seek in Excel. • It’s best to use near-the-money options to estimate volatility. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -42 Finance 457 Implied Volatility • The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price • The is a one-to-one correspondence between prices and implied volatilities • Traders and brokers often quote implied volatilities rather than dollar prices Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -43 Finance 457 12. 12 The Causes of Volatility • Trading itself can be said to be a cause. • When implied volatilities are calculated, the life of an option should be measured in trading days. • Furthermore, if daily data are used to provide a historical volatility estimate, day when the exchange are closed should be ignored and the volatility per annum should be calculated from the volatility per trading day using this formula: • The normal assumption is that there are 252 trading days per year. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -44 Finance 457 Causes of Volatility • Volatility is usually much greater when the market is open (i. e. the asset is trading) than when it is closed • For this reason time is usually measured in “trading days” not calendar days when options are valued Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -45 Finance 457 Warrants & Dilution (pages 249 -50) • When a regular call option is exercised the stock that is delivered must be purchased in the open market • When a warrant is exercised new Treasury stock is issued by the company • This will dilute the value of the existing stock • One valuation approach is to assume that all equity (warrants + stock) follows geometric Brownian motion Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -46 Finance 457 12. 13 Dividends • European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes • Only dividends with ex-dividend dates during life of option should be included • The “dividend” should be the expected reduction in the stock price anticipated. • Elton and Gruber estimate this as 72% of the dividend. Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -47 Finance 457 American Calls • An American call on a non-dividend-paying stock should never be exercised early • An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -48 Finance 457 Black’s Approach to Dealing with Dividends in American Call Options Set the American price equal to the maximum of two European prices: 1. The first European price is for an option maturing at the same time as the American option 2. The second European price is for an option maturing just before the final exdividend date Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -49 Finance 457 Black’s Approach to Dealing with Dividends in American Call Options Set the American price equal to the maximum of two European prices: K 1 ST K 1 + c. American – c 1 Buy a long-lived option strike K 1 for c 1 Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
12 -50 Finance 457 12. 14 Summary • This chapter covers important material: – – – The lognormality of stock prices The calculation of volatility from historical data Risk-neutral valuation The Black-Scholes option pricing formulas Implied volatilities The impact of dividends Mc. Graw-Hill/Irwin Copyright © 2002 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
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