Chapter 18 Continuous Time Option Pricing Models Assumptions of the Black-Scholes Option Pricing Model (BSOPM): No taxes No transactions costs Unrestricted short-selling of stock, with full use of short-sale proceeds Shares are infinitely divisible Constant riskless interest rate for borrowing/lending No dividends European options (or American calls on non-dividend paying stocks) Continuous trading The stock price evolves via a specific process through time (more on this later.)

David Dubofsky and 18-1 Thomas W. Miller, Jr. Derivation of the BSOPM Specify a process that the stock price will follow (i.e., all possible paths) Construct a riskless portfolio of: Long Call Short Shares (or, long Shares and write one call, or long 1 share and write 1/calls) As delta changes (because time passes and/or S changes), one must maintain this risk-free portfolio over time This is accomplished by purchasing or selling the appropriate number of shares. David Dubofsky and 18-2 Thomas W. Miller, Jr. The BSOPM Formula C SN d1 Ke rTN d2 where N(di) = the cumulative standard normal distribution function, evaluated at di, and: d1 ln(S/K) (r 2 /2)T

T d2 d1 T N(-di) = 1-N(di) David Dubofsky and 18-3 Thomas W. Miller, Jr. The Standard Normal Curve The standard normal curve is a member of the family of normal curves with = 0.0 and = 1.0. The X-axis on a standard normal curve is often relabeled and called Z scores. The area under the curve equals 1.0. The cumulative probability measures the area to the left of a value of Z. E.g., N(0) = Prob(Z) < 0.0 = 0.50, because the normal distribution is symmetric. David Dubofsky and 18-4 Thomas W. Miller, Jr. The Standard Normal Curve The area between Z-scores of -1.00 and +1.00 is 0.68 or 68%. Note also that the area to the left of Z = -1 equals the area to the right of Z = 1. This

holds for any Z. What is N(1)? What is N(-1)? The area between Z-scores of -1.96 and +1.96 is 0.95 or 95%. What is the area in each tail? What is N(1.96)? What is N(-1.96)? David Dubofsky and 18-5 Thomas W. Miller, Jr. Cumulative Normal Table (Pages 560-561) Cumulative Probability for the Standard Normal Distribution 2nd digit of Z z 0.00 0.01 0.02

0.03 0.04 0.05 0.06 0.07 0.08 0.09 -1.3 0.0968 0.0951 0.0934 0.0918 0.0901

0.0885 0.0869 0.0853 0.0838 0.0823 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131

0.9147 0.9162 0.9177 N(-d) = 1-N(d) N(-1.34) = 1-N(1.34) David Dubofsky and 18-6 Thomas W. Miller, Jr. Example Calculation of the BSOPM Value S = $92 K = $95 T = 50 days (50/365 year = 0.137 year) r = 7% (per annum) = 35% (per annum) What is the value of the call?

David Dubofsky and 18-7 Thomas W. Miller, Jr. Solving for the Call Price, I. Calculate the PV of the Strike Price: Ke-rT = 95e(-0.07)(50/365) = (95)(0.9905) = $94.093. Calculate d1 and d2: d1 ln(S/K) (r 2 /2)T T ln(92/95) (0.07 0.1225/2)0.137 0.35 0.137 ln(0.96842) 0.01798 0.03209 0.01798 0.1089 (0.35)(0.3701)

0.12955 d2 = d1 T.5 = -0.1089 (0.35)(0.137).5 = -0.2385 Solving for N(d1) and N(d2) Choices: Standard Normal Probability Tables (pp. 560-561) Excel Function NORMSDIST N(-0.1089) = 0.4566; N(-0.2385) = 0.4058. There are several approximations; e.g. the one in fn 8 (pg. 549) is accurate to 0.01 if 0 < d < 2.20: N(d) ~ 0.5 + (d)(4.4-d)/10 N(0.1089) ~ 0.5 + (0.1089)(4.4-0.1089)/10 = 0.5467 Thus, N(-0.1089) ~ 0.4533 David Dubofsky and 18-9 Thomas W. Miller, Jr. Solving for the Call Value C = S N(d1) Ke-rT N(d2) = (92)(0.4566) (94.0934)(0.4058) = $3.8307. Applying Put-Call Parity, the put price is: P = C S + Ke-rT = 3.8307 92 + 94.0934 = $5.92. David Dubofsky and 18-10 Thomas W. Miller, Jr.

Lognormal Distribution The BSOPM assumes that the stock price follows a Geometric Brownian Motion (see http://www.stat.umn.edu/~charlie/Stoch/brown.html for a depiction of Brownian Motion). In turn, this implies that the distribution of the returns of the stock, at any future date, will be lognormally distributed. Lognormal returns are realistic for two reasons: if returns are lognormally distributed, then the lowest possible return in any period is -100%. lognormal returns distributions are "positively skewed," that is, skewed to the right. Thus, a realistic depiction of a stock's returns distribution would have a minimum return of -100% and a maximum return well beyond 100%. This is particularly important if T is long. David Dubofsky and 18-11 Thomas W. Miller, Jr.

The Lognormal Distribution E ( ST ) S0 e T 2 2 T var ( ST ) S0 e (e 2T 1) David Dubofsky and 18-12 Thomas W. Miller, Jr. BSOPM Properties and Questions What happens to the Black-Scholes call price when the call gets deep-deep-deep in the money? How about the corresponding put price in the case above? [Can you verify this using put-call parity?] Suppose gets very, very close to zero. What happens to the call price? What happens to the put price? Hint: Suppose the stock price will not change from time 0 to time T. How much are you willing to pay for an out of the money option? An in the money option?

David Dubofsky and 18-13 Thomas W. Miller, Jr. Volatility Volatility is the key to pricing options. Believing that an option is undervalued is tantamount to believing that the volatility of the rate of return on the stock will be less than what the market believes. The volatility is the standard deviation of the continuously compounded rate of return of the stock, per year. David Dubofsky and 18-14 Thomas W. Miller, Jr. Estimating Volatility from Historical Data 1. Take observations S0, S1, . . . , Sn at intervals of t (fractional years); e.g., = 1/52 if we are dealing with weeks; = 1/12 if we

are dealing with months. 2. Define the continuously compounded return as: S div Si ; on an ex-div day: ri ln i ri ln S i 1 Si 1 3. Calculate the average rate of return 4. Calculate the standard deviation, , of the ris 5. Annualize the computed (see next slide). David Dubofsky and 18-15 Thomas W. Miller, Jr. Annualizing 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, when valuing options, time is usually measured in trading days not calendar days.

The general convention is to use 252 trading days per year. What is most important is to be consistent. annual per trading day 252 annual weekly 52 annual monthly 12 Note that yr > mo > week > day David Dubofsky and 18-16 Thomas W. Miller, Jr. Implied Volatility The implied volatility of an option is the volatility for which the BSOPM value 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. Note that the volatility is assumed to be the same across strikes, but it often is not. In practice, there is a volatility smile, where implied volatility is often u-shaped when plotted as a function of the strike price. David Dubofsky and 18-17 Thomas W. Miller, Jr.

Using Observed Call (or Put) Option Prices to Estimate Implied Volatility Take the observed option price as given. Plug C, S, K, r, T into the BSOPM (or other model). Solving this is the tricky part) requires either an iterative technique, or using one of several approximations. The Iterative Way: Plug S, K, r, T, and into the BSOPM. Calculate (theoretical value) Compare c (theoretical) to C (the actual call price). If they are really close, stop. If not, change the value of and start over again.

David Dubofsky and 18-18 Thomas W. Miller, Jr. Volatility Smiles In this table, the option premium column is the average bid-ask price for June 2000 S&P 500 Index call and put options at the close of trading on May 16, 2000. The May 16, 2000 closing S&P 500 Index level was 1466.04, a riskless interest rate of 5.75%, an estimated dividend yield of 1.5%, and T = 0.08493 year. Strike 1225 1250 1275 1300 1325 1350 1375 1400 1425 1450 1475 1500 1525 1550 1575 1600

1625 1650 Call price 129.5 107.625 86.625 67.625 50.6875 35.625 22.5 14 7.875 4.375 2.28125 1.21875 0.625 Call IV 0.2834 0.2689 0.2534 0.2423 0.2324

0.2201 0.2036 0.1982 0.1912 0.1896 0.1881 0.1897 0.1912 call price with = 0.22 Put Price Put IV 1.5625 0.336 2.34375 0.3283 2.625 0.3021 3.8125 0.292 5.75 0.2855 124.335 8.25 0.2762

102.51 11.375 0.2641 82.353 14.875 0.2463 64.288 20.0625 0.2315 48.648 29 0.2285 35.612 38.625 0.2152 25.178 50.875 0.2016 17.173 66.75 0.1923 11.291 85.375 0.1826 7.154 106.875

0.1788 4.367 129.5 0.1668 2.569 153.375 0.1512 1.457 put price with = 0.22 0.054 0.153 0.39 0.904 1.919 3.753 6.809 11.533 18.35 27.592 39.437 53.885 70.762 89.761 110.505

132.6 155.684 David Dubofsky and 18-19 Thomas W. Miller, Jr. Graphing Implied Volatility versus Strike (S = 1466): Im p lie d V o la tility Volatility Smiles 0.4 0.3 Call IV 0.2 Put IV 0.1 0 1200 1400 1600

Strike Price 2 S S IV 1ln 2 ln X X Using the call data: = 2.17; 1 = -4.47; 2 = 2.52; R2 = 0.988 David Dubofsky and 18-20 Thomas W. Miller, Jr. Dividends European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes. * C =S 0N(d1 ) - Ke -rT N(d2 )

Where: S* = S PV(divs) between today and T. Also use S* when computing d1 and d2. Only dividends with ex-dividend dates during life of option should be included. The dividend should be the expected reduction in the stock price expected. For continuous dividends, use S* = Se-dT, where d is the annual constant dividend yield, and T is in years. David Dubofsky and 18-21 Thomas W. Miller, Jr. The BOPM and the BSOPM, I. Note the analogous structures of the BOPM and the BSOPM: C = S + B (0< <1; B < 0) C = SN(d1) Ke-rTN(d2) = N(d1) B = -Ke-rTN(d2)

David Dubofsky and 18-22 Thomas W. Miller, Jr. The BOPM and the BSOPM, II. The BOPM actually becomes the BSOPM as the number of periods approaches , and the length of each period approaches 0. In addition, there is a relationship between u and d, and , so that the stock will follow a Geometric Brownian Motion. If you carve T years into n periods, then: u e d e T/n 1 T/n 1 and, the probabilit y of an uptick, q, must equal q 0.5 0.5

T n David Dubofsky and 18-23 Thomas W. Miller, Jr. The BOPM and the BSOPM: An Example T = 7 months = 0.5833, and n = 7. Let = 14% and = 0.40 Then choosing the following u, d, and q will make the stock follow the right process for making the BOPM and BSOPM consistent: u e 0.40 d e 0.40 0.0833 0.833 1 0.1224 1 0.10905 and, the probabilit y of an uptick, q, must equal 0.14 q 0.5 0.5 0.0833 0.5505 0.40

David Dubofsky and 18-24 Thomas W. Miller, Jr. Generalizing the BSOPM It is important to understand that many option pricing models are related. For example, you will see that many exotic options (see chap. 20) use parts of the BSOPM. The BSOPM itself, however, can be generalized to encapsulate several important pricing models. All that is needed is to change things a bit by adding a new term, denoted by b. David Dubofsky and 18-25 Thomas W. Miller, Jr.

The Generalized Model Can Price European Options on: 1. Non-dividend paying stocks [Black and Scholes (1973)] 2. Options on stocks (or stock indices) that pay a continuous dividend [Merton (1973)] 3. Currency options [Garman and Kohlhagen (1983)] 4. Options on futures [Black (1976)] David Dubofsky and 18-26 Thomas W. Miller, Jr. The Generalized Formulae are: C gen =Se

b-r T N d1 - Ke-rTN d2 Pgen =Ke-rTN -d2 - Se d1 d2 b-r T N -d1 ln(S/K) (b 2 /2)T T ln(S/K) (b 2/2)T T d1 T David Dubofsky and 18-27 Thomas W. Miller, Jr.

By Altering the b term in These Equations, Four Option-Pricing Models Emerge. By Setting: Yields this European Option Pricing Model: b=r Stock option model, i.e., the BSOPM b=r-d Stock option model with continuous dividend yield, d (see section 18.5.2) b = r rf Currency option model where rf is the foreign risk-free rate b=0 Futures option model David Dubofsky and 18-28 Thomas W. Miller, Jr.

American Options It is computationally difficult to value an American option (a call on a dividend paying stock, or any put). Methods: Pseudo-American Model (Sect. 18.10.1) BOPM (Sects. 17.3.3, 17.4) Numerical methods Approximations Of course, computer programs are most often used. David Dubofsky and 18-29 Thomas W. Miller, Jr. Some Extra Slides on this Material Note: In some chapters, we try to include some extra slides in an effort to allow for a deeper (or different) treatment of the material in the chapter. If you have created some slides that you would like to share with the community of educators that use our book, please send

them to us! David Dubofsky and 18-30 Thomas W. Miller, Jr. The Stock Price Process, I. The percent change in the stock prices does not depend on the price of the stock. Over a small interval, the size of the change in the stock price is small (i.e., no jumps) Over a single period, there are only TWO possible outcomes. David Dubofsky and 18-31 Thomas W. Miller, Jr. The Stock Price Process, II. Consider a stock whose current price is S In a short period of time of length t, the change in the stock price is assumed to be normal with mean of S t and standard deviation,

is expected return and is volatility S t David Dubofsky and 18-32 Thomas W. Miller, Jr. That is, the Black-Scholes-Merton model assumes that the stock price, S, follows a Geometric Brownian motion through time: dS Sdt Sdz dS dt dz S 2 dt dz dlnS 2 David Dubofsky and 18-33 Thomas W. Miller, Jr.

The Discrete-Time Process S t z S S S S t t S t ln t t S St St t a unit of time z =a normally distributed random variable with a mean of zero, and a variance of t z =is distributed normally, with E(z) =0, and VAR(z)=t David Dubofsky and 18-34 Thomas W. Miller, Jr. Example: Suppose t is one day. A stock has an expected return of = 0.0005 per day. NB: (1.0005)365 1 = 0.20016, 20% The standard deviation of the stock's daily return distribution is 0.0261725

NB: This is a variance of 0.000685 Annualized Variance: (365)(0.000685) = 0.250025 Annualized STDEV: (0.250025)0.5 = 0.500025, or 50% (0.0261725)(365)0.5 = 0.500025, or 50% David Dubofsky and 18-35 Thomas W. Miller, Jr. Example, II. Then, the return generating process is such that each day, the return consists of: a non-stochastic component, 0.0005 or 0.05% a random component consisting of: The stock's daily standard deviation times the realization of z, z is drawn from a normal probability distribution with a mean of zero and a variance of one. Then, we can create the following table: David Dubofsky and 18-36 Thomas W. Miller, Jr. The First 10 Days of a Stochastic Process Creating Stock Returns:

DAY z 0 1 2.48007 2 0.87537 0.914623 3 0.80587 4 1.03927 5 0.10523 6 0.66993 7 0.21137 8 2.19733 9 0.82807 10 0.58783

NONSTOCHASTIC STOCHASTIC S(T)=S(T1)[1.0+R] TREND PRICE S(T)=S(T1)[1.00050] 1.000000 1.000500 1.001000 COMPONENT (0.0261725)z R = t + z t = 1 DAY 1.000000 0.935590 1.001501 1.002002 1.002503 1.003004 1.003505 1.004007

1.004509 1.005011 0.064910 0.022911 0.021092 0.027200 0.002754 0.017534 0.005532 0.057510 0.021673 0.015385 0.895789 0.871871 0.874709 0.890483 0.886002 0.937398 0.917551 0.932126 David Dubofsky and 18-37 Thomas W. Miller, Jr.

The Result (Multiplying by 100): A Graph of a Stock Price Following a Geometric Brownian Motion Process Price (times 100) 110.000 100.000 90.000 80.000 1 6 11 16 21 26 31 36 41 46 51 56 61 Day David Dubofsky and 18-38 Thomas W. Miller, Jr. Risk-Neutral Valuation The variable does not appear in the Black-Scholes equation. The equation is independent of all variables affected by risk preference. The solution to the differential equation is therefore the same in a riskfree world as it is in the real world. This leads to the principle of risk-neutral valuation. Applying Risk-Neutral Valuation:

1. Assume that the expected return from the stock price is the risk-free rate. 2. Calculate the expected payoff from the option. 3. Discount at the risk-free rate. David Dubofsky and 18-39 Thomas W. Miller, Jr. Two Simple and Accurate Approximations for Estimating Implied Volatility 1. Brenner-Subramanyam formula. Suppose: S Ke rT then, 2 C T S David Dubofsky and 18-40 Thomas W. Miller, Jr. 2. Corrado and Miller Quadratic formula. If S Ke rT If S Ke rT

S Ke rT C 2 8 : T 3S Ke rT S Ke rT C 2 8 : T 3Ke rT S David Dubofsky and 18-41 Thomas W. Miller, Jr.

Lets Have a Horse Race Data: S = 54, K = 55, C = 1.4375, 29 days to expiration (T=29/365), r = 3%. We can verify: The actual ISD is 30.06%. Brenner-Subramanyam estimated ISD is 23.67%. Corrado-Miller estimated ISD is 30.09%. David Dubofsky and 18-42 Thomas W. Miller, Jr.