h 45 2F 4 PL g292 ofaf = (Ffus+ at art 29st? S° } dt +—oSdz (12.9)The discrete versions of equations (12.8) and (12.9) are88 — pS8t + oSsz (12.10)andf BF oe) 5 ofSf= (i ps4 2 1 Lisa S* ) it +a50S5z (12.11)
_ 5 We discuss the rebalancing of portfolios in more detail in Chapter 14.The Black-Scholes Model 243
where 5S and 8f are the changes in f and S in a small time interval 5t. Recall from the discussionof It6’s lemma in Section 11.6 that the Wiener processes underlying f and S are the same. In otherwords, the 5z (= € /8t) in equations (12.10) and (12.11) are the same. It follows that, by choosing aportfolio of the stock and the derivative, the Wiener process can be eliminated.
The appropriate portfolio is as follows:
—1: derivative
0+ of : sharesThe holder of this portfolio is short one derivative and long an amount 0 f/@S of shares. Define I
as the value of the portfolio. By definition,of=—-ft+x5 12.12f+558 (12.12)
The change 61] in the value of the portfolio in the time interval dt is given by
sm n=-sf +54 58 (12.13)
Substituting equations (12.10) and (12.11) into equation (12.13) yields
2m= (-4-1e *) a (12.14)
Because this equation does not involve 6z, the portfolio must be riskless during time ét. Theassumptions listed in the preceding section imply that the portfolio must instantaneously earn thesame 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 followsthat
éi = rl dt
where r is the risk-free interest rate. Substituting from equations (12.12) and (12.14), we obtain
(fess) anes)
of of 120ay TTS ag FATS gar rl (12.15)
so that
Equation (12.15) is the Black-Scholes~Merton differential equation. It has many solutions, corres-ponding to all the different derivatives that can be defined with S as the underlying variable. Theparticular derivative that is obtained when the equation is solved depends on the boundaryeauditinas that ara nead Thace anacify the vahice of the desivative st the houndarics of aocsible
coneiions iat ate Usa. ~NneSse Speciry tie values OF LIC OCrivative at tac COUMGATICS OL POSSiOIc
values of S and ¢. In the case of a European call option, the key boundary condition is
f =max(S—K,0) whent=T244 CHAPTER 12
In the case of a European put option, it isf =max(K —S,0) whent=T
One point that should be emphasized about the portfolio used in the derivation of equation (12.15)is that it is not permanently riskless. It is riskless only for an infinitesimally short period of time.As S and t change, 0f/9S also changes. To keep the portfolio riskless, it is therefore necessary tofrequently change the relative proportions of the derivative and the stock in the portfolio.
Example12.6 A forward contract on a non-dividend-paying stock is a derivative dependent on thestock. As such, it should satisfy equation (12.15). From equation (3.9), we know that the value of theforward contract, f, at a general time tis given in terms of the stack price S at this time by
f=S-Ke™
where K is the delivery price. This means that
2of =-rkert, Fy IF 4
This equals rf, showing that equation (12.15) is indeed satisfied.
The Prices of Tradeable Derivatives ,
Any function f(S, t) that is a solution of the differential equation (12.15) is the theoretical price ofa derivative that could be traded. If a derivative with that price existed, it would not create anyarbitrage opportunities. Conversely, if a function f(S,t) does not satisfy the differential equa-tion (12.15), it cannot be the price of a derivative without creating arbitrage opportunities fortraders.
To illustrate this point, consider first the function e*. This does not satisfy the differentialequation (12.15). It is, therefore, not a candidate for being the price of a derivative dependent onthe stock price. If an instrument whose price was always e° existed, there would be an arbitrageopportunity. As a second example, consider the function
(PT)Ss
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/S; at time 7.) For other examples of tradeablederivatives, see Problems 12.11, 12.12, 12.23, and 12.26.
12.7 RISK-NEUTRAL VALUATION
 
 
 
We introduced risk-neutral valuation in connection with the binomial model in Chapter 10. It iswithout doubt the single most important tool for the analysis of derivatives. It arises from one keyproperty of the Black—-Scholes—Merton differential equation (12.15). This property is that theThe Black-Scholes Model 245
equation does not involve any variable that is affected by the risk preferences of investors. Thevariables that do appear in the equation are the current stock price, time, stock price volatility, andthe risk-free rate of interest. All are independent of risk preferences.
The Black-Scholes—Merton differential equation would not be independent of risk preferences ifit involved the expected return on the stock, 2. This is because the value of 4. does depend on riskpreferences. The higher the level of risk aversion by investors, the higher yu will be for any givenstock. It is fortunate that 4. happens to drop out in the derivation of the differential equation.
Because the Black—Scholes—Merton differential equation is independent of risk preferences, aningenious argument can be used. If risk preferences do not enter the equation, they cannot affectits 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 securities is the risk-freerate of interest, r. The reason is that risk-neutral investors do not require a premium to induce themto take risks. It is also true that the present value of any cash flow in a risk-neutral world can beobtained by discounting its expected value at the risk-free rate. The assumption that the world isrisk neutral, therefore, considerably simplifies 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(ie., assume uw =r).
2. Calculate the expects’ payoff from the ¢ option t its maturity.
"3
5ri.t
It is important to appreciate that risk-neutral valuation (or the assumption that all investors are riskneutral) is merely an artificial device for obtaining solutions Lo the Black-Scholes differentialequation. The solutions that are obtained are valid in all worlds, not just those where investors aretisk neutral. When we move from a risk-neutral world to a risk-averse world, two things happen. Theexpected growth rate in the stock price changes and the discount rate that must be used for anypayoffs from the derivative changes. It happens that these two changes always offset each otherexactly.
Application to Forward Contracts on a Stock
We valued forward contracts on a non-dividend-paying stock in Section 3.5. In Example 12.6 weverified that the pricing formula satisfies the Black-Scholes differential equation. In this section wederive the pricing formula from risk-neutral valuation. We make the assumption that interest ratesare constant and equal to r. This is somewhat more restrictive than the assumption in Chapter 3.
Consider a long forward contract that matures at time T with delivery price K. As explained inChapter |, the value of the contract at maturity is
Sp-K
where Sy is the stock price at time 7. From the risk-neutral valuation argument, the value of theforward contract at time zero is its expected value at time T in a risk-neutral world discounted atthe risk-free rate of interest. If we denote the value of the forward contract at time zero by f, thismeans that
f =e E(S; — K) (12.16)246 CHAPTER 12
where £ denotes the expected value in a risk-neutral world. Because K is a constant, equation (12.16)becomesfae E(Sp)— Keo" (12.17)
The expected growth rate of the stock price, 4, becomes r in a risk-neutral world. Hence, fromequation (12.4),E(Sy) = Sye™™ (12.18)
Substituting equation (12.18) into equation (12.17) givesf=Sy-Ke™ (12.19)
This is in agreement with equation (3.9).
12.8 BLACK-SCHOLES PRICING FORMULAS
 
 
The Black-Scholes formulas for the prices at time zero of a European call option on a non-dividend-paying stock and a European put option on a non-dividend paying stock are
c= SyN(d)) — Ke"? N(dy) (12.20)andp = Ke" N(—d,) — SyN(—d)) (12.21)whereg, <n SolK) + +07 /2)T1= oJT24 = Biro o [OT _ 4 _ oT
The function N(x) is the cumulative probability distribution function for a standardized normaldistribution. In other words, it is the probability that a variable with a standard normaldistribution, @(0, !) will be less than x. It is illustrated in Figure 12.3. The remaining variablesshould be familiar. The variables ¢ and p are the European call and European put price, Sp is thestock price at time zero, K is the strike price, r is the continuously compounded risk-free rate, o isthe stock price volatility, and 7 is the time to maturity of the option.
 
 
x
Figure 12.3 Shaded area represents N(x)The Black-Scholes Mode! 247
One way of deriving the Black-Scholes formulas is by solving the differential equation (12.15)subject to the boundary conditions mentioned in Section 12.6.° Another approach is to use risk-neutral valuation. Consider a European call option. The expected value of the option at maturityin a risk-neutral world is
Elmax(S; — K, 0)
where, as before, E denotes the expected value in a risk-neutral world. From the risk-neutralvaluation argument, the European call option price, c, is this expected value discounted at the risk-free rate of interest, that is,
=e” Elmax(S; — K, 0)] (12.22)
Appendix 12A shows that this equation leads to the result in (12.20).To provide an interpretation of the terms in equation (12.20), we note that it can be written
cae [S)N(d)e™ — KN(d)| (12.23)
The expression M(d;) is the probability that the option will be exercised in a risk-neutral world, sothat KN(d>) is the strike price times the probability that the strike price will be paid. The expressionSpN(d,)e™ is the expected value of a variable that equals S; if Sr > K and is zero otherwise in arisk-neutral world.
Because the European price equals the American price when there are no dividends, equa-tion (12.20) also gives the value of an American call option on a non-dividend-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 and analytic approximations forcalculating American put values are discussed in Chapter 18.
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 theoreticallycorrect when ¢ is a Known function of time. It is also theoretically correct when the interest rate isstochastic providing the stock price at time T is lognormal and the volatility parameter is chosenappropriately.
Properties of the Black-Scholes Formulas
We now show that the Black-Scholes formulas have the right general properties by consideringwhat happens when some of the parameters take extreme values.
When the stock price, Sp, becomes very large, a call option is almost certain to be exercised. Itthen becomes very similar to a forward contract with delivery price K. From equation (3.9), weexpect the call price to be
So — Ke'™
This is, in fact, the call price given by equation (12.20) because, when So becomes very large, both d,and d, become very large, and M(d;) and N(d,) are both close to 1.0. When the stock price becomes
6 The differential equation gives the call and put prices at a general time t. For example, the call price that satisfiesthe differential equation is c = SN(d)) — Ke~"?~ N(dy), whereIn(S/K) + (+ 07/2)(T — 1)~ evT—1and d, = d, — o/T — 1. See Problem 12.17 to prove that the differential equation is satisfied.
a248 CHAPTER 12
very large, the price of a European put option, p, approaches zero. This is consistent with equation(12.21) because N(—d,) and N(—d>) are both close to zero.
Consider next what happens when the volatility ¢ approaches zero. Because the stock is virtuallyriskless, its price will grow at rate r to Sye”’ at time T and the payoff from a call option is
max(Sye”’ — K, 0)If we discount at rate r, then the value of the call today ise-’ max(Sge"” — K, 0) = max(Sy — Ke’, 0)To show that this is consistent with equation (12.20), consider first the case where Sy > Ke~’’. Thisimplies that In(Sy/K)+rT > 0. As o tends to zero, d; and d tend to +00, so that N(d;) and N(d)tend to t.0 and equation (12.20) becomesc=Sy—Ke™
When Sp < Ke~’’, it follows that In(S)/K)+ rT < 0. Aso tends to zero, d, and d, tend to —o0, sothat N(d)) and N(d3) tend to zero and equation (12.20) gives a call price of zero. The call price is
therefore always max(Sy — Ke’, 0) as o tends to zero. Similarly, it can be shown that the putprice is always max(Ke™”” — Sp, 0) as o tends to zero.
12.9 CUMULATIVE NORMAL DISTRIBUTION FUNCTION
 
The only problem in implementing equations (12.20) and (12.21) is in calculating the cumulativenormal distribution function, N. Tables for N(x) are provided at the end of this book. Apolynomial approximation that gives six-decimal-place accuracy is’
NG) = L—N(xayk + ak? + ask + agkt +.a5h) if x 201 — N(—x) ifx <0
where1l+yxa, = 0.319381530, a, = —0.356563782, a; = 1.781477937,
ag = —1.821255978, as = 1.330274429
 
k= y =0.2316419
’
] 2N(x) =——e™ {2
fd
Example12.7 The stock price six months from the expiration of an option is $42, the exercise priceof the option is $40, the risk-free interest rate is 10% per annum, and the volatility is 20% per
7 See M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1972.The function NORMSDIST in Excel can also be used.The Black-Scholes Model 249
annum. This means that Sy = 42, K = 40, r=0.1, 0= 0.2, T=0.5,
_ In(42/40) + 0.1 + 0.27/2).x 0.5
d = 0.7693} 0.2/0.5In(42/40) + (0.1 — 0.27/2) x 0.52 02/05 6278
andKew = 40¢% = 38.049
Hence, if the option is a European call, its value, c, is given byc¢ = 42.N(0.7693) — 38.049. N(0.6278)If the option is a European put, its value, p, is given byp = 38.049 N(—0.6278) — 42(—0.7693)Using the polynomial approximation, we haveN(0.7693) = 0.7791, N(—0.7693) = 0.2209, N(0.6278) = 0.7349, N(—0.6278) = 0.2651
so thatc=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 tobreak even. Similarly, the stock price has to fall by $2.81 for the purchaser of the put to break even.
12.10 WARRANTS ISSUED BY A COMPANY ON ITS OWN STOCK
 
The Black-Scholes formula, with some adjustments for the impact of dilution, can be used to valueEuropean warrants issued by a company on its own stock.* Consider a company with N out-standing shares and M outstanding European warrants. Suppose that each warrant entitles theholder to purchase y shares from the company at time T at a price of K per share.
If V; is the value of the company’s equity (including the warrants) at time 7 and the warrantholders exercise, the company receives a cash inflow from the payment of the exercise price of MyKand the value of the company’s equity increases to Vr + MyK. This value is distributed amongN+ My shares, so that the share price immediately after exercise becomes
8 See F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy,81 (May/June 1973), 637-59; D. Galai and M. Schneller, “Pricing Warrants and the Value of the Firm,” Journal ofFinance, 33 (1978), 1339-42; B. Lauterbach and P. Schultz, ‘‘Pricing Warrants: An Empirical Study of the Black—Scholes Model and Its Alternatives,” Journal of Finance, 45 (1990), 1181-1209.250 “CHAPTER 12
orN_v_ (Yr _rxN+My\N
The warrants should be exercised only if this payoff is positive. The payoff to the warrant holder istherefore
N ‘V; ‘
—Y max at K, 0N+My N
This shows that the value of the warrant is the value of
NyN+My
 
regular call options on V/N, where V is the value of the company’s equity.The value of V at time zero is given by
Vo = NSy + MWwhere Sp is the stock price at time zero and W is the warrant price at that time, so that
Vo MNa Sota Ww
The Black-Scholes formula in equation (12.20) therefore gives the warrant price W if:
1. The stock price Sp is replaced by So + (M4/N)W.2. The volatility o is the volatility of the equity of the company (i.-e., it is the volatility of thevalue of the shares plus the warrants, not just the shares).
3. The formula is multiplied by Ny/(N + My).
When these adjustments are made, we end up with a formula for W as a function of W. This can besolved numerically.
12.11 IMPLIED VOLATILITIES
 
The one parameter in the Black-Scholes pricing formulas that cannot be directly observed is thevolatility of the stock price. In Section 12.3, we discussed how this can be estimated from a historyof 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 call option onanon-dividend-paying stock is 1.875 when Sy = 21, K = 20, r=0.1, and 7 =0.25. The impliedvolatility is the value of o that, when substituted into equation (12.20), gives c = 1.875. Unfortu-nately, it is not possible to invert equation (12.20) so that o is expressed as a function of So, K, r, T,and c. However, an iterative search procedure can be used to find the implied o. For example, wecan start by trying o = 0.20. This gives a value of c equal to 1.76, which is too low. Because c is anincreasing function of o, a higher value of o is required. We can next try a value of 0.30 for o. Thisgives a value of c equal to 2.10, which is too high and means that o must lie between 0.20 and 0,30.The Black-Scholes Model 251
 
Next, a value of 0.25 can be tried for o. This also proves to be too high, showing that o lies between0.20 and 0.25. In this way, the range for o can be halved at each iteration and the correct value of ocan be calculated to any required accuracy.’ In this example, the implied volatility is 0.235,or 23.5% per annum,
Implied volatilities are used to monitor the market’s opinion about the volatility of a particularstack. Traders like to calculate implied volatilities from actively traded options on a certain assetand interpolate between them to calculate the appropriate volatility for pricing a less activelytraded option on the same stock. We explain this procedure in Chapter 15. It is important tonote that the prices of deep-in-the-money and deep-out-of-the-money options are relativelyinsensitive to volatility. Implied volatilities calculated from these options tend, therefore, to beunreliable.
12.12 THE CAUSES OF VOLATILITY
 
 
Some analysts have claimed that the volatility of a stock price is caused solely by the randomarrival of new information about the future returns from the stock. Others have claimed thatvolatility is caused largely by trading. An interesting question, therefore, is whether the volatility ofan exchange-traded instrument is the same when the exchange is open as when it is closed.
Both Fama and K. R. French have tested this question empirically.'° They collected data on thestock price at the close of each trading day over a long period of time, and then calculated:
1, The variance of stock price returns between the close of trading on one day and the close oftrading on the next trading day when there are no intervening nontrading days
2. The variance of the stock price returns between the.close of trading on Fridays and the closeof trading on Mondays
If trading and nontrading days are equivalent, the variance in the second case should be threetimes as great as the variance in the first case. Fama found that it was only 22% higher. French’sresults were similar: he found that it was 19% higher.
These results suggest that volatility is far larger when the exchange is open than when it isclosed. Proponents of the view that volatility is caused only by new information might be temptedto argue that most new information on stocks arrives during trading days.!! However, studies of- futures prices on agricultural commodities, which depend largely on the weather, have shown thatthey exhibit much the same behavior as stock prices; that is, they are much more volatile duringtrading hours. Presumably, news about the weather is equally likely to arise on any day. The onlyreasonable conclusion seems to be that volatility is largely caused by trading itself.'*
> This method is presented for illustration. Other more powerful methods, such as the Newton-Raphson method,are often used in practice (see footnote 2 of Chapter 5). DerivaGem can be used to calculate implied volatilities.
'° See E. E. Fama, “The Behavior of Stock Market Prices,” Journal of Business, 38 (January 1965), 34-105; K. R.French, “Stock Returns and the Weekend Effect,” Journal of Financial Economics, 8 (March 1980), 55-69.1! In fact, this is questionable. Frequently, important announcements (e.g., those concerned with sales and earnings)are made when exchanges are closed.
, 2 For a discussion of this, see K. R. French and R. Roll, “Stock Return Variances: The Arrival of Information and
© the Reaction of Traders.” Journal of Financial Economics, 17 (September 1986), 5-26. We consider one way in whichtrading can generate volatility when we discuss portfolio insurance schemes in Chapter 14.252 CHAPTER 12
What are the implications of all of this for the measurement of volatility and the Black-Scholesmodel? When implied volatilities are calculated, the life of an option should be measured in tradingdays. Furthermore, if daily data are used to provide a historical volatility estimate, days when theexchange is closed should be ignored and the volatility per annum should be calculated from thevolatility per trading day using the formula
volatility per annum = volatility per irading day x ./number of trading days per annum
This is what we did in Example 12.5. The normal assumption in equity markets is that there are252 trading days per year.
Although volatility appears to be a phenomenon that is related largely to trading days, interest ispaid by the calendar day. This has led D. W. French to suggest that, when options are beingvalued, two time measures should be calculated:!
trading days until maturitytrading days per year
calendar days until maturity
ce calendar days per year
and that the Black-Scholes formulas should be adjusted to
¢ = SyN(d,) — Ke N(dp)
andp = Ke" N(—dy) — SgN(—d})wherea, = Sol K) +2 +0°%/21= ofIn (Sp/K) tr, — ot /2b= SONATE =d,-oft
In practice, this adjustment makes little difference except for very short life options.
12.13 DIVIDENDS
 
 
Up to now, we have assumed that the stock upon which the option is written pays no dividends. Inthis section, we modify the Black-Scholes model to take account of dividends. We assume that theamount and timing of the dividends during the life of an option can be predicted with certainty.For short-life options, it is usually possible to estimate dividends during the life of the optionreasonably accurately. For options lasting several years, there is likely to be uncertainty aboutdividend growth rates making option pricing much more difficult.
A dividend-paying stock can reasonably be expected to follow the stochastic process developedin Chapter 11 except when the stock goes ex-dividend. At this point, the stock’s price goes down by
3 See D. W. French, “The Weekend Effect on the Distribution of Stock Prices: Implications for Option Pricing,”Journal of Financial Economics, 13 (September 1984), 547 59.The Black-Scholes Model 253
an amount reflecting the dividend paid per share. For tax reasons, the stock price may go down bysomewhat less than the cash amount of the dividend. To take account of this, the word dividend inthis section should be interpreted as the reduction in the stock price on the ex-dividend date causedby the dividend. Thus, if a dividend of $1 per share is anticipated and the share price normallygoes down by 80% of the dividend on the ex-dividend date, the dividend should be assumed to be$0.80 for the purposes of the analysis.
European Options
European options can be analyzed by assuming that the stock price is the sum of two components:ariskless component that corresponds to the known dividends during the life of the option and arisky component. The riskless component, at any given time, is the present value of all thedividends during the life of the option discounted from the ex-dividend dates to the present atthe risk-free rate. By the time the option matures, the dividends will have been paid and the risklesscomponent will no longer exist. The Black-Scholes formula is therefore correct if Sg is equal to therisky component of the stock price and a is the volatility of the process followed by the riskycomponent.'* Operationally, this means that the Black-Scholes formula can be used provided thatthe stock price is reduced by the present value of all the dividends during the life of the option, thediscounting being done from the ex-dividend dates at the risk-free rate. A dividend is counted asbeing during the life of the option only if its ex-dividend date occurs during the life of the option.
Example12.8 Consider a European call option on a stock when there are ex-dividend dates in twomonths and five months. The dividend on each ex-dividend date is expected to be $0.50. The currentshare price is $40, the exercise price is $40, the stock price volatility is 30% per annum, the risk-freerate of interest is 9% per annum, and the time to maturity is six months. The present value of thedividends is
().5¢0-16670.09 1 4 54-0-4167%0.09 _ 9 g74]
The option price can therefore be calculated from the Black-Scholes formula with Sp = 39.0259,K=40,r=0.09, o = 0.3, and 7 = 0.5. We have
1n(39.0259/40) + (0.09 + 0.3/2) x 0.5
 
 
 
 
 
d 0.2017' 0.3/0.5In(39.0259/40) + (0.09 — 0.37/2) x 0.5d= ~0.0