central to a fullunderstanding of the theory underlying the pricing of derivatives.
11.1 THE MARKOV PROPERTY
 
 
A Markov process is a particular type of stochastic process where only the present value of a variableis relevant for predicting the future. The past history of the variable and the way that the present hasemerged from the past are irrelevant. Stock prices are usually assumed to follow a Markov process.Suppose that the price of IBM stock is $100 now. If the stock price follows a Markov process, ourpredictions for the future should be unaffected by the price one week ago, one month ago, or oneyear ago. The only relevant piece of information is that the price is now $100,' Predictions for the
' Statistical properties of the stock price history of IBM may be useful in determining the characteristics of thestochastic process followed by the stock price (e.g., its volatility). The point being made here is that the particularpath followed by the stock in the past is irrelevant.
216Model of the Behavior of Stock Prices 217
future are uncertain and must be expressed in terms of probability distributions. The Markovproperty implies that the probability distribution of the price at any particular future time is notdependent on the particular path followed by the price in the past.
The Markov property of stock prices is consistent with the weak form of market efficiency. Thisstates that the present price of a stock impounds all the information contained in a record of pastprices, If the weak form of market efficiency were not true, technical analysts could make above-average returns by interpreting charts of the past history of stock prices. There is very littleevidence that they are in fact able to do this.
Tt is competition in the marketplace that tends to ensure that weak-form market efficiency holds.There are many, many investors watching the stock market closely. Trying to make a profit from itleads to a situation where a stock price, at any given time, reflects the information in past prices.Suppose that it was discovered that a particular pattern in stock prices always gave a 65% chance ofsubsequent steep price rises. Investors would attempt to buy a stock as soon as the pattern wasobserved, and demand for the stock would immediately rise. This would lead to an immediate rise inits price and the observed effect would be eliminated, as would any profitable trading opportunities.
11.2. CONTINUOUS-TIME STOCHASTIC PROCESSES
 
 
Consider a variable that follows a Markov stochastic process. Suppose that its current value is 10and that the change in its value during one year is ¢(0, 1), where (4, o) denotes a probabilitydistribution that is normally distributed with mean yz and standard deviation o. What is theprobability distribution of the change in the value of the variable during two years?
The change in two years is the sum of two normal distributions, each of which has a mean ofzero and standard deviation of 1.0. Because the variable is Markov, the two probability distribu-tions are independent. When we add two independent normal distributions, the result is a normaldistribution in which the mean is the sum of the means and the variance is the sum of thevariances.” The mean of the change during two years in the variable we are considering is thereforezero, and the variance of this change is 2.0. The change in the variable over two years is therefore(0, V2).
Consider next the change in the variable during six months. The variance of the change in thevalue of the variable during one year equals the variance of the change during the first six monthsplus the variance of the change during the second six months. We assume these are the same. Itfollows that the variance of the change during a six month period must be 0.5. Equivalently, thestandard deviation of the change is /0.5, so that the probability distribution for the change in thevalue of the variable during six months is ¢(0, V0.5).
A similar argument shows thai the change in the value of the variable during three mounihs is(0, 0.25). More generally, the change during any time period of length T is ¢(0, VT). Inparticular, the change during a very short time period of length 81 is o(0, /8t).
The square root signs in these results may seem strange. They arise because, when Markovprocesses are considered, the variance of the changes in successive time periods are additive. Thestandard deviations of the changes in successive time periods are not additive. The variance of thechange in the variable in our example is 1.0 per year, so that the variance of the change in twoyears is 2.0 and the variance of the change in three years is 3.0. The standard deviation of the
? The variance of a probability distribution is the square of its standard deviation. The variance of a one-year changein the value of the variable we are considering is therefore 1.0.218 CHAPTER 11
change in two and three years is /2 and V3, respectively. Strictly speaking, we should not refer tothe standard deviation of the variable as 1.0 per year. It should be “1.0 per square root of years”.The results explain why uncertainty is often referred to as being proportional to the square root oflime.
Wiener Processes
The process followed by the variable we have been considering is known as a Wiener process. It isaparticular type of Markov stochastic process with a mean change of zero and a variance rate of1.0 per year. It has been used in physics to describe the motion of a particle that is subject to alarge number of small molecular shocks and is sometimes referred to as Brownian motion.Expressed formally, a variable z follows a Wiener process if it has the following two properties:
Property 1. The change 6z during a small period of time 6t iséz = eV8t did)where ¢ is a random drawing from a standardized normal distribution, (0, 1).Property 2. The values of 5z for any two different short intervals of time é¢ are independent.It follows from the first property that 5z itself has a normal distribution with
cae ofi
me of Se oaMcan o1 ol —vu
standard deviation of 8z = /6tvariance of $z = 8¢The second property implies that z follows a Markov process.Consider the increase in the value of z during a relatively long period of time, T. This can be
denoted by 2(T) — 2(0). It can be regarded as the sum of the increases in z in N small time intervalsof length dt, where
TNaeThus,N2(T) — 200) = )e,vot (11.2)i=1
where the ¢; (i = 1,2,..., N) are random drawings from ¢(0, 1). From second property of Wienerprocesses, the €,’s are independent of each other. It follows from equation (11.2) that 2(T) — 2(0) isnormally distributed with
mean of [z(T) — z(0)] = 0
variance of [z(T) ~ 2(0)] = Nét = Tstandard deviation of [2(T) ~ 2(0)] = /TThis is consistent with the discussion earlier in this section.
Example11.1 Suppose that the value, z, of a variable that follows a Wiener process is initially 25and that time is measured in years, At the end of one year, the value of the variable is normallyModel of the Behavior of Stock Prices 219
 
distributed with a mean of 25 and a standard deviation of 1.0. At the end of five years, it is normallydistributed with a mean of 25 and a standard deviation of /5, or 2.236. Our uncertainty about thevalue of the variable at a certain time in the future, as measured by its standard deviation, increasesas the square root of how far we are looking ahead.
In ordinary calculus it is usual to proceed from small changes to the limit as the small changesbecome closer to zero. Thus 5y/8x becomes dy/dx in the limit, and so on. We can proceed similarlywhen dealing with stochastic processes. A Wiener process is the limit as é¢ > 0 of the processdescribed above for z.
Figure 11.1 illustrates what happens to the path followed by z as the limit 6 > 0 is approached.Note that the path is quite ‘‘jagged’’. This is because the size of a movement in z in time 6ét isproportional to /ét and, when 6¢ is small, /St is much bigger than 6t. Two intriguing properties ofWiener processes, related to this /St property, are:
1. The expected length of the path followed by z in any time interval is infinite.2. The expected number of times z equals any particular value in any time interval is infinite.
Generalized Wiener Process
The basic Wiener process, dz, that has been developed so far has a drift rate of zero and a variancerate of 1.0. The drift rate of zero means that the expected value of z at any future time is equal to itscurrent value. The variance rate of 1.0 means that the variance of the change in z in a time intervalof iengih T equais T. A generaiized Wiener process for a variabie x can be defined in terms of dz asfollows:dx =adt+bdz (11.3)where a and b are constants.To understand equation (11.3), it is useful to consider the two components on the right-handside separately. The a dt term implies that x has an expected drift rate of a per unit of time. Withoutthe bdz term, the equation is
dx =adtwhich implies thatdx _a7Integrating with respect to time, we getX=Xy tat
where xq is the value of x at time zero. In a period of time of length T, the value of x increases byan amount aT. The bdz term on the right-hand side of equation (11.3) can be regarded as addingnoise or variability to the path followed by x. The amount of this noise or variability is b times aWiener process. A Wiener process has a standard deviation of 1.0. It follows that b times a Wienerprocess has a standard deviation of b. In a small time interval 8¢, the change &x in the value of x isgiven by equations (11.1) and (11.3) as
6x = adt + beV 5
where, as before, € is a random drawing from a standardized normal distribution. Thus 8x has a220 CHAPTER 11
Relatively large value of
me
Smaller value of 5¢
 
The true process obtained as §t > 0
Figure 11.1 How a Wiener process is obtained when St > 0 in equation (11.1)Model of the Behavior of Stock Prices 221
 
‘ Monee x GeneralizedWiener processdx=a dt + bdz
  
Wiener process, dz
  
 
 
Figure 11.2 Generalized Wiener process: a = 0.3, b= 1.5
normal distribution withmean of 8x = adt
standard deviation of dx = bV8t
variance of 5x = b’ st
Similar arguments to those given for a Wiener process show that the change in the value of x in anytime interval T is normally distributed with
mean of change in x = aTstandard deviation of change in x = b./T
variance of change in x = b°T
Thus, the generalized Wiener process given in equation (11.3) has an expected drift rate(ie., average drift per unit of time) of a and a variance rate (i.e., variance per unit of time)of 6°. It is illustrated in Figure 11.2.
Example 11.2 Consider the situation where the cash position of a company, measured inthousands of dollars, follows a generalized Wiener process with a drift of 20 per year and a variancerate of 900 per year. Initially, the cash position is 50. At the end of one year, the cash position willhave a normal distribution with a mean of 70 and a standard deviation of 900, or 30. At the end ofsix months, it will have a normal distribution with a mean of 60 and a standard deviation of300.5 = 21.21. Our uncertainty about the cash position at some time in the future, as measured byits standard deviation, increases as the square root of how far ahead we are looking. Note that thecash position can become negative (we can interpret this as a situation where the company isborrowing funds).222 CHAPTER 11
It6 Process
A further type of stochastic process can be defined. This is known as an /té process. This is ageneralized Wiener process in which the parameters a and 6 are functions of the value of theunderlying variable x and time t. Algebraically, an Ité process can be written
dx = a(x, t)dt + b(x, dz d1.4)
Both the expected drift rate and variance rate of an Ité process are liable to change over time. Inasmall time interval between t and t+ dt, the variable changes from x to x + 6x, where
dx = a(x, t) dt + b(x, Dev Et
This relationship involves a small approximation. It assumes that the drift and variance rate of xremain constant, equal to a(x, t) and A(x, #)’, respectively, during the time interval between tand t+ 61.
11.3. THE PROCESS FOR STOCK PRICES
 
In this section we discuss the stochastic process usually assumed for the price of a non-dividend-paying stock.
It is tempting to suggest that a stock price follows a generalized Wiener process, that is, that ithas a constant expected drift rate and a constant variance rate. However, this model fails to capturea key aspect of stock prices. This is that the expected percentage return required by investors fromastock is independent of the stock’s price. If investors require a 14% per annum expected returnwhen the stock price is $10, then, ceteris paribus, they will also require a 14% per annum expected
return when it is $450rewurin Wen it is oo,
Clearly, the constant expected drift-rate assumption is inappropriate and needs to be replaced bythe assumption that the expected return (i.e., expected drift divided by the stock price) is constant.If S is the stock price at time ¢, the expected drift rate in S should be assumed to be wS for someconstant parameter jz. This means that in a short interval of time, dt, the expected increase in S ispS ét. The parameter yz is the expected rate of return on the stock, expressed in decimal form.
If the volatility of the stock price is always zero, this model implies that
6S = wS dtIn the limit as 6¢ > 0,dS = wS dtordS—=pdS$ hb
Integrating between time zero and time T, we getSp = Spet™ (11.5)
where S) and S; are the stock price at time zero and time T. Equation (11.5) shows that, whenthe variance rate is zero, the stock price grows at a continuously compounded rate of y per unitof time.Model of the Behavior of Stock Prices 223
In practice, of course, a stock price does exhibit volatility. A reasonable assumption is that thevariability of the percentage return in a short period of time, 6t, is the same regardless of the stockprice. In other words, an investor is just as uncertain of the percentage return when the stock priceis $50 as when it is $10. This suggests that the standard deviation of the change in a short period oftime St should be proportional to the stock price and leads to the model
dS = uSdt+oSdzorGa uditod: (11.6)
Equation (11.6) is the most widely used model of stock price behavior. The variable o is thevolatility of the stock price. The variable yz is its expected rate of return.
Example 11.3 Consider a stock that pays no dividends, has a volatility of 30% per annum, andprovides an expected return of 15% per annum with continuous compounding. In this case, 2 = 0.15and o = 0.30. The process for the stock price is
e= 0.15 dt + 0.30 dz
If S is the stock price at a particular time and 8S is the increase in the stock price in the next smallinterval of time, then> = 0.158t + 0.30e/8twhere € is a random drawing from a standardized normal distribution. Consider a time interval ofone week, or 0.0192 years, and suppose that the initial stock price is $100. Then 6t = 0.0192, 5 = 100,and5S = 100(0.00288 + 0.041 6¢)or6S = 0.288 + 4.16
showing that the price increase is a random drawing from a normal distribution with mean $0.288and standard deviation $4.16.
11.4 REVIEW OF THE MODEL
 
The model of stock price behavior developed in the previous section is known as geometricBrownian motion. The discrete-time version of the model is
5S
 
tHE t oeJ/5t (11.7)or6S = wS bt + oSeV/8t (11.8)
The variable 5S is the change in the stock price S in a small time interval 6t, and « is a randomdrawing from a standardized normal distribution (i.e., a normal distribution with a mean of zeroand standard deviation of 1.0). The parameter jz is the expected rate of return per unit of time from224 CHAPTER 11
the stock, and the parameter o is the volatility of the stock price. Both of these parameters areassumed constant.
The left-hand side of equation (11.7) is the return provided by the stock in a short period oftime dt. The term dt is the expected value of this return, and the term oe/at is the stochasticcomponent of the return. The variance of the stochastic component (and therefore of the wholereturn) is o° dt. This is consistent with the definition of the volatility o given in Section 10.7; that is,o is such that o4/ét is the standard deviation of the return in a short time period ét.
Equation (11.7) shows that 5S/S is normally distributed with mean yz 5t and standard deviationost. In other words,
* ~ gust, air) (11.9)
Monte Carlo Simulation
A Monte Carlo simulation of a stochastic process is a procedure for sampling random outcomesfor the process. We will use it as a way of developing some understanding of the nature of the stockprice process in equation (11.6).
Suppose that the expected return from a stock is 14% per annum and that the standarddeviation of the return (i.e., the volatility) is 20% per annum. This means that p = 0.14 ando = 0.20. Suppose that dt = 0.01, so that we are considering changes in the stock price in timeintervals of length 0.01 years (or 3.65 days). From equation (11.8), we have
85 = 0.14 x 0.018 +0.2V0.01 Seor8S = 0.00145 + 0.02Se (11.10)
A path for the stock price can be simulated by sampling repeatedly for ¢ from (0,1) andsubstituting into equation (11.10). Table 11.1 shows one particular set of results from doing this.
Table 11.1 Simulation of stock price when p=0.14 ando = 0.20 during periods of length 0.01 year
Stock price at Random sample Change in stock pricestart of period fore during period20.000 0.52 0.23620.236 1.44 0.61120.847 —0.86 —0.32920.518 1.46 0.62821.146 —0.69 —0.26220.883 —0.74 —0.28020.603 0.21 0.11520.719 —  -1.10 —0.42720.292 0.73 0.32520.617 1.16 0.507
21.124 2.56 1.111Model of the Behavior of Stock Prices 225
The initial stock price is assumed to be $20. For the first period, € is sampled as 0.52. Fromequation (11.10), the change during the first time period is
5S = 0.0014 x 20 4+ 0.02 x 20 x 0.52 = 0,236
Therefore, at the beginning of the second time period, the stock price is $20.236. The value of €sampled for the next period is 1.44. From equation (11.10), the change during the second timeperiod is :
5S = 0.0014 x 20.236 + 0.02 x 20.236 x 1.44 = 0.611
Therefore, at the beginning of the next period, the stock price is $20.847. And so on. Note that,because the process we are simulating is Markov, the samples for € should be independent of eachother.
Table 11.1 assumes that stock prices are measured to the nearest 0.001. It is important to realizethat the table shows only one possible pattern of stock price movements. Different random sampleswould lead to different price movements. Any smail time interval $t can be used in the simulation.In the limit as 8t > 0, a perfect description of the stochastic process is obtained. The final stockprice of 21.124 in Table 11.1 can be regarded as a random sample from the distribution of stockprices at the end of 10 time intervals {ie., at the end of one-tenth of a year). By repeatedlysimulating movements in the stock price, as in Table 11.1, a complete probability distribution ofthe stock price at the end of this time is obtained.
11.5 THE PARAMETERS
 
The process for stock prices developed in this chapter involves two parameters: w and o. Theparameter yz is the expected continuously compounded return earned by an investor per year. Mostinvestors require higher expected returns to induce them to take higher risks. It follows that thevalue of x should depend on the risk of the return from the stock.* It should also depend on thelevel of interest rates in the economy. The higher the level of interest rates, the higher the expectedreturn required on any given stock.
Fortunately, we do not have to concern ourselves with the determinants of y in any detailbecause the value of a derivative dependent on a stock is, in general, independent of uw. Theparameter a, the stock price volatility, is, by contrast, critically important to the determination ofthe value of most derivatives. We will discuss procedures for estimating o in Chapter 12. Typicalvalues of o for a stock are in the range 0.20 to 0.50 (i.e., 20% to 50%).
The standard deviation of the proportional change in the stock price in a small interval oftime &f is o./St. As a rough approximation, the standard deviation of the proportional changein the stock price over a relatively long period of time T is o./T. This means that, as anapproximation, volatility can be interpreted as the standard deviation of the change in the stockprice in one year. In Chapter 12 we will show that the volatility of a stock price is exactly equalto the standard deviation of the continuously compounded return provided by the stock inone year.
* More precisely, jz depends on that part of the risk that cannot be diversified away by the investor.226 CHAPTER 11
11.6 ITO’S LEMMA
 
The price of a stock option is a function of the underlying stock’s price and time. More generally,we can say that the price of any derivative is a function of the stochastic variables underlying thederivative and time. A serious student of derivatives must, therefore, acquire some understandingof the behavior of functions of stochastic variables. An important result in this area was discoveredby the mathematician Kiyosi It6 in 1951.* It is known as /td’s lemma.
Suppose that the value of a variable x follows the It6 process
dx = a(x, dt + W(x, dz (11.1)
where dz is a Wiener process and a and b are functions of x and t. The variable x has a drift rateof a and a variance rate of ”. Ité’s lemma shows that a function G of x and t follows the process
aa= (ar 4 oor) a+ ae (11.12)
where the dz is the same Wiener process as in equation (11.11). Thus, G also follows an Ité process.It has a drift rate ofaG acs,ae tart?
aGY >(=)A completely rigorous proof of It6’s lemma is beyond the scope of this book. In Appendix I1A, we
show that the lemma can be viewed as an extension of well-known results in differential calculus.Earlier, we argued that
vG >a
and a variance rate of
dS = pSdt +oSdz (11.13)
with uw and o constant, is a reasonable model of stock price movements. From It6é’s lemma, itfollows that the process followed by a function G of S and t is
aG aG 1G ats? aG= S* ) dt +— 11.14dG (Sous +5 p+ 2 5gFe ) t+ ayosdz ( )
Note that both S and G are affected by the same underlying source of uncertainty, dz. This provesto be very important in the derivation of the Black-Scholes results.
Application to Forward Contracts
To illustrate It6’s lemma, consider a forward contract on a non-dividend-paying stock. Assumethat the risk-free rate of interest is constant and equal to r for all maturities. From equation (3.5),
Fy = Spe’™
4 See K. 1t6, “On Stochastic Differential Equations,” Memoirs, American Mathematical Society, 4 (1951), 1-51.Model of the Behavior of Stock Prices 227
where Fy is the forward price at time zero, Symaturity of the forward contract.
We are interested in what happens to the forward price as time passes. We define F and S as theforward price and the stock price, respectively, at a general time t, with t < T. The relationship
between F and S is
3oof7)
Ss
F = SéT~ (11.15)
Assuming that the process for S is given by equation (11.13), we can use Ité’s lemma to determinethe process for F.From equation (11.15),
aF ory OF _ aF T-1)ag 7° : gsr = WT rSe
From equation (11.14), the process for F is given bydF = [ews — rSe 9) dt + eT GS deSubstituting F = Se?—, we obtaindF =(u—r)Fdt+oaFdz (11.16)
Like the stock price S, the forward price F follows geometric Brownian motion. It has anexpected growth rate of 4 —r rather than yw. The growth rate in F is the excess return of S$ overthe risk-free rate.
11.7. THE LOGNORMAL PROPERTY
 
We now use Ité’s lemma to derive the process followed by In S when S follows the process inequation (11.13). Define
G=InSBecauseaG_i rat aG_as SSeS ot
it follows from equation (11.14) that the process followed by G iseaa =( -$) a+ ods (11.17)
Because w and o are constant, this equation indicates that G = In S follows a generalized Wienerprocess. It has constant drift rate — 7/2 and constant variance rate a”. The change in In Sbetween time zero and some future time, T, is therefore normally distributed with mean
th
oT
and variance228 CHAPTER 11
This means thatins, —In)~ af (uF), VT (11.18)
or
2inp ~ afin) + («-5)r, oT | (11.19)
where Sy is the stock price at a future time T, Sp is the stock price at time zero, and ¢(m, s) denotesa normal distribution with mean m and standard deviation s.
Equation (11.19) shows that In S; is normally distributed. A variable has a lognormal distribu-tion if the natural logarithm of the variable is normally distributed. The model of stock pricebehavior we have developed in this chapter therefore implies that a stock’s price at time T, given itsprice today, is lognormally distributed. The standard deviation of the logarithm of the stock priceis oVT. It is proportional to the square root of how far ahead we are looking.
SUMMARY
 
Stochastic processes describe the probabilistic evolution of the value of a variable through time. AMarkov process is one where only the present value of the variable is relevant for predicting thefuture. The past history of the variable and the way in which the present has emerged from the pastare irrelevant.
A Wiener process, dz, is a process describing the evolution of a normally distributed variable.The drift of the process is zero and the variance rate is 1.0 per unit time. This means that, if thevalue of the variable is xp at time zero, then at time T it is normally distributed with mean x andstandard deviation JT. .
A generalized Wiener process describes the evolution of a normally distributed variable with adrift of a per unit time and a variance rate of LY per unit time, where a and b are constants. Thismeans that if, as before, the value of the variable is xg at time zero then it is normally distributedwith a mean of x) + aT and a standard deviation of b/T at time T.
An Ité process is a process where the drift and variance rate of x can be a function of both xitself and time. The change in x in a very short period of time is, to a good approximation,normally distributed, but its change over longer periods of time is liable to be nonnormal.
One way of gaining an intuitive understanding of a stochastic process for a variable is to simulatethe behavior of the variable. This involves dividing a time interval into many small time steps andrandomly sampling possible paths for the variable. The future probability distribution for thevariable can then be calculated. Monte Carlo simulation is discussed further in Chapter 18.
It6’s lemma is a way of calculating the stochastic process followed by a function of a variablefrom the stochastic process followed by the variable itself. As we will see in Chapter 12, Ité’slemma is very important in the pricing of derivatives. A key point is that the Wiener process, dz,underlying the stochastic process for the variable is exactly the same as the Wiener processunderlying the stochastic process for the function of the variable. Both are subject to the sameunderlying source of uncertainty.
The stochastic process usually assumed for a stock price is geometric Brownian motion. Underthis process the return to the holder of the stock in a small period of time is normally distributedand the returns in two nonoverlapping periods are independent. The value of the stock price at aModel of the Behavior of Stock Prices 229
future time has a lognormal distribution. The Black-Scholes model, which we cover in the nextchapter, is based on the geometric Brownian motion assumption.
SUGGESTIONS FOR FURTHER READING
 
 
On Efficient Markets and the Markav Property of Stack Prices
Brealey, R. A., An Introduction to Risk and Return from Common Stock, 2nd edn., MIT Press,Cambridge, MA, 1983.
Cootner, P. H. (ed.), The Random Character of Stock Market Prices, MIT Press, Cambridge, MA, 1964.
On Stochastic ProcessesCox, D. R., and H. D. Miller, The Theory of Stochastic Processes, Chapman & Hall, London, 1965.Feller, W., Probability Theory and Its Applications, vols. | and 2. Wiley, New York, 1950.
Karlin, S., and H. M. Taylor, A First Course in Stochastic Processes, Ind edn., Academic Press, NewYork, 1975.
Neftci, S., Introduction to Mathematics of Financial Derivatives, Academic Press, New York, 1996.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
 
 
11.1. What would it mean to assert that the temperature at a certain place follows a Markov process?Do you think that