The Lognormal Distribution PDF

The Lognormal Distribution McDonald, Robert L. Derivatives Markets. Available from: VitalSource Bookshelf, (3rd Edition). Pearson Education (US), 2012. Printed by: [email protected]. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 18 The Lognormal Distribution W e have seen that it is common in option pricing to assume the lognormality of asset prices. The purpose of this chapter is to explain the meaning of this assumption. We ﬁrst review the normal distribution, which gives rise to the lognormal distribution. We then deﬁne lognormality and illustrate some common calculations based on lognormality. These calculations result in terms that look much like the parts of the Black-Scholes formula. Finally, we examine stock returns to see whether stock price data seem consistent with lognormality. We will ﬁnd that stock prices are not exactly lognormal. Nevertheless, the lognormal assumption is the basis for many frequently used pricing formulas. Moreover, it is difﬁcult to understand more realistic models used in practice without ﬁrst understanding models based on the lognormal distribution. 18.1 THE NORMAL DISTRIBUTION A random variable, x̃, obeys the normal distribution—or is normally distributed—if the probability that x̃ takes on a particular value is described by the normal density function, which we represent by φ. The formula for the normal density function is1 x−μ 2 1 −1 φ(x; μ, σ ) ≡ √ e 2 σ (18.1) σ 2π Notice in equation (18.1) that in order to calculate a value for φ, in addition to x, you need to supply two numbers: a mean, μ, and a standard deviation, σ. For this reason, the normal distribution is said to be a two-parameter distribution; it is completely described by the mean and the standard deviation. Figure 18.1 graphs equation (18.1) for two different standard deviations (1 and 1.5), and for the same mean (0). The normal density with μ = 0 and σ = 1 is called the standard normal density. When working with the standard normal density, we will write φ(x), without a mean and standard deviation. 1. You can calculate the normal density in Excel using NormDist(x , μ, σ ,False). 545 546 Chapter 18. The Lognormal Distribution FIGURE 18.1 0.40 Two normal densities with mean 0, one with σ = 1 (the 0.35 standard normal), the other with σ = 1.5. 0.30 φ(x; 0, 1) 0.25 0.20 0.15 φ(x; 0, 1.5) 0.10 0.05 0 –6 –4 –2 0 2 4 6 Compared to the standard normal density, the normal density with σ = 1.5 assigns lower probabilities to values of x close to 0, and greater probabilities for x farther from 0. Increasing the variance spreads out the distribution. The mean locates the center of the distribution, and the standard deviation tells you how spread out it is. The normal density is symmetric about the mean, μ, meaning that φ(μ + a; μ, σ ) = φ(μ − a; μ, σ ) If a random variable x is normally distributed with mean μ and standard deviation σ , we write this as x ∼ N (μ, σ 2) We will use z to represent a random variable that has the standard normal distribution: z ∼ N (0, 1) We can use the normal distribution to compute the probability of different events, but we have to be careful about what we mean by an event. Since the distribution is continuous, there are an inﬁnite number of events that can occur when we randomly draw a number from the distribution. (This is unlike the binomial distribution, in which an event can have only one of two values.) The probability of any particular number being drawn from the normal distribution is zero. Thus, we use the normal distribution to describe the probability that a number randomly selected from the normal distribution will be in a particular range. We could ask, for example, what is the probability that if we draw a number from the standard normal distribution, it will be less than some number a? The area under the curve to the left of a, denoted N (a), equals this probability, Pr(z < a). We call N (a) the cumulative normal distribution function. The integral from −∞ to a is the area under the density over that range; it is cumulative in that it sums the probabilities from −∞ to a. 18.1 The Normal Distribution 547 FIGURE 18.2 Standard Normal Density, φ(z) Top panel: Area under the normal curve to the left of 0.4 0.3. Bottom panel: Cumula- φ(z) tive normal distribution. The 0.3 height at x = 0.3, given by 0.2 N (0.3), is 0.6179. 0.1 Area = 0.6179 z = 0.3 0 –4 –3 –2 –1 0 1 2 3 4 z Standard Normal Distribution, N(z) 1.0 0.8 N(x) Height = N(0.3) = 0.6179 0.6 0.4 z = 0.3 0.2 0 –4 –3 –2 –1 0 1 2 3 4 z Mathematically, this is accomplished by integrating the standard normal density, equation (18.1) with μ = 0 and σ = 1, from −∞ to a: a 1 1 2 N (a) ≡ √ e− 2 x dx (18.2) −∞ 2π As an example, N (0.3) is shown in Figure 18.2. In the top panel, N (0.3) is the area under the normal density curve between −∞ and 0.3. In the bottom panel, N (0.3) is a point on the cumulative distribution. The range −∞ to +∞ covers all possible outcomes for a single draw from a normal distribution. The probability that a randomly drawn number will be less than ∞ is 1; hence, N (∞) = 1. As you may already have surmised, the N (a) deﬁned above is the same N ( ) used in computing the Black-Scholes formula. There is no simple formula for the cumulative normal distribution function, equation (18.2), but as we mentioned in Chapter 12, it is a frequent-enough calculation that modern spreadsheets have it as a built-in function. (In Excel the function is called NormSDist.) The area under the normal density from −∞ to 0.3 is 0.6179. Thus, if you draw a number from the standard normal distribution, 61.79% of the time the number you draw will be less than 0.3. Suppose that we wish to know the probability that a number drawn from the standard normal distribution will be between a and −a. We have 548 Chapter 18. The Lognormal Distribution Pr(z < −a) = N (−a) Pr(z < a) = N (a) These relationships imply that Pr(−a < z < a) = N (a) − N (−a) The area under the curve between −a and a equals the difference between the area below a and the area below −a. Since the standard normal distribution is symmetric about 0, the area under the curve above a equals the area under the curve below −a. Thus, N (−a) = 1 − N (a) (18.3) Example 18.1 The probability that a number drawn from the standard normal distribution will be between −0.3 and +0.3 is Pr(−0.3 < z < 0.3) = N (0.3) − N (−0.3) = N (0.3) − [1 − N (0.3)] = 2 × 0.6179 − 1 = 0.2358 Finally, if a variable obeys the standard normal distribution, it is extremely unlikely to take on large positive or negative values. The probability that a single draw will be below −3 or above 3 is only 0.0027. If you drew from a standard normal distribution every day, you would draw above 3 or below −3 only about once a year. The probability of being below −4 or above 4 is 0.000063, which, with daily draws, would occur on average about once every 43.25 years. Converting a Normal Random Variable to Standard Normal If we have an arbitrary normal random variable, it is easy to convert it to standard normal. Suppose x ∼ N (μ, σ 2) Then we can create a standard normal random variable, z, by subtracting the mean and dividing by the standard deviation: x−μ z= (18.4) σ Using this fact, we can compute the probability that x is less than some number b: x−μ b−μ Pr(x < b) = Pr < σ σ (18.5) b−μ =N σ 18.1 The Normal Distribution 549 Using equation (18.3), the complementary probability is Pr(x > b) = 1 − Pr(x < b) b−μ = 1− N σ (18.6) μ−b =N σ This result will be helpful in interpreting the Black-Scholes formula. If we have a standard normal random variable z, we can generate a variable x ∼ N (μ, σ 2), using the following: x = μ + σz (18.7) Example 18.2 Suppose that x ∼ N (3, 25) and z ∼ N (0, 1). Then x−3 ∼ N (0, 1) 5 and 3 + 5 × z ∼ N (3, 25) Sums of Normal Random Variables Suppose we have n jointly distributed random variables xi , i = 1,... , n, with mean and variance E(xi ) = μi , Var(xi ) = σi2, and covariance Cov(xi , xj ) = σij. (The covariance between two random variables measures their tendency to move together. We can also write the covariance in terms of ρij , the correlation between xi and xj : σij = ρij σi σj.) Then the weighted sum of the n random variables has mean n n E ωi xi = ωi μi (18.8) i=1 i=1 and variance n n n Var ωi x i = ωi ωj σij (18.9) i=1 i=1 j =1 where ωi and ωj represent arbitrary weights. These formulas for the mean and variance are true for any distribution of the xi. In general, the distribution of a sum of random variables is different from the distribu- tion of the individual random variables. However, the normal distribution is an example of a stable distribution. A distribution is stable if sums of random variables have the same distri- bution as the original random variables. In this case, the sum of jointly-normally distributed random variables is normal. Thus, if xi are jointly-normally distributed, ⎛ ⎞ n n n n ωi xi ∼ N ⎝ ωi μi , ωi ωj σij ⎠ (18.10) i=1 i=1 i=1 j =1 550 Chapter 18. The Lognormal Distribution A familiar special case of this occurs with the sum of two random variables: ax1 + bx2 ∼ N aμ1 + bμ2 , a 2σ12 + b2σ22 + 2abρσ1σ2 The Central Limit Theorem. Why does the normal distribution appear in option pricing (and frequently in other contexts)? The normal distribution is important because it arises naturally when random variables are added. The normal distribution was originally discov- ered by mathematicians studying series of random events, such as gambling outcomes and observational errors.2 Suppose, for example, that a surveyor is making observations to draft a map. The measurements will always have some error, and the error will differ from mea- surement to measurement. Errors can arise from observational error, imprecise use of the instruments, or simply from recording the wrong number. Whatever the reason, the errors will in general be accidental and, hence, uncorrelated. If you were using such error-prone data, you would like to know the statistical distribution of these errors in order to assess the reliability of your conclusions for a given number of observations, and also to decide how many observations to make to achieve a given degree of reliability. It would seem that the nature of the errors would differ depending on who made them, the kind of equipment used, and so forth. The remarkable result is that sums of such errors are approximately normal. The normal distribution is therefore not just a convenient, aesthetically pleasing dis- tribution, but it arises in nature when outcomes can be characterized as sums of independent random variables with a ﬁnite variance. The distribution of such a sum approaches normality. This result is known as the central limit theorem.3 In the context of asset returns, the continuously compounded stock return over a year is the sum of the daily continuously compounded returns. If news and other factors are the shocks that cause asset prices to change, and if these changes are independent, then it is natural to think that longer-period continuously compounded returns are normally dis- tributed. Since the central limit theorem is a theorem about what happens in the limit, sums of just a few random variables may not appear normal. But the normality of continuously compounded returns is a reasonable starting point for thinking about stock returns. 18.2 THE LOGNORMAL DISTRIBUTION A random variable, y, is said to be lognormally distributed if ln(y) is normally distributed. Put another way, if x is normally distributed, y is lognormal if it can be written in either of two equivalent ways: ln(y) = x or y = ex This last equation is the link between normally distributed continuously compounded returns and lognormality of the stock price. 2. The history of statistics—including the story of the normal distribution—is entertainingly related in Bernstein (1996). 3. Most statistics books discuss one or more versions of the central limit theorem. See, for example, Casella and Berger (2002, pp. 236–239), DeGroot (1975, pp. 227–231), or Mood et al. (1974, pp. 233–236). 18.2 The Lognormal Distribution 551 By deﬁnition, the continuously compounded return from 0 to t is R(0, t) = ln(St /S0) (18.11) Suppose R(0, t) is normally distributed. By exponentiating both sides, we obtain St = S0eR(0, t) (18.12) Equation (18.12) shows that if continuously compounded returns are normally distributed, then the stock price is lognormally distributed. Exponentiation converts the continuously compounded return, R(0, t), into one plus the effective total return from 0 to t, eR(0, t). Notice that because St is created by exponentiation of R(0, t), a lognormal stock price cannot be negative. We saw that the sum of normal variables is normal. For this reason, the product of lognormal random variables is lognormal. If x1 and x2 are normal, then y1 = ex1 and y2 = ex2 are lognormal. The product of y1 and y2 is y1 × y2 = ex1 × ex2 = ex1+x2 Since x1 + x2 is normal, ex1+x2 is lognormal. Thus, because normality is preserved by addition, lognormality is preserved by multiplication. However, just as the product of normal random variables is not normal, the sum of lognormal random variables is not lognormal. We saw in Section 11.3 that the binomial model generates a stock price distribution that appears lognormal; this was an example of the central limit theorem. In the binomial model, the continuously compounded stock return is binomially distributed. Sums of bino- mial random variables approach normality. Thus, in the binomial model, the continuously compounded return approaches normality and the stock price distribution approaches log- normality. If ln(y) ∼ N (m, v 2), the lognormal density function is given by ln(y)−m 2 1 − 21 g(y; m, v) ≡ √ e v yv 2π Figure 18.3 displays three different lognormal densities for various values of m and ν. Notice that the lognormal distribution is nonnegative and skewed to the right. The distribution for which the underlying normal distribution has a high mean (1.5) and low standard deviation (0.2) most resembles the normal distribution. It is still bounded below by zero and skewed to the right. Two of the distributions in Figure 18.3 are based upon exponentiating the distributions in Figure 18.1. We can compute the mean and variance of a lognormally distributed random variable. If x ∼ N (m, v 2), then the expected value of ex is given by 1 2 E(ex ) = em+ 2 v (18.13) We prove this in Appendix 18.A, but it is intuitive that the mean of the exponentiated variable will be greater than the exponentiated mean of the underlying normal variable. Exponentiation is asymmetric: A positive random draw generates a bigger increase than an identical negative random draw does a decrease. To see this, consider a mean zero binomial random variable that is 0.5 with probability 0.5 and −0.5 with probability 0.5. You can verify 552 Chapter 18. The Lognormal Distribution FIGURE 18.3 Probability Density Graph of the lognormal ln(y) ~ N (0, 1) density for y, where ln(y) ∼ 0.8 ln(y) ~ N (0, 1.5) N (0, 1), ln(y) ∼ N (0, 1.5), ln(y) ~ N (1.5, 0.2) and ln(y) ∼ N (1.5, 0.2). 0.6 0.4 0.2 0.0 0 2 4 6 8 10 y −0.5 that e0.5 = 1.6487. Thus, e +e 0.5 2 = 1.6487+0.6065 2 = 1.128, which is obviously greater than e = 1. 0 This is a speciﬁc example of Jensen’s inequality (see Appendix C at the end of this book): The expectation of a function of a random variable is not generally equal to the function evaluated at the expectation of the random variable. In the context of this example, E(ex ) = eE(x). Since the exponential function is convex, Jensen’s inequality implies that E(ex ) > eE(x). Derivatives theory is replete with examples of Jensen’s inequality. The variance of a lognormal random variable is 2 2 Var(ex ) = e2m+v ev − 1 (18.14) While we can compute the variance of a lognormal variable, it is much more convenient to use only the variance of ln(y), which is normal. We will not use equation (18.14) in the rest of this book. 18.3 A LOGNORMAL MODEL OF STOCK PRICES How do we implement lognormality as a model for the stock price? If the stock price St is lognormal, we can write St = ex S0 where x, the continuously compounded return from 0 to t, is normally distributed. We want to ﬁnd a speciﬁcation for x that provides a useful way to think about stock prices. Let the continuously compounded return from time t to some later time s be R(t , s). Suppose we have times t0 < t1 < t2. By the deﬁnition of the continuously compounded return, we have 18.3 A Lognormal Model of Stock Prices 553 St1 = St0 eR(t0 , t1) St2 = St1eR(t1, t2) The stock price at t2 can therefore be expressed as St2 = St1eR(t1, t2) = St0 eR(t0 , t1)eR(t1, t2) = St0 eR(t0 , t1)+R(t1, t2) Thus, the continuously compounded return from t0 to t2, R(t0 , t2), is the sum of the continuously compounded returns over the shorter periods: R(t0 , t2) = R(t0 , t1) + R(t1, t2) (18.15) Example 18.3 Suppose the stock price is initially $100 and the continuously compounded return on a stock is 15% one year and 3% the next year. The price after 1 year is $100e0.15 = $116.1834, and after 2 years is $116.1834e0.03 = $119.722. This equals 100e0.15+0.03 = 100e0.18. As we saw in Section 11.3, equation (18.15), together with the assumption that returns are independent and identically distributed over time, implies that the mean and variance of returns over different horizons are proportional to the length of the horizon. Take the period of time from 0 to T and carve it up into n intervals of length h, where h = T /n. We can then write the continuously compounded return from 0 to T as the sum of the n returns over the shorter periods: R(0, T ) = R(0, h) + R(h, 2h) +... + R[(n − 1)h, T ] n = R[(i − 1)h, ih] i=1 Let E(R[(i − 1)h, ih]) = αh and Var(R[(i − 1)h, ih]) = σh2. Then over the entire period, the mean and variance are E[R(0, T )] = nαh (18.16) Var[R(0, T )] = nσh2 (18.17) Thus, if returns are independent and identically distributed, the mean and variance of the continuously compounded returns are proportional to time. This result corresponds with the intuition that both the mean and variance of the return should be greater over long horizons than over short horizons. Now we have enough background to present an explicit lognormal model of the stock price. Generally we will let t be denominated in years and α and σ be the annual mean and standard deviation, with δ the annual dividend yield on the stock. We will assume that the continuously compounded capital gain from 0 to t, ln(St /S0), is normally distributed with mean (α − δ − 0.5σ 2)t and variance σ 2t: ln(St /S0) ∼ N [(α − δ − 0.5σ 2)t , σ 2t] (18.18) 554 Chapter 18. The Lognormal Distribution This gives us two equivalent ways to write an expression for the stock price. First, recall from equation (18.7) that we can convert a standard normal random variable, Z, into one with an arbitrary mean or variance by multiplying by the standard deviation and adding the mean. We can write 1 √ ln(St /S0) = (α − δ − σ 2)t + σ tZ (18.19) 2 Second, we can exponentiate equation (18.19) to obtain an expression for the stock price: 1 2 √ St = S0e(α−δ− 2 σ )t+σ tZ (18.20) We will use equation (18.20) often in what follows. You may be wondering how to interpret equations (18.18), (18.19), and (18.20). The subtraction of the dividend yield, δ, is necessary since, other things equal, a higher dividend yield means a lower future stock price. But why do we subtract 21 σ 2 in the mean? To understand equation (18.20) it helps to compute the expected stock price. We can do this by breaking up the right-hand side of equation (18.20) into two terms, one of which contains the random variable Z and the other of which does not: 1 2 √ St = S0e(α−δ− 2 σ )t eσ tZ √ Next, evaluate the expectation of eσ using equation (18.13). Since z ∼ N (0, 1), we have tZ √ 1 2 E eσ tZ = e 2 σ t This gives us 1 2 1 2 E(St ) = S0e(α−δ− 2 σ )t e 2 σ t = S0e(α−δ)t (18.21) We therefore have St ln E = (α − δ)t (18.22) S0 The expression α − δ is the continuously compounded expected rate of appreciation on the stock. If we did not subtract 21 σ 2 in equation (18.20), then the expected rate of appreciation would be α − δ + 21 σ 2. This is ﬁne (we can deﬁne things as we like), except that it renders α difﬁcult to interpret. Thus, the issue is purely one of creating an expression where it is easy to interpret the parameters. If we want α − δ to have an interpretation as the continuously compounded expected capital gain on the stock, then because of equation (18.13), we need to sub- tract 21 σ 2. The median stock price—the value such that 50% of the time prices will be above or below that value—is obtained by setting Z = 0 in equation (18.20). The median is thus 1 2 1 2 S0e(α−δ− 2 σ )t = E(St )e− 2 σ t This equation demonstrates that the median is below the mean. More than 50% of the time, a lognormally distributed stock will earn below its expected return.Perhaps more surprisingly, 18.3 A Lognormal Model of Stock Prices 555 if σ is large, a lognormally distributed stock will lose money (St < S0) more than half the time! Example 18.4 Suppose that the stock price today is $100, the expected rate of return on the stock is α = 10%/year, and the standard deviation (volatility) is σ = 30%/year. If the stock is lognormally distributed, √ the continuously compounded 2-year return is 20% and the 2-year volatility is 0.30 × 2 = 0.4243. Thus, we have 1 2 √ S2 = $100e(0.1− 2 0.3 )×2+σ 2Z The expected value of S2 is E(S2) = $100e(0.1×2) = $122.14 The median stock price is 2 $100e(0.1−0.5×0.3 )×2 = $111.63 If the volatility were 60%, the expected value would still be $122.14, but the median would be 2 )×2 $100e(0.1−0.5×0.6 = $85.21 Half the time, after 2 years the stock price would be below this value. We can also deﬁne a “one standard deviation move” in the stock price. Since z has the standard normal distribution, then if Z = 1, the continuously compounded stock return is the mean plus one standard deviation, and if Z = −1, the continuously compounded stock return is the mean minus one standard deviation. Example 18.5 Using the same assumptions as in Example 18.4, a one standard deviation move up over 2 years is given by 1 2 √ S2 = $100e(0.1− 2 0.3 )×2+σ 2×1 = $170.62 A one standard deviation move down is given by 1 2 √ S2 = $100e(0.1− 2 0.3 )×2−σ 2×1 = $73.03 We can think of these prices as logarithmically centered around the mean price of $122.14. This discussion also shows us where the binomial models in Chapter 11 come from. In Section 11.3, we presented three different ways to construct a binomial model. All had up and down stock price moves of the form √ √ Su = Seαh+σ h ; Sd = Seαh−σ h where α differed for the three models. In all cases, we generated up and down moves by setting Z = ±1. As h → 0 the three models converge; the effects of the different α’s in each case are offset by the different risk-neutral probabilities.

The Lognormal Distribution PDF

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