Stock Price Distribution

3 Stock Price Distribution 3.1 STOCK PRICE MOVEMENTS (i) Consider the evolution of a stock price: the prices observed at successive moments in time δt apart are: S 0 , S 1 , .,S N . For simplicity, it is assumed that no dividend is paid in this period. The “price relative” in period n is defined as the ratio R n = S n /S n−1 . This quantity is a random variable and we make the following apparently innocuous but far-reaching assumption: the price relatives are independently and identically distributed. If the price relatives are independently distributed, the next move does not depend on what happened at the last move. This is a statement of the so-called weak form market efficiency hypothesis, which maintains that the entire history of a stock is summed up in its present price; future movements depend only on new information and on changes in sentiment or the environment. People who believe in charts or concepts such as momentum for predicting stock prices clearly do not believe in this hypothesis; in consequence, they should not believe in the results of modern option theory. If price relatives are identically distributed, then their expected means and variances are constant, i.e. E  S n S n−1  = E  S n+1 S n  ;var  S n S n−1  = var  S n+1 S n  The first equation says that the expected growth of the stock remains constant. It will be seen below that the second equation is a statement that the volatility of the stock remains constant. (ii) Consider the following simple identity: S N S 0 ≡ S 1 S 0 S 2 S 1 ··· S N S N −1 = R 1 R 2 ···R N Taking the logarithm of each side gives x N = r 1 + r 2 +···+r N where x N = ln S N S 0 ; r n = ln R n = ln S n S n−1 (3.1) Some very profound conclusions emerge from this trivial-looking equation: if the price relatives R n are identically distributed and independent, then so are their logarithms, r n . They can be treated as independent random variables, drawn from the same infinite population. It follows that E[x N ] = N  n=1 E[r n ] = N E[r n ] = Nm δ T var[x N ] = N  n=1 var[r n ] = N var[r n ] = N σ 2 δ T where m δ T and σ 2 δ T are the mean and variance of the logarithm of the price relatives. 3 Stock Price Distribution Suppose that two people perform this analysis on the same stock price evolution, but one person considers N successive time increments of length δT while the other considers N/2 time increments of length 2δT . Slicing up the stock price path is completely arbitrary, but the expectation E [ x T ] must be the same in both cases. Explicitly writing out each example: E[x T ] = Nm δ T and E[x T ] = N 2 m 2 δ T From which it follows that m 2 δ T = 2m δ T , or in other words m δ T is proportional to the time interval δT . We may therefore write m δ T = mδT ; identical reasoning gives σ 2 δ T = σ 2 δT . The mean and variance of x T may now be written E[x T ] = mNδT = mT var[x T ] = σ 2 N δT = σ 2 T (3.2) Both mean and variance are equal to a constant multiplied by T. (iii) Central Limit Theorem: Before going on to derive the key conclusion of this section, we need to make a detour back to the reader’s earliest encounters with statistics: suppose we have a random sample y 1 , .,y N taken from an infinite population with mean m and variance σ 2 . The mean of this sample is ¯ y N = (1/N )  n i=1 y i . This sample mean is of course itself a ran- dom variable, with expectation m and variance σ 2 /N . Furthermore, the central limit theorem states that whatever the distribution of y i might be, the distribution of ¯ y N tends to a normal distribution as N →∞. These results may be summarized as lim N→∞ ¯ y N ∼ N (m,σ 2 /N ) or lim N→∞ N ¯ y N = lim N→∞ (y 1 +···+y N ) ∼ N (mN,σ 2 N ) the sign ∼ means “is distributed as” while N(m,ν 2 ) indicates the normal distribution with mean m and variance ν 2 . Applying this last result to equation (3.1) gives the very general result x T = lim N→∞; δ T→0 (r 1 +···+r N ) ∼ N  m δ T N,σ 2 δ T N  = N (mδTN,σ 2 δTN) = N(mT,σ 2 T ) We can reassure ourselves that this result holds simply by notionally slicing the time period into an arbitrarily large number of arbitrarily small segments so that N →∞and the central limit theorem holds true. Finally, an important piece of jargon: if a random variable takes its various values r 1 , r 2 , .,r N at successive points in time, it is referred to as a stochastic variable. 3.2 PROPERTIES OF STOCK PRICE DISTRIBUTION In the last section we defined x T = ln S T /S 0 and showed that x T is normally distributed with mean mT and variance σ 2 T . S T is then said to be lognormally distributed. Some of the more useful properties of the normal and lognormal distributions are explained in Appendix A.1. (i) In terms of x t , the stock price is given by S t = S 0 e x t and the explicit probability distribution function for x t is n(x t ) = 1 σ √ 2πt exp  − 1 2  x t − mt σ √ t  2  30 3.2 PROPERTIES OF STOCK PRICE DISTRIBUTION The various moments for the distribution of S t may be written E   S t S 0  λ  = E[e λx t ] =  +∞ −∞ e λx t n(x t )dx t = e λmt+ 1 2 λ 2 σ 2 t (3.3) This result is proved in Appendix A.1(v), item (D). (ii) Mean and Variance of S T : The rate of return µ of a non-dividend-paying stock over a time interval T is defined by E  S T  = S 0 e µT . Using equation (3.3) with λ = 1gives E[S T ] = S 0 e µT = S 0 e (m+ 1 2 σ 2 )T (3.4) A relationship which simply falls out of the last equation and which is used repeatedly through- out this book is µ = m + 1 2 σ 2 (3.5) Again using equation (3.3), this time with λ = 2, gives var  S T S 0  = E  S T S 0  2 − E 2  S T S 0  = e 2µT  e σ 2 T − 1  (3.6) (iii) Variance and Volatility: x T is a stochastic variable with mean mT and variance σ 2 T . T is measured in units of a year and σ is referred to as the (annual) volatility of the stock. The reader would be quite right to comment that σ should be called the volatility of the logarithm of the stock price; but the two are closely related and for most practical purposes, the same. To see this, consider a small time interval δT (maybe a day or a week) and write S δ T − S 0 = δS. Then var  ln S δ T S 0  = var  ln  1 + δS S 0  ≈ var  δS S 0  = var  1 + δS S 0  = var  S δ T S 0  where the approximation uses the following standard Taylor expansion for small a: ln(1 + a) = a − 1 2 a 2 + 1 3 a 3 −···and we use var[constant + x] = varx. The conclusion is that for small time steps, the variance of S T /S 0 and of ln(S T /S 0 ) are the same. In practical terms this means that if we estimate volatility from historical stock price data, measuring either daily or weekly price movements, then we get more or less the same result by using either the stock prices themselves or their logarithms. However, these two measures are likely to produce appreciably different results if we use quarterly data. (iv) In Section 3.1 it was established that x t = ln (S t /S 0 ) ∼ N (mT,σ 2 T ) where m = µ − 1 2 σ 2 . We define a standard normal variate z t by z t = (x t − mt)/σ √ t so that z t is distributed as z t ∼ N (0, 1). This can be inverted as x t = mt + σ √ tz t , or equivalently S t = S 0 e mt+σ √ tz t = S 0 e mt+σ W t (3.7) (v) It is worth pondering this last equation for a moment to get an appreciation of what is really meant. The quantity √ tz t is usually written W t and the study of its mathematical properties 31 3 Stock Price Distribution has filled many textbooks; but for the moment, it is enough just to understand the big picture. W t is said to describe a one-dimensional Brownian motion. Brown was the nineteenth century botanist who, studying pollen grains suspended in a liquid, was surprised to observe their erratic movements, caused (as was later discovered) by the buffeting they received from the impact of individual molecules. Most people find it helpful to think of W t in terms of the physical movement of a particle, rather than the more abstract movement of the logarithm of a stock price. Strictly speaking, we should think of a particle moving backwards and forwards along a one-dimensional line. Suppose that at time t = 0 the particle starts at position W 0 = 0. The movement is random so we do not know where the particle will be after time T. We merely know the probability distribution of W t . Let us look at a couple of examples. z t is a standard normal variate, i.e. with mean 0 and variance 1. From standard tables we know that Pr[−2 < z T < +2] ≈ 95% r Example 1: T = 2 years Pr  −2 < W 2 years √ 2 < +2  = 95% Pr[−2.8 < W 2 years < +2.8] = 95% r Example 2: T = 1 week Pr  −2 < W 1 week √ 1/52 < +2  = 95% Pr[−0.28 < W 1 week < +0.28] = 95% The quantity W t could take more or less any value. We know that it is proportional to √ t, but the value could be positive or negative, large or small. Remember that z t is a standard normal variate: z 1000 and z 0.0001 are merely two readings from the same distribution and there is nothing to suggest that one should be larger or smaller than the other. (vi) Continuous Dividends: In Section 1.1 it was seen that a continuous dividend could be accounted for by making the substitution S 0 → S 0 e −qt for the stock price. The distribution for S t at the beginning of this section can therefore be described by ln S t S 0 → ln S t S 0 e −qt = ln S t S 0 + qt ∼ N(mt,σ 2 t) But if ln(S t /S 0 ) + qt is normally distributed with mean mt and variance σ 2 t, the term qt merely shifts the mean of the distribution so that ln(S t /S 0 ) ∼ N ((m − q)t,σ 2 t) The effect of a continuous dividend rate q can therefore be taken into account simply by making the substitutions m → m − q and µ → µ − q. 32 3.3 INFINITESIMAL PRICE MOVEMENTS 3.3 INFINITESIMAL PRICE MOVEMENTS (i) Let us return to equation (3.7) for the stock price evolution, and consider only small time intervals δt. We may write that equation as S t + δS t = S t e m δ t+σ δ W t where δW t = √ δtz δ t We are dealing with small time periods and small price changes so that we may use the standard expansion e a = 1 + a + 1 2! a 2 +···in the last equation, giving S t + δS t = S t  1 + (mδt + σ δW t ) + 1 2 (mδt + σ δW t ) 2 +···  Normally, one might expect to drop the squared and higher terms in this equation; but recall the definition δW t = √ δtz δ t . The term z δ t is a standard normal variate, taking the values −1 to +1 for about 67% of the time; values−2to+2 for about 95% of the time; values−3to+3 for about 99.5% of the time, etc. δW t is therefore not of the same order as δt (written O[δt]); it is O[ √ δt]. To be consistent in the last equation then, we need to retain terms up to δt together with terms up to δW 2 t . This gives us δS t S t = mδt + σ δW t + 1 2 σ 2 δW 2 t (ii) An appreciation of the significance of the last term in this equation is obtained by analyzing the following expectations and variances of powers of δW t . First, recall from Appendix A.1(ii) that the moment generating function for a standard normal distribution is M[] = e  2 /2 ; the various moments are given by µ λ = Ez λ = ∂ λ M[] ∂ λ ] =0 . Using this procedure we get E[δW t ] = √ δt E[z δ t ] = 0 var[δW t ] = E  δW 2 t  = δt E  z 2 δ t  = δt var  δW 2 t  = E  δW 4 t  − E 2  δW 2 t  = δt 2 E  z 4 δ t  − δt 2 = 2δt 2 The quantity δW 2 t has expected value δt and variance proportional to δt 2 . Thus as δt → 0 the variance of δW 2 t approaches zero much faster than δt itself. But as the variance of δW 2 t approaches zero, δW 2 t approaches its expected value with greater and greater certainty, i.e. it ceases to behave like a random variable at all. This permits us to make the substitution lim δ t→0 δW 2 t → E  δW 2 t  = δt (iii) Using equation (3.5), the process for the evolution of the stock price over a very small time interval can be written δS t S t = mδt + σ δW t + 1 2 σ 2 δt = µδt + σ δW t (3.8) With a continuous proportional dividend q this becomes δS t S t = (µ − q)δt + σ δW t (3.9) This representation has a great deal of appeal. The model has the stock price growing at a constant rate µ, with random fluctuations superimposed. These fluctuations are proportional to the standard deviation of the stock price and are dependent on a standard normal random variable. This type of process is known as a Wiener process. 33 3 Stock Price Distribution For future reference, we quote an often used result obtained by squaring the last result and dropping higher terms in δt:  δS t S t  2 = (µδt + σ δW t ) 2 ≈ σ 2 δW 2 t → σ 2 δt (3.10) 3.4 ITO’S LEMMA In the last section it was seen that an infinitesimal stock price movement δS t in an infinitesimal time interval δt could be described by the Wiener process δS t = S t (µ − q)δt + S t σ δW t .A more generalized Wiener process (also known as an Ito process) can be written δS t = a S t t δt + b S t t δW t where a S t t and b S t t are now functions of both S t and t. Consider any function f S t t of S t and t, which is reasonably well behaved (i.e. adequately differentiable with respect to S t and t). Taylor’s theorem states that δ f t = ∂ f t ∂ S t δS t + ∂ f t ∂t δt + 1 2  ∂ 2 f t ∂ S 2 t δS 2 t + ∂ 2 f t ∂ S t ∂t δS t δt + ∂ 2 f t ∂t 2 δt 2  +··· where the subscript notation has been lightened a little for the sake of legibility. Substitute for S t from the generalized Wiener process and retain only terms of order δt or lower, remembering that δW t ∼ O[ √ δt]: δ f t = ∂ f t ∂ S t δS t + ∂ f t ∂t δt + 1 2 ∂ 2 f t ∂ S 2 t b 2 t δW 2 t Put δW 2 t → δt as explained in Section 3.3, to give δ f t =  ∂ f t ∂t + a t ∂ f t ∂ S t + 1 2 b 2 t ∂ 2 f t ∂ S 2 t  δt + b t ∂ f t ∂ S t δW t (3.11) This result is known as Ito’s lemma and is one of the cornerstones of option theory. It basically says that if f t is any well-behaved function of an Ito process and of time, then f t itself follows an Ito process. The function of particular interest in this book is the price of a derivative. In the case of the simple Wiener process of equation (3.9), Ito’s lemma becomes δ f t =  ∂ f t ∂t + (µ − q)S t ∂ f t ∂ S t + 1 2 σ 2 S 2 t ∂ 2 f t ∂ S 2 t  δt + σ S t ∂ f t ∂ S t δW t (3.12) 34 . 3 Stock Price Distribution 3.1 STOCK PRICE MOVEMENTS (i) Consider the evolution of a stock price: the prices observed at successive. the logarithm of the price relatives. 3 Stock Price Distribution Suppose that two people perform this analysis on the same stock price evolution, but