Brownian Motion as a Limit of Simpler Models Let  be a small increment of time, and consider a process such that

every time units the value of the process either increases by the amount σ√

 with probability p or decreases by the amount σ√

 with

and where the successive changes in value are independent.

Thus, we are supposing that the process values change only at times that are integral multiples of, and that at each change point the value of the process either increases or decreases by the amountσ√

, with the change being an increase with probability p= ¹₂(1 + ^μ_σ√

).

As we take smaller and smaller, so that changes occur more and more frequently (though by amounts that become smaller and smaller), the process becomes a Brownian motion with drift parameter μ and variance parameterσ². Consequently, Brownian motion can be approx-imated by a relatively simple process that either increases or decreases by a fixed amount at regularly specified times.

We now verify that the preceding model becomes Brownian motion as we let become smaller and smaller. To begin, let

Xi =

 1, if the change at time i is an increase

−1, if the change at time i is a decrease

Hence, if X(0) is the process value at time 0, then its value after n changes is

X(n) = X(0) + σ√

 (X1+ . . . + Xn)

36 Brownian Motion and Geometric Brownian Motion

Because there would have been n = t/ changes by time t, this gives that there are more and more terms in the summationt/

i=1Xi, the central limit theorem suggests that this sum converges to a normal random vari-able. Consequently, as goes to 0, the process value at time t becomes a normal random variable. To compute its mean and variance, note first that

Var(Xi) (by independence)

= σ²t [1− (2p − 1)²]

Because p→ 1/2 as  → 0, the preceding shows that Var(X(t) − X (0)) → tσ² as  → 0

Brownian Motion as a Limit of Simpler Models 37 Consequently, as gets smaller and smaller, X(t) − X(0) converges to a normal random variable with meanμt and variance tσ². In addition, because successive process changes are independent and each has the same probability of being an increase, it follows that X(t + y) − X (y) has the same distribution as does X(t) − X (0) and is, in addition, in-dependent of earlier process changes before time y. Hence, it follows that as goes to 0, the collection of process values over time becomes a Brownian motion process with drift parameterμ and variance param-eterσ².

An important result about Brownian motion is that, conditional on the value of the process at time t, the joint distribution of the process values up to time t does not depend on the value of the drift parameter. This result is easily proven by using the approximating processes, as we now show.

Theorem 3.2.1 Given that X(t) = x, the conditional probability law of the collection of prices X(y), 0 ≤ y ≤ t, is the same for all values ofμ.

Proof. Let s = X (0) be the price at time 0. Now, consider the approxi-mating model where the price changes every time units by an amount equal, in absolute value, to c≡ σ√

, and note that c does not depend onμ. By time t, there would have been t/ changes. Hence, given that the price has increased from time 0 to time t by the amount x − s, it follows that, of the t/ changes, there have been a total of _2^t + ^x_2c^−s positive changes and a total of _2^t − ^x_2c^−s negative changes. (This fol-lows because if the preceding were so, then, of the first t/ changes, there would have been ^x−s_c more positive than negative changes, and so the price would have increased by c(^x−s_c ) = x − s.) Because each change is, independently, a positive change with the same probability p, it follows, conditional on there being a total of _2^t + ^x_2c^−s positive changes out of the first t/ changes, that all possible choices of the changes that were positive are equally likely. (That is, if a coin having probability p is flipped m times, then, given that k heads resulted, the subset of trials that resulted in heads is equally likely to be any of the _m

k

subsets of size k.) Thus, even though p depends on μ, the condi-tional distribution of the history of prices up to time t, given that X(t) = x, does not depend onμ. (It does, however, depend on σ because c, the size of a change, depends onσ, and so if σ changed, then so would the

38 Brownian Motion and Geometric Brownian Motion

number of the t/ changes that would have had to be positive for S(t) to equal x.) Letting go to 0 now completes the proof.

The Brownian motion process has a distinguished scientific pedigree.

It is named after the English botanist Robert Brown, who first described (in 1827) the unusual motion exhibited by a small particle that is totally immersed in a liquid or gas. The first explanation of this motion was given by Albert Einstein in 1905. He showed mathemati-cally that Brownian motion could be explained by assuming that the im-mersed particle was continually being subjected to bombardment by the molecules of the surrounding medium. A mathematically concise defi-nition, as well as an elucidation of some of the mathematical properties of Brownian motion, was given by the American applied mathematician Norbert Wiener in a series of papers originating in 1918.

Interestingly, Brownian motion was independently introduced in 1900 by the French mathematician Bachelier, who used it in his doctoral dis-sertation to model the price movements of stocks and commodities.

However, Brownian motion appears to have two major flaws when used to model stock or commodity prices. First, since the price of a stock is a normal random variable, it can theoretically become negative. Second, the assumption that a price difference over an interval of fixed length has the same normal distribution no matter what the price at the beginning of the interval does not seem totally reasonable. For instance, many peo-ple might not think that the probability a stock presently selling at $20 would drop to $15 (a loss of 25%) in one month would be the same as the probability that when the stock is at $10 it would drop to $5 (a loss of 50%) in one month.

A process often used to model the price of a security as it evolves over time is the geometric Brownian motion process.