In many investment contexts, we view a result as either a success or a failure, or as binary

(twofold) in some other way. When we make probability statements about a record of successes

and failures, or about anything with binary outcomes, we often use the binomial distribution.

What is a good model for how a stock price moves through time? Different models are

appropriate for different uses. Cox, Ross, and Rubinstein (1979) developed an option pricing

model based on binary moves, price up or price down, for the asset underlying the option.

Their binomial option pricing model was the first of a class of related option pricing models

that have played an important role in the development of the derivatives industry. That fact

alone would be sufficient reason for studying the binomial distribution, but the binomial

distribution has uses in decision-making as well.

The building block of the binomial distribution is the Bernoulli random variable,

named after the Swiss probabilist Jakob Bernoulli (1654–1704). Suppose we have a trial (an

event that may repeat) that produces one of two outcomes. Such a trial is a Bernoulli trial. If

we let Y equal 1 when the outcome is success and Y equal 0 when the outcome is failure, then

the probability function of the Bernoulli random variable Y is

p(1) = P(Y = 1) = p

p(0) = P(Y = 0) = 1 − p

where p is the probability that the trial is a success. Our next example is the very first step on

the road to understanding the binomial option pricing model.

3See Hillier and Lieberman (2000). Random numbers initially generated by computers are usually

random positive integer numbers that are converted to approximate continuous uniform random

numbers between 0 and 1. Then the continuous uniform random numbers are used to produce random

observations on other distributions, such as the normal, using various techniques. We will discuss random

observation generation further in the section on Monte Carlo simulation.

176 Quantitative Investment Analysis

EXAMPLE 5-2 One-Period Stock Price Movement as a

Bernoulli Random Variable

Suppose we describe stock price movement in the following way. Stock price today is

S. Next period stock price can move up or down. The probability of an up move is p,

and the probability of a down move is 1 − p. Thus, stock price is a Bernoulli random

variable with probability of success (an up move) equal to p. When the stock moves up,

ending price is uS, with u equal to 1 plus the rate of return if the stock moves up. For

example, if the stock earns 0.01 or 1 percent on an up move, u = 1.01. When the stock

moves down, ending price is dS, with d equal to 1 plus the rate of return if the stock

moves down. For example, if the stock earns −0.01 or −1 percent on a down move,

d = 0.99. Figure 5-1 shows a diagram of this model of stock price dynamics.

End-of-Period Stock Price

Probability = p Stock Price Moves Up:

Stock Price Equals uS

Stock Price

Today, S

Probability = 1 − p

Stock Price Moves Down:

Stock Price Equals dS

FIGURE 5-1 One-Period Stock Price as a Bernoulli Random Variable

We will continue with the above example later. In the model of stock price movement

in Example 5-2, success and failure at a given trial relate to up moves and down moves,

respectively. In the following example, success is a profitable trade and failure is an unprofitable

one.

EXAMPLE 5-3 A Trading Desk Evaluates Block Brokers (1)

You work in equities trading at an institutional money manager that regularly trades

with a number of block brokers. Blocks are orders to sell or buy that are too large for

the liquidity ordinarily available in dealer networks or stock exchanges. Your firm has

known interests in certain kinds of stock. Block brokers call your trading desk when

they want to sell blocks of stocks that they think your firm may be interested in buying.

You know that these transactions have definite risks. For example, if the broker’s client

(the seller of the shares) has unfavorable information on the stock, or if the total amount

Chapter 5 Common Probability Distributions 177

he is selling through all channels is not truthfully communicated to you, you may see

an immediate loss on the trade. From time to time, your firm audits the performance of

block brokers. Your firm calculates the post-trade, market-risk-adjusted dollar returns

on stocks purchased from block brokers. On that basis, you classify each trade as

unprofitable or profitable. You have summarized the performance of the brokers in a

spreadsheet, excerpted in Table 5-2 for November 2003. (The broker names are coded

BB001 and BB002.)

TABLE 5-2 Block Trading Gains and Losses

November 2003

Profitable Trades Losing Trades

BB001 3 9

BB002 5 3

View each trade as a Bernoulli trial. Calculate the percentage of profitable trades

with the two block brokers for November 2003. These are estimates of p, the underlying

probability of a successful (profitable) trade with each broker.

Your firm has logged 3 + 9 = 12 trades (the row total) with block broker BB001.

Because 3 of the 12 trades were profitable, the percentage of profitable trades was 3/12

or 25 percent. With broker BB002, the percentage of profitable trades was 5/8 or

62.5 percent. A trade is a Bernoulli trial, and the above calculations provide estimates of

the underlying probability of a profitable trade (success) with the two brokers. For broker

BB001, your estimate is pˆ = 0.25; for broker BB002, your estimate is pˆ = 0.625.4

In n Bernoulli trials, we can have 0 to n successes. If the outcome of an individual trial is

random, the total number of successes in n trials is also random. A binomial random variable

X is defined as the number of successes in n Bernoulli trials. A binomial random variable is

the sum of Bernoulli random variables Yi, i = 1, 2, … , n:

X = Y1 + Y2 +···+ Yn

where Yi is the outcome on the ith trial (1 if a success, 0 if a failure). We know that a Bernoulli

random variable is defined by the parameter p. The number of trials, n, is the second parameter

of a binomial random variable. The binomial distribution makes these assumptions:

• The probability, p, of success is constant for all trials.

• The trials are independent.

4The ‘‘hat’’ over p indicates that it is an estimate of p, the underlying probability of a profitable trade

with the broker.

178 Quantitative Investment Analysis

The second assumption has great simplifying force. If individual trials were correlated,

calculating the probability of a given number of successes in n trials would be much more

complicated.

Under the above two assumptions, a binomial random variable is completely described

by two parameters, n and p. We write

X ∼ B(n, p)

which we read as ‘‘X has a binomial distribution with parameters n and p.’’ You can see that a

Bernoulli random variable is a binomial random variable with n = 1: Y ∼ B(1, p).

Now we can find the general expression for the probability that a binomial random

variable shows x successes in n trials. We can think in terms of a model of stock price dynamics

that can be generalized to allow any possible stock price movements if the periods are made

extremely small. Each period is a Bernoulli trial: With probability p, the stock price moves

up; with probability 1 − p, the price moves down. A success is an up move, and x is the

number of up moves or successes in n periods (trials). With each period’s moves independent

and p constant, the number of up moves in n periods is a binomial random variable. We now

develop an expression for P(X = x), the probability function for a binomial random variable.

Any sequence of n periods that shows exactly x up moves must show n − x down moves.

We have many different ways to order the up moves and down moves to get a total of x

up moves, but given independent trials, any sequence with x up moves must occur with

probability px (1 − p)

n−x . Now we need to multiply this probability by the number of different

ways we can get a sequence with x up moves. Using a basic result in counting from the chapter

on probability concepts, there are

n!

(n − x)!x!

different sequences in n trials that result in x up moves (or successes) and n − x down moves

(or failures). Recall from the chapter on probability concepts that n factorial (n!) is defined as

n(n − 1)(n − 2) … 1 (and 0! = 1 by convention). For example, 5! = (5)(4)(3)(2)(1) = 120.

The combination formula n!/[(n − x)!x!] is denoted by

n

x

(read ‘‘n combination x’’ or ‘‘n choose x’’). For example, over three periods, exactly three

different sequences have two up moves: UUD, UDU, and DUU. We confirm this by

3

2

= 3!

(3 − 2)!2! = (3)(2)(1)

(1)(2)(1) = 3

If, hypothetically, each sequence with two up moves had a probability of 0.15, then the total

probability of two up moves in three periods would be 3 × 0.15 = 0.45. This example should

persuade you that for X distributed B(n, p), the probability of x successes in n trials is given by

p(x) = P(X = x) =

n

x

px

(1 − p)

n−x = n!

(n − x)!x!

px

(1 − p)

n−x (5-1)

Chapter 5 Common Probability Distributions 179

TABLE 5-3 Binomial Probabilities, p = 0.50 and n = 5

Number of

Possible Ways

Number of to Reach x Up Probability Probability

Up Moves, x Moves for Each Way for x, p(x) F(x) = P(X ≤ x)

(1) (2) (3) (4) = (2) × (3) (5)

0 10.500(1 − 0.50)5 = 0.03125 0.03125 0.03125

1 50.501(1 − 0.50)4 = 0.03125 0.15625 0.18750

2 10 0.502(1 − 0.50)3 = 0.03125 0.31250 0.50000

3 10 0.503(1 − 0.50)2 = 0.03125 0.31250 0.81250

4 50.504(1 − 0.50)1 = 0.03125 0.15625 0.96875

5 10.505(1 − 0.50)0 = 0.03125 0.03125 1.00000

Some distributions are always symmetric, such as the normal, and others are always asymmetric

or skewed, such as the lognormal. The binomial distribution is symmetric when the probability

of success on a trial is 0.50, but it is asymmetric or skewed otherwise.

We illustrate Equation 5-1 (the probability function) and the cdf through the symmetrical

case. Consider a random variable distributed B(n = 5, p = 0.50). Table 5-3 contains a

complete description of this random variable. The fourth column of Table 5-3 is Column 2, n

combination x, times Column 3, px (1 − p)

n−x ; Column 4 gives the probability for each value

of the number of up moves from the first column. The fifth column, cumulating the entries

in the fourth column, is the cumulative distribution function.

What would happen if we kept n = 5 but sharply lowered the probability of success

on a trial to 10 percent? ‘‘Probability for Each Way’’ for X = 0 (no up moves) would

then be about 59 percent: 0.100(1 − 0.10)5 = 0.59049. Because zero successes could still

happen one way (Column 2), p(0) = 59 percent. You may want to check that given

p = 0.10, P(X ≤ 2) = 99.14 percent: The probability of two or fewer up moves would be

more than 99 percent. The random variable’s probability would be massed on 0, 1, and 2 up

moves, and the probability of larger outcomes would be minute. The outcomes of 3 and larger

would be the long right tail, and the distribution would be right skewed. On the other hand,

if we set p = 0.90, we would have the mirror image of the distribution with p = 0.10. The

distribution would be left skewed.

With an understanding of the binomial probability function in hand, we can continue

with our example of block brokers