Excel
02 Dec 2008
Jun
07
Binomial distribution
by David Harper, CFA, FRM, CIPM
FRM 2007 learning outcome 2.3: Describe...a binomial random variable. Yesterday I reviewed the Poisson distribution. The Poisson has something in common with the Binomial distribution: they both describe discrete random variables. Recall that discrete variables can be counted (e.g., one, then two, then three); continuous variables must be measured (e.g., distance to the finished line, time until the 2007 FRM exam!).
Why the binomial distribution matters
As distributions go, the binomial is simple. But it shows up often. For example, as an alternative to the Black-Scholes option pricing model (OPM), we can use the binomial model to price a call option. In fact, it is better suited to some options, such as American-style options (i.e., options that can be exercised before expiration/maturity). At the heart of the binomial model is a giant tree with nodes; at each node, the simulated asset (stock) can move up or down. In the binomial option pricing model, therefore, an option valuation is achieved by way of the idea that the stock price is a function of a binomial random variable.
What is the pdf?
First, we need to know:
- The probability of success on a single trial, denoted by p. For a "fair coin" toss, this would be 50%. Conversely, the probability of failure is (1-p) or q.
- The total number of trials (denoted by n)
There is a familiar condition: the random variable must by i.i.d. which stands for independent and identically distributed. Independence means that an outcome has no influence on the next (if you flip a heads, you still think the next flip is 50/50, not more likely to be another heads). Identically distributed means that the expected value (and variance) of the random variable does not change during the trial (unlike financial assets, where often volatility is time-varying).
Given these rigid assumptions, the density function is given by:
What does this formula give us? Let's say our Bernoulli trial is a coin flip. We will flip the coin ten times. So n=10 because we are conducting ten trials. We know that p=50%. The P(X=3) is the probability that we will flip exactly three heads, out of ten total flips.
Note the first proportion is the formula for a combination. Recall FRM candidates: a combination does not care about the sequence of the outcomes. This explains why the binomial distribution, also, does not care about sequence. If we flip a coin ten times, P(X=3) is the probability of flipping three heads regardless of when they occur.
A binomial distribution example
I prepared an EditGrid spreadsheet below to illustrate this ten-trial experiment of coin flipping. Note the answer to P(X=3) is about 11.7%. That's the odds of flipping three heads and seven tails. Notice we get the same odds of flipping seven heads and three tails. If the probability is 50%, the binomial distribution will by symmetrical. But if it is unequal to 50%, it will be skewed.
Note I calculate the binomial using the built-in formula. But if you really want to learn, and I know you do, you should examine the expanded function. You can open your own read/write copy here.
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