How does the binomial distribution apply to finance? I want a practical example beyond coin flips.

The binomial distribution models scenarios with exactly two possible outcomes per trial — in finance, this translates directly into up or down price movements, which is the foundation of the binomial option pricing model.

Binomial Probability Formula:

P(X = x) = C(n,x) x p^x x (1-p)^(n-x)

Where:

• n = number of trials
• x = number of successes
• p = probability of success on each trial
• C(n,x) = n! / [x!(n-x)!] = combinations

Finance Application — Credit Defaults:

Portland Capital holds a portfolio of 8 corporate bonds. Historical data suggests each bond has a 5% probability of defaulting in the next year, independently.

What's the probability that exactly 2 bonds default?

P(X = 2) = C(8,2) x 0.05^2 x 0.95^6

= 28 x 0.0025 x 0.7351

= 0.0515 or 5.15%

What's the probability of zero defaults?

P(X = 0) = C(8,0) x 0.05^0 x 0.95^8 = 1 x 1 x 0.6634 = 66.34%

Binomial Stock Price Tree (Intro to Option Pricing):

A stock trades at $100. Each period, it can go up 10% (p = 0.55) or down 10% (1-p = 0.45).

After 2 periods:

Probability of each outcome:

• $121 (UU): 0.55 x 0.55 = 0.3025
• $99 (UD or DU): 2 x 0.55 x 0.45 = 0.4950
• $81 (DD): 0.45 x 0.45 = 0.2025

Expected stock price = $121(0.3025) + $99(0.4950) + $81(0.2025) = $36.60 + $49.01 + $16.40 = $102.01

Key properties:

• Mean = n x p
• Variance = n x p x (1-p)
• As n gets large, the binomial approaches the normal distribution

Exam tip: The CFA exam typically gives you n, p, and asks for the probability of exactly x successes or "at most" x successes (cumulative). For "at most," sum individual probabilities from 0 to x.

Explore quantitative methods in our CFA Level I course.