- What is a Probability Distribution
- Discrete Vs. Continuous Random Variable
- Cumulative Distribution Function
- Discrete Uniform Random Variable
- Bernoulli and Binomial Distribution
- Stock Price Movement Using a Binomial Tree
- Tracking Error and Tracking Risk
- Continuous Uniform Distribution
- Normal Distribution
- Univariate Vs. Multivariate Distribution
- Confidence Intervals for a Normal Distribution
- Standard Normal Distribution
- Calculating Probabilities Using Standard Normal Distribution
- Shortfall Risk
- Safety-first Ratio
- Lognormal Distribution and Stock Prices
- Discretely Compounded Rate of Return
- Continuously Compounded Rate of Return
- Option Pricing Using Monte Carlo Simulation
- Historical Simulation Vs Monte Carlo Simulation

# Stock Price Movement Using a Binomial Tree

The future price movement of a stock can be approximated using a binomial tree.

Let’s say the current stock price is S. The price of the stock can either move up or move down. We will refer to the up movement as u and the down movement as d. A down movement d will then be given by d where d is assumed to be equal to 1/u. We also need the probability of an up move (p) and the probability of a down move (1 - p).

The following diagram shows the price movement in a small time period Dt.

From each node at Su and Sd, the prices can again take two possible paths.

- Su can move to Suu or Sud. Note that Sud will be the same as S (S*u*1/u).
- Sd can move to Sdu (or just S) or Sdd.

This way a binomial tree can be built for multiple periods. The following diagram shows the binomial tree for 4 periods.

Let’s take an example to understand these values.

Initial stock price, S = $20

u = 1.02

d = 1/1.02

Probability of up move, p = 0.60

Probability of down move, (1-p) = 0.40

The possible stock values for period 1:

- Su = 20*1.02 = 20.40 with a probability of 0.60
- Sd = 19.60 with a probability of 0.40

The possible stock values for period 2:

- Suu = 20.81 with a probability of 0.60*0.60 = 0.36
- S = $20 with a probability of 2*0.60*0.40 = 0.48
- Sdd = 19.22 with a probability 0.40*0.40 = 0.16

Similarly we can calculate the values for period 3 and period 4. Binomial trees have their application in pricing options and other financial concepts.

The following diagram shows this binomial tree upto two periods.