# Binomial Tree

## What Is a Binomial Tree?

A binomial tree is a graphical representation of conceivable intrinsic values that an option might require some investment periods. The value of the option relies upon the underlying stock or bond, and the value of the option at any node relies upon the likelihood that the price of the underlying asset will either diminish or increase at some random node.

## How a Binomial Tree Works

A binomial tree is a helpful device while pricing American options and embedded options. Its simplicity is its advantage and disadvantage simultaneously. The tree is not difficult to model out precisely, yet the problem lies in the potential values the underlying asset can take in one period.

In a binomial tree model, the underlying asset must be worth precisely one of two potential values, which isn't practical, as assets can be worth quite a few values inside some random reach. A binomial tree permits investors to evaluate when and in the event that an option will be worked out. An option has a higher likelihood of being practiced in the event that the option has a positive value.

## Special Considerations

The binomial options pricing model (BOPM) is a method for esteeming options. The initial step of the BOPM is to build the binomial tree. The BOPM depends on the underlying asset throughout some stretch of time versus a single point in time.

There are a couple of major suppositions in a binomial option pricing model. To begin with, there are just two potential prices, one up and one down. Second, the underlying asset pays no dividends. Third, the interest rate is steady, and fourth, there are no taxes and transaction costs.

## Binomial Tree versus Black-Scholes Model

The Black Scholes model is one more method for esteeming options. Computing the price utilizing the binomial tree is more slow than the Black Scholes model. Notwithstanding, the binomial tree and BOPM are more accurate. This is especially true for options that are longer-dated and those securities with dividend payments.

The Black Scholes model is more solid with regards to confounded options and those with heaps of vulnerability. With regards to European options without dividends, the output of the binomial model and Black Scholes model meet as the time steps increase.

## Illustration of a Binomial Tree

Expect a stock has a price of $100, option strike price of $100, one-year expiration date, and interest rate (r) of 5%.

Toward the year's end, there is a half likelihood the stock will rise to $125 and half likelihood it will drop to $90. In the event that the stock rises to $125 the value of the option will be $25 ($125 stock price minus $100 strike price) and assuming it drops to $90 the option will be worthless.

The option value will be:

Option value = [(probability of rise * up value) + (likelihood of drop * down value)]/(1 + r) = [(0.50 * $25) + (0.50 * $0)]/(1 + 0.05) = $11.90.

## Features

- A binomial tree is a representation of the intrinsic values an option might take at various time spans.
- On the downside â€” an underlying asset must be worth precisely one of two potential values, which isn't practical.
- The value of the option at any node relies upon the likelihood that the price of the underlying asset will either diminish or increase at some random node.