## What is a lattice-based model?

A lattice-based model is used to price derivatives by using a binomial tree to calculate the various routes in which the price of an underlying asset, such as a stock, could take control of the life of the derivative. A binomial tree graphically plots the possible values that option prices can have at different time periods.

Examples of derivatives that can be traded using lattice models include stock options, as well as commodity and currency futures contracts. The lattice model is particularly well suited for pricing employee stock options (ESOs), which have a number of unique attributes.

Key takeaways

- A lattice model is used to value derivatives, which are financial instruments that are priced from an underlying asset.
- Lattice models use binomial trees to show the different routes that the price of an underlying asset could take over the life of the derivative.
- Lattice-based models can take into account expected changes in various parameters, such as volatility over the life of an option.

## Understanding a lattice-based model

Lattice-based models can take into account expected changes in various parameters, such as volatility over the life of options. Volatility is a measure of how much the price of an asset fluctuates during a particular period. As a result, lattice models can provide more accurate option price forecasts than the Black-Scholes model, which has been the standard mathematical model for price option contracts.

The flexibility of the lattice-based model to accommodate expected volatility changes is especially useful in certain circumstances, such as option pricing for employees in early-stage companies. These companies can expect less volatility in their share prices in the future as their businesses mature. The assumption can be factored into a lattice model, allowing more accurate option pricing than the Black-Scholes model, which assumes the same level of volatility over the life of the option.

The Binomial Option Pricing Model (BOPM) is a lattice method for pricing options. The first step of the BOPM is to build the binomial tree. The BOPM is based on the underlying asset over a period of time versus a single point in time. These patterns are called “lattice” because the various steps displayed in the model can appear interwoven like a lattice.

## Special Considerations

A lattice model is just one type of model that is used to price derivatives. The name of the model is derived from the appearance of the binomial tree that describes the possible paths that the price of the derivative can take. The Black-Scholes is considered a closed form model, which assumes that the derivative is exercised at the end of its life.

For example, the Black-Scholes model, when pricing stock options, assumes that employees who have options that expire in ten years will not exercise them until the expiration date. The assumption is seen as a weakness of the model since, in real life, option holders often exercise them long before they expire.

## Example of a binomial tree

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

At the end of the year, there is a 50% chance that the stock will rise to $ 125 and a 50% chance that it will fall to $ 90. If the stock rises to $ 125, the value of the option will be $ 25 ( $ 125 share price minus $ 100 strike price) and if it drops to $ 90, the option will be worthless.

The value of the option will be:

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