## What Is a Lattice-Based Model

A lattice-based model is used to value derivatives by employing a binomial tree to compute the various paths the price of an underlying asset, such as a stock, might take over the derivative's life. A binomial tree plots out the possible values graphically that option prices can have over different time periods.

Examples of derivatives that can be priced using lattice models include equity options as well as futures contracts for commodities and currencies. The lattice model is particularly suited to the pricing of employee stock options (ESO), which have a number of unique attributes.

### Key Takeaways

- A lattice-based model is used to value derivatives, which are financial instruments that derive their price from an underlying asset.
- Lattice models employ binomial trees to show the different paths the price of an underlying asset might take over the derivative's life.
- Lattice-based models can take into account expected changes in various parameters such as volatility during an option's life.

## Understanding a Lattice-Based Model

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

The lattice-based model's flexibility in incorporating expected volatility changes is especially useful in certain circumstances, such as pricing employee options at early-stage companies. Such companies may expect lower volatility in their stock prices in the future as their businesses mature. The assumption can be factored into a lattice model, enabling more accurate options pricing than the Black-Scholes model, which assumes the same level of volatility over the life of the option.

The binomial options pricing model (BOPM) is a lattice method for valuing 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 models are called "lattice" because the various steps visualized in the model can appear to be woven together 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 depicts the possible paths the derivative's price may 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 holding options expiring in ten years will not exercise them until the expiration date. The assumption is considered a weakness of the model since, in real life, option holders often exercise them well before they expire.

## Example of a Binomial Tree

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

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

The option value 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.