Lattice model (finance)

A Lattice Model, in finance, is a numerical technique used for pricing options and other derivatives. It works by discretizing the time to expiration into a finite number of time steps and modeling the possible paths the underlying asset's price can take over those steps. These paths are organized in a branching structure resembling a lattice or tree.

The core concept is to approximate the continuous price movements of the underlying asset with a series of discrete movements, either upwards or downwards, at each time step. The probabilities of these movements are chosen to match certain characteristics of the asset's price behavior, such as its volatility.

At each node of the lattice, representing a possible asset price at a specific time, the value of the derivative is calculated recursively, starting from the expiration date and working backwards to the present time. The value at the expiration date is determined by the payoff function of the derivative. Earlier nodes are valued based on the expected value of the derivative at the next time step, discounted back to the present. This process uses risk-neutral valuation, meaning the expected return on the underlying asset is assumed to be the risk-free rate.

Two common types of lattice models are the binomial tree model and the trinomial tree model. The binomial model allows for two possible price movements (up or down) at each time step, while the trinomial model allows for three (up, down, or stay the same).

Lattice models are particularly useful for pricing options with complex features such as American-style options (which can be exercised at any time before expiration), path-dependent options (whose payoff depends on the history of the underlying asset price), and options on assets with discrete dividends.

While more computationally intensive than closed-form solutions like the Black-Scholes model for simple European options, lattice models offer greater flexibility and accuracy for a wider range of derivative pricing problems. The accuracy of a lattice model generally increases with the number of time steps used, at the cost of increased computational time. Factors such as the choice of the step size, the probabilities of upward and downward movements, and the method of discounting can influence the accuracy and stability of the model.