# Lattice model

In physics, a lattice model is a physical model that is not defined on a continuum, but on a lattice, which is a graph or an n-complex approximating spacetime or space. The lattice is translationally invariant if it is generated by a subgroup of translations acting upon a finite number of points. Of course, any arbitrary triangulation would do as well.

In condensed matter physics, the atoms of a crystal automatically form a lattice and this is one application of lattice theory.

See Ising model, XY model.

Another is as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory to prevent divergences or to perform numerical computations.

Numerical computations for partial differential equations are lattice models because computers only work with discrete data.

In finance, a lattice model can be used to find the fair value of a stock option. The model divides time between now and the option's expiration into N discrete periods. At the specific time n, the model has an infinite number of outcomes at time n + 1 such that every possible change in the state of the world between n and n + 1 is captured in a branch. This process is iterated until every possible path between n = 0 and n = N is mapped. Probabilities are then estimated for every n to n + 1 path. The outcomes and probabilities flow backwards through the tree until a fair value of the option today is calculated. A simple lattice model for options is the binomial pricing model (see also binomial (disambiguation)).

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