Lattice Methods

Lattice methods (or trees) approximate the price path of underlying assets, by discretizing both the state space and time. This is done in a way that matches the characteristics of the corresponding asset diffusion. Such methods are primarily used as numerical methods to price American options and other exotic contracts.

Rather than allowing the asset price to take any (perhaps non-negative) value, we only permit a finite number of values. In addition, we assume that price moves can take place only on discrete points in time.

For example, say that the price today (at \fs4 t_0) is \fs4 S_0. We might allow the price at time \fs4 t_1 = t_0 + \delta_1 to be in the set \fs4 (S_1^1, S_1^2,\ldots,S_1^{N_1}); the price at time \fs4 t_2 = t_1 + \delta_2 to be in the set \fs4 (S_2^1, S_2^2,\ldots,S_2^{N_2}); etc. We also specify the probabilities of moving from each node to another.

The most widely used lattice models are

  • the binomial model, where each node connects to two other nodes, allowing the price to go up or down, and
  • the trinomial model, where each node connects to three nodes, allowing a middle scenario as well

Practitionars and academics often construct implied trees, lattices that agree with a set of observed European option prices, or a set of implied volatility skews. They subsequently use these lattices to price exotic options in a way that is consistent with their European counterparts.

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