The TrinomialTree(G, S, , opts) calling sequence constructs a recombining trinomial tree approximating a certain stochastic process. This can be a GeometricBrownianMotion or a mean-reverting process such as OrnsteinUhlenbeckProcess or GaussMarkovProcess. The constructed tree will be based on the discretizations of the time and the state spaces given by G and S.

Assume that the time grid G consists of  points , , ..., . Then the resulting trinomial tree will have  levels, each level representing possible states of the discretized process at time , . The parameter S contains all possible states of the discretized process. The number of elements of S should be equal to , and the elements of S must be sorted in descending order.

At level ,  the tree has  nodes, , ..., . Each node  has three descendants at level ,  (the upper descendant),  (the middle descendant), and  (the lower descendant). The initial state of the underlying process will be equal to . The states of the underlying at the level  are , ..., , , , ..., .

The transition probabilities (i.e. the probability of going from  to , the probability of going from  to , and the probability of going from  to ) are defined by ,  and . The parameter  is optional. By default, the middle probability is calculated so that the total sum of all three probabilities is equal to , that is,  is set to .

The parameters , , and  can be either non-negative real constants or one-parameter operators. If , , or  is given in the operator form the corresponding transition probability at level  will be calculated as , , or  respectively, where .

The TrinomialTree(T, S, , opts) calling sequence is similar except that in this case a uniform time grid with step size  is used instead of G. In this case  will be deduced from the size of the state array S.

Finally, TrinomialTree(T, N, , opts) calling sequence will construct a trinomial tree based on a uniform time grid with step size . Each tree node  will have two descendants  (the upper descendant) and  (the lower descendant). The transition probabilities will be calculated the same way as above.

The resulting data structure can be further manipulated using the Finance[SetUnderlying] and Finance[SetProbabilities] commands.

mutable = truefalse -- This option specifies whether the tree should be mutable or not. The default is true.

The Finance[TrinomialTree] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.

Brigo, D., Mercurio, F., Interest Rate Models: Theory and Practice, New York: Springer-Verlag, 2001.

Glasserman, P., Monte Carlo Methods in Financial Engineering, New York: Springer-Verlag, 2004.

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.

Hull, J., and White, A., Numerical Procedures for Implementing Term Structure Models I: Single-Factor Models, Journal of Derivatives, 1994, 7-16.

Hull, J., and White, A., Using Hull-White Interest Rate Trees, Journal of Derivatives, 1996, 26-36.