The BinomialTree(G, S, , , opts) calling sequence constructs a recombining binomial tree approximating a certain stochastic process, typically a GeometricBrownianMotion. 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 N points , , ..., . Then the resulting binomial tree will have N 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 two descendants at level ,  (the upper descendant), and  (the lower descendant). The initial state of the underlying process will be equal to . For odd , the states of the underlying at the level  are , , ..., , , , ..., , . For even , the states of the underlying at the level  are , , ..., , , ..., , .

The transition probabilities (i.e. the probability of going from  to  and the probability of going from  to ) are defineed by  and . Both  and  can be either non-negative real constants or one-parameter operators. If  and   are given in the operator form the corresponding transition probabilities at level  will be calculated as  and  respectively, where .

The BinomialTree(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 N will be deduced from the size of the state array S.

The BinomialTree(T, N, , , , , , opts) calling sequence will construct a binomial 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. By default  is set to  and  is set to .

The resulting data structure can be inspected using the GetUnderlying and GetProbabilities commands and can be further manipulated using the SetUnderlying and SetProbabilities commands.

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

The Finance[BinomialTree] 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.