return a trinomial tree approximating the evolution of the instantaneous rate in the given model
ShortRateTree(M, T, N)
ShortRateTree(X, T, N)
short-rate model data structure; short-rate model
time grid data structure; time grid
positive; stopping time
posint; number of times steps
stochastic process; process to be approximated
The ShortRateTree(M, G) calling sequence returns a trinomial tree approximating the stochastic process that represents the instantaneous spot rate in the given short-rate model. The constructed tree will be based on the time discretization given by G.
Assume that the time grid G consists of N points T1, T1, ..., TN. Then the resulting trinomial tree will have N levels, each level representing possible states of the discretized process at time Ti, i=1..N. At level i, i=1..N the tree has i nodes, Si,1, ..., Si,n, where n is the number of nodes at level i (see GetSize). Each node Si,j has three descendants at level i+1, Si+1,j (the upper descendant), Si+1,j+1 (the middle descendant) and Si+1,j+2 (the lower descendant). The initial state of the underlying process and the transition probabilities (i.e. the probability of going from Si,j to Si+1,j, the probability of going from Si,j to Si+1,j+1, and the probability of going from Si,j to Si+1,j+2) will be calculated based on the given model.
The ShortRateTree(M, T, N) calling sequence is similar except that in this case a uniform time grid with step size TN is used instead of G.
The ShortRateTree(X, G) and ShortRateTree(X, T, N) commands construct a trinomial tree approximating an Ito process X. This tree is constructed using the procedure proposed by Hull and White ,  (see also  and ). This construction requires that the diffusion term in the corresponding SDE is independent of the state variable X.
Construct a trinomial for the Vasicek model.
M ≔ VasicekModel⁡0.05,0.03,0.5,0.03
M ≔ moduleend module
T ≔ ShortRateTree⁡M,3,20
T ≔ moduleend module
Construct a trinomial tree approximating a given Ito process.
X ≔ ItoProcess⁡0.0,sin⁡t,0.05,x,t
T ≔ ShortRateTree⁡X,3.0,20
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.
The Finance[ShortRateTree] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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