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 $T_1$, $T_1$, ..., $T_N$. Then the resulting trinomial tree will have $N$ levels, each level representing possible states of the discretized process at time $T_i$, $i=1..N$. At level $i$, $i=1..N$ the tree has $i$ nodes, $S_{i,1}$, ..., $S_{i,n}$, where $n$ is the number of nodes at level $i$ (see GetSize). Each node $S_{i,j}$ has three descendants at level $i+1$, $S_{i+1,j}$ (the upper descendant), $S_{i+1,j+1}$ (the middle descendant) and $S_{i+1,j+2}$ (the lower descendant). The initial state of the underlying process and the transition probabilities (i.e. the probability of going from $S_{i,j}$ to $S_{i+1,j}$, the probability of going from $S_{i,j}$ to $S_{i+1,j+1}$, and the probability of going from $S_{i,j}$ to $S_{i+1,j+2}$) 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 $\frac{T}{N}$ 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 [4], [5] (see also [1] and [2]). This construction requires that the diffusion term in the corresponding SDE is independent of the state variable X.

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

$M≔\mathrm{VasicekModel}\left(0.05\,0.03\,0.5\,0.03\right)$

$M≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$T≔\mathrm{ShortRateTree}\left(M\,3\,20\right)$

$T≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{TreePlot}\left(T\,\mathrm{thickness}=2\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{GetSize}\left(T\,1\right)$

$1$

$\mathrm{GetSize}\left(T\,2\right)$

$3$

$\mathrm{GetSize}\left(T\,3\right)$

$5$

$\mathrm{GetSize}\left(T\,10\right)$

$15$

$\mathrm{GetSize}\left(T\,11\right)$

$15$

$X≔\mathrm{ItoProcess}\left(0.\,\sin \left(t\right)\,0.05\,x\,t\right)$

$X≔\mathrm{\_X1}$

$T≔\mathrm{ShortRateTree}\left(X\,3.0\,20\right)$

$T≔\mathbf{module}\left(\right)\mathbf{end\; module}$

$\mathrm{TreePlot}\left(T\,\mathrm{thickness}=2\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(X\left(t\right)\,t=0..3\,\mathrm{timesteps}=20\,\mathrm{replications}=10\,\mathrm{thickness}=2\,\mathrm{gridlines}=\mathrm{true}\,\mathrm{axes}=\mathrm{BOXED}\right)$

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.