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Finance

  

ShortRateTree

  

return a trinomial tree approximating the evolution of the instantaneous rate in the given model

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

ShortRateTree(M, G)

ShortRateTree(M, T, N)

ShortRateTree(X, G)

ShortRateTree(X, T, N)

Parameters

M

-

short-rate model data structure; short-rate model

G

-

time grid data structure; time grid

T

-

positive; stopping time

N

-

posint; number of times steps

X

-

stochastic process; process to be approximated

Description

• 

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 [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.

Examples

withFinance:

Construct a trinomial for the Vasicek model.

MVasicekModel0.05,0.03,0.5,0.03

M:=moduleend module

(1)

TShortRateTreeM,3,20

T:=moduleend module

(2)

TreePlotT,thickness=2,axes=BOXED,gridlines=true

GetSizeT,1

1

(3)

GetSizeT,2

3

(4)

GetSizeT,3

5

(5)

GetSizeT,10

15

(6)

GetSizeT,11

15

(7)

Construct a trinomial tree approximating a given Ito process.

XItoProcess0.0,sint,0.05,x,t

X:=_X0

(8)

TShortRateTreeX,3.0,20

T:=moduleend module

(9)

TreePlotT,thickness=2,axes=BOXED,gridlines=true

PathPlotXt,t=0..3,timesteps=20,replications=10,thickness=2,gridlines=true,axes=BOXED

References

  

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.

Compatibility

• 

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

• 

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

See Also

Finance[BinomialTree]

Finance[BlackScholesBinomialTree]

Finance[BlackScholesTrinomialTree]

Finance[CoxIngersollRossModel]

Finance[GetDescendants]

Finance[GetProbabilities]

Finance[GetUnderlying]

Finance[HullWhiteModel]

Finance[ImpliedBinomialTree]

Finance[ImpliedTrinomialTree]

Finance[LatticeMethods]

Finance[SetProbabilities]

Finance[SetUnderlying]

Finance[ShortRateProcess]

Finance[StochasticProcesses]

Finance[TreePlot]

 


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