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

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Finance[ShortRateTree] - return a trinomial tree approximating the evolution of the instantaneous rate in the given model

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.

M:=VasicekModel0.05,0.03,0.5,0.03

M:=moduleend module

(1)

T:=ShortRateTreeM,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.

X:=ItoProcess0.0,sint,0.05,x,t

X:=_X0

(8)

T:=ShortRateTreeX,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

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]

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.


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