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Finance

  

BinomialTree

  

construct a recombining binomial tree

 

Calling Sequence

Parameters

Options

Description

Examples

References

Compatibility

Calling Sequence

BinomialTree(G, S, Pu, Pd, opts)

BinomialTree(T, S, Pu, Pd, opts)

BinomialTree(T, N, S0, Su, Pu, Sd, Pd, opts)

Parameters

G

-

time grid data structure; time grid

S

-

Array or list; state space of the discretized process

Pu

-

non-negative constant or operator; probability of going up

Pd

-

(optional) non-negative constant or operator; probability of going down

T

-

positive; stopping time

N

-

posint; number of times steps

S0

-

positive constant; initial value

Su

-

positive constant; upward movement

Sd

-

(optional) positive constant; downward movement

opts

-

(optional) equation(s) of the form option = value where option is mutable; specify options for the BinomialTree command

Options

• 

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

Description

• 

The BinomialTree(G, S, Pu, Pd, 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 T1, T1, ..., TN. Then the resulting binomial tree will have N levels, each level representing possible states of the discretized process at time Ti, i=1..N. The parameter S contains all possible states of the discretized process. The number of elements of S should be equal to 2N1, and the elements of S must be sorted in descending order.

• 

At level i, i=1..N the tree has i nodes, Si,1, ..., Si,i. Each node Si,j has two descendants at level i+1, Si+1,j (the upper descendant), and Si+1,j+1 (the lower descendant). The initial state of the underlying process will be equal to SN. For odd i, the states of the underlying at the level i are SNi, SNi+2, ..., SN2, SN, SN+2, ..., SN+i2, SN+i. For even i, the states of the underlying at the level i are SNi, SNi+2, ..., SN1, SN+1, ..., SN+i2, SN+i.

• 

The transition probabilities (i.e. the probability of going from Si,j to Si+1,j and the probability of going from Si,j to Si+1,j+1) are defined by Pu and Pd. Both Pu and Pd can be either non-negative real constants or one-parameter operators. If Pu and  Pd are given in the operator form the corresponding transition probabilities at level i will be calculated as P[u]dt and P[d]dt respectively, where dt=Ti+1Ti.

• 

The BinomialTree(T, S, Pu, Pd, opts) calling sequence is similar except that in this case a uniform time grid with step size TN is used instead of G. In this case N will be deduced from the size of the state array S.

• 

The BinomialTree(T, N, S0, Su, Pu, Sd, Pd, opts) calling sequence will construct a binomial tree based on a uniform time grid with step size TN. Each tree node Si,j will have two descendants Si+1,j=Si,jSu (the upper descendant) and Si+1,j+1=Si,jSd (the lower descendant). The transition probabilities will be calculated the same way as above. By default Sd is set to 1Su and Pd is set to 1Pu.

• 

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

Examples

withFinance:

S7.9,7.5,7.1,6.5,5.,3.7,3.3,2.95,2.8

S:=7.9,7.5,7.1,6.5,5.,3.7,3.3,2.95,2.8

(1)

TBinomialTree3,S,0.3:

Here are two different views of the same tree. The first one uses the standard scale, the second one uses the logarithmic scale.

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

TreePlotT,thickness=2,axes=BOXED,gridlines=true,color=red..blue,scale=logarithmic

Inspect the tree.

GetUnderlyingT,5,1

7.900000000

(2)

GetProbabilitiesT,3,1

0.3000000000,0.7000000000

(3)

Change the value of the underlying at the uppermost node on level 5 and compare the two trees.

P1TreePlotT,thickness=2,axes=BOXED,gridlines=true:

SetUnderlyingT,5,1,8.5

P2TreePlotT,thickness=2,axes=BOXED,color=red,gridlines=true:

plots[display]P1,P2

Here is the same example as above but using a non-homogeneous time grid.

GTimeGrid0,1.5,2.0,2.5,3.0

G:=moduleend module

(4)

S7.9,7.5,7.1,6.5,5.,3.7,3.3,2.95,2.8

S:=7.9,7.5,7.1,6.5,5.,3.7,3.3,2.95,2.8

(5)

TBinomialTreeG,S,0.3:

Here are two different views of the same tree. The first one uses the standard scale, the second one uses the logarithmic scale.

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

TreePlotT,thickness=2,axes=BOXED,gridlines=true,color=red..blue,scale=logarithmic

Inspect the tree.

GetUnderlyingT,5,1

7.900000000

(6)

GetProbabilitiesT,3,1

0.3000000000,0.7000000000

(7)

Change the value of the underlying at the uppermost node on level 5 and compare the two trees.

P1TreePlotT,thickness=2,axes=BOXED,gridlines=true:

SetUnderlyingT,5,1,8.5

P2TreePlotT,thickness=2,axes=BOXED,color=red,gridlines=true:

plots[display]P1,P2

In this example you will use the third construction.

Su1.1

Su:=1.1

(8)

Sd0.95

Sd:=0.95

(9)

TBinomialTree3,20,100,Su,0.3,Sd,0.7:

Here are two different views of the same tree. The first one uses the standard scale, the second one uses the logarithmic scale.

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

TreePlotT,thickness=2,axes=BOXED,gridlines=true,color=red..blue,scale=logarithmic

Construct an immutable tree.

T2BinomialTree3,20,100,Su,0.3,Sd,0.7,mutable=false:

plots[display]TreePlotT2,thickness=2,axes=BOXED,gridlines=true,color=red,TreePlotT,thickness=2,axes=BOXED,gridlines=true,color=blue

Use the default values for Sd and Pd.

T3BinomialTree3,20,100,Su,0.3,mutable=false:

plots[display]TreePlotT3,thickness=2,axes=BOXED,gridlines=true,color=red,TreePlotT,thickness=2,axes=BOXED,gridlines=true,color=blue

Inspect the tree.

GetUnderlyingT,5,1

146.4100000

(10)

GetProbabilitiesT,3,1

0.3000000000,0.7000000000

(11)

Change the value of the underlying at the uppermost node on level 5 and compare the two trees.

P1TreePlotT,thickness=2,axes=BOXED,gridlines=true:

forito20doSetUnderlyingT,i,i,1000.9i1end do

P2TreePlotT,thickness=2,axes=BOXED,color=red,gridlines=true:

plots[display]P1,P2

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.

Compatibility

• 

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

• 

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

See Also

Finance[BlackScholesBinomialTree]

Finance[BlackScholesTrinomialTree]

Finance[GetDescendants]

Finance[GetProbabilities]

Finance[GetUnderlying]

Finance[ImpliedBinomialTree]

Finance[ImpliedTrinomialTree]

Finance[LatticeMethods]

Finance[SetProbabilities]

Finance[SetUnderlying]

Finance[StochasticProcesses]

Finance[TreePlot]

Finance[TrinomialTree]

 


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