construct a recombining binomial tree
BinomialTree(G, S, Pu, Pd, opts)
BinomialTree(T, S, Pu, Pd, opts)
BinomialTree(T, N, S0, Su, Pu, Sd, Pd, opts)
time grid data structure; time grid
Array or list; state space of the discretized process
non-negative constant or operator; probability of going up
(optional) non-negative constant or operator; probability of going down
positive; stopping time
posint; number of times steps
positive constant; initial value
positive constant; upward movement
(optional) positive constant; downward movement
(optional) equation(s) of the form option = value where option is mutable; specify options for the BinomialTree command
mutable = truefalse -- This option specifies whether the tree should be mutable or not. The default is true.
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 2⁢N−1, 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 SN−i, SN−i+2, ..., SN−2, SN, SN+2, ..., SN+i−2, SN+i. For even i, the states of the underlying at the level i are SN−i, SN−i+2, ..., SN−1, SN+1, ..., SN+i−2, 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 Pu⁡dt and Pd⁡dt respectively, where dt=Ti+1−Ti.
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,j⁢Su (the upper descendant) and Si+1,j+1=Si,j⁢Sd (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 1−Pu.
The resulting data structure can be inspected using the GetUnderlying and GetProbabilities commands and can be further manipulated using the SetUnderlying and SetProbabilities commands.
S ≔ 7.9,7.5,7.1,6.5,5.,3.7,3.3,2.95,2.8
T ≔ BinomialTree⁡3,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.
Inspect the tree.
Change the value of the underlying at the uppermost node on level 5 and compare the two trees.
P1 ≔ TreePlot⁡T,thickness=2,axes=BOXED,gridlines=true:
P2 ≔ TreePlot⁡T,thickness=2,axes=BOXED,color=red,gridlines=true:
Here is the same example as above but using a non-homogeneous time grid.
G ≔ TimeGrid⁡0,1.5,2.0,2.5,3.0
T ≔ BinomialTree⁡G,S,0.3:
In this example you will use the third construction.
Su ≔ 1.1
Sd ≔ 0.95
T ≔ BinomialTree⁡3,20,100,Su,0.3,Sd,0.7:
Construct an immutable tree.
T2 ≔ BinomialTree⁡3,20,100,Su,0.3,Sd,0.7,mutable=false:
Use the default values for Sd and Pd.
T3 ≔ BinomialTree⁡3,20,100,Su,0.3,mutable=false:
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.
The Finance[BinomialTree] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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