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Finance[BlackScholesBinomialTree] - create a binomial tree approximating a Black-Scholes process

Calling Sequence

BlackScholesBinomialTree(S0, r, d, v, T, N)

BlackScholesBinomialTree(S0, r, d, v, G)

Parameters

S0

-

positive constant; inital value of the underlying asset

r

-

non-negative constant or yield term structure; annual risk-free rate function for the underlying asset

d

-

non-negative constant or yield term structure; annual dividend rate function for the underlying asset

v

-

non-negative constant or a local volatility term structure; local volatility

T

-

positive constant; time to maturity date (in years)

N

-

positive integer; number of steps

G

-

the number of steps used in the binomial tree

Description

• 

The BlackScholesBinomialTree(S0, r, d, v, G) calling sequence returns a binomial tree approximating a Black-Scholes process with the specified parameters. When r, d, and v are constant and the time grid is homogeneous, the BlackScholesBinomialTree constructs the standard Cox, Ross, and Rubinstein binomial tree. In the general case the binomial tree is constructed as follows:

• 

Assume that the time grid G consists of N points T1, T2, ..., 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. At level i, i=1..N the tree has i nodes, Si,1, ..., Si,i. The initial state of the discretized process will be equal to S0. Each node Si,j has two descendants at level i+1, Si+1,j=Si,jSu (the upper descendant), and Si+1,j+1=Si,jSd (the lower descendant), where Su=ⅇvTidt and Sd=1Su. Note that the value of the local volatility must be independent of the value of the underlying process.

• 

The transition probabilities Pu=ⅇrTiSuSuSd (the probability of going from Si,j to Si+1,j) and Pd=1Pu (the probability of going from Si,j to Si+1,j+1).

• 

The BlackScholesBinomialTree(S0, r, d, v, T, N) calling sequence is similar except that in this case a uniform time grid with step size TN is used instead of G.

Examples

withFinance:

First construct a binomial tree for a Black-Scholes process with constant drift and volatility.

S0:=100:

r:=0.1:

d:=0.05:

v:=0.15:

T0:=BlackScholesBinomialTreeS0,r,d,v,3,10:

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

TreePlotT0,thickness=2,axes=BOXED,gridlines=true

TreePlotT0,thickness=2,axes=BOXED,gridlines=true,color=red,scale=logarithmic

Inspect the tree.

GetUnderlyingT0,2,1

108.5627742

(1)

GetUnderlyingT0,2,2

92.11260557

(2)

GetProbabilitiesT0,1,1

0.5713437437,0.4286562563

(3)

Here is an example of a Black-Scholes process with time-dependent drift and volatility.

v:=LocalVolatilitySurface0.15t0.01,t,K:

T1:=BlackScholesBinomialTreeS0,r,d,v,3,10:

Again, here are two different views of the same tree; the first one uses the standard scale, the second one uses the logarithmic scale.

TreePlotT1,thickness=2,axes=BOXED,gridlines=true

TreePlotT1,thickness=2,axes=BOXED,gridlines=true,color=red,scale=logarithmic

Inspect the second tree.

GetUnderlyingT1,2,1

108.3845338

(4)

GetUnderlyingT1,2,2

92.26408648

(5)

GetProbabilitiesT1,1,1

0.5713437437,0.4286562563

(6)

GetProbabilitiesT1,2,2

0.5736329667,0.4263670333

(7)

Compare the two trees.

P1:=TreePlotT0,thickness=2,axes=BOXED,gridlines=true,color=blue:

P2:=TreePlotT1,thickness=2,axes=BOXED,gridlines=true,color=red:

plots[display]P1,P2

See Also

Finance[BinomialTree], 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]

References

  

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.


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