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Finance

 BlackScholesBinomialTree
 create a binomial tree approximating a Black-Scholes process

 Calling Sequence BlackScholesBinomialTree(${S}_{0}$, r, d, v, T, N) BlackScholesBinomialTree(${S}_{0}$, r, d, v, G)

Parameters

 ${S}_{0}$ - 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(${S}_{0}$, 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 ${T}_{1}$, ${T}_{2}$, ..., ${T}_{N}$. Then the resulting binomial tree will have $N$ levels, each level representing possible states of the discretized process at time ${T}_{i}$, $i=1..N$. At level $i$, $i=1..N$ the tree has $i$ nodes, ${S}_{i,1}$, ..., ${S}_{i,i}$. The initial state of the discretized process will be equal to ${S}_{0}$. Each node ${S}_{i,j}$ has two descendants at level $i+1$, ${S}_{i+1,j}={S}_{i,j}\mathrm{Su}$ (the upper descendant), and ${S}_{i+1,j+1}={S}_{i,j}\mathrm{Sd}$ (the lower descendant), where $\mathrm{Su}={ⅇ}^{v\left({T}_{i}\right)\sqrt{\mathrm{dt}}}$ and $\mathrm{Sd}=\frac{1}{\mathrm{Su}}$. Note that the value of the local volatility must be independent of the value of the underlying process.
 • The transition probabilities ${P}_{u}=\frac{{ⅇ}^{r\left({T}_{i}\right)}-\mathrm{Su}}{\mathrm{Su}-\mathrm{Sd}}$ (the probability of going from ${S}_{i,j}$ to ${S}_{i+1,j}$) and ${P}_{d}=1-{P}_{u}$ (the probability of going from ${S}_{i,j}$ to ${S}_{i+1,j+1}$).
 • The BlackScholesBinomialTree(${S}_{0}$, r, d, v, T, N) calling sequence is similar except that in this case a uniform time grid with step size $\frac{T}{N}$ is used instead of G.

Examples

 > $\mathrm{with}\left(\mathrm{Finance}\right):$

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

 > $\mathrm{S0}≔100:$
 > $r≔0.1:$
 > $d≔0.05:$
 > $v≔0.15:$
 > $\mathrm{T0}≔\mathrm{BlackScholesBinomialTree}\left(\mathrm{S0},r,d,v,3,10\right):$

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

 > $\mathrm{TreePlot}\left(\mathrm{T0},\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)$
 > $\mathrm{TreePlot}\left(\mathrm{T0},\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{red},\mathrm{scale}=\mathrm{logarithmic}\right)$

Inspect the tree.

 > $\mathrm{GetUnderlying}\left(\mathrm{T0},2,1\right)$
 ${108.5627742}$ (1)
 > $\mathrm{GetUnderlying}\left(\mathrm{T0},2,2\right)$
 ${92.11260557}$ (2)
 > $\mathrm{GetProbabilities}\left(\mathrm{T0},1,1\right)$
 $\left[{0.5713437437}{,}{0.4286562563}\right]$ (3)

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

 > $v≔\mathrm{LocalVolatilitySurface}\left(0.15-t\cdot 0.01,t,K\right):$
 > $\mathrm{T1}≔\mathrm{BlackScholesBinomialTree}\left(\mathrm{S0},r,d,v,3,10\right):$

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

 > $\mathrm{TreePlot}\left(\mathrm{T1},\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)$
 > $\mathrm{TreePlot}\left(\mathrm{T1},\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{red},\mathrm{scale}=\mathrm{logarithmic}\right)$

Inspect the second tree.

 > $\mathrm{GetUnderlying}\left(\mathrm{T1},2,1\right)$
 ${108.3845338}$ (4)
 > $\mathrm{GetUnderlying}\left(\mathrm{T1},2,2\right)$
 ${92.26408648}$ (5)
 > $\mathrm{GetProbabilities}\left(\mathrm{T1},1,1\right)$
 $\left[{0.5713437437}{,}{0.4286562563}\right]$ (6)
 > $\mathrm{GetProbabilities}\left(\mathrm{T1},2,2\right)$
 $\left[{0.5736329667}{,}{0.4263670333}\right]$ (7)

Compare the two trees.

 > $\mathrm{P1}≔\mathrm{TreePlot}\left(\mathrm{T0},\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{blue}\right):$
 > $\mathrm{P2}≔\mathrm{TreePlot}\left(\mathrm{T1},\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{red}\right):$
 > $\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{P1},\mathrm{P2}\right)$

References

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

Compatibility

 • The Finance[BlackScholesBinomialTree] command was introduced in Maple 15.