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.

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

$\mathrm{S0}≔100\:$

$r≔0.1\:$

$d≔0.05\:$

$v≔0.15\:$

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

$\mathrm{TreePlot}\left(\mathrm{T0}\,\mathrm{thickness}=2\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

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

$\mathrm{GetUnderlying}\left(\mathrm{T0}\,2\,1\right)$

$108.5627742$

$\mathrm{GetUnderlying}\left(\mathrm{T0}\,2\,2\right)$

$92.11260557$

$\mathrm{GetProbabilities}\left(\mathrm{T0}\,1\,1\right)$

$\left[0.5713437437\,0.4286562563\right]$

$v≔\mathrm{LocalVolatilitySurface}\left(0.15-t\cdot 0.01\,t\,K\right)\:$

$\mathrm{T1}≔\mathrm{BlackScholesBinomialTree}\left(\mathrm{S0}\,r\,d\,v\,3\,10\right)\:$

$\mathrm{TreePlot}\left(\mathrm{T1}\,\mathrm{thickness}=2\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

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

$\mathrm{GetUnderlying}\left(\mathrm{T1}\,2\,1\right)$

$108.3845338$

$\mathrm{GetUnderlying}\left(\mathrm{T1}\,2\,2\right)$

$92.26408648$

$\mathrm{GetProbabilities}\left(\mathrm{T1}\,1\,1\right)$

$\left[0.5713437437\,0.4286562563\right]$

$\mathrm{GetProbabilities}\left(\mathrm{T1}\,2\,2\right)$

$\left[0.5736329667\,0.4263670333\right]$

$\mathrm{P1}≔\mathrm{TreePlot}\left(\mathrm{T0}\,\mathrm{thickness}=2\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\,\mathrm{color}=\mathrm{blue}\right)\:$

$\mathrm{P2}≔\mathrm{TreePlot}\left(\mathrm{T1}\,\mathrm{thickness}=2\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\,\mathrm{color}=\mathrm{red}\right)\:$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{P1}\,\mathrm{P2}\right)$

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

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

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