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 ImpliedBinomialTree
 construct an implied binomial tree

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

Parameters

 ${S}_{0}$ - positive constant; initial value of the underlying asset r - non-negative constant; annual risk-free rate function for the underlying asset d - non-negative constant; annual dividend rate function for the underlying asset v - implied volatility term structure; implied volatility G - time grid data structure; time grid T - positive constant; time to maturity date (in years) N - positive integer; number of steps p - procedure; a procedure for calculating the price of a European put option c - procedure; a procedure for calculating the price of a European call option

Description

 • The ImpliedBinomialTree(${S}_{0}$, r, d, v, G) command constructs an implied binomial tree based on the given and implied volatility term structure. This is an implementation of an algorithm proposed by E. Derman and I. Kani (1994) in which the state space of the implied binomial tree is decided by any method for building constant volatility binomial trees (we use the CRR binomial tree). Once we have already fixed the state space of the implied binomial tree, we use induction to infer the transition probabilities and local volatilities.
 • The ImpliedBinomialTree(${S}_{0}$, r, d, v, T, N) calling sequence is similar except that in this case a uniform time grid with time step $\frac{T}{N}$ is used.
 • The ImpliedBinomialTree(${S}_{0}$, r, d, v, p, c, G) and ImpliedBinomialTree(${S}_{0}$, r, d, v, p, c, T, N) calling sequences construct an implied binomial tree given two pricing functions: p, which, given a strike price and time to maturity, computes the price of a European put option for the underlying asset; and c, which computes the price of a European call option.

Examples

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

r is the risk-free interest rate, compounded annually:

 > $r≔0.11$
 ${r}{≔}{0.11}$ (1)

d is the annual dividend rate:

 > $d≔0.04$
 ${d}{≔}{0.04}$ (2)

The following implied volatility surface uses the volatility implied by the market price, computed using the Newton-Raphson method:

 > $\mathrm{\sigma }≔\mathrm{ImpliedVolatilitySurface}\left(0.11-\frac{K-100}{10}\cdot 0.001,t,K\right):$

The implied binomial tree is then computed using time to maturity, 3, and number of steps, 7:

 > $T≔\mathrm{ImpliedBinomialTree}\left(100,r,d,\mathrm{\sigma },3,7\right):$

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

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

Inspect the tree.

 > $\mathrm{GetProbabilities}\left(T,1,1\right)$
 $\left[{0.5000000000}{,}{0.5000000000}\right]$ (3)
 > $\mathrm{GetProbabilities}\left(T,2,1\right)$
 $\left[{0.3817424623}{,}{0.6182575377}\right]$ (4)
 > $\mathrm{GetProbabilities}\left(T,2,2\right)$
 $\left[{0.6409710334}{,}{0.3590289666}\right]$ (5)
 > $\mathrm{GetUnderlying}\left(T,2,2\right)$
 ${109.1225607}$ (6)
 > $\mathrm{GetLocalVolatility}\left(T,2,2,0.01\right)$
 ${0.1232257814}$ (7)
 > $\mathrm{GetLocalVolatility}\left(T,2,2,0.05\right)$
 ${0.1153946887}$ (8)

Compare this tree with the standard Cox-Ross-Rubinstein binomial tree constructed for the volatility equal to sigma(0, 100).

 > $\mathrm{T2}≔\mathrm{BlackScholesBinomialTree}\left(100,r,d,\mathrm{\sigma }\left(0,100\right),3,7\right):$
 > $\mathrm{P1}≔\mathrm{TreePlot}\left(T,\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{blue}\right):$
 > $\mathrm{P2}≔\mathrm{TreePlot}\left(\mathrm{T2},\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{red}\right):$
 > $\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{P1},\mathrm{P2}\right)$
 > $\mathrm{P3}≔\mathrm{TreePlot}\left(T,\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{blue},\mathrm{scale}=\mathrm{logarithmic}\right):$
 > $\mathrm{P4}≔\mathrm{TreePlot}\left(\mathrm{T2},\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true},\mathrm{color}=\mathrm{red},\mathrm{scale}=\mathrm{logarithmic}\right):$
 > $\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{P3},\mathrm{P4}\right)$

References

 Cizek, P., and Komorad, K., Implied Trinomial Trees, SFB 649 Economic Risk, Berlin, 2005-07.
 Derman, E., and Kani, I., The Volatility Smile and Its Implied Tree, Goldman Sachs Quantitative Strategies Research Notes, January 1994.
 Derman, E., Kani, I., Chriss, N., Implied Trinomial Trees of the Volatility Smile, Goldman Sachs Quantitative Strategies Research Notes, February 1996.
 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.
 Jackwerth, J.C., Option-Implied Risk-Neutral Distributions and Implied Binomial Trees: A Literature Review, 1999.
 Rubinstein, M., Implied binomial trees, J. Finance, 49 ,1994, pp. 771--818.

Compatibility

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