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 ImpliedTrinomialTree
 construct an implied trinomial tree

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

Parameters

 S - vector or list; state space of the discretized process 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 ${S}_{0}$ - positive constant; inital value of the underlying asset

Description

 • The ImpliedTrinomialTree command constructs an implied trinomial tree based on the given an implied volatility term structure. This is an implementation of an algorithm proposed by E. Derman, I. Kani and N. Chriss in Implied Trinomial Trees of the Volatility Smile, in which the state space of the implied trinomial tree is decided by any method for building constant volatility trinomial trees (we use the method of combining two steps of a CRR binomial tree). Once we have already fixed the state space of the implied trinomial tree, we use induction to infer the transition probabilities, Arrow-Debreu prices, and local volatilities.
 • The ImpliedTrinomialTree(${S}_{0}$, r, d, v, T, N) command is similar except that in this case a uniform time grid with time step $\frac{T}{N}$ is used.
 • The ImpliedTrinomialTree(${S}_{0}$, r, d, v, p, c, G) and ImpliedTrinomialTree(${S}_{0}$, r, d, v, p, c, T, N) commands construct an implied trinomial 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:

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

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

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

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.

 > $\mathrm{GetProbabilities}\left(T,1,1\right)$
 $\left[{0.1746125989}{,}{0.3354287273}{,}{0.4899586738}\right]$ (3)
 > $\mathrm{GetProbabilities}\left(T,2,1\right)$
 $\left[{0.1872660575}{,}{0.3113469958}{,}{0.5013869467}\right]$ (4)
 > $\mathrm{GetProbabilities}\left(T,2,2\right)$
 $\left[{0.2093420029}{,}{0.2693326334}{,}{0.5213253636}\right]$ (5)
 > $\mathrm{GetUnderlying}\left(T,2,3\right)$
 ${110.7206538}$ (6)
 > $\mathrm{GetLocalVolatility}\left(T,2,3,0.01\right)$
 ${0.1278295575}$ (7)
 > $\mathrm{GetLocalVolatility}\left(T,2,3,0.05\right)$
 ${0.1188253679}$ (8)

Here is an implied binomial tree for the same process.

 > $\mathrm{T2}≔\mathrm{ImpliedBinomialTree}\left(100,r,d,\mathrm{σ},3,14\right):$
 > $\mathrm{plots}[\mathrm{display}]\left(\left[\mathrm{TreePlot}\left(T,\mathrm{thickness}=2,\mathrm{color}=\mathrm{red}\right),\mathrm{TreePlot}\left(\mathrm{T2},\mathrm{thickness}=2,\mathrm{color}=\mathrm{blue}\right)\right],\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)$

Compare this tree with a simple trinomial tree based on joining two steps of the standard Cox-Ross-Rubinstein tree constructed for the volatility equal to sigma(0, 100).

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

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[ImpliedTrinomialTree] command was introduced in Maple 15.