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.

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

$r≔0.11$

$r≔0.11$

$d≔0.04$

$d≔0.04$

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

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

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

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

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

$\left[0.1746125989\,0.3354287273\,0.4899586738\right]$

$\mathrm{GetProbabilities}\left(T\,2\,1\right)$

$\left[0.1872660575\,0.3113469958\,0.5013869467\right]$

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

$\left[0.2093420029\,0.2693326334\,0.5213253636\right]$

$\mathrm{GetUnderlying}\left(T\,2\,3\right)$

$110.7206538$

$\mathrm{GetLocalVolatility}\left(T\,2\,3\,0.01\right)$

$0.1278295575$

$\mathrm{GetLocalVolatility}\left(T\,2\,3\,0.05\right)$

$0.1188253679$

$\mathrm{T2}≔\mathrm{ImpliedBinomialTree}\left(100\,r\,d\,\mathrm{\sigma }\,3\,14\right)\:$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\left[\mathrm{TreePlot}\left(T\,\mathrm{thickness}=2\,\mathrm{color}=\mathrm{red}\right)\,\mathrm{TreePlot}\left(\mathrm{T2}\,\mathrm{thickness}=2\,\mathrm{color}=\mathrm{blue}\right)\right]\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{T2}≔\mathrm{BlackScholesTrinomialTree}\left(100\,r\,d\,\mathrm{\sigma }\left(0\,100\right)\,3\,7\right)\:$

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

$\mathrm{P2}≔\mathrm{TreePlot}\left(\mathrm{T2}\,\mathrm{thickness}=2\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\,\mathrm{color}=\mathrm{red}\right)\:$

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

$\mathrm{P3}≔\mathrm{TreePlot}\left(T\,\mathrm{thickness}=2\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\,\mathrm{color}=\mathrm{blue}\,\mathrm{scale}=\mathrm{logarithmic}\right)\:$

$\mathrm{P4}≔\mathrm{TreePlot}\left(\mathrm{T2}\,\mathrm{thickness}=2\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\,\mathrm{color}=\mathrm{red}\,\mathrm{scale}=\mathrm{logarithmic}\right)\:$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{P3}\,\mathrm{P4}\right)$

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.

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

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