construct an implied trinomial tree

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Finance[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(, r, d, v, G) ImpliedTrinomialTree(, r, d, v, T, N) ImpliedTrinomialTree(, r, d, p, c, G) ImpliedTrinomialTree(, 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
	-	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(, r, d, v, T, N) command is similar except that in this case a uniform time grid with time step is used.

•

The ImpliedTrinomialTree(, r, d, v, p, c, G) and ImpliedTrinomialTree(, 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.

Compatibility

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

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

Examples

(1)

(2)

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

Inspect the tree.

(3)

(4)

(5)

(6)

(7)

(8)

Here is an implied binomial tree for the same process.

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).