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Implied Trinomial Trees of the Volatility Smile (The Journal of Derivatives)

In options markets where there is a significant or persis- tent volatility smile, implied tree models can ensure the consistency of exotic options prices with the market prices of liquid standard options.

Implied trees can be constructed in a variety of ways. Implied binomial trees are minimal: they have just enough parameters – node prices and transition probabil- ities – to fit the smile. In this paper we show how to build implied trinomial tree models of the volatility smile. Tri- nomial trees have inherently more parameters than bino- mial trees. We can use these additional parameters to conveniently choose the “state space” of all node prices in the trinomial tree, and let only the transition probabili- ties be constrained by market options prices. This free- dom of state space provides a flexibility that is sometimes advantageous in matching trees to smiles.

A judicially chosen state space is needed to obtain a rea- sonable fit to the smile. We discuss a simple method for building “skewed” state spaces which fit typical index option smiles rather well.

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