Implied Trinomial Trees

Abstract

Implied trinomial trees (ITTs) present an analogous extension of trinomial trees proposed by Derman, Kani, and Chriss (1996). Like their binomial counterparts, they can fit the market volatility smile and actually converge to the same continuous limit as binomial trees. In addition, they allow for a free choice of the underlying prices at each node of a tree, the so-called state space. This feature of ITTs allows to improve the fit of the volatility smile under some circumstances such as inconsistent, arbitrage-violating, or other market prices leading to implausible or degenerated probability distributions in binomial trees. We introduce ITTs in several steps. We first review main concepts regarding option pricing (Section 1) and implied models (Section 2). Later, we discuss the construction of ITTs (Section 3) and provide some illustrative examples (Section 4).