Options are financial derivatives that, conditional on the price of an underlying asset, constitute a right to transfer the ownership of this underlying. More specifically, a European call and put options give their owner the right to buy and sell, respectively, at a fixed strike price at a given date. Options are important financial instruments used for hedging since they can be included into a portfolio to reduce risk. Corporate securities (e.g., bonds or stocks) may include option features as well. Last, but not least, some new financing techniques, such as contingent value rights, are straightforward applications of options. Thus, option pricing has become one of the basic techniques in finance.
The boom in research on the use of options started after Black and Scholes (1973) published an option-pricing formula based on geometric Brownian motion. Option prices computed by the Black-Scholes formula and the market prices of options exhibit a discrepancy though. Whereas the volatility of market option prices varies with the price (or moneyness) - the dependency referred to as the volatility smile, the Black-Scholes model is based on the assumption of a constant volatility. Therefore, to model option prices consistent with the market many new approaches were proposed. Probably the most commonly used and rather intuitive procedure for option pricing is based on binomial trees, which represent a discrete form of the Black-Scholes model. To fit the market data, Derman and Kani (1994) proposed an extension of binomial trees: the so-called implied binomial trees, which are able to model the market volatility smile.
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 6.1) and implied models (Section 6.2). Later, we discuss the construction of ITTs (Section 6.3) and provide some illustrative examples (Section 6.4).