Trinomial tree

The trinomial tree is a lattice-based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also be shown that the approach is equivalent to the explicit finite difference method for option pricing. For fixed income and interest rate derivatives see Lattice model (finance) #Interest rate derivatives.
Formula.
Under the trinomial method, the underlying stock price is modeled as a recombining tree, where, at each node the price has three possible paths: an up, down and stable or middle path. These values are found by multiplying the value at the current node by the appropriate factor formula_1, formula_2 or formula_3 where
and the corresponding probabilities are: 
In the above formulae: formula_10 is the length of time per step in the tree and is simply time to maturity divided by the number of time steps; formula_11 is the risk-free interest rate over this maturity; formula_12 is the corresponding volatility of the underlying; formula_13 is its corresponding dividend yield.
As with the binomial model, these factors and probabilities are specified so as to ensure that the price of the underlying evolves as a martingale, while the moments - considering node spacing and probabilities - are matched to those of the log-normal distribution (and with increasing accuracy for smaller time-steps). Note that for formula_14, formula_15, and formula_16 to be in the interval formula_17 the following condition on formula_18 has to be satisfied formula_19.
Once the tree of prices has been calculated, the option price is found at each node largely as for the binomial model, by working backwards from the final nodes to the present node (formula_20). The difference being that the option value at each non-final node is determined based on the threeas opposed to "two" later nodes and their corresponding probabilities.
If the length of time-steps formula_18 is taken as an exponentially distributed random variable and interpreted as the waiting time between two movements of the stock price then the resulting stochastic process is a birth–death process. The resulting model is soluble and there exist analytic pricing and hedging formulae for various options.
Application.
The trinomial model is considered to produce more accurate results than the binomial model when fewer time steps are modelled, and is therefore used when computational speed or resources may be an issue. For vanilla options, as the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation. For exotic options the trinomial model (or adaptations) is sometimes more stable and accurate, regardless of step-size.