This page discusses all inputs for calculating option prices with binomial models.
Characteristics of the Option
Call or Put Option
A call option represents the right to buy the underlying security. A put option represents a right to sell it.
American or European Option
This refers to the condition of exercise, not to the geographic area where the options are traded. European options can be exercised only at expiration, while American options can be exercised at any time before or at expiration. Unlike Black-Scholes, which can’t model early exercise and therefore can be inaccurate for some American options, binomial models can be accurately used for all European and American options.
The exercise price of the option is the price for which you would buy the underlying security when exercising a call, or for which you would sell the underlying security when exercising a put.
Time to Expiration
Longer time to expiration generally makes options more valuable (though not always for some European options). In binomial models, the period from valuation to expiration is divided into a certain number of steps of equal length. Binomial trees then model how underlying price and option price change from one step to another.
Characteristics of the Underlying Security
Underlying Security Type
Binomial models can price options on equities (stocks or ETFs), indexes, currencies or futures. Different security types may require small adjustments to the way some inputs are treated in the formulas.
This is the current price of the underlying security. When building binomial trees, this price is the origin (first node) of the underlying price tree. When pricing options on futures, it is the futures price, not the price of the ultimate underlying for the futures contract.
Volatility is a measure of how much the underlying price is expected to move between now and the option’s expiration. Higher volatility and bigger expected moves usually make options more valuable. In binomial trees, higher volatility input leads to more widely dispersed underlying prices at different nodes. Like in the Black-Scholes model, volatility is entered as annualized standard deviation of returns.
Typically dividends enter the calculations as continuous percentage yield, but binomial models can also be adjusted to work with discrete dividends (although that can be quite inaccurate with small number of model steps). Dividends generally decrease the values of call options and increase the values of put options.
Like in Black-Scholes, the risk-free interest rate enters binomial models as the cost of financing a position, or as the return on cash. Different interest rates tend to be used for this input, as opinions about which rates are tryly risk-free tend to vary. Examples include the very simple but inaccurate Fed funds rate, LIBOR rates or overnight indexed swap (OIS) rates.
Besides the above listed inputs, the resulting option price also depends on the particular model and the model’s number of steps.
Individual binomial models (e.g. Cox-Ross-Rubinstein, Jarrow-Rudd, Leisen-Reimer) differ in the way up and down moves and probabilities are calculated, and thereby in the prices and probabilities in binomial trees.
Number of Steps
Even with the same model and same inputs, changing number of steps can lead to slightly different results due to the discrete nature of binomial models. In general, higher number of steps produces more accurate option prices, although it is also more computationally expensive. The higher the number of steps, the more binomial models approach continuous models such as Black-Scholes.