Binomial Option Pricing Model

Another options valuation model developed by Cox, et al, in 1979.  The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option's expiration date.

The model  makes following assumptions:

  • No price changes,
  • No arbitrage,
  • Efficient market, short duration of options.

Under these assumptions, it is able to provide a mathematical valuation of the option via a binomial lattice (tree), at each point in time specified. A simplified example of a binomial tree might look something like this:


Each node in the lattice, represents a possible price of the underlying, at a particular point in time.

The valuation process is iterative, starting at each final node, and then working backwards through the tree to the first node (valuation date), where the calculated result is the value of the option.

Option valuation using this method is, as described, a three step process:

  1. Price tree generation
  2. Calculation of option value at each final node
  3. Progressive calculation of option value at each earlier node; the value at the first node is the value of the option.

The tree of prices is produced by working forward from valuation date to expiration

At each step, it is assumed that the underlying instrument will move up or down by a specific factor (u or d) per step of the tree (where, by definition, image and image ). So, if S is the current price, then in the next period the price will either be S_{up} = S \cdot u or S_{down} = S \cdot d.

The up and down factors are calculated using the underlying volatility, σ and the time duration of a step, t, measured in years (using the day count convention of the underlying instrument). From the condition that the variance of the log of the price is σ2t, we have:

u = e^{\sigma\sqrt t}
d = e^{-\sigma\sqrt t} = \frac{1}{u}.

Due to its simple and iterative structure, the model presents certain unique advantages. For example, since it provides a stream of valuations for a derivative for each node in a span of time, it is useful for valuing derivatives such as American options which allow the owner to exercise the option at any point in time until expiration (unlike European options which are exercisable only at expiration). The model is also somewhat simple mathematically when compared to counterparts such as the Black-Scholes model, and is therefore relatively easy to build and implement with a computer spreadsheet.

Although slower than the Black-Scholes model, it is considered more accurate, particularly for longer-dated options, and options on securities with dividend payments. For these reasons, various versions of the binomial model are widely used by practitioners in the options markets.


1 comment:

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