Binomial options pricing model

Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting, which in general does not exist for the BOPM.

[1] The binomial model was first proposed by William Sharpe in the 1978 edition of Investments (ISBN 013504605X),[2] and formalized by Cox, Ross and Rubinstein in 1979[3] and by Rendleman and Bartter in that same year.

This is largely because the BOPM is based on the description of an underlying instrument over a period of time rather than a single point.

Being relatively simple, the model is readily implementable in computer software (including a spreadsheet).

For these reasons, various versions of the binomial model are widely used by practitioners in the options markets.

Monte Carlo simulations are also less susceptible to sampling errors, since binomial techniques use discrete time units.

The binomial pricing model traces the evolution of the option's key underlying variables in discrete-time.

This is done by means of a binomial lattice (Tree), for a number of time steps between the valuation and expiration dates.

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

At each step, it is assumed that the underlying instrument will move up or down by a specific factor (

, measured in years (using the day count convention of the underlying instrument).

, we have: Above is the original Cox, Ross, & Rubinstein (CRR) method; there are various other techniques for generating the lattice, such as "the equal probabilities" tree, see.

This property reduces the number of tree nodes, and thus accelerates the computation of the option price.

This property also allows the value of the underlying asset at each node to be calculated directly via formula, and does not require that the tree be built first.

at expiration of the option—the option value is simply its intrinsic, or exercise, value: Where K is the strike price and

Once the above step is complete, the option value is then found for each node, starting at the penultimate time step, and working back to the first node of the tree (the valuation date) where the calculated result is the value of the option.

In this case then, for European options without dividends, the binomial model value converges on the Black–Scholes formula value as the number of time steps increases.