Editing Binomial options pricing model (section)

===Step 1: Create the binomial price tree===
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 (<math>u</math> or <math>d</math>) per step of the tree (where, by definition, <math>u \ge 1</math> and <math>0 < d \le 1 </math>). So, if <math>S</math> is the current price, then in the next period the price will either be <math>S_{up} = S \cdot u</math> or <math>S_{down} = S \cdot d</math>.

The up and down factors are calculated using the underlying (fixed) [[Volatility (finance)|volatility]], <math>\sigma</math>, and the time duration of a step, <math>t</math>, 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 <math>\sigma^2 t</math>, we have:

:<math>u = e^{\sigma\sqrt \Delta t}</math>
:<math>d = e^{-\sigma\sqrt \Delta t} = \frac{1}{u}.</math>

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.<ref name="Joshi">Mark s. Joshi (2008). [http://fbe.unimelb.edu.au/__data/assets/pdf_file/0006/806280/170.pdf The Convergence of Binomial Trees for Pricing the American Put]</ref><ref name="Chance"/>

The CRR method ensures that the tree is recombinant, i.e. if the underlying asset moves up and then down (u,d), the price will be the same as if it had moved down and then up (d,u)—here the two paths merge or recombine. 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. The node-value will be: 
:<math>S_n = S_0 \times u ^{N_u - N_d},</math>

Where <math>N_u</math> is the number of up ticks and <math>N_d</math> is the number of down ticks.