Editing Binomial options pricing model (section)

===Step 3: Find option value at earlier nodes===
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 overview: the "binomial value" is found at each node, using the [[risk-neutral measure|risk neutrality]] assumption; see [[Rational pricing#Risk neutral valuation|Risk neutral valuation]]. If exercise is permitted at the node, then the model takes the greater of binomial and exercise value at the node.

The steps are as follows:

{{ordered list|1= Under the risk neutrality assumption, today's [[fair value|fair price]] of a [[derivative (finance)|derivative]] is equal to the [[expected value]] of its future payoff discounted by the [[Risk-free interest rate|risk free rate]]. Therefore, expected value is calculated using the option values from the later two nodes (''Option up'' and ''Option down'') weighted by their respective (fixed) probabilities—"probability" '''p''' of an up move in the underlying, and "probability" '''(1−p)''' of a down move. The expected value is then discounted at '''r''', the [[Risk-free interest rate|risk free rate]] corresponding to the life of the option.

:The following formula to compute the [[expectation value]] is applied at each node:

:<math>\text { Binomial Value }=[p \times \text { Option up }+(1-p) \times \text { Option down] } \times \exp (-r \times \Delta t)</math>, or

:<math>C_{t-\Delta t,i} = e^{-r \Delta t}(pC_{t,i} + (1-p)C_{t,i+1}) \,</math>

:where
:<math>C_{t,i} \,</math> is the option's value for the <math>i^{th} \,</math> node at time {{mvar|t}},

:<math>p = \frac{e^{(r-q) \Delta t} - d}{u - d}</math> is chosen such that the related [[binomial distribution]] simulates the [[geometric Brownian motion]] of the underlying stock with parameters '''r''' and '''σ''',

:{{mvar|q}} is the [[dividend yield]] of the underlying corresponding to the life of the option. It follows that in a risk-neutral world futures price should have an expected growth rate of zero and therefore we can consider <math>q=r</math> for futures.

:Note that for {{mvar|p}} to be in the interval <math>(0,1)</math> the following condition on <math>\Delta t</math> has to be satisfied <math>\Delta t < \frac{\sigma^2}{(r-q)^2}</math>.

:(Note that the alternative valuation approach, [[arbitrage-free]] pricing, yields identical results; see “[[Rational pricing#Delta hedging|delta-hedging]]”.)

|2= This result is the "Binomial Value". It represents the fair price of the derivative at a particular point in time (i.e. at each node), given the evolution in the price of the underlying to that point. It is the value of the option if it were to be held—as opposed to exercised at that point.

|3= Depending on the style of the option, evaluate the possibility of early exercise at each node: if (1) the option can be exercised, and (2) the exercise value exceeds the Binomial Value, then (3) the value at the node is the exercise value.
* For a [[European option]], there is no option of early exercise, and the binomial value applies at all nodes.
* For an [[American option]], since the option may either be held or exercised prior to expiry, the value at each node is: Max (Binomial Value, Exercise Value).
* For a [[Bermudan option]], the value at nodes where early exercise is allowed is: Max (Binomial Value, Exercise Value); at nodes where early exercise is not allowed, only the binomial value applies.
}}
In calculating the value at the next time step calculated—i.e. one step closer to valuation—the model must use the value selected here, for "Option up"/"Option down" as appropriate, in the formula at the node.
The aside [[algorithm]] demonstrates the approach computing the price of an American put option, although is easily generalized for calls and for European and Bermudan options: