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Source: Binomial options pricing model

In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. 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.
The binomial model was first proposed by William Sharpe in the 1978 edition of Investments (ISBN 013504605X), and formalized by Cox, Ross and Rubinstein in 1979 and by Rendleman and Bartter in that same year.
For binomial trees as applied to fixed income and interest rate derivatives see Lattice model (finance) § Interest rate derivatives.

Use of the model

Binomial

Method

= Step 1: Create the binomial price tree

). So, if

S

{\displaystyle S}

is the current price, then in the next period the price will either be

S

u
p

=
S
⋅
u

{\displaystyle S_{up}=S\cdot u}

or

S

d
o
w
n

=
S
⋅
d

{\displaystyle S_{down}=S\cdot d}

.
The up and down factors are calculated using the underlying volatility,

σ

{\displaystyle \sigma }

, and the time duration of a step,

t

{\displaystyle 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

σ

2

t

{\displaystyle \sigma ^{2}t}

, we have:

u
=

e

σ

Δ

t

{\displaystyle u=e^{\sigma {\sqrt {\Delta }}t}}

d
=

e

−
σ

Δ

t

=

1
u

.

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

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.
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:

S

n

=

S

0

×

u

N

u

−

N

d

,

{\displaystyle S_{n}=S_{0}\times u^{N_{u}-N_{d}},}

Where

N

u

{\displaystyle N_{u}}

is the number of up ticks and

N

d

{\displaystyle N_{d}}

is the number of down ticks.

= Step 2: Find option value at each final node

=
At each final node of the tree—i.e. at expiration of the option—the option value is simply its intrinsic, or exercise, value:

Max [ (Sn − K), 0 ], for a call option
Max [ (K − Sn), 0 ], for a put option,
Where K is the strike price and

S

n

{\displaystyle S_{n}}

is the spot price of the underlying asset at the nth period.

= 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 neutrality assumption; see 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:

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:

Relationship with Black–Scholes

Similar assumptions underpin both the binomial model and the Black–Scholes model, and the binomial model thus provides a discrete time approximation to the continuous process underlying the Black–Scholes model. The binomial model assumes that movements in the price follow a binomial distribution; for many trials, this binomial distribution approaches the log-normal distribution assumed by Black–Scholes. 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.
In addition, when analyzed as a numerical procedure, the CRR binomial method can be viewed as a special case of the explicit finite difference method for the Black–Scholes PDE; see finite difference methods for option pricing.

References

External links

The Binomial Model for Pricing Options, Prof. Thayer Watkins
Binomial Option Pricing (PDF), Prof. Robert M. Conroy
Binomial Option Pricing Model by Fiona Maclachlan, The Wolfram Demonstrations Project
On the Irrelevance of Expected Stock Returns in the Pricing of Options in the Binomial Model: A Pedagogical Note by Valeri Zakamouline
A Simple Derivation of Risk-Neutral Probability in the Binomial Option Pricing Model by Greg Orosi