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, which in general does not exist for the BOPM.

The binomial model was first proposed by William Sharpe in the 1978 edition of Investments (), 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 .

Use of the model

The Binomial options pricing model approach has been widely used since it is able to handle a variety of conditions for which other models cannot easily be applied. 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. As a consequence, it is used to value American options that are exercisable at any time in a given interval as well as Bermudan options that are exercisable at specific instances of time. Being relatively simple, the model is readily implementable in computer software (including a spreadsheet).

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

For options with several sources of uncertainty (e.g., real options) and for options with complicated features (e.g., Asian options), binomial methods are less practical due to several difficulties, and Monte Carlo option models are commonly used instead. When simulating a small number of time steps Monte Carlo simulation will be more computationally time-consuming than BOPM (cf. Monte Carlo methods in finance). However, the worst-case runtime of BOPM will be O(2<sup>n</sup>), where n is the number of time steps in the simulation. Monte Carlo simulations will generally have a polynomial time complexity, and will be faster for large numbers of simulation steps. Monte Carlo simulations are also less susceptible to sampling errors, since binomial techniques use discrete time units. This becomes more true the smaller the discrete units become.

Method

right|Binomial Lattice with CRR formulae

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Incorrect code, p[i] is overwritten used in calculation p[i-1], introduce another temp vector to save

function americanPut(T, S, K, r, sigma, q, n)

{

deltaT := T / n;

up := exp(sigma * sqrt(deltaT));

p0 := (up * exp(-q * deltaT) - exp(-r * deltaT)) / (up^2 - 1);

p1 := exp(-r * deltaT) - p0;

for i := 0 to n {

p[i] := K - S * up^(2*i - n+1);

if p[i] < 0 then p[i] := 0;

}

for j := n-1 down to 0 {

for i := 0 to j {

p[i] := p0 * p[i+1] + p1 * p[i];

exercise := K - S * up^(2*i - j);

if p[i] < exercise then p[i] := exercise;

}

return americanPut := p[0];

}

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.

Valuation is performed iteratively, starting at each of the final nodes (those that may be reached at the time of expiration), and then working backwards through the tree towards the first node (valuation date). The value computed at each stage is the value of the option at that point in time.

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

Price tree generation,
Calculation of option value at each final node,
Sequential calculation of the option value at each preceding node.

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, <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.

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

Use of the model

Method

Step 1: Create the binomial price tree

See also

References

External links