Binomial Option 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.

Method

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.

The two assets, which the valuation depends upon, are the call option and the underlying stock. There is an agreement among participants that the underlying stock price can move from the current $100 to either $110 or $90 in one year and there are no other price moves possible.

In an arbitrage-free world, if you have to create a portfolio comprised of these two assets, call option and underlying stock, such that regardless of where the underlying price goes – $110 or $90 – the net return on the portfolio always remains the same. Suppose you buy “d” shares of underlying and short one call options to create this portfolio.

If the price goes to $110, your shares will be worth $110*d, and you’ll lose $10 on the short call payoff. The net value of your portfolio will be (110d – 10).

If the price goes down to $90, your shares will be worth $90*d, and the option will expire worthlessly. The net value of your portfolio will be (90d).

If you want your portfolio’s value to remain the same regardless of where the underlying stock price goes, then your portfolio value should remain the same in either case:

h(d)−m=l(d)

where:

h = Highest potential underlying price

d = Number of underlying shares

m = Money lost on short call payoff

l = Lowest potential underlying price

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 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(2n), 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.