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Binomial options pricing model

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In finance, the binomial options pricing model provides a generalisable numerical method for the valuation of options. The binomial model was first proposed by Cox, Ross and Rubinstein (1979). Essentially, the model uses a "discrete-time" model of the varying price over time of the underlying financial instrument. Option valuation is then via application of the risk neutrality assumption over the life of the option, as the price of the underlying instrument evolves.

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Use of the model

The Binomial options pricing model approach is widely used as it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM models the underlying instrument over time - as opposed to at a particular point. For example, the model is used to value American options which can be exercised at any point and Bermudan options which can be exercised at various points. The model is also relatively simple, mathematically, and can therefore be readily implemented in a software (or even spreadsheet) environment.

Although slower than the Black-Scholes model, it is considered more accurate, particularly for longer-dated options, and 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), or for options with complicated features (e.g. Asian options), lattice methods face several difficulties and are not practical. Monte Carlo option models are generally used in these cases. Monte Carlo simulation is, however, time-consuming in terms of computation, and is not used when the Lattice approach (or a formula) will suffice. See Monte Carlo methods in finance.

Methodology

The binomial pricing model uses a "discrete-time framework" to trace the evolution of the option's key underlying variable via a binomial lattice (tree), for a given number of time steps between valuation date and option expiration.

Each node in the lattice, represents a possible price of the underlying, at a particular point in time. This price evolution forms the basis for the option valuation.

The valuation process is iterative, starting at each final node, and then working backwards through the tree to the first node (valuation date), where the calculated result is the value of the option.

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

  1. price tree generation
  2. calculation of option value at each final node
  3. progressive calculation of option value at each earlier node; the value at the first node is the value of the option.
The methodology is best illustrated via example. Link here for a graphical [Binomial Tree Option Calculator].

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 - u or d - per step of the tree. (The Binomial model allows for only two states.)

If S is the current price, then in the next period the price will either be [Sup] or [Sdown], where S up =S x u and S down =S x d. The up and down factors are calculated using the underlying volatility, σ, and years per time step, t:

[u = e^]
[d = e^ = \frac.]
The above is the original Cox, Ross, & Rubinstein (CRR) method; there are other techniques for generating the lattice, such as "the equal probabilities" tree.

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 [ (S – K), 0 ], for a call option
Max [ (K – S), 0 ], for a put option:
(Where K is the Strike price and S is the spot price of the underlying asset)

Option value at earlier nodes

At each earlier node, the value of the option is calculated using the risk neutrality assumption.

Under this assumption, today's fair price of a derivative is equal to the discounted expected value of its future payoff. See Risk neutral valuation.

Expected value is therefore calculated using the option values from the later two nodes (Option up and Option down) weighted by their respective 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 rate corresponding to the life of the option. This result, the "Binomial Value", is thus 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.

The Binomial Value is 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. 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).

The Binomial Value is calculated as follows.

Binomial Value = [ p × Option up + (1-p)× Option down] × exp (- r × t)
[p = \frac - d}]
q is the dividend yield of the underlying corresponding to the life of the option.
Note that the alternative valuation approach, arbitrage-free pricing ("delta-hedging"), yields identical results; see Rational pricing.

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. In fact, for European options without dividends, the binomial model value converges on the Black-Scholes formula value as the number of time steps increases.

See also

References

External links

 

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