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Home > Binomial options model

1 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.

1) 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 S up or S down, 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 = exp ( σ √ t )

d = exp ( - σ √ t ) = 1 / u

The above is the original Cox, Ross, & Rubinstein (CRR) method; there are other techniques for generating the lattice, such as "the equal probabilities" tree.

2) 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.

For a call: value = Max (S – Exercise price, 0)

For a put: value = Max ( Exercise price – S, 0)

3) 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 security is equal to the discounted expected value of its future payoff. See Risk neutral valuation.

Expected value here is 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 rateThe risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk. Though a truly "risk-free" asset exists only in theory, in practice most professionals and academics use short-dated go 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 = [ exp((r-q) × t) - d ] ÷ [ u - d ]

q is the dividend yieldThe dividend yield is a term used among investors in the financial markets as an evaluation of the possible return on an equity investment, such as in the stock market. The dividend yield is the return on a stock purchase in the form of a dividend as a pe 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.

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Topics: Binomial Options Model Option Value Price Node Valuation...

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