Binomial Option Pricing Model

The binomial option pricing model is a method for valuing an option in which the time until expiry is cut into a number of short steps. In each step the price of the underlying asset can do one of two things: rise by a set factor or fall by a set factor. The possible prices form a branching tree, and the option's value is found by starting at the end of the tree and working back to the present.

The model was published in 1979 by John Cox, Stephen Ross and Mark Rubinstein and is often called the CRR model. It rests on the same no-arbitrage reasoning as the Black-Scholes model but uses only simple arithmetic. Its main practical advantage is that it can value contracts with early exercise, which the Black-Scholes formula cannot.

The central idea is replication. Over one step, a portfolio holding a certain number of shares and a certain amount of borrowing can be built so that it is worth exactly what the option is worth in both the up state and the down state. Since the portfolio and the option have the same future value in every case, they must cost the same today. Otherwise a trader could buy the cheaper and sell the dearer for a riskless profit, which is arbitrage.

The number of shares in the replicating portfolio is the option's delta. The argument does not depend on the true probability of an up move or on how much risk investors are willing to bear. The result can be written as if investors were indifferent to risk and all assets earned the risk-free rate, an approach known as risk-neutral valuation.

For one step of length $\Delta t$, the option value is:

where the risk-neutral probability of an up move is:

• $V_u$ and $V_d$ are the option's values after an up move and a down move.
• $u$ and $d$ are the factors by which the price is multiplied in an up or down move.
• $r$ is the risk-free interest rate, continuously compounded.
• $e^{-r \Delta t}$ discounts the expected value back one step.

In the CRR version the factors are set from the asset's volatility $\sigma$:

Consider a one-year call option with a strike price of $100 on a share priced at $100. Use a single step with $u = 1.1$, $d = 0.9$ and $r = 5$ percent. The share ends at $110 or $90, so the call is worth $10 or $0. The growth factor $e^{0.05}$ is 1.0513, giving $p = (1.0513 - 0.9) \div 0.2 = 0.756$. The expected payoff is 0.756 × 10 = 7.56, and discounting by $e^{-0.05} = 0.9512$ gives a value of about $7.19.

The replication check agrees. Delta is (10 − 0) ÷ (110 − 90) = 0.5. Half a share costs $50. It will be worth $45 in the down state, so $45 can be repaid then, and the amount borrowed today is 45 × 0.9512 = $42.80. The net cost is 50 − 42.80 = $7.20, the same figure up to rounding.

One step is too coarse for real use. In practice the life of the option is split into dozens or hundreds of steps. Because an up move followed by a down move returns to the starting price, the tree recombines and has only $n + 1$ end points after $n$ steps. The option's payoff is computed at each end point, and the one-step formula is applied repeatedly back to the start.

This backward pass is what makes the model suited to an American option. At every node the model compares the value of keeping the option alive with the value of exercising it at once, and records the larger. A European option skips that comparison. Known cash payments such as a dividend can be built into the tree by lowering the share price at the relevant step.

As the number of steps grows, the distribution of end prices in the tree approaches the lognormal distribution assumed by Black and Scholes, and the binomial value of a European option converges to the Black-Scholes value. The binomial model can therefore be seen as a discrete-time version of the same theory. The convergence is not smooth. The value oscillates as steps are added, and practitioners average results or use adjusted trees to steady it. Sensitivities, the option Greeks, are read from differences between neighboring nodes.

The standard tree assumes constant volatility and a constant interest rate over the option's life, as Black-Scholes does. Market prices show that implied volatility differs by strike and expiry, so one tree with one volatility cannot match all traded options at once. Extended versions, called implied trees, let volatility vary by node.

The method becomes slow when many steps are needed, and it copes poorly with options that depend on several assets or on the path taken by the price. For those, Monte Carlo simulation or finite-difference methods are generally preferred. Like every pricing model, it gives a value that is only as good as its inputs, and the volatility input is an estimate.