Binomial Option Pricing Calculator

Methodology, equations and example

A binomial tree prices an option by letting the underlying move only up or down over each of n short steps, then working back from the payoffs at expiration. Unlike the Black-Scholes formula it handles American options, which can be exercised at any time: at every node the tree compares holding the option with exercising it. With enough steps the European price from the tree converges to the Black-Scholes price.

The Cox-Ross-Rubinstein tree

The time to expiration T is split into n steps of length Δt = T/n. Over each step the price moves up by a factor u or down by a factor d:

u = e^(σ√Δt), d = 1/u

The up probability p is the one that makes the expected return over a step the risk-free rate less the dividend yield. It is a risk-neutral probability, not a forecast:

p = (e^((r − q)Δt) − d) / (u − d)

After i steps with j up moves the price is S u^j d^(i − j). At expiration (step n) each node's value is the payoff, max(0, S_T − K) for a call and max(0, K − S_T) for a put. Each earlier node is valued from the two nodes it leads to:

f = e^(−rΔt) [p f_up + (1 − p) f_down]

For an American option the node's value is the larger of that and the payoff from exercising now. A node where exercising is larger is an early-exercise node; the line through the first such node at each step is the early-exercise boundary.

• S – Underlying price
• K – Strike price
• T – Time to expiration in years
• σ – Volatility, per year
• r – Risk-free rate, continuously compounded
• q – Dividend yield, continuously compounded
• n – Number of steps

The greeks from the tree

The tree gives the greeks from its first nodes, without a formula. With f_u and f_d the values one step in, and f_uu, f_ud and f_dd two steps in:

Delta = (f_u − f_d) / (S u − S d)

Gamma = [(f_uu − f_ud)/(S u² − S) − (f_ud − f_dd)/(S − S d²)] / [(S u² − S d²)/2]

Theta = (f_ud − f) / (2Δt), shown per calendar day, where f is the price today

The middle node two steps in has the same stock price as today, so theta compares the same price 2Δt apart.

Example

Price an American put with a strike of $50 on a $50 stock that pays no dividend, with 5 months to expiration, 40 % volatility and a 10 % risk-free rate, on a five-step tree (one step a month). This is the American put example in Hull, Options, Futures, and Other Derivatives.

With Δt = 1/12 year:

u = e^(0.40 √(1/12)) = 1.1224, d = 1/1.1224 = 0.8909

p = (e^(0.10/12) − 0.8909) / (1.1224 − 0.8909) = 0.5073

At expiration the lowest nodes are $28.07 and $35.36, with put payoffs of $21.93 and $14.64, and the next one up is $44.55, a payoff of $5.45. One step earlier, at the $39.69 node, holding the put is worth

e^(−0.10/12) × (0.5073 × $5.4526 + 0.4927 × $14.6389) = $9.8957

but exercising it is worth $50 − $39.69 = $10.31, so the put is exercised there and the node is worth $10.31. Working back through every node in the same way gives the price today:

American put = $4.49

The same tree without early exercise gives a European put of $4.32, so the right to exercise early is worth $0.17 here. Five steps is coarse: with 30, 50, 100 and 500 steps the American price is $4.263, $4.272, $4.278 and $4.283. The European price converges in the same way to the Black-Scholes price of $4.076.

Accuracy and limits

The tree price zigzags as n increases, alternating between slightly high and slightly low for odd and even n, and the error shrinks roughly in proportion to 1/n. A few hundred steps is enough for most uses; the chart in the results shows how the price settles for your inputs. The tree assumes a constant volatility, rate and dividend yield over the life of the option, and a dividend paid continuously rather than in discrete amounts.

An American call on an underlying that pays no dividend is never worth exercising early, because the time value it gives up is always positive. Its price equals the European call's, and the calculator shows a zero early-exercise premium. With a dividend yield, early exercise of a deep in-the-money call can be worth it. For comparing options and for study; not financial advice.