What is the binomial option pricing model?

The binomial option pricing model values options by assuming that, over each small time step, the underlying asset price can move to one of two possible values: up or down. By building a recombining tree of possible prices and discounting expected payoffs under the risk-neutral probability measure, the model produces a fair value for European and American call and put options.

What is the difference between European and American options in the binomial tree?

European options can only be exercised at maturity, so the binomial algorithm only considers the payoff at the final time step and works backward. American options can be exercised at any node before maturity, so at each step the model compares the value of holding the option with the intrinsic value from exercising immediately and takes the maximum.

How many steps should I use in the binomial tree?

More steps generally give a more accurate approximation of the continuous-time models such as Black–Scholes, but they also increase computation time. For most equity options, 50–200 steps are often sufficient. You can start with 50 steps and increase until the price stabilizes if you need higher precision.

What interest rate should I enter?

Use the continuously compounded risk-free rate that matches the option's maturity, typically derived from government bond yields or OIS curves. If you only have an annual simple rate, you can still enter it here as an approximation; the calculator will convert it to a per-step rate internally.

Does this calculator support dividends?

Yes. You can enter either a continuous dividend yield (q) or a discrete cash dividend amount per step. For stocks with regular percentage yields, use the continuous yield. For known fixed cash dividends, use the discrete dividend field and choose the number of dividend steps.

Binomial Option Pricing Calculator

Price European and American call/put options using a recombining binomial tree. See risk‑neutral probabilities, node values, and early-exercise decisions step-by-step.

Option Inputs

Spot price S₀

Strike price K

Time to maturity T (years)

Risk-free rate r (annual, %)

Volatility σ (annual, %)

Number of steps N

Higher N → smoother convergence, but slower.

Option type

Call Put

Exercise style

European American

Dividend treatment

Results

Option Value

–

Up factor u–

Down factor d–

Risk‑neutral probability p–

Per‑step discount factor–

Binomial Tree (Node Table)

Each row shows a node in the tree: time step, number of up moves, underlying price, option value, and whether early exercise is optimal (for American options).

Step t	Up moves	S(t)	Option value	Intrinsic	Early exercise?
Run a calculation to see the full binomial tree.

How the binomial option pricing model works

The binomial option pricing model values an option by modeling the underlying asset price as a recombining tree of up and down moves over many small time steps. At each node, the option value is the discounted expected value of its future payoffs under the risk‑neutral probability measure.

1. Time step and price dynamics

Suppose the current stock price is \(S_0\), the time to maturity is \(T\) years, and we use \(N\) steps. The length of each step is:

\[ \Delta t = \frac{T}{N} \]

In the Cox–Ross–Rubinstein (CRR) specification, the up and down factors are:

\[ u = e^{\sigma \sqrt{\Delta t}}, \quad d = e^{-\sigma \sqrt{\Delta t}} = \frac{1}{u} \]

where \(\sigma\) is the annual volatility. After \(i\) up moves and \(j\) down moves (\(i + j = n\)), the stock price is:

\[ S_{n,i} = S_0 \, u^{i} d^{n-i} \]

2. Risk‑neutral probability and discounting

Under the risk‑neutral measure, the expected growth rate of the stock is the risk‑free rate \(r\) (adjusted for dividends). For a continuous dividend yield \(q\), the risk‑neutral up‑move probability is:

\[ p = \frac{e^{(r - q)\Delta t} - d}{u - d}, \quad 0 < p < 1 \]

The per‑step discount factor is:

\[ \text{disc} = e^{-r \Delta t} \]

3. Payoff at maturity

At the final step \(N\), the option value equals its intrinsic payoff:

Call: \(\; C_{N,i} = \max(S_{N,i} - K, 0)\)
Put: \(\; P_{N,i} = \max(K - S_{N,i}, 0)\)

4. Backward induction

Working backward from maturity to today, the option value at each node is the discounted expected value of the two possible next‑step values:

\[ V_{n,i}^{\text{hold}} = \text{disc} \cdot \big( p \, V_{n+1,i+1} + (1-p) \, V_{n+1,i} \big) \]

For a European option, the node value is simply \(V_{n,i} = V_{n,i}^{\text{hold}}\).

For an American option, early exercise is allowed. The intrinsic value at node \((n,i)\) is:

Call: \(\; \text{intrinsic} = \max(S_{n,i} - K, 0)\)
Put: \(\; \text{intrinsic} = \max(K - S_{n,i}, 0)\)

The node value is the maximum of holding and exercising:

\[ V_{n,i} = \max\big( V_{n,i}^{\text{hold}}, \; \text{intrinsic} \big) \]

5. Discrete dividends

For discrete cash dividends, the stock price is reduced by the dividend amount at ex‑dividend nodes. In this calculator, if you choose a fixed dividend per step, the model subtracts that amount from the stock price at the specified dividend steps before computing payoffs.

Worked example

Consider a non‑dividend‑paying stock with:

\(S_0 = 100\)
\(K = 100\)
\(T = 1\) year
\(r = 5\%\)
\(\sigma = 20\%\)
\(N = 3\) steps

Then:

\(\Delta t = 1/3\)
\(u = e^{0.2\sqrt{1/3}} \approx 1.1224\)
\(d = 1/u \approx 0.8910\)
\(p = \dfrac{e^{0.05/3} - d}{u - d} \approx 0.5438\)
\(\text{disc} = e^{-0.05/3} \approx 0.9835\)

The calculator reproduces this tree and shows the option value at each node, highlighting where early exercise is optimal for American options.

Tips for using this binomial option pricing calculator

Convergence: Increase the number of steps until the price stabilizes if you need high precision.
Dividends: For typical equity indices, a continuous dividend yield is often a good approximation. For single stocks with known cash dividends, use the discrete dividend mode.
American vs European: American calls on non‑dividend‑paying stocks should rarely be exercised early; American puts and dividend‑paying calls can have significant early‑exercise value.

Binomial option pricing – FAQ

When should I use the binomial model instead of Black–Scholes?

The binomial model is especially useful for American options (which allow early exercise) and for options on dividend‑paying stocks with complex dividend schedules. Black–Scholes only gives closed‑form prices for European options under restrictive assumptions. Binomial trees are more flexible and can approximate many real‑world features.

Why does the risk‑neutral probability p sometimes show as invalid?

For a valid binomial model, the up and down factors must satisfy \(d < e^{(r-q)\Delta t} < u\). If this condition fails, the implied risk‑neutral probability would be outside the 0–1 range. This can happen if volatility is extremely low, the number of steps is too small, or the dividend yield is unrealistically high relative to the risk‑free rate.

How does the calculator mark early‑exercise nodes?

For American options, at each node the calculator compares the intrinsic value to the discounted continuation value. If intrinsic value is higher, the node is flagged as “Yes” in the Early exercise? column, indicating that immediate exercise is optimal under the model assumptions.