Excel Models

American Options – Binomial Trees

January 24, 2026neoinsight5 min read

One method used to value an American option is the construction of a binomial tree. I have written about this before (Binomial Trees), so here I will focus on the more important nuances.

Let us start with the Excel-based option calculator presented in John C. Hull’s book Options, Futures & Other Derivatives, which constructs binomial trees (DerivaGem Software).

For example, consider a 5-month American PUT option on a non-dividend-paying stock with the following parameters:

Stock price – $50
Strike price – $50
Annual risk-free interest rate – 10%
Number of steps – 5
Volatility – 40%

DerivaGem generates the following tree (DG400a):

In each paired cell, the upper cell represents the evolution of the stock price, while the lower cell is used to calculate the option value. Stock prices evolve based on volatility, while option valuation starts at the terminal nodes and is calculated backward.

At the terminal nodes, in those scenarios where the stock price exceeds the strike price, the PUT option naturally becomes worthless. In the other scenarios, the option value equals the corresponding price difference. These values are then discounted backward using the risk-free rate. The weighting is done according to risk-neutral probabilities.

For reference, the formulas and coefficients are as follows:

One important nuance: when rolling the option value backward, the weighted discounted value is compared with the intrinsic value (the difference between the current stock price and the price in the corresponding upper node), and the option node records the maximum of the two. This reflects the fact that in some cases early exercise of the option is rational.

Number of steps

The more steps we use to construct the tree, the smaller the valuation error becomes. Absolute accuracy is achieved when the number of steps approaches infinity, but a 30-step tree already provides sufficiently good accuracy in practice.

The diagram shows how the valuation of the European version of the option discussed converges to the Black–Scholes value as the number of time steps increases.

DG400 Applications

This file also contains formulas for calculating the Greeks for American options.

Control Variate Technique – a refinement method

To refine the price of an American option, the same tree is used to calculate the price of a European option, which is then compared to the value obtained from the Black–Scholes formula. The difference is added to the American option price obtained from the tree. In other words, it is assumed that the deviation of the tree-based result from the true value is the same for both American and European options:

Indices, futures, currencies

The binomial tree method can also be applied to other types of options. In these cases, a tree construction technique based on a “known dividend yield” is used, which is also implemented in DerivaGem.

For index options, the known dividend yield corresponds to the yield generated by the stocks in the index portfolio.
For currency options, the foreign risk-free interest rate should be used.
For futures options, the relevant domestic risk-free interest rate is applied.

Alternative trees

There are alternative ways to construct binomial trees in which the probabilities of upward and downward price movements are not based on the risk-neutral framework but are instead set to 50/50, with volatility ranges adjusted accordingly. Such trees simplify option valuation calculations, but they have the drawback that calculating the Greeks becomes more difficult.

Trinomial (three-branch) trees are also used. In this case, there are probabilities for price increases, decreases, and remaining unchanged. Trinomial trees converge to reasonable valuation accuracy with fewer steps. For example, a 200-step trinomial tree can achieve the accuracy of a 1,000-step binomial tree. For this reason, they are often more suitable for exotic options.

Time-dependent variables

In the analysis above, we assumed that the parameters are time-independent. In practice, however, interest rates, dividend yields, and volatility are all time-dependent variables.

Interest rates are typically taken from forward rates. This does not change the geometry of the tree (the volatility bounds), but it does change the risk-neutral probabilities at each step.

A more complex issue is time-varying volatility. Volatility affects the up and down factors (u and d), which complicates tree construction. One technical approach to preserving constant u and d across all nodes is to adjust the time intervals between steps inversely to changes in volatility (see Appendix 1).

Source:
Options, Futures & Other Derivatives, John C. Hull

Step-by-step explanation

1. Volatility is time‑dependent: $α (t)$

Cumulative variance up to time $t$ is:

$α (t)^{2} t$

This is just “volatility squared × time”.

2. Define total variance over the life of the option:

$V = α (T)^{2} T$

This is the total variance from 0 to maturity $T$ .

3. You want N steps, each with equal variance

So each step should have variance:

$\frac{V}{N}$

4. Choose the time grid so that cumulative variance grows linearly

Hull defines the time points $t_{i}$ by:

$α (t_{i})^{2} t_{i} = \frac{i V}{N}$

This means:

Step 1 ends when cumulative variance = $V / N$
Step 2 ends when cumulative variance = $2 V / N$
…
Step N ends when cumulative variance = $V$

This ensures:

$α (t_{i})^{2} t_{i} - α (t_{i - 1})^{2} t_{i - 1} = \frac{V}{N}$

So each step has the same variance, even though the time lengths $Δ t_{i} = t_{i} - t_{i - 1}$ are different.

🎯 Why this works

If each step has the same variance, then:

$u = e^{σ \sqrt{Δ t_{i}}} = e^{\sqrt{V / N}}$

is constant for all steps.

So the tree recombines perfectly.

✔ Example with 20% volatility

Let’s assume:

$σ = 20 %$
$T = 1$ year
Choose $N = 500$ steps (a common choice)

Then:

$V = {0.2}^{2} \cdot 1 = 0.04$

Each step must have variance:

$\frac{V}{N} = \frac{0.04}{500} = 0.00008$

So:

$u = e^{\sqrt{0.00008}}$

and the time grid $t_{i}$ is chosen so that:

${0.2}^{2} t_{i} = \frac{i}{500} \cdot 0.04$

which simplifies to:

$t_{i} = \frac{i}{500}$

In this special case (constant volatility), the grid becomes uniform again.

🧩 Final takeaway

Hull’s method is needed only when volatility changes with time.
The number of steps N is your choice, based on accuracy.
Volatility does not determine N.
Uneven time steps ensure each step has equal variance, so u and d stay constant.