Binomial trees

The Cox–Ross–Rubinstein (CRR) binomial tree is a flexible way to approximate the evolution of an underlying asset and the value of derivative claims at every node. The interactive visualizer below lets you explore how the tree responds when you change market inputs, underlying type, or exercise style.

CRR Binomial Framework

CRR discretizes the continuous price dynamics of an asset into N recombining steps of length \(\Delta t\). Each step either moves the asset up by a factor \(u\) or down by \(d\), and the risk-neutral probability \(p\) links the tree to the absence of arbitrage. All parameters flow into the backward induction that produces the time-zero option price.

Parameter Equations

The model ingredients satisfy:

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

\[ p = \frac{a - d}{u - d} \]

\[ \text{Discount factor per step} = e^{-r \Delta t} \]

\[ \text{Intrinsic payoff at maturity} = \begin{cases} \max(0, S_T - K) & \text{Call} \\ \max(0, K - S_T) & \text{Put} \end{cases} \]

The growth term \(a\) adapts to the underlying asset:

\[ \begin{aligned} a & = e^{r \Delta t} && \text{non-dividend stock} \\ a & = e^{(r-q)\Delta t} && \text{stock index with dividend yield } q \\ a & = e^{(r-r_f)\Delta t} && \text{currency with foreign rate } r_f \\ a & = 1 && \text{futures contract} \end{aligned} \]

The growth factor for futures is 1 because futures prices are already forward-looking; in a risk-neutral world, the expected next-step price is the current price, implying no drift.

Keeping \(p\) inside \([0,1]\) ensures a valid risk-neutral measure and guarantees that the discounted tree is arbitrage-free.

As the number of steps \(N\) increases, the price calculated by the CRR binomial model converges to the celebrated Black-Scholes-Merton formula, bridging the gap between discrete-time and continuous-time models.

Interactive Binomial Tree

Use the controls to change the number of steps, underlying specification, option style, and payoff type. The visualization recomputes prices and node values with the CRR formulas and displays the current parameter values beneath their equations.

\(\Delta t = \frac{T}{N}\) Current \(\Delta t\): –
\(u = e^{\sigma\sqrt{\Delta t}}\) Current \(u\): –
\(d = \frac{1}{u}\) Current \(d\): –
\(a\) per definitions above Current \(a\): –
\(p = \dfrac{a-d}{u-d}\) Current \(p\): –
Option value today: –

Understanding Option Delta

Delta (Δ) is one of the most important “Greeks,” measuring the rate of change of the option’s price with respect to a change in the underlying asset’s price. It represents the option’s price sensitivity to the underlying and is a key metric for hedging.

Delta in the Binomial Tree

In the context of a binomial tree, the delta at any given node (i, j) is calculated by taking the difference in the option’s value in the two possible subsequent nodes (up and down) and dividing it by the difference in the underlying asset’s price at those same nodes.

The formula at node (i, j) is:

\[ \Delta_{i,j} = \frac{f_{i+1, j+1} - f_{i+1, j}}{S_{i+1, j+1} - S_{i+1, j}} \]

Where:
- \(f_{i+1, j+1}\) is the option value at the subsequent up node.
- \(f_{i+1, j}\) is the option value at the subsequent down node.
- \(S_{i+1, j+1}\) is the asset price at the subsequent up node.
- \(S_{i+1, j}\) is the asset price at the subsequent down node.

You can observe this calculation in the tooltip for any non-terminal node in the interactive tree above.

Why Delta Changes

Delta is not a constant value; it changes with both the underlying asset’s price and time to expiry.

  • For a call option, the delta will range from 0 to 1. When the option is deep out-of-the-money, its price is not very sensitive to changes in the underlying, so delta is close to 0. As the option becomes at-the-money and then in-the-money, its price moves more in lockstep with the underlying, and delta approaches 1.
  • For a put option, the delta ranges from -1 to 0, reflecting the inverse relationship between the put option’s price and the underlying asset’s price.

This dynamic nature is why delta hedging strategies require constant rebalancing. The binomial tree provides an intuitive visualization of how an option’s delta can evolve over its lifetime.

Option Valuation Steps

  1. Discretize time: Choose the number of steps \(N\) and compute \(\Delta t = T/N\), then set the CRR up and down multipliers \(u\) and \(d\).
  2. Set the growth factor: Use the appropriate \(a\) definition for the underlying (stock, index, currency, or futures) so the risk-neutral drift matches carry costs or benefits.
  3. Compute the probability: Evaluate \(p = \dfrac{a-d}{u-d}\) and verify it lies in \([0,1]\); otherwise the calibration violates no-arbitrage conditions.
  4. Propagate asset prices: Build the stock price tree for all time steps. The price at any node (i, j) is given by \(S_{i,j} = S_0 u^j d^{i-j}\), where j is the number of up-steps.
  5. Apply payoffs at maturity: At step \(N\), compute the intrinsic value (call or put) at each node.
  6. Backward induction: Discount the expected continuation value \(e^{-r\Delta t}[p V_{\text{up}} + (1-p)V_{\text{down}}]\) at every earlier node, comparing it with the immediate exercise value when the option is American.
  7. Read the option price: The value at the root node \((i=0, j=0)\) is today’s option price, and the entire lattice provides exercise policies and hedge ratios along the path.