Binomial Treesintermediate

One-step Binomial Tree

The one-step binomial model prices an option by restricting the underlying asset to two possible values at expiry, an up state $S_0 u$ and a down state $S_0 d$ . A delta-hedged portfolio of the underlying and the option is riskless, so it must earn the risk-free rate. This no-arbitrage argument uniquely determines the option price. Equivalently, the same price emerges by discounting the expected payoff under the risk-neutral probability measure, which assigns probabilities $p$ and $1-p$ so that the stock's expected return equals the risk-free rate. The one-step tree is the building block for multi-step lattices, and in the continuous limit it recovers the Black–Scholes–Merton formula.

Why it matters

Think of an ASX SPI 200 futures contract one quarter from expiry. You don't know whether the index will be up 20% or down 20%, and you don't need to. What you do know is that you can construct a portfolio (long $\Delta$ units of the futures and short one call) that pays exactly the same amount regardless of which branch is realised. Because that portfolio carries no risk, it must yield exactly the 3-month BBSW rate. Pinning down $\Delta$ from the two payoff equations gives you the hedge ratio, and substituting back gives the option price. The risk-neutral probability $p$ is not a forecast of where the index will go. It is the implied probability that, if every asset earns the risk-free rate, makes the model internally consistent.

Formulas

Risk-neutral probability

p = \frac{e^{rT} - d}{u - d}

Requires

d < e^{rT} < u

to ensure $0 < p < 1$ and rule out arbitrage. Hull (2022) §13.2 derives this from the condition that the delta-hedged portfolio earns the risk-free rate

r

continuously compounded over horizon

T

Option value

f = e^{-rT}\bigl[p\,f_u + (1-p)\,f_d\bigr]

Risk-neutral valuation: discount the expected payoff at the risk-free rate. The real-world drift of the stock does not appear — a result formalised by Harrison & Kreps (1979) as the equivalence between no-arbitrage and the existence of a risk-neutral (equivalent martingale) measure.

Hedge ratio (delta)

\Delta = \frac{f_u - f_d}{S_0 u - S_0 d}

The number of shares held long per option written short to create the riskless portfolio. Delta links the binomial model directly to the delta-hedging concept used in continuous-time models.

Worked examples

Scenario

Stock now worth A$50. Over 1 year, it can rise by factor $u = 1.2$ or fall by $d = 0.8$ . Risk-free rate $r = 5\%$ p.a. continuously compounded. Price a European call with strike $K = 50$ .

Solution

In the up state, $S_u = 50 \times 1.2 = 60$ , so $f_u = \max(60 - 50, 0) = 10$ . In the down state, $S_d = 50 \times 0.8 = 40$ , so $f_d = \max(40 - 50, 0) = 0$ . The no-arbitrage condition holds since $d = 0.8 < e^{0.05} = 1.0513 < u = 1.2$ ✓. Solve for the risk-neutral probability $p = (e^{0.05} - 0.8) / (1.2 - 0.8) = 0.2513 / 0.4 = 0.6282$ . The option value is then $f = e^{-0.05} \bigl[0.6282 \times 10 + 0.3718 \times 0\bigr] = 0.9512 \times 6.282 = 5.98$ . To verify via delta hedge, $\Delta = (10 - 0)/(60 - 40) = 0.50$ . The riskless portfolio at expiry equals $0.5 \times 60 - 10 = 0.5 \times 40 - 0 = 20 $, with present value$ 20 \, e^{-0.05} = 19.02$. So $f = 50 \times 0.5 - 19.02 = 5.98$ ✓.

Common mistakes

✗Risk-neutral probabilities are the market's true probabilities of an up move. They are not. They are artificial probabilities under which every asset is priced as if it grows at the risk-free rate. Real-world probabilities depend on investor risk preferences and have no role in the pricing formula.
✗A larger $u$ (more optimistic up move) raises the option price. Holding $S_0, d, r, T$ fixed, a larger $u$ increases $f_u$ but simultaneously lowers $p$ so the risk-neutral expected payoff stays unchanged. Only the discounting matters.
✗The binomial model requires knowing the stock's expected return. The no-arbitrage argument eliminates the expected return entirely. The option price depends only on $S_0, u, d, r, T$ , which is one of the most counterintuitive results in derivatives pricing.

Revision bullets

•Stock moves to $S_0 u$ or $S_0 d$ at expiry
•No-arbitrage requires $d < e^{rT} < u$
•Risk-neutral probability $p = (e^{rT} - d) / (u - d)$
•Option value $f = e^{-rT}\bigl[p\,f_u + (1-p)\,f_d\bigr]$
•Hedge ratio $\Delta = (f_u - f_d) / (S_0 u - S_0 d)$
•Real-world drift does not enter the price
•Foundation for multi-step trees and BSM

Quick check

In a one-step binomial model, the risk-neutral probability $p$ is:

The no-arbitrage condition in a one-step binomial model requires:

Connected topics

Risk-Neutral p Backward Ind.Delta Δ

In learning paths

Pricing Models Path

Sources

Hull (2022), §13.1–13.2
Hull, John C. Options, Futures, and Other Derivatives, 11th ed. Pearson, 2022. ISBN 978-0-13-693949-7.
Primary textbook reference for the one-step binomial tree: derives the hedge ratio, risk-neutral probability, and option pricing formula via both the no-arbitrage portfolio and risk-neutral valuation approaches.
Cox, Ross & Rubinstein (1979)
Cox, John C., Stephen A. Ross, and Mark Rubinstein. 'Option Pricing: A Simplified Approach.' Journal of Financial Economics 7, no. 3 (1979): 229–263.
Seminal paper introducing the binomial option pricing model, establishing the risk-neutral valuation approach for discrete-time trees and demonstrating that it converges to the Black–Scholes formula in the continuous limit.
Harrison & Kreps (1979)
Harrison, J. Michael, and David M. Kreps. 'Martingales and Arbitrage in Multiperiod Securities Markets.' Journal of Economic Theory 20, no. 3 (1979): 381–408.
Provides the mathematical foundation for risk-neutral pricing: the no-arbitrage condition in a complete market is equivalent to the existence of a unique equivalent martingale measure, underpinning the theoretical validity of discounting at the risk-free rate.
ASX Equity Derivatives — Index Derivatives
Australian Securities Exchange (ASX). 'Index Derivatives: Settlement History.' ASX, 2026.
Provides real-world context for applying the one-step binomial model: ASX SPI 200 futures and XJO options are cash-settled exchange-traded contracts suitable for classroom examples of discrete-time option pricing.

How to cite this page

Dr. Phil's Quant Lab. (2026). One-step Binomial Tree. Derivatives Atlas. https://phucnguyenvan.com/concept/binomial-one-step

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