The One-Period Binomial Model

With just an up-state and a down-state, replication pins down a unique option price via a risk-neutral probability, no real-world probabilities required.

1. The idea

Suppose a stock at $S_{0}$ moves to $S_{0} u$ (up) or $S_{0} d$ (down) after one period, with $d < e^{r T} < u$ to preclude arbitrage. An option with known payoffs $C_{u}$ and $C_{d}$ in each state can be replicated by holding $Δ$ shares of stock and $B$ in the risk-free bond.

Solving the two replication equations $Δ S_{0} u + B e^{r T} = C_{u}$ and $Δ S_{0} d + B e^{r T} = C_{d}$ gives $Δ = \frac{C _{u} - C _{d}}{S _{0} ( u - d )}$ , the option's hedge ratio. The option price is the cost of this replicating portfolio, $Δ S_{0} + B$ .

This price can be rewritten as a discounted expectation under a risk-neutral probability $p = \frac{e ^{r T} - d}{u - d}$ : $C_{0} = e^{- r T} (p C_{u} + (1 - p) C_{d})$ . Crucially, $p$ is not the real-world probability of an up move — it is whatever probability makes the stock's expected return equal the risk-free rate, purely a pricing device from replication.

2. Why it matters

The binomial model is the discrete-time ancestor of Black-Scholes and the clearest illustration that option pricing does not require knowing real-world probabilities, only the ability to replicate and the absence of arbitrage.

3. Core mathematics

Risk-neutral probability

p = \frac{e ^{r T} - d}{u - d}

Option price

C_{0} = e^{- r T} (p C_{u} + (1 - p) C_{d})

Replicating delta

Δ = \frac{C _{u} - C _{d}}{S _{0} ( u - d )}

No-arbitrage condition

d < e^{r T} < u

4. How to recognize it

Think of this when…

A stock given two possible future prices and asked to price an option or derivative
'Find the hedge ratio' or 'replicating portfolio' language
Any 'risk-neutral probability' or 'risk-neutral valuation' phrasing

5. Worked example

Worked example

A stock at $100 moves to $120 (up) or $90 (down) after one period. The risk-free rate over the period is 5% (i.e. $e^{r T} = 1.05$ ). Price a call with strike $100$ .

1
Compute payoffs.
$C_{u} = max (120 - 100, 0) = 20, C_{d} = max (90 - 100, 0) = 0$
2
Compute u, d and risk-neutral probability.
$u = 1.2, d = 0.9, p = \frac{1.05 - 0.9}{1.2 - 0.9} = 0.5$
3
Discount the expected payoff.
$C_{0} = \frac{1}{1.05} (0.5 (20) + 0.5 (0)) = \frac{10}{1.05} \approx 9.52$
Answer: Call price ≈ $9.52

6. Common mistakes

Using the real-world probability of an up move instead of the risk-neutral probability
Forgetting the no-arbitrage bound $d < e^{r T} < u$ , which if violated means the model is inconsistent
Discounting at the wrong rate or forgetting to discount at all

7. Quick check

The risk-neutral probability $p$ is used because it makes:

The replicating delta $Δ$ equals:

8. Practice

8 QuantSolve questions use this concept.

Practice this concept

9. Related methods

No-Arbitrage Replication

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