One-Period Binomial Option Value

Builds onExpected Value · Forward Price — if this page feels steep, start there.

c_0 = \frac{\pi c_u + (1-\pi) c_d}{1+r}, \qquad \pi = \frac{(1+r) - d}{u - d}

$u, d$: the up and down factors: the only two things the stock can multiply by this period
$c_u, c_d$: the option's payoff in the up state and the down state
$\pi$: the risk-neutral probability — the fake odds under which the stock earns exactly r
$(1+r)-d, \; u-d$: π's recipe: how far the riskless growth sits between the down and up factors
$c_0$: the option's value today: π-weighted payoff, discounted at the risk-free rate

Why it must be true

Price an option without knowing the probability the stock goes up — that is the binomial model's magic trick. In a world with only two futures (stock up by factor $u$ or down by factor $d$ ), a dealer can hedge the option perfectly with stock and borrowing. A perfectly hedged position must earn the risk-free rate, whatever the true odds are.

The bookkeeping shortcut for that hedge is the risk-neutral probability $\pi$ : the fake probability under which the stock itself would earn exactly the risk-free rate. Weight the option's two payoffs with $\pi$ , discount at the risk-free rate, done. $\pi$ is not anyone's forecast — it is the probability the hedge implies, and real-world optimism appears nowhere in the price.

The derivation

Force the stock to be "fair" under fake probabilities: demand its π-expected growth equal the risk-free rate:

\pi u + (1-\pi) d = 1 + r \quad\Rightarrow\quad \pi = \frac{(1+r) - d}{u - d}

The option pays $c_u = \max(uS - X, 0)$ up and $c_d = \max(dS - X, 0)$ down. Under the fake measure, discount its expected payoff like any riskless claim:

c_0 = \frac{\pi c_u + (1-\pi)c_d}{1+r}

Why legal? Because the same numbers fall out of an explicit hedge (buy $\Delta$ shares, borrow the rest) with no probabilities at all — π is just the tidy way to write the hedge's answer. Chain many small periods and this converges to Black–Scholes.

When to reach for it

Valuing an option in a two-state (one-period binomial) world given up/down factors, the risk-free rate and the strike.

Listen for

“one-period binomial model”“up factor / down factor (u and d)”“risk-neutral probability”“the stock will either rise to … or fall to …”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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π first, always: how far does (1+r) sit between d and u? With u=1.25, d=0.8, r=5%: π = (1.05−0.8)/0.45 ≈ 0.556. It must land strictly between 0 and 1 — outside means an arbitrage (or a typo).
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Payoffs before probabilities: cu = max(uS − X, 0), cd likewise. For an out-of-the-money down state cd = 0 and the formula collapses to π·cu/(1+r).
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The REAL probability of an up-move is a decoy if given — risk-neutral valuation never touches it.

Traps in applying it

✗Using the actual/physical probability of an up-move instead of π — the model's whole point is that the hedge sets the odds.
✗Forgetting to discount the expected payoff by (1+r).
✗Computing π with the factors swapped — the numerator is (1+r) − d, the DOWN factor.
✗Forgetting the max(·, 0): an option's payoff can't go negative.

Limits & criticisms

Two states per period is a cartoon — accuracy comes only from chaining many steps, at which point the tree is a numerical method, not a formula. The hedge argument also needs frictionless rebalancing and no jumps; real markets gap (crashes are one big step the lattice didn't allow), and hedging costs money. As always, the elegance holds exactly in the model and approximately in life.

Where it came from

Cox, Ross and Rubinstein (1979) built the binomial lattice as a teaching-friendly, discrete route to the Black–Scholes–Merton (1973) result — and it promptly became more than pedagogy: the lattice handles American early exercise, dividends and real options where the famous formula cannot. Trading desks, employee-stock-option valuers and real-options analysts still run binomial trees daily, and "risk-neutral valuation" — the fake-probability trick this formula teaches — is the master key to all of modern derivatives pricing.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

The fake odds

\pi = \frac{(1+r) - d}{u - d}

The measure-change face: the probability that makes the STOCK fair at the risk-free rate — the hedge's implied odds, no forecast involved.

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Discounted risk-neutral payoff

c_0 = \frac{\pi c_u + (1-\pi)c_d}{1+r}

The valuation face: weight the two payoffs by the fake odds and discount risklessly — the master recipe of derivatives pricing.

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On the BA II Plus

Worked example: One-period binomial: S = $85.00, u = 1.2, d = 0.85, r = 5%, strike $40.00. Value the call using risk-neutral probabilities.

1.1 [+] 0.05 [−] 0.85 [=] [÷] [(] 1.2 [−] 0.85 [)] [=] [STO] 1π, the risk-neutral probability
2.62 [×] [RCL] 1 [=] [STO] 2π × up-payoff (cu = 62)
3.1 [−] [RCL] 1 [=] [×] 32.25 [+] [RCL] 2 [=]add (1−π) × down-payoff (cd = 32.25)
4.[÷] [(] 1 [+] 0.05 [)] [=]discount one period

→ $46.90

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