Price an option the honest way: build a lattice of up/down moves, pick the one probability that removes all arbitrage, and roll the payoff back to today. Shrink the time step and the lattice becomes Black–Scholes before your eyes. Down the center, data flows: the market inputs go in, the tree computes by backward induction, the price comes out. The blue team builds and defends it; the red team tries to break it. This is a model, not investment advice.
source Cox, Ross & Rubinstein, Option Pricing: A Simplified Approach, J. Financial Economics 7(3) 229–263 (1979) — ideas.repec.org/a/eee/jfinec/v7y1979i3p229-263 · doi:10.1016/0304-405X(79)90015-1. Rendered, not quoted.
Over one step of length dt the stock either multiplies by u = e^(σ√dt) or by d = 1/u. The recombining tree of these moves is the whole state space.
The valuation probability is not the real-world odds — it is the unique risk-neutral weight p = (e^(r·dt) − d) / (u − d) that makes today's stock its own discounted expectation. Then every option is
price = e^(−r·dt) · [ p·Vup + (1−p)·Vdown ]
applied backward from the terminal payoff. Live values for the current inputs:
This sphere is the-risk-neutral-pricing made discrete. One replicating step — a stock-plus-bond hedge — pins the price with no probabilities at all; iterate the step and the risk-neutral measure appears as bookkeeping.
Now let n → ∞. The binomial log-return is a sum of tiny ±σ√dt shocks; the Central Limit Theorem carries it to a normal, and the lattice price converges to the-black-scholes closed form. Each sphere is the next one's limit.
The blue team's live check: recompute the 500-step lattice price and compare it to the Black–Scholes closed form. If red swaps the risk-neutral p for a coin flip, convergence breaks and this badge turns red.
A European option needs five numbers plus a step count. Each is fed straight into the lattice below:
| symbol | meaning |
|---|---|
| S | spot price of the stock today |
| K | strike — the agreed exercise price |
| r | risk-free rate (continuously compounded) |
| σ | volatility assumed constant |
| T | time to expiry, in years |
| n | number of lattice steps (dt = T/n) |
From these, the engine derives u, d, p and rolls the payoff back. Feed them into the panel below.
Change any input — u, d, p and the price are recomputed from the closed-form lattice on the spot, never looked up.
What the machine produces, proven: the arbitrage-free option price, the no-arbitrage band d < e^(r·dt) < u, and the gap to Black–Scholes shrinking toward zero as steps grow. Backward induction and the discounted risk-neutral expectation agree to 1e-9; a one-step tree is exactly replicated by a stock-plus-bond hedge.
The blue team's witness (left) re-checks convergence live; the red team (right) tries to make the lattice lie.
So the "no-arbitrage price" is only exact inside the model. It is a disciplined benchmark, not a promise — and never advice to buy or sell anything.
"p is the real probability the stock goes up." Cut. p is a pricing weight, not a forecast — it depends on r, not on any view of returns. The lattice prices correctly even when the true drift is nothing like it.
"The binomial model is just an approximation of Black–Scholes." Corrected. It is a self-contained no-arbitrage model in its own right; BS is its limit, and the lattice also prices American and path-dependent options BS cannot.
"More steps always means a smoother, better price." Kept, corrected. Convergence is real but oscillates with n's parity (odd/even) around the strike — it settles, it does not slide straight in.
The red team's move: replace the risk-neutral p with a naïve real-world coin flip p = 0.5. The lattice still returns a number — but it is the wrong number, it no longer converges to Black–Scholes, and it admits arbitrage. The witness (window 7) is watching.
Swap the risk-neutral weight for 0.5 and the 500-step price drifts far from the closed form — the witness recomputes, sees the gap blow past tolerance, and turns red. Nothing is faked; the attack is real and it is caught.