The formula that gave every option a price. Assume a stock drifts and shakes as geometric Brownian motion, let a portfolio hedge away the shake, and one number falls out: what a European call is worth today. Down the center the terms go in, the closed form computes, the price comes out. The blue team builds and defends it; the red team attacks its assumptions. These are models, not investment advice.
source Black & Scholes, The Pricing of Options and Corporate Liabilities (1973), Journal of Political Economy 81(3), 637–654 — stable id jstor.org/stable/1831029 (paywalled — AMBER). Rendered, not quoted.
Price a European call with
C = S·N(d₁) − K·e−rT·N(d₂)
d₁ = [ ln(S/K) + (r + σ²/2)·T ] / (σ√T)
d₂ = d₁ − σ√T
N is the standard-normal CDF, computed here from erf (Abramowitz–Stegun 7.1.26). For the current inputs, the pieces the panel is standing on:
| quantity | value |
|---|
The price rides on the-geometric-brownian-motion: the stock is S·e(μ−σ²/2)t+σW. Hedging cancels the real drift μ, so pricing happens under the risk-neutral measure where the drift is r. Black, Scholes and Merton, 1973.
The equation that opened the listed options market. It assumes constant volatility σ, log-normal prices, no jumps, and continuous frictionless hedging — every one of those is AMBER, and the red team is standing on each.
The blue team's live check: re-derive put-call parity, the T→0 and deep in/out limits, vega > 0, and a 700-step binomial that must converge on the formula. If red tampers, this badge is where it shows.
Everything the formula needs is five numbers — four you can read off the market, one you must estimate:
| term | is | from |
|---|---|---|
| S | spot price of the stock | market |
| K | strike of the option | contract |
| r | risk-free rate | market |
| T | time to expiry (years) | contract |
| σ | volatility | estimated ▲ |
Four are observable; σ is not. The whole argument the red team makes lives in that last row — you feed a guess about the future into the panel below.
Move any slider — C, P and the d-values are computed from the closed form on the spot, never looked up.
What the machine produces, proven: a call price C and a put price P that satisfy put-call parity C − P = S − K·e−rT to 1e-9, that reach the right limits as T→0 and as the option goes deep in/out of the money, and that a 700-step binomial lattice reproduces to within the stated tolerance.
The blue team's witness (left) re-derives these live; the red team (right) tries to make the price lie.
It also assumes continuous, frictionless hedging — no transaction costs, no gaps, infinite divisibility. Real hedging is discrete and costs money, so the perfect replicating portfolio is a limit, not a trade. The formula is not the price; it is the first proof that an option's price has a computable shape.
"Black-Scholes gives the true price of an option." Cut. It gives the arbitrage-free price under its assumptions. Traders quote in implied vol precisely because the flat-σ assumption is wrong.
"σ is just the stock's historical volatility." Cut. The input is expected future volatility over the option's life. Historical σ is one estimate; the market's implied σ is another, and they disagree.
"It's a recommendation to buy or sell." Cut. It is a model, not advice. It prices a contract given inputs; it says nothing about whether you should hold it.
"Merton was left out." Kept, corrected. Merton's continuous-hedging derivation is why it is Black-Scholes-Merton; the 1997 Nobel named Scholes and Merton (Black had died).
The red team's move: drop the discount on the strike — use K instead of K·e−rT in d₂ and the price. At r > 0 the price is now wrong and put-call parity breaks. The blue team's witness (window 7) is watching.
Remove the strike discount and the call is over-priced while parity C − P = S − K·e−rT no longer holds — the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.