THE GREEKS

An option is not just a price — it is a bundle of sensitivities. The Greeks are the dials a trader turns to hedge: delta (how price moves with the stock), gamma (how delta itself drifts), vega (exposure to volatility), theta (bleed with time). Rendered, not quoted — every number below is computed live from Black–Scholes and checked against finite differences.

source Option Greeks — derived from Black & Scholes, “The Pricing of Options and Corporate Liabilities”, J. Political Economy 81(3), 1973 (DOI 10.1086/260062); systematised in Hull, Options, Futures, and Other Derivatives. AMBER: the Greeks have no single canonical paper — the closed forms are textbook. These are models with named assumptions, not investment advice.

Blue Team · builds & defends
3

THE MODEL

Under Black–Scholes, the call price is a smooth function of the stock S, strike K, rate r, vol σ, time T. The Greeks are just its partial derivatives.

d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T) d2 = d1 − σ√T Δ = ∂C/∂S = N(d1) Γ = ∂²C/∂S² = n(d1)/(Sσ√T) vega = ∂C/∂σ = S·n(d1)√T Θ = ∂C/∂t

N is the normal CDF (built here from an erf approximation), n its bell curve. Delta lives strictly in (0,1) for a call: it is the number of shares that replicate it.

5

THE LINEAGE

The Greeks are the derivatives of the-black-scholes. Where that sphere prices the option, this one differentiates it. Delta is exactly the replication weight the-risk-neutral-pricing implies — hold Δ shares and you clone the option to first order. Gamma is the reason that clone is never perfect: as S moves, Δ moves, so the hedge must be continually rebalanced.

7

THE WITNESS

Live re-check of the panel's core invariant — delta must equal N(d1), and the delta-hedge must kill the first-order stock move. Trip the tamper in window 6 and this flips red.

auditing…
The Machine
4

in ↓DATA IN

Contract & market state. Move the stock or the volatility and watch every Greek respond.

Fixed: K = 100 · r = 5% · T = 1yr

↓
0

LITTHE PANEL

Delta Δ
—
shares to hedge
Gamma Γ
—
Δ drift / $1
Vega
—
per 1% vol
Theta Θ
—
per day
Call price — · Put delta — = Δ−1
Portfolio: short 1 call, long Δ shares. Nudge S by ±$1 → value change: — (first order neutralised; residual is ½ΓΔS²).
↓
8

out ↓DATA OUT

The proven result at boot — delta equals N(d1), matches finite-difference dC/dS to 1e-4, and the delta-hedge zeroes the first-order move.

—
Red Team · attacks & breaks
1

THE ADVERSARY

wall The Greeks inherit every Black–Scholes fiction. Constant volatility — real vega is itself unstable (vol smiles). Log-normal prices, no jumps — a gap through the strike blows past any delta-hedge; gamma explodes near expiry. Frictionless, continuous rebalancing — real hedging is discrete and costs spread, so the “riskless” portfolio leaks. These are AMBER assumptions, not laws of nature.
2

THE GRAVEYARD

“Delta-hedging makes the position riskless, full stop.”

→ Only to first order. Gamma means delta moves as S moves; between rebalances the portfolio carries curvature risk. Riskless is the limit of continuous rebalancing, never the reality.

“A high delta means the option is likely to finish in the money.”

→ Delta = N(d1) is the hedge ratio, not that probability. N(d2) is the risk-neutral in-the-money probability — a different number (this is exactly the tamper).

“Vega is a Greek letter.”

→ It is not — there is no Greek letter “vega.” It is trader slang kept for the family.

6

THE TAMPER

Disclosed planted void: swap the hedge weight to Δ = N(d2) instead of N(d1). The short-call / long-share portfolio no longer neutralises the first-order stock move — the witness in window 7 catches it live.