greeks-calculator

Option Greeks Calculator

Compute the standard Greek family for European options under BSM.

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Step 1: Extract Inputs

Same as black-scholes skill: S, K, T (years), r, q, σ, option type. For multi-leg positions, gather all legs and the signed quantity (+ long, − short, multiplier 100).

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Step 2: Compute Greeks (per-share, then scale)

d1 = [ln(S/K) + (r − q + σ²/2)·T] / (σ·√T)
d2 = d1 − σ·√T
n(d1) = (1/√(2π)) · exp(−d1²/2)        # standard normal PDF

First-Order

Greek	Call	Put
Delta Δ	e^(−qT)·N(d1)	e^(−qT)·(N(d1) − 1)
Vega ν	S·e^(−qT)·n(d1)·√T (per 1.00 vol; divide by 100 for per-vol-point)	same as call
Theta Θ	−S·e^(−qT)·n(d1)·σ/(2√T) − r·K·e^(−rT)·N(d2) + q·S·e^(−qT)·N(d1)	−S·e^(−qT)·n(d1)·σ/(2√T) + r·K·e^(−rT)·N(−d2) − q·S·e^(−qT)·N(−d1)
Rho ρ	K·T·e^(−rT)·N(d2)	−K·T·e^(−rT)·N(−d2)

Second-Order

Greek	Formula
Gamma Γ	e^(−qT)·n(d1) / (S·σ·√T) — same for call and put
Vanna (DδDσ)	−e^(−qT)·n(d1) · d2/σ
Volga (DνDσ)	ν · d1·d2/σ
Charm (DδDT)	call: −e^(−qT)·n(d1)·[2(r−q)T − d2·σ·√T]/(2T·σ·√T) + q·e^(−qT)·N(d1); put: similar with sign flips
Color (DΓDT)	−e^(−qT)·n(d1)/(2·S·T·σ·√T) · [2qT + 1 + (2(r−q)T − d2·σ·√T)·d1/(σ·√T)]

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Step 3: Conventions — Always State Them

Practitioners disagree; state your scaling explicitly:

Greek	Common scaling	This skill's default
Delta	per $1 of spot	per $1 of spot
Gamma	per $1 (raw) or per $1 of spot per $1 (raw)	raw (1.00 = position re-deltas at this rate)
Vega	per 1.00 vol (raw) vs per 0.01 ("per vol point")	per vol point (raw / 100)
Theta	per year vs per calendar day	per calendar day (raw / 365)
Rho	per 1.00 rate vs per 0.01 (1bp × 100)	per 1% rate move (raw / 100)

Show the scaled numbers AND label them.

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Step 4: Position-Level Aggregation

For a multi-leg book:

net_greek = Σ (sign_i · qty_i · multiplier_i · greek_i)

• Multiplier = 100 for US equity options.
• Sign: +1 for long, −1 for short.

Use the table from references/greeks_reference.md for a worked 4-leg iron condor example.

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Step 5: Sanity Checks

1. Call Δ ∈ [0, e^(−qT)], Put Δ ∈ [−e^(−qT), 0]. Outside → bug.
2. Gamma ≥ 0 always for long options. Negative implies short positioning.
3. Vega and Gamma peak near ATM, asymmetric for high T or high σ.
4. Put-Call parity check on Δ: Δ_call − Δ_put = e^(−qT). If off by > 1e-6, your d1 has drift.

If any check fails, stop and report — don't paper over.

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Step 6: Respond to User

For a single option: a 2-column table of (greek, value, units, "per X move").

For a book: a per-leg table PLUS a "net" row, then a one-line interpretation: e.g., "Net Δ = +37 → equivalent to long 37 shares of underlying. Net vega = −$420/vol-pt → you LOSE $420 if IV climbs 1 point. Theta = +$58/day → you collect $58/day if nothing moves."

Recommend delta-hedging if net Δ is large, or iv-surface / vol-skew if vega exposure dominates.

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Reference Files

• references/greeks_reference.md — Worked examples, position aggregation table, "what does X exposure feel like" intuition guide