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The Greeks

Difficulty advanced

Overview

The Greeks are the partial derivatives of option price with respect to its inputs. They translate the Black-Scholes formula into actionable risk dimensions: how much you make or lose when the underlying moves, time passes, or implied vol shifts.

Delta and gamma across strike

K price → 1.0 0.5 0.0 OTM ITM delta — call K price → Γ peak max at ATM gamma
delta is the sigmoid · gamma is its derivative (bell-shaped, peaks ATM)

First-Order Greeks

Greek Symbol Sensitivity To Sign for long call Sign for long put
Delta Δ Underlying price (S) + (0 to +1) − (−1 to 0)
Vega ν Implied volatility (σ) + +
Theta θ Time decay (−t) − −
Rho ρ Interest rate (r) + −

Delta

Δ_call = N(d1)
Δ_put  = N(d1) − 1

where: N(d1) standard-normal CDF at d1 · call delta lives in [0, 1], put delta in [−1, 0]. does: ∂Option/∂S — share-equivalent hedge ratio. A market maker who buys 100 calls at Δ = 0.45 sells 45 shares to be delta-neutral. Also the risk-neutral probability of finishing ITM (approximately). Delta dominates risk for deep-ITM positions and any non-hedged directional book.

Approximate probability the option finishes in the money (under risk-neutral measure). Used for hedge ratios — a market maker who buys 100 calls with delta 0.45 hedges by shorting 45 shares.

Vega

ν = S · N′(d1) · √T

where: S spot · N′(d1) standard-normal PDF at d1 · √T square-root-of-time scaling. does: ∂Option/∂σ — change in option price per 1.0 (100 vol-point) move in implied vol; divide by 100 for the per-vol-point quote. Hedged by trading offsetting options across strikes or expiries. Vega dominates risk for long-dated and ATM positions; near expiry it collapses toward zero.

Sensitivity to a 1-percentage-point change in implied vol. Highest for ATM options with intermediate maturity.

Theta

θ_call = − S · N′(d1) · σ / (2√T) − r·K·e^(−rT)·N(d2)
θ_put  = − S · N′(d1) · σ / (2√T) + r·K·e^(−rT)·N(−d2)

where: first term = time-decay of optionality (always negative) · second term = interest-rate effect on the discounted strike (negative for call, positive for put) · usually quoted per day (divide by 365). does: ∂Option/∂t — daily bleed from time passing alone. Long options pay theta; short options collect it. Theta dominates risk for short-dated ATM short-options books; gamma is the offsetting hedge — you sell theta to earn the carry but pay it back on realized moves.

Daily P&L from passage of time alone, holding everything else constant. Long options pay theta; short options collect it.

Rho

ρ_call =  K · T · e^(−rT) · N(d2)
ρ_put  = −K · T · e^(−rT) · N(−d2)

where: K · T · e^(−rT) PV of strike scaled by time · N(d2) risk-neutral exercise probability. does: ∂Option/∂r — sensitivity to a 1.0 change in the risk-free rate. Negligible for short-dated equity options; rho dominates risk for long-dated options, LEAPS, and fixed-income derivatives where rate moves compound over years.

Usually a second-order concern for short-dated equity options. Matters for long-dated options and fixed-income derivatives.

Second-Order Greeks

Greek What it measures Why it matters
Gamma (Γ) ∂Δ / ∂S — convexity in S Hedge re-balancing frequency
Vanna ∂Δ / ∂σ — Δ change per vol move Skew trading, FX options
Vomma / Volga ∂ν / ∂σ — vega change per vol move Vol-of-vol exposure
Charm ∂Δ / ∂t — Δ decay over time Delta hedges before expiry
Color ∂Γ / ∂t — gamma decay over time Short-dated gamma books

Gamma

Γ = N′(d1) / (S · σ · √T)

where: N′(d1) standard-normal PDF · denominator scales with spot, vol, and √T. does: ∂Δ/∂S — convexity of option value in spot. Identical for puts and calls of same strike/expiry. Long gamma rebalances spot exposure into a "buy-low, sell-high" hedge; short gamma is the carry trade with blow-up tail. Gamma dominates risk for short-dated ATM positions and near expiry.

Same magnitude for puts and calls of identical strike/expiry. Peaks for ATM, near expiry.

Long gamma ⇒ delta increases when S rises and decreases when S falls ⇒ buying low / selling high mechanically. Short gamma is the opposite — and the source of "blow-up" risk for options sellers around large moves.

Greeks Across Strike and Time

Position Δ at ATM Γ at ATM ν at ATM θ at ATM
Long short-dated call ~0.5 High Low Most negative
Long long-dated call ~0.55 Low High Mildly negative
Long deep ITM call ~1.0 Low Low Slightly negative
Long deep OTM call ~0.05 Low Low Slightly negative

Portfolio-Level Greeks

Aggregating across a book:

A delta-neutral, long-gamma book makes money on realized vol; loses theta if the underlying doesn't move.

Practical Hedging

Goal Action
Hedge directional risk Trade shares to flatten net delta
Hedge gamma risk Trade other options to flatten net gamma (often via opposite-strike)
Hedge vega risk Calendar spreads or different-strike options
Hedge theta drag Sell shorter-dated options against longer-dated longs

Rebalance frequency is a tradeoff: - More frequent ⇒ tighter hedge, higher transaction cost - Less frequent ⇒ residual gamma P&L (positive if long gamma, negative if short)

Greek-Driven Strategies

Strategy Greek profile
Long straddle Long gamma, long vega, short theta
Short strangle Short gamma, short vega, long theta
Covered call Long delta (reduced), short gamma & vega
Calendar spread ~Flat delta, long vega, mostly flat theta
Risk reversal Long delta, short skew

See the strategies subdirectory for detailed payoff structures.

Common Pitfalls

Pitfall Problem
Hedging only delta Gamma & vega exposures unaccounted for
Static Greeks They change with S, σ, t — re-measure
Ignoring smile Single-vol Black-Scholes greeks understate skew risk
Holding gamma short into events Catalysts produce realized > implied moves

q&a

Why do market makers care so much about gamma?

Gamma controls how often you need to re-hedge. High gamma = small spot move requires re-hedging = transaction cost. Low gamma = you can hold a delta hedge longer. Short gamma is the structural risk of options selling — losses scale convexly with realized moves, which is why volatility events disproportionately hurt vol-sellers.

What's the intuition for delta?

Delta is "share equivalent" — a 0.4 delta call moves like 40 shares for small spot changes. It's also approximately the risk-neutral probability of finishing ITM. Both interpretations are useful and roughly consistent.

Why does theta accelerate near expiration?

Time value scales with √T. As T → 0, the derivative ∂C/∂t blows up. Near expiration, ATM options bleed time value rapidly while OTM options decay smoothly (they have little time value left to lose).

What is 'gamma scalping' actually?

A delta-neutral, long-gamma position that re-hedges as spot moves. Each re-hedge captures a small P&L from the curvature — buy lower, sell higher mechanically. Profitable if realized variance > implied variance. The classic long-gamma trade.

Why do FX traders care about vanna and volga, but equity traders don't usually?

FX skew is roughly symmetric and stable; vol-of-vol matters. Equity skew is steep, sticky, and often more impactful than the second-order effects. Different desks weight Greeks differently based on which surface dynamics dominate their book.

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