Options Greeks Guide

Δ Delta — Price Sensitivity

Delta is how much the option's price changes per $1 move in the underlying. Calls range 0 → +1, puts range −1 → 0. ATM ≈ ±0.50. As price moves, gamma causes delta to shift — giving options their convex payoff. Long options gain faster on winning moves and lose slower on losing moves.

Deep ITM call → δ near +1.0 ATM call → δ ≈ +0.50 Far OTM call → δ near 0 100 delta ≈ 1 share equivalent δ also ≈ probability of expiring ITM Rising price → call δ increases (gamma effect)

Long Call · Price Rising

Delta climbs toward 1 — gains accelerate (convexity)

↑ Delta rising · P&L accelerates — convexity works for you

Long Call · Price Falling

Delta shrinks toward 0 — losses decelerate

↓ Delta falling · Losses slow — convexity cushions the fall

Long Put · Price Falling

Put delta deepens toward −1 — gains accelerate

↓ Delta deepening · P&L rises faster as price falls

Long Put · Price Rising

Put delta approaches 0 — put bleeds value

↑ Delta fading · Put loses value as it moves OTM

Underlying Price

Delta (|value|)

Option P&L (gain)

Option P&L (loss)

Γ Gamma — Rate of Delta Change

Gamma is the rate at which delta changes per $1 move. Highest at ATM, near zero deep ITM/OTM. Explodes as expiration approaches — a small move near 0 DTE can flip delta dramatically. Long gamma: you benefit from large moves. Short gamma: you profit from stillness but bleed on big moves. See the Gamma Scenarios page for dealer flow detail.

Gamma peaks at ATM strike Gamma spikes near 0 DTE (pin risk!) Long gamma = convex P&L Short gamma = concave P&L (range wins) Long gamma: want big moves either way Short gamma: want price pinned near strike

Gamma Profile — 30 DTE

Broad bell curve — risk spread across nearby strikes

Manageable gamma at 30 DTE — moves are measured

Gamma Profile — Near Expiry (2 DTE)

Spike at ATM — tiny move = massive delta shift

⚠ Extreme gamma near expiry — pin risk is real

Long Gamma P&L Profile

Convex curve — profits on large moves either direction

✓ You want the underlying to move far from entry

Short Gamma P&L Profile

Concave curve — max profit at ATM, heavy losses on wings

✗ You need the underlying to stay near entry

Gamma Intensity

Profit zone

Loss zone

ATM / Entry

Θ Theta — Time Decay

Theta is the daily loss in option value due to time passing — all else equal. Long options lose theta every day. Short options collect theta. Decay is non-linear: accelerates sharply in the last 30–45 DTE. ATM options have the highest theta; deep ITM/OTM options have far less.

ATM options have the highest theta Theta accelerates under 30 DTE Weekends: 3 days of theta in one close Long option: theta is your enemy Short option: theta is your friend Theta + Gamma: always a trade-off

Long ATM Option — Theta Bleed

Value erodes daily, decay rate accelerates near expiry

📉 Each day the clock ticks against you

Short Option — Theta Collection

Premium you keep accelerates as expiry approaches

📈 Each day the clock ticks in your favor

ATM vs OTM Theta Comparison

ATM decays fastest — OTM has far less daily theta

Sell ATM for most theta; buy OTM for less daily bleed

Daily Theta Rate by DTE

Theta is gentle far out, then becomes aggressive near expiry

⚡ The 30 DTE cliff — where decay really kicks in

Option Value (long)

Premium collected (short)

Theta loss rate

OTM comparison

ν Vega — Volatility Sensitivity (IV Expansion & Crush)

Vega is how much option value changes per 1% move in Implied Volatility (IV). Long options are long vega — you want IV to rise. Short options are short vega — you want IV to fall. IV crush after earnings can destroy long option value even if you predicted the direction correctly. Vega is highest ATM and declines for deep ITM/OTM options.

Vega highest at ATM Longer-dated options have more vega IV crush = vol collapses post-event Vol expansion = spike in uncertainty (VIX ↑) Long options: want IV ↑ Short options: want IV ↓ (sell pre-earnings)

Long Option · IV Expansion

Volatility spikes — long option value rises sharply

📈 Long vega profits — rising IV lifts all options

Long Option · IV Crush (Post-Earnings)

IV collapses after event — option tanks despite correct direction

💀 IV crush kills longs — even being right isn't enough

Short Option · IV Expansion

IV spike hurts short options — mark-to-market loss

⚠ Short vega gets crushed by sudden vol spikes

Short Option · IV Crush

Sell premium before event, buy back cheap after vol drops

✓ The earnings premium-selling play — short vega wins

Implied Volatility (IV)

Underlying Price

Option P&L (gain)

Option P&L (loss)

Event marker

∂Δ/∂σ Vanna — Delta's Sensitivity to Volatility

Vanna measures how delta changes when IV moves (equivalently, how vega changes when the underlying price moves). When IV rises, OTM options gain delta — they become more likely to expire ITM. When IV falls, that delta evaporates. Dealer vanna flows matter: dealers short OTM calls must buy more delta as IV rises (amplifying moves), and sell delta as IV falls. Vanna is largest for near-ATM, short-dated options.

Vanna = dDelta/dVol = dVega/dSpot IV rises → OTM call delta increases IV falls → OTM call delta contracts Vanna flows peak near ATM, short DTE Price up + IV up = vanna amplification IV crush = delta AND vega loss simultaneously

IV Rising → OTM Call Delta Increases

Vanna lifts delta of OTM calls as vol expands

IV ↑ pulls OTM delta upward — calls get more sensitive

IV Crush → Delta Contracts (Double Hit)

OTM calls lose both vega value and delta simultaneously

IV ↓ drains delta + vega — long OTM options crushed

Vanna Flow · Price Up + IV Up → Dealers Buy

Dealers short OTM calls must buy delta as vanna rises it

⚡ Vanna amplifies: rising IV forces dealer buying

Vanna Flow · Price Down + IV Down → Dealers Sell

IV drop reduces dealer delta needs — systematic selling

⚡ Vanna amplifies: falling IV forces dealer selling

Implied Volatility

OTM Call Delta

Underlying Price

Dealer buy arrows

Dealer sell arrows

∂Δ/∂t Charm — Delta Decay Over Time

Charm is the rate at which delta changes as time passes — think of it as "theta of delta." OTM option delta bleeds toward 0 every day (they become less likely to go ITM). ITM option delta firms toward 1 (binary outcome becomes clearer). ATM options stay near 0.50 but become explosively binary right at expiry. Dealers must rebalance hedges daily as charm shifts their delta exposure — creating systematic pressure.

Charm = dDelta/dTime OTM call delta drifts toward 0 over time ITM call delta drifts toward 1 over time ATM delta stays ~0.50 but becomes binary near expiry Dealers rebalance daily as charm shifts deltas Largest effect: near ATM, last 30 DTE

OTM Call — Delta Fades Toward 0

Price unchanged — but OTM delta bleeds away daily

Charm erodes OTM delta even if price stays still

ITM Call — Delta Firms Toward 1

ITM call becomes more certain — delta rises to 1

ITM call delta strengthens as expiry approaches

ATM Call — Binary Flip Near Expiry

ATM delta stays ≈ 0.50, then snaps to 0 or 1 at 0 DTE

ATM charm: stable until the last moment — then chaos

Charm Rebalancing — Dealer Delta Drift

OTM deltas fade, ITM deltas firm — dealers must adjust daily

Daily charm flows create systematic buy/sell pressure

Delta (ATM)

Delta (OTM)

Delta (ITM)

DTE countdown →

∂V/∂σ² Vomma — Vega's Sensitivity to Volatility (Vol of Vol)

Vomma (also called Volga) measures how vega itself changes as IV changes. High positive vomma means each additional % of IV rise makes the option more and more vega-sensitive — an accelerating effect. OTM options have high vomma: vega surges when vol spikes. ATM options have low vomma: vega changes nearly linearly. Long vomma positions (OTM options) are bets on vol-of-vol — that IV itself becomes more volatile. Short vomma (short OTM strangles) profit from vol stability but are exposed to vol acceleration.

Vomma = dVega/dVol (2nd deriv of price w.r.t. vol) OTM options: high vomma (vega surges with IV) ATM options: low vomma (vega is more stable) Long vomma = bet on vol-of-vol rising Short strangle = short vomma (danger in vol spikes) VIX spike events = vomma triggers

OTM Option — High Vomma (Vega Accelerates)

Each % of IV rise adds more vega than the last

OTM vega curves upward — doubly benefits from vol expansion

ATM Option — Low Vomma (Vega Nearly Linear)

ATM vega rises with IV but without the curvature

ATM vega is more predictable — lower vol-of-vol exposure

Long OTM Straddle · Vol Spike — Vomma Payoff

Vega gains as IV rises AND vomma accelerates the gain further

📈 Vol spike: vomma creates exponential P&L growth

Short Strangle — Vomma Risk in Vol Expansion

Short OTM options: losses accelerate as IV rises (negative vomma)

⚠ Negative vomma: losses grow faster than expected in vol spikes

Vega (OTM — curved)

Implied Volatility

Vega (ATM — linear)

P&L gain

P&L loss

ρ Rho — Interest Rate Sensitivity

Rho measures sensitivity to changes in the risk-free interest rate. Calls have positive rho — higher rates make calls slightly more valuable (cost-of-carry rises, owning the call is cheaper than owning stock). Puts have negative rho — higher rates make puts slightly less valuable. Rho matters most for long-dated LEAPS; for weekly options it is nearly negligible.

Calls: positive rho (rate rises → call value ↑) Puts: negative rho (rate rises → put value ↓) LEAPS most sensitive to rate changes Short-dated options: rho effect minimal Fed hikes → tailwind for calls, headwind for puts

Calls · Rising Interest Rates

Positive rho — call value rises with rates (esp. LEAPS)

Call value benefits modestly from rising rates

Puts · Rising Interest Rates

Negative rho — put value falls as rates rise

Put value is pressured by rising rates

Interest Rate

Call Value

Put Value