The Greeks
The Greeks measure how an option's price responds to changes in its inputs. Delta tells you directional exposure, gamma tells you how fast your delta changes, theta tells you how fast time erodes your premium, and vega tells you how sensitive you are to volatility swings. Every options trader tracks these four numbers. Every DeFi options AMM computes them on-chain.
Interactive Greeks Calculator
Adjust spot, strike, DTE, and IV to see all four Greeks update in real time. This is the core computation every options AMM runs on-chain.
? Delta - Position Sensitivity
Delta shows how much the option price moves per $1 change in spot. It ranges from 0 (deep OTM) to 1 (deep ITM) for calls. ATM options have delta ? 0.50. Toggle strikes and DTE to see delta shift.
? Gamma - Rate of Delta Change
Gamma measures how fast delta changes per $1 move in spot. ATM options have the highest gamma - delta changes most rapidly near the strike. As expiry approaches, gamma spikes near ATM. Slide to see how gamma peaks at ATM and flattens for ITM/OTM.
? Theta - Time Decay
Theta shows daily value erosion. ATM options lose the most per day. In the last 14 days, theta accelerates steeply. Use the slider to see how daily theta changes at different DTE.
? Vega - Volatility Sensitivity
Vega shows how much the option price changes per 1% change in implied volatility. ATM options have the highest vega - they are most sensitive to vol moves. OTM options have lower vega but more proportional sensitivity (a vol increase can double the price of an OTM option).
Black-Scholes Formula - The Source of All Greeks
Implied Volatility (IV) - Back-Solving from Price
Implied volatility is the volatility the market expects, extracted by running the BS formula in reverse: given a market price, what vol must the market be pricing in? High IV = expensive options, low IV = cheap options.
What are the Greeks?
The Greeks are partial derivatives - they tell you how sensitive an option's price is to small changes in each input variable. They are the primary tool for managing options risk. If you know your position's delta, gamma, theta, and vega, you know exactly how your portfolio behaves when ETH moves, time passes, or volatility changes.
Each Greek is named after a Greek letter: Delta (?) for spot sensitivity, Gamma (?) for the rate of delta change, Theta (?) for time decay, and Vega (?) for volatility sensitivity. Together they describe the full risk profile of any options position. A position that is delta-neutral (balanced) might still have large gamma or vega exposure - which matters enormously during volatile markets.
Delta: your spot sensitivity
Delta measures how much an option's price moves per $1 move in the underlying. A call with delta 0.65 means if ETH rises $1, the call gains about $0.65. Delta is also approximately the probability the option expires in-the-money - a 0.70 delta put has about a 30% chance of finishing ITM.
For a book of options, delta tells you your net directional exposure. If you have 10 long calls each with delta 0.50, your total delta is +5 - equivalent to owning 5 ETH. To make this position delta-neutral, you'd need to sell 5 ETH or short 5 ETH worth of perps. This is exactly what Lyra's AMM does continuously: every trader order shifts the portfolio's delta, and the protocol re-hedges by trading the underlying.
Delta changes as spot moves. For a long call, delta increases as ETH rises (you become more directional). This is gamma - the rate at which your delta exposure changes. Delta can never exceed 1 for a call or go below -1 for a put. Delta ranges: Deep ITM calls -> delta ? 1 (acts like the underlying). Deep OTM calls -> delta ? 0 (almost no spot sensitivity). ATM calls -> delta ? 0.50.
Gamma: the delta accelerator
Gamma measures how fast your delta changes per $1 move in spot. If you own a call with delta 0.50 and gamma 0.05, then a $1 rise in ETH increases your delta by 0.05 to 0.55 - now your position is more directional. Conversely, if ETH drops $1, delta falls to 0.45.
Gamma is highest for ATM options and lowest for deep ITM/OTM options. It's also highest for short-dated options - as expiry approaches, small spot moves near the strike cause enormous delta swings. This is why buying weeklies is a high-variance bet: you get huge gamma exposure, which means if your directional bet is right you make a lot, but if it's wrong you lose fast.
Long options positions have positive gamma (your exposure grows in your favor when you're right, shrinks when you're wrong). Short options positions have negative gamma (your directional exposure grows against you when you're wrong). This asymmetry is fundamental to options risk management.
Theta: time decay working against you
Theta measures how much value the option loses per day due to time passing, holding all else equal. A theta of -0.10 means the option is worth $0.10 less today than yesterday, assuming spot and vol didn't change. Over 30 days, that's $3 of time value erosion from theta alone.
Theta is negative for long option holders (time is your enemy) and positive for short sellers (time is your friend). ATM options have the highest theta burn because they have the most time value to lose. As expiry approaches, theta accelerates - the last 14 days see the steepest decay. A 7 DTE option might lose 5-10% of its remaining value per day in theta.
The practical implication: if you buy options and ETH doesn't move in your direction quickly, theta erodes your position. This is why directional options traders need to be right AND timely - even a correct directional bet can lose money if it takes 60 days and theta burns 30% of the position along the way.
Vega: volatility risk
Vega measures sensitivity to changes in implied volatility. A vega of 0.12 means a 1% increase in IV (say, from 80% to 81%) increases the option price by $0.12. Vega is highest for ATM options with moderate DTE - exactly the options most retail traders buy.
Long option holders are long vega - they benefit when IV rises and suffer when IV falls. Short option sellers are short vega - they benefit when IV falls (their sold options get cheaper to buy back) and suffer when IV spikes (their short positions blow up). During a market crash, IV spikes and long options surge in value while short option sellers get crushed - this is why selling puts during a panic is so dangerous despite seeming like a "safe" income strategy.
The IV surface shows how implied volatility varies across strikes (skew) and expirations (term structure). Typically, OTM options have higher IV than ATM (the volatility "smile" or "skew"), reflecting the market's demand for downside protection. On-chain AMMs manage this skew by widening spreads for OTM strikes where the market demands more premium.
How DeFi protocols use the Greeks
Lyra's AMM computes delta for every position using Black-Scholes, then immediately hedges by trading Synthetix perps to keep the overall book delta-neutral. The protocol tracks portfolio delta across all open positions and aggregates hedging trades to minimize slippage. LPs effectively sell gamma to traders - the protocol hedges the delta, leaving LPs exposed to the gamma and vega of the underlying option book.
Dopex uses a similar approach but with Atlanticstraddle architecture - option decks combine multiple strikes and expiries, and the vault system manages Greek exposure passively across the entire book. The protocol uses external vol oracle feeds to compute d? and the Greeks, with competitive market makers narrowing spreads as modelVol updates.
Premia's concentrated liquidity model lets LPs choose their own strike ranges, essentially deciding which parts of the volatility surface to provide liquidity in. An LP in a tight ATM range earns more fees but has higher gamma exposure; an LP in a wide OTM range earns less but has lower Greek risk. This gives LPs granular control over their Greek exposure - a sophistication that doesn't exist in traditional options market making.
Frequently asked questions
- What is delta and how does it work for options?
- Delta (?) measures how much an option's price changes for a $1 move in the underlying. A call option with delta 0.50 moves $0.50 for every $1 move in ETH. Delta ranges from 0 to 1 for calls (deep OTM ? 0, deep ITM ? 1) and -1 to 0 for puts. An at-the-money option has delta ? 0.50 for calls and -0.50 for puts. Delta is also approximately the probability that an option expires ITM - a 0.70 delta call is roughly 70% likely to finish in the money.
- Why does gamma matter more near expiry?
- Gamma measures how fast delta changes per $1 move in spot. ATM options have the highest gamma because small spot moves cross the strike, flipping delta from 0.50 toward 1 (call) or toward 0. As expiry approaches, the time value shrinks and the option's price becomes extremely sensitive to spot moves near the strike - gamma spikes. This is why options near expiry can suddenly spike or crash on small news. Gamma is highest for ATM options and approaches zero for deep ITM/OTM options.
- How does vega affect my option's price?
- Vega measures sensitivity to implied volatility. A vega of 0.15 means a 1% increase in IV increases your option's price by $0.15. ATM options have the highest vega (most to lose if IV drops, most to gain if IV rises). When you're long options and IV rises, you benefit. When you're short options and IV falls, you benefit. This is why selling options in high-IV environments (like pre-major-news periods) is profitable - you're selling expensive options that have high vega exposure.
- What is theta and how does it erode option value?
- Theta (?) is the daily time decay of an option - how much value the option loses per day assuming spot doesn't move. A theta of -0.05 means the option loses $0.05 per day. Theta is negative for long option holders (you lose money daily) and positive for short sellers (you gain from time decay). ATM options have the highest theta burn. Theta accelerates in the last 2-3 weeks of an option's life as the time value rapidly disappears. This is why selling 30-60 DTE options can be more profitable than selling 7 DTE - you collect more time value upfront with less gamma risk.
- How does Black-Scholes relate to the Greeks?
- Black-Scholes produces a closed-form option price, and the Greeks are simply the partial derivatives of that price with respect to each input variable. d? = (ln(S/K) + (r + ?)T) / (??T) is the core intermediate variable. N(d?) is the cumulative normal distribution (delta). N'(d?) is the standard normal PDF (used in gamma, vega, theta). All four Greeks - delta, gamma, theta, and vega - are derived directly from BS, which is why they are all computable from the same four inputs (S, K, T, ?).
- How do DeFi protocols use Greeks for pricing?
- Lyra's AMM uses Black-Scholes with a protocol-controlled modelVol (internal volatility estimate) to compute delta for every quoted option. The AMM then immediately hedges its delta exposure by trading the underlying or perps, keeping the overall portfolio delta-neutral. The gamma and vega exposure accumulate in the LP pool - LPs are effectively short gamma (they lose if prices make big moves outside their hedged range). Dopex uses a similar BS framework but with external vol oracle feeds. Premia's concentrated liquidity model requires LPs to manage their own Greek exposure within their chosen strike range.
- Why do option sellers have 'short gamma'?
- When you sell an option, you're short gamma - meaning your delta becomes less favorable as the underlying moves in either direction (for calls). If you sell a $2,000 call and ETH rallies to $2,500, your short call delta becomes increasingly negative and you lose more with each additional $1 move. You're exposed to large directional swings. A gamma-short position is the opposite of a long option holder's gamma-long position, where big moves work in your favor. This is why selling naked calls is dangerous - theoretically unlimited loss, compressed into a short premium receipt.