The Complete Taxonomy of Options Greeks by Order

Introduction: Greeks Are Partial Derivatives¶

Every options trader has heard of Delta. Fewer trade around Gamma. Almost nobody looks at Zomma. That isn’t an accident. The Greeks form a hierarchy, organized by how many times you have differentiated the Black-Scholes pricing function. Delta is a first derivative. Gamma is a second. Zomma is a third. Each additional layer of differentiation captures a new, smaller effect in the option’s price behavior, and the further down the hierarchy you go, the more specialized and marginal the Greek becomes.

Understanding this hierarchy is useful because it tells you, at a glance, which Greeks matter for your strategy, which matter only for market-makers running large books, and which are monitoring curiosities rather than hedging tools. This post walks through all sixteen options Greeks, organized by order, and points you to where you can learn each one in depth.

What “Order” Means¶

The Black-Scholes formula prices a European option as a function of six inputs: spot price, strike price, volatility, time to expiration, the risk-free rate, and dividend yield. Every one of the options Greeks is a partial derivative of that price function with respect to one or more of those inputs.

A first-order Greek is a single partial derivative. It answers the question, “If one input changes by a small amount, how much does the option price move?” Delta, the partial of price with respect to spot, is the most familiar example. These are linear approximations, which work well for small perturbations and break down for larger moves.

A second-order Greek is a second partial derivative. It comes in two flavors: pure second derivatives like Gamma, which measure curvature with respect to a single variable, and cross-partials like Vanna, which measure how the sensitivity to one variable changes when another variable moves (Garman 1976).

A third-order Greek is a third partial derivative, the curvature of the curvature. The Taylor expansion of option profit and loss makes the logic explicit: each additional order captures the next, smaller correction term in the approximation. Third-order Greeks matter only in extreme scenarios or for very large portfolios.

First-Order Greeks¶

Delta (Δ)¶

Delta is the hedge ratio. It measures how much an option’s price changes when the underlying moves by one dollar. For a call, Delta ranges from 0 to 1; for a put, from 0 to negative 1. Market-makers hedge every trade they execute by buying or selling the underlying to stay delta-neutral. For retail traders, Delta doubles as a rough proxy for the probability the option finishes in the money. It is the single most important of the options Greeks, and every strategy begins with a view on Delta.

Learn Delta in depth at the Skavinski Academy.

Theta (Θ)¶

Theta is the time decay. It measures how much an option loses in value for each day that passes, holding all other inputs constant. Long options bleed theta; short options collect it. The gamma-theta tradeoff is the central equation of market-making: being long gamma lets you profit from realized volatility but forces you to pay theta, while being short gamma does the opposite. Understanding this tradeoff is what separates directional traders from volatility traders, a theme explored at length in our volatility risk premium post.

Learn Theta in depth at the Skavinski Academy.

Vega (ν)¶

Vega is the sensitivity to implied volatility. It tells you how much an option’s price changes for each one-percentage-point move in implied vol. Unlike Delta, Vega cannot be hedged with the underlying; only other options can offset it. Vega peaks for at-the-money options with substantial time to expiration and shrinks to zero at expiration. Traders who think options are rich or cheap are expressing a view on Vega, which is why the volatility risk premium literature focuses on it.

Learn Vega in depth at the Skavinski Academy.

Rho (ρ)¶

Rho measures sensitivity to interest rates. Calls rise in value when rates move up; puts fall. For short-dated options, Rho is close to zero and can be ignored. For long-dated options like LEAPS, Rho becomes meaningful: a one-percentage-point move in rates can shift a two-year LEAPS call by several percentage points. Rho is the Greek that most traders forget until rates become volatile, and the one fixed-income derivatives desks think about first (Merton 1973).

Learn Rho in depth at the Skavinski Academy.

Lambda (λ)¶

Lambda, sometimes called elasticity or omega, is option leverage. It measures the percentage change in option price for a one-percent change in the underlying. A deep out-of-the-money call can carry a Lambda above 20, meaning a one-percent move in the stock produces a twenty-percent move in the option. Lambda translates Delta into a percentage-return framework that matches how retail traders think about position sizing and capital efficiency.

Lambda is coming to the Skavinski Academy in an upcoming module.

Epsilon (ε)¶

Epsilon, occasionally called Psi, measures sensitivity to the dividend yield. It tells you how much an option’s price changes when the assumed dividend yield moves. For single-name equities near ex-dividend dates, Epsilon matters because the expected dividend stream is priced into the option. For indices and non-dividend-paying stocks, it is typically negligible. Most traders only think about Epsilon when they are forced to, usually when a company announces a special dividend.

Epsilon is coming to the Skavinski Academy in an upcoming module.

Second-Order Greeks¶

Gamma (Γ)¶

Gamma measures how Delta changes as the underlying moves. It is the rate of change of the rate of change, which is another way of saying it is the curvature of the option price with respect to spot. Gamma is highest at-the-money and spikes dramatically as expiration approaches. For market-makers hedging large books, Gamma is the Greek that drives daily profit and loss through the practice of gamma scalping. Aggregated across an index, dealer Gamma shapes market microstructure itself, a phenomenon explored in depth in our gamma exposure post and documented in academic research on expiration-day pinning (Ni, Pearson, and Poteshman 2005; Baltussen, Da, Lammers, and Martens 2021).

Learn Gamma in depth at the Skavinski Academy, where we also walk through the full family of options Greeks that drive dealer flows.

Vanna¶

Vanna is the cross partial derivative of option price with respect to both spot and volatility. Equivalently, it measures how Delta changes as implied volatility moves, or how Vega changes as spot moves. When a large options market-maker is net short puts and implied vol drops after a scheduled event, the deltas of those puts shrink, leaving the dealer over-hedged and mechanically forced to buy stock back. This rally dynamic is estimated to explain a meaningful share of post-FOMC afternoon drift (Barbon and Buraschi 2021).

Charm¶

Charm is the cross partial of price with respect to spot and time. In plain English, it is the rate at which Delta decays simply from the passage of time. If a call has a Delta of 0.40 today and a Charm of negative 0.05 per day, its Delta will drift to roughly 0.35 tomorrow even if nothing else changes. For delta-hedgers managing overnight and weekend exposure, and for 0DTE traders watching expiration-day flows, Charm is the Greek that determines how much rehedging is required purely from the clock ticking. Our 0DTE options guide unpacks the expiration-day mechanics.

Vomma¶

Vomma, sometimes called Volga, is the second derivative of price with respect to volatility. It measures the convexity of Vega itself, answering the question, “how much does Vega change when implied vol changes?” Vomma peaks for far out-of-the-money options, especially puts, which is why far-OTM puts have a double benefit in a crash: the volatility spike raises their price through Vega, and their positive Vomma means that Vega itself grows as vol continues rising. This is why crash protection via far-OTM puts often pays asymmetrically.

Vanna, Charm, Vomma, and the third-order Greek Speed are all taught together in the Skavinski Academy’s Advanced Greeks module.

Veta¶

Veta is the cross partial of price with respect to volatility and time. It measures how Vega decays as expiration approaches. Long-dated volatility positions, such as VIX-targeting strategies, use Veta to anticipate how their Vega exposure will shrink over the life of the trade. For short-dated options, Veta is small enough that most traders ignore it entirely.

Veta is coming to the Skavinski Academy in an upcoming module.

Vera¶

Vera is the cross partial of price with respect to volatility and the interest rate. It captures how Vega changes as rates move. Vera is among the most obscure of the Greeks and is rarely monitored outside fixed-income derivatives desks, where long-dated options have meaningful exposure to both vol and rates simultaneously.

Vera is coming to the Skavinski Academy in an upcoming module.

Third-Order Greeks¶

Speed¶

Speed is the third derivative of price with respect to spot. It measures how Gamma itself changes as the underlying moves. Below a given strike, a long option position gains Gamma as spot rises toward that strike; above it, Gamma sheds. For small retail positions, Speed is irrelevant. For portfolios running billions in notional exposure, Speed is the Greek that determines how badly a gamma hedge will break down during a large, fast move. Speed is taught alongside Vanna, Charm, and Vomma in the Academy’s Advanced Greeks module linked above.

Zomma¶

Zomma is the cross partial of Gamma with respect to volatility, or equivalently, the third derivative of price with respect to spot twice and volatility once. In plain terms, it measures how Gamma changes when implied vol changes. A portfolio that is short Zomma will see its gamma hedge deteriorate exactly when volatility spikes, which is usually the worst possible timing. Stochastic volatility desks monitor Zomma closely.

Zomma is coming to the Skavinski Academy in an upcoming module.

Color¶

Color is the rate of change of Gamma with respect to time. It tells a gamma-hedged portfolio manager how their Gamma will decay over the life of the position, separately from any price moves. Color is most relevant when running a delta-hedged book that also needs to stay within strict gamma limits, a situation common at institutional options desks but essentially never at the retail level.

Color is coming to the Skavinski Academy in an upcoming module.

Ultima¶

Ultima is the third derivative of price with respect to volatility. It measures how Vomma, the convexity of Vega, itself changes as volatility moves. Ultima is the province of stress-testing under extreme VIX scenarios and exotic option pricing. For the overwhelming majority of traders, Ultima is a curiosity, not a tool.

Ultima is coming to the Skavinski Academy in an upcoming module.

Practical Importance Hierarchy¶

Put all sixteen options Greeks in priority order and a simple chain emerges:

Delta >> Gamma ≈ Theta ≈ Vega > Vanna ≈ Charm > Vomma >> Speed ≈ Color ≈ Zomma >> Ultima

Most traders only ever need the first five Greeks. Professional market-makers pay close attention through Vanna and Charm. Quants and stress-testers think about Vomma, Speed, and Zomma. Ultima is, for nearly everyone, a footnote.

Reference Table¶

The table below lists all sixteen options Greeks with their derivative notation, closed-form Black-Scholes formula, and the traders who actually use them.

where Φ is the standard normal cumulative distribution function, φ is the standard normal probability density, S is spot, K is strike, τ is time to expiration, σ is volatility, r is the risk-free rate, q is the dividend yield, and d₁, d₂ are the Black-Scholes auxiliary variables.

What Comes Next¶

Nine of the sixteen options Greeks listed above, Delta, Gamma, Theta, Vega, Rho, Vanna, Charm, Vomma, and Speed, are taught in depth at the Skavinski Academy right now, with interactive widgets, worked examples, and quiz questions on each. The remaining seven options Greeks, Lambda, Epsilon, Veta, Vera, Zomma, Color, and Ultima, are coming to the Academy in upcoming modules. These are the Greeks that most textbooks gloss over and most traders never need, but when you do need them, you want a reference that treats them seriously.

Explore real-time gamma exposure on TeploMap for SPX, NDX, and thousands of individual stocks. Monitor your positions on the Skavinski Dashboard, or learn the mechanics in depth at the Skavinski Academy.

References¶

• Black, F. and Scholes, M. (1973). “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81(3), 637-654.
• Merton, R. C. (1973). “Theory of Rational Option Pricing.” Bell Journal of Economics and Management Science, 4(1), 141-183.
• Garman, M. B. (1976). “An Algebra for Evaluating Hedge Portfolios.” Research Paper No. 299, Graduate School of Business, Stanford University.
• Ni, S. X., Pearson, N. D., and Poteshman, A. M. (2005). “Stock Price Clustering on Option Expiration Dates.” Journal of Financial Economics, 78(1), 49-87.
• Baltussen, G., Da, Z., Lammers, S., and Martens, M. (2021). “Hedging Demand and Market Intraday Momentum.” Journal of Financial Economics, 142(1), 377-403.
• Barbon, A. and Buraschi, A. (2021). “Gamma Fragility.” SSRN Working Paper No. 3725454.
• Buis, B., Pieterse-Bloem, M., Verschoor, W. F. C., and Zwinkels, R. C. J. (2024). “Gamma Positioning and Market Quality.” Journal of Economic Dynamics and Control, 164, Article 104880.
• Dim, C., Eraker, B., and Vilkov, G. (2024). “0DTEs: Trading, Gamma Risk, and Volatility Propagation.” SSRN Working Paper.