A Greek is nothing more than a partial derivative of the option price with respect to one input, holding everything else fixed, the exact rate at which your P&L moves if that one thing moves. Five inputs, five Greeks, and (with a dividend yield q) closed forms for each, all built from the same d1, d2, and the standard normal density n(x) = e^(−x²/2)/√(2π):
Delta_call = e^(−qT)·N(d1) Delta_put = e^(−qT)·[N(d1) − 1]
Gamma = e^(−qT)·n(d1) / (S·σ·√T) [identical for call and put]
Vega = S·e^(−qT)·n(d1)·√T [identical for call and put; per 1.00 = 100 vol points]
Theta_call = −[S·e^(−qT)·n(d1)·σ / (2√T)] − r·K·e^(−rT)·N(d2) + q·S·e^(−qT)·N(d1)
Theta_put = −[S·e^(−qT)·n(d1)·σ / (2√T)] + r·K·e^(−rT)·N(−d2) − q·S·e^(−qT)·N(−d1)
Rho_call = K·T·e^(−rT)·N(d2) Rho_put = −K·T·e^(−rT)·N(−d2)Advertisement
(Theta above is stated in the trader's convention: the rate of change per unit of calendar time passing, already negative for a typical long option, because time decay erodes value as expiry approaches. Divide the annual figure by 365 to read it per calendar day.) In plain language, each one answers a specific question a risk manager actually asks:
- Delta: "if the underlying moves ₹1/$1, how much does my option move?" Also the number of shares that exactly hedges one option (DV1.03's starting point for every hedge).
- Gamma: "how fast does my delta itself change?" The curvature/convexity term; a delta hedge without a gamma check only protects against small moves (Worked Example 8 proves this directly).
- Vega: "if implied volatility moves one point, how much does my option move?" Every options position is, whether intended or not, a bet on volatility.
- Theta: "how much value do I lose (or gain, if short) purely from one day passing, nothing else moving?"
- Rho: "if the risk-free rate moves one point, how much does my option move?" Usually the smallest of the five for short-dated equity options, but never zero, and never assume it away for long-dated ones.
How to check Greeks with finite differences and parity
Every closed-form Greek below is cross-checked by finite difference, bump the input by a small amount, reprice with the same BSM function, and confirm the numerical derivative matches the formula. This is the module's central verification habit, applied five times over. Three Greek parity relationships, direct consequences of differentiating the put-call parity identity itself, so they must hold for any correctly built pricer, independent of the BSM model:
Delta_call − Delta_put = e^(−qT)
Gamma_call = Gamma_put (always, exactly)
Rho_call − Rho_put = K·T·e^(−rT)
Theta_call − Theta_put = q·S·e^(−qT) − r·K·e^(−rT)Worked Example 7 verifies all four numerically, a second, independent check on the Greeks beyond the finite-difference one, and a genuinely useful bug-catcher: if your pricer's Greeks fail these identities, something in the code is wrong, full stop, regardless of how plausible any single number looks.
{
"skill": "Explaining a Greek's economic meaning to a non-quant stakeholder, and drafting an implied-volatility solver",
"use": "Have an AI assistant turn a raw Greek (say, 'gamma = 0.00025') into a plain-language, decision-relevant sentence for someone who has never seen the word, and separately have it draft a bisection or Newton root-finder that backs out implied volatility from a quoted price — then verify both outputs independently before repeating either to anyone or trusting the solver.",
"tools": ["Chat assistants — Claude, ChatGPT", "AI coding assistants for the root-finder — Claude Code, Copilot"],
"prompt": "Explain what a gamma of <VALUE> means for a position of <N> lots/contracts of <OPTION>, in one sentence a non-quant portfolio manager would act on. Separately, write a bisection root-finder in Python that recovers implied volatility from a quoted option price, and show me the convergence at each iteration.",
"verify": "For the plain-language explanation: recompute the actual dollar/rupee P&L for a stated underlying move yourself (Worked Example 8's Taylor-decomposition pattern) and confirm the AI's sentence matches the number, not just the vibe. For the solver: confirm it reprices the ORIGINAL quoted price back exactly when you feed its solved volatility back into the same BSM formula (Worked Example 6's pattern) — a solver that 'looks converged' but doesn't reprice exactly has a bug, most often a sign error or a stale bracket in the bisection search.",
"diy": "You must already be able to state what each Greek measures in your own words and reprice an option from a solved volatility by hand — this Core section and Worked Example 6 build exactly that. An AI-drafted explanation or solver is a first draft to be checked, never a citable answer on its own.",
"market": "US"
}Each Greek in words, before any arithmetic
Formulas make a Greek computable. They do not make it usable, and a risk report full of correctly computed Greeks that nobody can read is worth about as much as no risk report. Take each one in turn: what it is, what sign it carries and why, where it is largest, and the decision it drives.
Delta is the first derivative of the option price with respect to the underlying, and it does three jobs at once. It is a P&L estimate, the amount the option makes for a one-unit move. It is a hedge ratio, the number of shares that neutralises one option's directional exposure. And it is a rough, deliberately loose reading of moneyness, since a call's delta runs from near zero far out of the money to near e^(−qT) deep in. A long call carries positive delta and a long put negative; short those positions and the signs flip, which is the first line of every position report. Delta is largest in absolute value for deep in-the-money options and moves fastest for at-the-money ones near expiry, which is the same statement as saying gamma peaks there.
Gamma is the second derivative with respect to the underlying, the rate at which delta itself changes. Its sign belongs to the position, not to the option type: long any option, call or put, is long gamma; short any option is short gamma. A long-gamma book gets longer as the market rises and shorter as it falls, so its hedge errors work in its favour and it pays for that privilege in theta. A short-gamma book does the opposite, collects the theta and loses on every large move, which is the arithmetic behind the observation that selling options feels like free income until it does not. Gamma is largest at the money and grows without bound as expiry approaches, which is why a short-gamma book that is comfortable on Monday can be unmanageable on expiry Thursday with nothing having changed except the calendar.
Vega is the sensitivity to implied volatility, and it is the Greek most often mis-scaled. The closed form gives the change per 1.00 of volatility, meaning a move from 20% to 120%, so the per-point figure a trader actually quotes is the closed form divided by 100. Every option is long vega, and every option seller is short it, whether or not volatility was any part of the intended trade. Vega is largest for at-the-money options with long time to expiry, which is why a one-year straddle is a volatility position with a directional footnote while a one-week straddle is a gamma position with a volatility footnote. Vega also dies at expiry: at T = 0 there is nothing left for volatility to act on.
Theta is the rate of change with the passage of calendar time, quoted per year in the closed form and per day after dividing by 365. It is negative for almost every long option, because time value only shrinks. The two exceptions are worth knowing: a deep in-the-money European put on a non-dividend payer can have positive theta, because holding it means waiting to receive the strike, and a deep in-the-money call on a high-dividend-yield underlying can too. Theta and gamma are two readings of one position. A long-gamma book pays theta; a short-gamma book collects it. If a strategy pitch describes theta income without naming the gamma that is being sold to earn it, the pitch is incomplete.
Rho is the sensitivity to the risk-free rate, positive for calls and negative for puts, and it scales with K·T. For a one-month equity option it is usually the smallest of the five and can be ignored for a day's risk management. For a two-year option, or for anything on a rates underlying, it is not small at all, and treating it as noise is how a book acquires an unhedged rate position by accident. Rho is also the Greek that most obviously depends on the discounting convention, so a rho that looks wrong is often a convention error rather than a pricing error.
How to use the Greeks in practice
Two habits make the whole set useful rather than decorative. First, always state the units and the position scaling in the same breath as the number: "gamma 0.00025 per index point, on 40 lots of 65, is 0.65 index points of delta per point of index". A Greek without units is a number nobody can act on, and mis-scaled vega is the single most common version of that error in practice. Second, read the five together rather than one at a time, because they trade against each other. There is no position that is long gamma, long vega and long theta; the market prices exactly that impossibility.