The Greeks I: Delta, Gamma, Vega, Theta
Objectives
- State and interpret the four primary Greeks — delta, gamma, vega, and theta — as partial derivatives of the Black–Scholes price.
- Compute delta as N(d1) and gamma as n(d1)/(S·sigma·sqrt(T)) from the standard normal CDF and PDF, and read vega and theta off the same d1, d2 machinery.
- Distinguish first-order exposure (delta) from curvature (gamma) and use both to judge whether an option's price move is genuine evidence about the underlying.
In the last chapter you derived the Black–Scholes price. A clean, closed-form number sits there on the screen, and it feels like an answer. But a price is a snapshot, and the world does not hold still. The underlying moves, volatility breathes, and the clock runs down. What you actually need to know — what every desk needs to know — is how that price responds when the world shifts under it. The Greeks are that answer. Each Greek is a single number that tells you the sensitivity of the option price to one of the things that can change.
Let us build the intuition before the calculus, because the calculus is just bookkeeping once the picture is clear.
Imagine you hold a call option. The most pressing question is: if the underlying moves up by one dollar, how much does my option gain? That is delta. Delta is the slope of the option price as a function of the underlying. If delta is zero-point-six, then a one-dollar rise in the stock lifts the option by roughly sixty cents. Delta also doubles as a hedge ratio — it tells you how many shares you would short to neutralize your directional exposure — and, loosely, as the market's implied probability that the option finishes in the money. Three readings of one number, and they all come from the same slope.
But a slope is only a local truth. The option price is a curve, not a line, and as the underlying moves, the slope itself moves. Gamma measures that. Gamma is the rate at which delta changes as the underlying changes — the curvature of the price. High gamma means your delta is unstable: hedge it now and a small move in the underlying leaves you mis-hedged a moment later. Gamma is largest for options near the money and near expiry, where the price kinks most sharply around the strike. When you hear a trader say a position is "long gamma," they mean they profit from movement itself, in either direction, because their delta keeps re-aligning in their favor.
Two more sensitivities round out the primary set. Vega measures how the price responds to a change in volatility — specifically, how many points the option gains if implied volatility rises by one percentage point. Vega is always positive for vanilla options: more uncertainty is more valuable to the holder, because the downside is capped and the upside is not. And theta measures the passage of time. Theta is almost always negative for a long option position — each day that passes, with nothing else changing, bleeds a little value out, because there is less time left for the favorable move to arrive. Theta is the rent you pay for holding optionality.
Now the formalism. Each Greek is simply a partial derivative of the Black–Scholes price with respect to one input, holding the others fixed. Recall the two quantities from the pricing formula: d-one and d-two, where d-one is the log of spot over strike, plus the term involving the rate and half the variance times time, all divided by sigma times the square root of T; and d-two is just d-one minus sigma times the square root of T.
For a European call, delta equals N of d-one — the standard normal cumulative distribution function evaluated at d-one. Because a CDF lives strictly between zero and one, a call's delta is always between zero and one, exactly as the hedge-ratio and probability intuitions demand. Differentiate the price once more with respect to spot and you get gamma, which equals the standard normal probability density at d-one, divided by the product of spot, sigma, and the square root of T. Notice gamma uses little-n, the density, where delta used big-N, the cumulative — and notice gamma is identical for a call and a put at the same strike, because curvature does not care about direction.
Vega is the density at d-one times spot times the square root of T, and theta carries the more elaborate expression with the discounting and decay terms — but both are read off the very same d-one and d-two you already computed for the price. That is the quiet elegance of Black–Scholes: price and all its sensitivities share one engine.
SVQ's primary Greeks panel sits on every option. The platform computes delta as N(d1) and gamma as n(d1) divided by S·sigma·sqrt(T) — exactly the formulas this chapter derived — alongside vega and theta from the same d1, d2 pair. For an at-the-money call, delta sits near zero-point-five while gamma peaks. As the strike steps deep in the money, delta climbs toward one, but gamma collapses toward zero, because a deep-in-the-money option behaves almost like the stock itself — pure slope, no curvature left. Delta and gamma read together reveal whether a price move in the option is genuine evidence about the underlying, or merely the mechanical re-pricing of a position whose slope was always going to drag it along.