Black-Scholes-Mertonintermediate

The Greeks Dashboard

The Greeks measure how an option's value responds to each of its inputs. Delta ( $\Delta$ ) is sensitivity to the underlying, gamma ( $\Gamma$ ) is how fast delta itself moves, vega ( $\mathcal{V}$ ) is sensitivity to volatility, theta ( $\Theta$ ) is time decay, and rho ( $\rho$ ) is sensitivity to the interest rate. Seen together they let a desk read, and hedge, every dimension of an option's risk at once, not just the first-order stock exposure that delta alone captures.

Try it yourself

Greeks explorer

Move the inputs and watch the five Black-Scholes Greeks update. Each Greek is a sensitivity of the option price: delta to the spot, gamma to delta itself, vega to volatility, theta to the passage of time, and rho to the rate. They all build on d₁ and d₂.

Curve Δ DeltaAt spot 0.5840

Δ DeltaN(d₁)0.5840

Γ Gammaφ(d₁)/(S·σ·√T)0.02758

𝒱 Vegaper 1.00 vol · per 1% = 0.275827.582

Θ Thetaper year · per day = −0.0208−7.587

ρ Rhoper 1.00 rate · per 1% = 0.258925.886

d₁0.2121

d₂0.0707

Vega and gamma are identical for a call and a put; delta, theta and rho differ by side.

Spot S100

Strike K100

Volatility σ20%

Maturity T0.50 yr

Rate r4.0%

Why it matters

Delta tells you how much the option moves for a A$1 move in the stock, but it is only the first slice. Gamma warns you that delta will itself shift, so a delta hedge has to be rebalanced. Theta is the rent the option pays for every day it is held. Vega is what you gain or lose when the market's view of volatility changes. Rho, usually the quietest, wakes up for long-dated options. Putting all five side by side shows which risks light up in a given situation: an at-the-money option near expiry is dominated by gamma and theta, while rho barely stirs.

Formulas

Gamma (curvature of the price)

\Gamma = \frac{\varphi(d_1)}{S\,\sigma\,\sqrt{T}}

Same for a call and a put. Largest at the money and as expiry nears (Hull 2022, §19.6).

Vega (sensitivity to volatility)

\mathcal{V} = S\,\varphi(d_1)\,\sqrt{T}

Per 1.00 change in volatility; divide by 100 for the change per one percentage point. Same for call and put.

Theta (time decay), call

\Theta_{\text{call}} = -\frac{S\,\varphi(d_1)\,\sigma}{2\sqrt{T}} - rKe^{-rT}N(d_2)

Usually negative: the option bleeds value as time passes. Per year; divide by 365 for per-day decay.

Rho (sensitivity to the rate), call

\rho_{\text{call}} = KTe^{-rT}N(d_2)

Per 1.00 change in r; divide by 100 for one percentage point. Matters most for long-dated options.

Worked examples

Scenario

An at-the-money call with $S =$ A$100, $K =$ A$100, $\sigma = 0.20$ , $r = 0.05$ , and $T = 0.5$ years.

Solution

With $d_1 \approx 0.25$ and $d_2 \approx 0.11$ : delta $\approx 0.60$ (per A$1), gamma $\approx 0.027$ (per A$1), vega $\approx$ A$0.27 per 1% of volatility, theta $\approx -$ A$0.022 per day, and rho $\approx$ A$0.26 per 1% of rate. The call gains about A$0.60 for a A$1 rise, but that delta is itself climbing by 0.027 per further A$1 (gamma); it loses roughly two cents a day to time decay (theta); and here it is far more sensitive to a change in volatility than to a change in the interest rate.

Common mistakes

✗The Greeks are constant. They are not. Every Greek is itself a function of $S$ , $\sigma$ , $T$ and $r$ . Gamma and theta in particular spike for at-the-money options as expiry approaches, which is exactly why a hedge set today can be badly wrong tomorrow.
✗Vega is a fully consistent Greek inside Black–Scholes. Strictly, Black–Scholes assumes volatility is constant, so vega measures sensitivity to a parameter the model treats as fixed. Desks use it anyway as the practical measure of volatility risk (Hull 2022, §19.8).
✗A bigger Greek always means more risk. Each Greek is a sensitivity per unit of a different input, with different units (delta per A$1, theta per day, vega per 1% of vol). Their raw magnitudes are not comparable; you scale each one by the size of the move you actually expect.

Revision bullets

•Five Greeks: $\Delta$ (spot), $\Gamma$ (delta's change), $\mathcal{V}$ (vol), $\Theta$ (time), $\rho$ (rate)
•Gamma $\Gamma = \varphi(d_1)/(S\sigma\sqrt{T})$ , largest at the money near expiry
•Vega $\mathcal{V} = S\varphi(d_1)\sqrt{T}$ , identical for call and put
•Theta is usually negative: options lose value as time passes
•Each Greek has its own units, so never compare raw magnitudes

Quick check

For an at-the-money option, which two Greeks become largest as expiry approaches?

A dashboard reports vega as A$0.30 "per 1% of volatility". If implied volatility rises by 2 percentage points, the option value changes by approximately:

Connected topics

Delta Δ Delta Hedge BSM Intro Impl. Vol

Sources

Hull (2022), Ch. 19
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
The Greek letters chapter: definitions, signs, and the at-the-money behaviour of gamma, vega and theta.
Black & Scholes (1973)
Black, Fischer and Myron Scholes. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3), 1973, pp. 637–654.
The model whose closed form yields the analytic Greeks summarised here.

How to cite this page

Dr. Phil's Quant Lab. (2026). The Greeks Dashboard. Derivatives Atlas. https://phucnguyenvan.com/concept/greeks-dashboard

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