The Greeks, what are they ?
The Greeks are the sensitivities of a derivative's value to the parameters it depends on: the price of the underlying, volatility, time and interest rates. Mathematically they are partial derivatives of the pricing function, which is why they come in orders: first-order Greeks measure the direct sensitivity, second-order Greeks measure how those sensitivities change, and third-order Greeks measure how the second-order ones change.
Each Greek is a local approximation. It describes the effect of a small change in one input, with the others held constant, and under the assumptions of a pricing model. The higher the order, the more it tells you about where that approximation stops being reliable. The tables below give each Greek with its Black–Scholes formula and what it is used for.
Options risk · First order · 1 of 3
First-order Greeks: where the position stands today
Five numbers that say what a book gains or loses from a small move in price, volatility, time and rates.
| Greek | Derivative | Formula | What it is used for |
|---|
| Delta(Δ) | ∂V / ∂S | e−qτ Φ(d₁) | Sets the hedge ratio: how much of the underlying offsets the position today. Its absolute value is also a rough read on the odds of finishing in the money. |
| Vega(ν) | ∂V / ∂σ | S e−qτ φ(d₁) √τ | Exposure to volatility. Long options gain when vol rises, whichever way spot goes. On a book of options it is often the bigger risk than delta. |
| Theta(Θ) | ∂V / ∂t, quoted as decay | − e−qτ S φ(d₁) σ / (2√τ) − r K e−rτ Φ(d₂) + q S e−qτ Φ(d₁) | The carrying cost of optionality: what a long option bleeds each day if nothing moves. The mirror image of gamma, which pays for it. |
| Rho(ρ) | ∂V / ∂r | K τ e−rτ Φ(d₂) | Sensitivity to the rate curve. Marginal on short maturities, material on long-dated ones. The nearest relative of DV01 on the bond side. |
| Lambda(Λ) | (∂V / ∂S) · S / V | Δ · S / V | Leverage: the percentage move in the option for a one percent move in the underlying. Puts positions of very different size on a single scale. |
Delta(Δ)
- Derivative
- ∂V / ∂S
- Formula
- e−qτ Φ(d₁)
- What it is used for
- Sets the hedge ratio: how much of the underlying offsets the position today. Its absolute value is also a rough read on the odds of finishing in the money.
Vega(ν)
- Derivative
- ∂V / ∂σ
- Formula
- S e−qτ φ(d₁) √τ
- What it is used for
- Exposure to volatility. Long options gain when vol rises, whichever way spot goes. On a book of options it is often the bigger risk than delta.
Theta(Θ)
- Derivative
- ∂V / ∂t, quoted as decay
- Formula
- − e−qτ S φ(d₁) σ / (2√τ) − r K e−rτ Φ(d₂) + q S e−qτ Φ(d₁)
- What it is used for
- The carrying cost of optionality: what a long option bleeds each day if nothing moves. The mirror image of gamma, which pays for it.
Rho(ρ)
- Derivative
- ∂V / ∂r
- Formula
- K τ e−rτ Φ(d₂)
- What it is used for
- Sensitivity to the rate curve. Marginal on short maturities, material on long-dated ones. The nearest relative of DV01 on the bond side.
Lambda(Λ)
- Derivative
- (∂V / ∂S) · S / V
- Formula
- Δ · S / V
- What it is used for
- Leverage: the percentage move in the option for a one percent move in the underlying. Puts positions of very different size on a single scale.
Black–Scholes, European call, continuous dividend yield q. Vega, theta and rho are usually quoted per vol point, per day and per rate point. For puts, delta, theta and rho change; vega is identical.
Options risk · Second order · 2 of 3
Second-order Greeks: how fast the hedge goes stale
The sensitivities of the sensitivities. They explain why a book that is flat today rarely stays flat.
| Greek | Derivative | Formula | What it is used for |
|---|
| Gamma(Γ) | ∂²V / ∂S² = ∂Δ / ∂S | e−qτ φ(d₁) / (S σ √τ) | How fast delta changes as spot moves. Long gamma buys dips and sells rallies; short gamma has you chasing the market with your hedge. Peaks near the strike. Convexity, in bond language. |
| Vanna | ∂²V / ∂S∂σ = ∂Δ / ∂σ | − e−qτ φ(d₁) d₂ / σ | How delta moves when vol moves, or how vega moves when spot does. Spot and vol rarely move independently, so a delta hedge drifts as vol reprices. Matters most on skewed books. |
| Charm | ∂²V / ∂S∂t = ∂Δ / ∂t | q e−qτ Φ(d₁) − e−qτ φ(d₁) · [2(r − q)τ − d₂σ√τ] / (2τσ√τ) | Delta decay: the hedge changes simply because time passes. This is the Friday-to-Monday number. Flat on the close does not mean flat on the open. |
| Vomma | ∂²V / ∂σ² = ∂ν / ∂σ | ν · d₁ d₂ / σ | Vega convexity (also called volga). Out-of-the-money options get longer vega as vol rises, which is why the wings behave differently from at-the-money in a volatility spike. |
| Veta | ∂²V / ∂σ∂t = ∂ν / ∂t | − S e−qτ φ(d₁) √τ · [ q + (r − q)d₁ / (σ√τ) − (1 + d₁d₂) / (2τ) ] | Vega decay: how quickly the volatility exposure fades with time. Tells you how long a vol hedge keeps working before it has to be rolled. |
Gamma(Γ)
- Derivative
- ∂²V / ∂S² = ∂Δ / ∂S
- Formula
- e−qτ φ(d₁) / (S σ √τ)
- What it is used for
- How fast delta changes as spot moves. Long gamma buys dips and sells rallies; short gamma has you chasing the market with your hedge. Peaks near the strike. Convexity, in bond language.
Vanna
- Derivative
- ∂²V / ∂S∂σ = ∂Δ / ∂σ
- Formula
- − e−qτ φ(d₁) d₂ / σ
- What it is used for
- How delta moves when vol moves, or how vega moves when spot does. Spot and vol rarely move independently, so a delta hedge drifts as vol reprices. Matters most on skewed books.
Charm
- Derivative
- ∂²V / ∂S∂t = ∂Δ / ∂t
- Formula
- q e−qτ Φ(d₁) − e−qτ φ(d₁) · [2(r − q)τ − d₂σ√τ] / (2τσ√τ)
- What it is used for
- Delta decay: the hedge changes simply because time passes. This is the Friday-to-Monday number. Flat on the close does not mean flat on the open.
Vomma
- Derivative
- ∂²V / ∂σ² = ∂ν / ∂σ
- Formula
- ν · d₁ d₂ / σ
- What it is used for
- Vega convexity (also called volga). Out-of-the-money options get longer vega as vol rises, which is why the wings behave differently from at-the-money in a volatility spike.
Veta
- Derivative
- ∂²V / ∂σ∂t = ∂ν / ∂t
- Formula
- − S e−qτ φ(d₁) √τ · [ q + (r − q)d₁ / (σ√τ) − (1 + d₁d₂) / (2τ) ]
- What it is used for
- Vega decay: how quickly the volatility exposure fades with time. Tells you how long a vol hedge keeps working before it has to be rolled.
Black–Scholes, European call, continuous dividend yield q. For puts only charm changes; gamma, vanna, vomma and veta are identical.
Options risk · Third order · 3 of 3
Third-order Greeks: what breaks the second-order hedge
Abstract on a small book. Expensive on a large one, close to expiry, or in a volatility shock.
| Greek | Derivative | Formula | What it is used for |
|---|
| Speed | ∂³V / ∂S³ = ∂Γ / ∂S | − (Γ / S) · [ d₁ / (σ√τ) + 1 ] | How gamma changes as spot moves. In a large move, the gamma you hedged is not the gamma you end up with. Speed shows how far you can trust a gamma hedge before it needs redoing. |
| Zomma | ∂³V / ∂S²∂σ = ∂Γ / ∂σ | Γ · (d₁d₂ − 1) / σ | How gamma changes with volatility. In a vol shock, the rebalancing you planned for a calm market turns out to be the wrong size. |
| Color | ∂³V / ∂S²∂t = ∂Γ / ∂t | − e−qτ φ(d₁) / (2Sτσ√τ) · [ 2qτ + 1 + d₁ (2(r − q)τ − d₂σ√τ) / (σ√τ) ] | Gamma decay: how gamma changes as time passes. Gamma piles up around the strike into expiry, which is behind pinning and sudden hedging pressure in the last days. |
| Ultima | ∂³V / ∂σ³ = ∂Vomma / ∂σ | − (ν / σ²) · [ d₁d₂(1 − d₁d₂) + d₁² + d₂² ] | How vega convexity itself changes with vol. Relevant for books heavy in the wings, where the shape of the vol exposure shifts in a stressed market. |
Speed
- Derivative
- ∂³V / ∂S³ = ∂Γ / ∂S
- Formula
- − (Γ / S) · [ d₁ / (σ√τ) + 1 ]
- What it is used for
- How gamma changes as spot moves. In a large move, the gamma you hedged is not the gamma you end up with. Speed shows how far you can trust a gamma hedge before it needs redoing.
Zomma
- Derivative
- ∂³V / ∂S²∂σ = ∂Γ / ∂σ
- Formula
- Γ · (d₁d₂ − 1) / σ
- What it is used for
- How gamma changes with volatility. In a vol shock, the rebalancing you planned for a calm market turns out to be the wrong size.
Color
- Derivative
- ∂³V / ∂S²∂t = ∂Γ / ∂t
- Formula
- − e−qτ φ(d₁) / (2Sτσ√τ) · [ 2qτ + 1 + d₁ (2(r − q)τ − d₂σ√τ) / (σ√τ) ]
- What it is used for
- Gamma decay: how gamma changes as time passes. Gamma piles up around the strike into expiry, which is behind pinning and sudden hedging pressure in the last days.
Ultima
- Derivative
- ∂³V / ∂σ³ = ∂Vomma / ∂σ
- Formula
- − (ν / σ²) · [ d₁d₂(1 − d₁d₂) + d₁² + d₂² ]
- What it is used for
- How vega convexity itself changes with vol. Relevant for books heavy in the wings, where the shape of the vol exposure shifts in a stressed market.
Black–Scholes, European call, continuous dividend yield q. Identical for calls and puts.
Notation: d₁ = [ln(S/K) + (r − q + σ²/2)τ] / (σ√τ) · d₂ = d₁ − σ√τ · φ standard normal density · Φ standard normal cumulative · ν = vega = S e−qτ φ(d₁) √τ · Γ = gamma = e−qτ φ(d₁) / (S σ √τ) · S spot, K strike, σ volatility, τ time to expiry, r rate, q dividend yield.
Source: Wikipedia, "Greeks (finance)".