1. What Greeks are, in one sentence

Greeks are sensitivities of the option price to key inputs:

  • “If I nudge this input a bit, how much does price change?”

They are the language of risk for options desks.

2. Key Greeks (practitioner view)

For a call price C(S,K,σ,r,T)C(S, K, \sigma, r, T):

  • Delta Δ=∂C/∂S\Delta = \partial C / \partial S
    • Roughly: change in option price for a small change in underlying.
    • Long call: 0<Δ<10 < \Delta < 1.
    • Interpretation:
      • Hedge: “How many units of underlying do I need to offset this option?”
      • If Δ=0.4\Delta = 0.4, 1 option ≈ 0.4 units of underlying (for small moves).
  • Gamma Γ=∂2C/∂S2\Gamma = \partial^2 C / \partial S^2
    • How fast delta changes with spot.
    • Long options (both calls and puts) have positive gamma.
    • Interpretation:
      • Positive gamma → you like volatility (your delta adjusts in your favour).
      • Short gamma → you lose from large underlying moves.
  • Vega =∂C/∂σ=\partial C / \partial \sigma
    • Sensitivity to volatility.
    • Long options: positive vega.
    • Interpretation:
      • “If implied vol goes up 1 vol point (e.g. 20% → 21%), how much P&L?”
  • Theta =∂C/∂t=\partial C / \partial t (often quoted as ∂C/∂T\partial C / \partial T with minus sign)
    • Time decay.
    • Long option typically has negative theta:
      • Every day that passes (with nothing else moving) you lose some premium.
    • Short option: positive theta (you earn carry, but are short gamma/vega).
  • Rho =∂C/∂r=\partial C / \partial r
    • Sensitivity to interest rates.
    • Often smaller for short-dated options, more relevant for long tenors, rates products.

Practitioner framing:

“Delta/Gamma = spot risk, Vega = vol risk, Theta = time decay, Rho = rate risk. Long option: long gamma, long vega, short theta. Short option: opposite.”

3. Analytic vs finite-difference (FD) Greeks

3.1 Analytic Greeks

When you have a closed-form model (BS/Black-76), you can derive Greeks:

  • Example (Black–Scholes call on spot, no dividends):
    • Δ=N(d1)\Delta = N(d_1)
    • Γ=N′(d1)SσT\Gamma = \frac{N'(d_1)}{S \sigma \sqrt{T}}
    • Vega=STN′(d1)\text{Vega} = S \sqrt{T} N'(d_1), etc.

Pros:

  • Fast and smooth.
  • No numerical noise.
  • Great for vanilla products.

Cons:

  • Only available when the model has closed form.
  • If implementation is wrong, errors are systematic but often hidden.

3.2 Finite-difference Greeks (bump-and-revalue)

For general products (path-dependent, Monte Carlo, trees):

  • Compute price at base inputs: V0V_0.
  • Bump one input, recompute: V+V_+.
  • Approximate derivative:
    • Forward difference (simple): ∂V∂x≈V+−V0h \frac{\partial V}{\partial x} \approx \frac{V_+ - V_0}{h}
    • Central difference (more accurate): ∂V∂x≈V*+−V*−2h \frac{\partial V}{\partial x} \approx \frac{V*+ - V*-}{2h}

Pros:

  • Works for anything you can price.
  • Easy to understand and implement.

Cons:

  • Needs multiple valuations per Greek.
  • Sensitive to bump size and Monte Carlo noise.
  • Can be unstable near kinks (e.g. digital payoffs, barriers).

4. Bump sizing and stability: how to think like a practitioner

4.1 Trade-off

  • If bump too big:
    • You are not approximating a derivative; you are computing a big finite step.
    • Greeks are biased (non-linear effects show up).
  • If bump too small:
    • Subtraction V+−V0V_+ - V_0 suffers from numerical noise (MC error, rounding).
    • Greeks become noisy / unstable.

You want a middle ground where:

  • Bump is small enough to approximate local slope,
  • Large enough that signal >> numerical noise.

4.2 Typical rules of thumb

(These are ballpark values, not universal truths.)

  • Delta (spot bump):
    • Absolute bump in spot: 0.1%–1%0.1\% – 1\% of SS, or
    • “1 underlying unit” for discrete products.
    • Use central difference if possible.
  • Vega (vol bump):
    • Bump vol by 0.5–1 vol point (0.005–0.01 in decimals).
    • Again, central difference if feasible.
  • Rho (rate bump):
    • Bump curve by 1 bp (0.0001) or 10 bps depending on scale.
    • For full curves: bump parallel or by buckets.
  • Theta (time bump):
    • Move valuation date by 1 day and shift all relevant dates.

For Monte Carlo:

  • Use common random numbers (same random seed) between base and bump → variance reduction.
  • Otherwise, FD Greeks will be dominated by MC noise.

5. Stability checks / sanity rules

Practitioner “smell tests”:

  • Long call:
    • Delta ∈ (0, 1),
    • Gamma ≥ 0,
    • Vega ≥ 0,
    • Theta ≤ 0 (usually).
  • Put–call parity consistency:
    • Delta_put = Delta_call − 1 (for same K,T),
    • Check FD Greeks respect parity to numerical tolerance.
  • Smoothness:
    • Deltas should generally be monotone in strike and maturity.
    • If you see crazy zig-zags, bump size / MC noise / interpolation is wrong.
  • Scale vs price:
    • If price is small (deep OTM short-dated), Greeks should also be small.
    • Huge gamma on a nearly worthless option is suspicious.

6. How to explain this calmly in an interview

You can say something like:

“Greeks are just price sensitivities: delta and gamma for spot, vega for volatility, theta for time, rho for rates. For vanillas under Black-76 / Black–Scholes we have clean analytic Greeks, which are fast and smooth. For path-dependent or Monte Carlo products we use bump-and-revalue finite differences. There is a trade-off in bump size: too big and we bias the derivative, too small and we amplify noise. I usually use central differences, common random numbers, and bump sizes like 0.5–1% in spot and 0.5–1 vol point for volatility.”