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Option Greeks Interview Questions: Delta, Gamma, Vega, and Scenario Reasoning
Practice option Greeks interview questions with delta, gamma, vega units, signed exposures, hedge limits, finite-difference checks, and scenario repricing.
Option Greeks interview questions test whether you can explain how an option's value changes when its inputs change. Delta measures a local price sensitivity, gamma describes its curvature, and vega measures sensitivity to volatility. The difficult follow-up is usually about units, simultaneous changes, or the risk left after a hedge.
State the instrument, model, units, and position sign before multiplying any Greek. This guide develops an original unit ledger and a reproducible scenario calculation that compares a Greek approximation with a full model reprice.
Primary exchange and academic sources support the concepts. The numbers below are hypothetical interview exercises, not market quotes, candidate reports, or trading recommendations.
What variable does each Greek differentiate?
Official educational definitions: CME's delta lesson relates option value to the underlying's price movement. Its gamma lesson describes the change in delta as that underlying moves. Its vega lesson covers sensitivity to implied volatility.
For a smooth price function V(S, sigma, t), the mathematical definitions are Delta = ∂V/∂S, Gamma = ∂²V/∂S², and Vega = ∂V/∂sigma. Here t is calendar time; with a fixed expiry, time remaining decreases as t increases. Specify what other inputs are held fixed.
Do not apply a spot-equity derivative without checking an option-on-futures question. The underlying variable, pricing model, rate convention and contract units may differ. A quote described as “40 delta” often means 0.40 in sensitivity arithmetic, not a multiplier of 40.
An interview answer should name the local approximation and its limit: a small underlying change produces approximately Delta*dS of option-value change, before other factors and curvature. Delta itself can change during the move.
Which units make the calculation meaningful?
Our original unit ledger uses an option premium quoted per unit of underlying. Other contracts require their own quote-to-currency conversion.
| Quantity | Convention in this exercise | Common mismatch |
|---|---|---|
| Delta | Premium change per one-unit underlying move | Multiplying by a percentage move without converting it to price units |
| Gamma | Delta change per one-unit underlying move | Omitting the half in the second-order premium term |
| Vega, decimal | Premium change per 1.00 change in sigma | Treating 25%→28% as a sigma change of 3 |
| Vega, vol point | Premium change per one percentage-point volatility move | Using decimal-volatility change with this quoted vega |
| Theta | Premium change per calendar year in our model | Multiplying annual theta by an unconverted day count |
| Position exposure | Signed contracts × multiplier × per-unit Greek | Applying a per-unit sensitivity to the portfolio without scaling |
CME's vega lesson and the Options Industry Council's Greek overview use the practical convention of a one-point volatility change. A formula differentiating with respect to decimal sigma uses a different scale: divide that derivative by 100 to obtain vega per volatility point.
In an independent hypothetical arithmetic exercise, 12 long contracts have multiplier 100 and vega 0.25 per volatility point. A move from 25% to 28% is three percentage points, so the local vega contribution is 12*100*0.25*3 = 900 currency units. For 12 short contracts, the sign reverses. The calculation is an approximation, not a promised realized profit.
Why does gamma change a delta-only answer?
For an underlying move, a second-order expansion adds 0.5*Gamma*dS² to Delta*dS. The square means the curvature contribution has the same sign for either direction when gamma is positive. That does not mean a long call must gain for both directions: its first-order delta term also matters.
Official scope: CME describes gamma as dynamic and particularly important near the strike. In the ordinary European vanilla model used below, long calls and long puts have positive gamma. Position sign matters: shorting the option reverses its contribution to portfolio gamma.
Our rehearsal question is: “Two positions have equal current delta; are their risks equal?” Not necessarily. Different gamma means their deltas respond differently to the next move. Expiry, volatility and other inputs can change those sensitivities too.
Avoid universal statements about every instrument named an option. Digitals, barriers and multi-leg portfolios can behave differently from a single vanilla option. Identify the payoff and model before transferring a sign rule.
Does delta equal the probability of finishing in the money?
It can serve as a market shorthand or approximation, but it is not a general probability identity. For the non-dividend Black-Scholes call, delta is N(d1), whereas the risk-neutral probability of finishing above strike is N(d2). Those expressions differ.
The Columbia Black-Scholes-Merton notes provide the model foundation. A risk-neutral probability is also not automatically a real-world forecast probability. An interviewer may be checking whether you distinguish pricing sensitivity, pricing-measure probability and a forecast under another distribution.
For preparation, answer the sensitivity definition first. If probability is requested, state the model and measure before choosing a formula. Calling delta a precise success probability without either assumption gives the interviewer a misleading interpretation of the number.
Reprice an original two-factor-and-time scenario
Our original model fixture is a European call on a non-dividend-paying underlying: S=105, K=100, remaining time tau=0.5 years, continuously compounded r=0.03, and sigma=0.25. We use a 365-day calendar conversion, constant inputs within each reprice, and no transaction costs.
The executed baseline premium is 10.871469 per unit. Its sensitivities are approximately delta 0.673371, gamma 0.019430, vega 26.776910 per decimal sigma or 0.267769 per vol point, and calendar theta −8.489201 per year or −0.023258 per day.
Our selected approximation is:
dV ≈ Delta*dS + 0.5*Gamma*dS² + Vega_decimal*d_sigma
+ Theta_calendar_per_year*d_calendar_years
Rate and dividend inputs remain fixed. The approximation includes curvature in the underlying, but not volatility curvature or cross derivatives. It is not a complete second-order multivariable expansion.
| Original scenario | Full model premium change | Selected Greek approximation | Full reprice minus approximation |
|---|---|---|---|
| S rises by 1; volatility 25%→26%; one day passes | +0.923377 | +0.927597 | −0.004220 |
| S falls by 15; volatility 25%→35%; 30 days pass | −6.046885 | −5.934742 | −0.112143 |
These values come from the same model, with all changed inputs applied together. “Full model” means evaluating the selected pricing formula again; it does not establish an exact traded price or validate that model against market data.
The larger residual is a reason to inspect approximation limits, not to relabel the whole residual as gamma. Missing terms can include volatility curvature and interactions between price, volatility and time. Greek values also change as the scenario moves away from the baseline.
What remains after a delta hedge?
An underlying position can offset current first-order price sensitivity under a compatible unit convention. It does not generally offset option gamma, vega, time sensitivity, execution costs or jump exposure.
In the original fixture, four short calls with multiplier 100 have initial delta exposure approximately −400*0.673371 = −269.3484 underlying units. Adding that many long underlying units offsets this initial delta in an idealized divisible-unit model. Actual instruments may require discrete hedge sizes and different conversion factors.
After an underlying move, the call's delta changes. The unchanged hedge can therefore leave residual directional exposure. Underlying shares have no option gamma in this simple model, so their linear payoff does not cancel the short call's curvature. Similarly, a volatility move can change option value despite the initial delta offset.
This is scenario reasoning, not a complete hedging strategy. A path-dependent realized PnL calculation also needs the hedge schedule, fills, financing, fees and model assumptions. Matching a Greek at one timestamp does not prove risk elimination over an interval.
How do theta and time remaining change the sign?
Official convention warning: CME's theta lesson emphasizes the time unit and notes that time decay is not constant. For this fixed-expiry fixture, calendar theta is −∂V/∂tau: one positive day of elapsed calendar time corresponds to a decrease of 1/365 in remaining years.
Our baseline call has negative calendar theta. Multiplying ∂V/∂tau by a positive elapsed-time increment would produce the wrong sign. Keep the derivative convention and the input increment consistent rather than deciding the sign from memory.
Do not claim that all long options always have negative theta under every rate, carry and payoff specification. The fixture's sign is a calculated result under its assumptions. Near expiry, a small-looking calendar bump may be large relative to remaining time, making a local approximation less useful.
Verify the sensitivities with finite differences
We implemented the non-dividend European call formula and analytic Greeks, then checked them against central finite differences. For delta and gamma, underlying bumps of 0.01 and 0.001 each agreed with the analytic values within absolute tolerance 1e-6. Volatility and remaining-time bumps of 1e-4 and 1e-5 agreed with vega and calendar theta within 1e-4.
This executable delta check shows the idea; call_price is the validated formula implementation:
h = 0.01
delta_fd = (call_price(S+h, K, tau, r, sigma)
- call_price(S-h, K, tau, r, sigma)) / (2*h)
assert abs(delta_fd - delta_analytic) < 1e-6
We also rejected zero underlying price, zero remaining time, zero volatility and nonfinite input in this deliberately restricted helper. An expiry-capable production function should handle the boundary payoff explicitly; our helper rejects that case instead of dividing by zero.
A smaller bump is not automatically better. Large bumps estimate an average response across nonlinear prices; tiny bumps can amplify floating-point cancellation, especially for second derivatives. Explain how you compare several admissible bumps and state an error tolerance. If the underlying price function is estimated by simulation, numerical noise needs separate treatment.
Aggregate exposures without hiding the scenario
For compatible instruments, sum signed per-contract exposures after applying each multiplier and quote conversion. Keep price sensitivity, volatility sensitivity and time sensitivity in their own units. Adding delta directly to vega has no meaningful unit interpretation.
Different strikes and expiries may respond to different volatility changes. A net vega total often corresponds to an assumed common volatility shift; it does not describe every change in a volatility surface. Preserve maturity or strike buckets when the scenario needs them.
Our preparation recommendation: report the proposed input changes, local estimate, full model reprice, and residual. Then explain which assumptions could fail. This follows the multi-input perspective in CME's premium-and-Greeks lesson without copying its numerical example or treating sensitivities as constants over arbitrary moves.