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FRM Part 1 · Worked study notes

Options Greeks Intuition

Delta measures an option’s local price response to its underlying; gamma measures how delta changes; vega measures sensitivity to implied volatility; theta measures the effect of time passing. Each holds other inputs fixed.

The method at a glance

  • 01Delta and gamma

    How it works: ΔV ≈ delta × ΔS + ½ × gamma × (ΔS)² for a small underlying move.

    Watch for: Assuming a delta hedge remains neutral after prices move.

  • 02Vega

    How it works: Confirm whether the quoted sensitivity is per volatility percentage point.

    Watch for: Mixing a one-point volatility change with a decimal change.

  • 03Theta

    How it works: Check whether the quotation is per day or per year and the time convention.

    Watch for: Assuming every option position has negative theta.

Original worked example

A small move with curvature

One option has delta 0.60 and gamma 0.04 per unit of underlying price. The underlying rises by 2. Hold volatility and time fixed; ignore higher-order effects.

  1. STEP 1

    Delta effect = 0.60 × 2 = 1.20.

  2. STEP 2

    Gamma effect = 0.5 × 0.04 × 2² = 0.08.

  3. STEP 3

    Approximate option-price change = 1.20 + 0.08 = 1.28.

The estimate is +1.28 per option unit. Multiply by the contract size only after the per-unit calculation.

Illustrative assumptions and calculations for learning. Not observed market data or an official exam item.

Apply the method and check its limits

Before you start: Option price versus payoff, contract size and percentage-point changes.

Aggregate Greeks using signed quantities

Position delta = contracts × multiplier × option delta

Ten long calls with multiplier 100 and delta 0.60 have delta 600 underlying units. Selling 600 shares creates local delta neutrality. If gamma is 0.04 per option and the underlying rises by 2, delta increases by about 0.08 per option, leaving the hedged position roughly 80 units long. Gamma explains why rebalancing is necessary.

Keep vega and theta quotation units visible

If vega is 0.12 per volatility percentage point, a rise from 20% to 23% contributes approximately +0.36 per option. Do not multiply 0.12 by 0.03 unless the vega quotation is per unit of decimal volatility. Similarly, a theta of -0.02 per day contributes about -0.10 over five days under a local constant-theta approximation.

Read a Greek as a local slope

Greeks describe small changes around the current state. Large moves, changing volatility and the passage of time can change the sensitivities themselves. A delta-neutral position can retain gamma and vega risk. For portfolios, signs and contract multipliers matter: a short option reverses the long option’s exposures.

Work through the topic

Turn local option sensitivities into a consistent portfolio change estimate, including contract size, volatility units and the limits of delta hedging.

Identify the variable and quotation unit behind each Greek

Delta describes the first-order change in option value for a change in the underlying price. Gamma describes how delta changes with that price. Vega describes sensitivity to implied volatility, and theta describes sensitivity to time under the stated convention. A Greek is a local sensitivity at a particular market state. It is not a promise that the same slope applies after a large movement or a substantial passage of time.

Quotation units are especially important for vega and theta. If vega is 0.12 of option premium per one percentage-point increase in volatility, a move from 20% to 22% contributes approximately 0.24, not 0.0024. If another system quotes sensitivity per unit decimal volatility, the numerical vega differs by a factor of 100. Likewise, theta per day must not be multiplied by a year fraction without conversion.

The signs must reflect the position. Buying a standard call normally gives positive delta; selling it reverses the position sensitivities. A portfolio can combine positive and negative exposures, so the word “option” alone does not determine its overall delta, gamma or vega. Record whether each quoted Greek is per underlying unit, per contract or already aggregated across the position before adding anything.

A units check before using option sensitivities

Delta
Input change: Underlying price change
Typical mistake: Treating a 0.60 delta as 60 currency units without a multiplier
Gamma
Input change: Squared underlying price change in the price approximation
Typical mistake: Omitting the one-half term
Vega
Input change: Volatility change in the quoted unit
Typical mistake: Mixing percentage points and decimal volatility
Theta
Input change: Elapsed time in the quoted unit
Typical mistake: Combining per-day theta with an annual time input

Scale the option and then calculate the hedge

Consider 50 long call contracts, each representing 100 underlying units, with delta 0.60 and gamma 0.04 per unit. Position delta is 50 × 100 × 0.60 = 3,000 underlying units. Position gamma is 50 × 100 × 0.04 = 200 per unit of underlying-price change. The multiplier applies to both terms. Leaving it out can produce a correct-looking calculation that is off by two orders of magnitude.

For an immediate underlying price increase of 2, holding other variables constant, the approximate option-position gain is 3,000 × 2 + 0.5 × 200 × 2² = 6,400. Shorting 3,000 underlying units initially offsets the delta term and loses 6,000 on that price move. The combined approximation is therefore a gain of 400 from curvature, before financing, costs, time decay and volatility changes.

This is a local scenario calculation, not a guarantee of profitable hedging. Gamma changes delta as the underlying moves, so an initially neutral position generally needs rebalancing to remain neutral. Transaction costs and gaps matter, and the option can lose value through other sensitivities. State what is held constant whenever using the example to explain the benefit or cost of curvature.

Original 50-contract delta hedge under a +2 price move

Option delta
Calculation: 3,000 × 2
Approximate value change: +6,000
Option gamma
Calculation: 0.5 × 200 × 2²
Approximate value change: +400
Short underlying hedge
Calculation: -3,000 × 2
Approximate value change: -6,000
Combined
Calculation: 6,000 + 400 - 6,000
Approximate value change: +400 before other effects and costs

Combine changes carefully and recognize omitted terms

A common local approximation combines delta, one-half gamma times squared price change, vega times volatility change and theta times elapsed time. It can be informative when movements are small and the quotation conventions are consistent. It is still a truncated expansion. Cross-effects and changes in the sensitivities themselves can become material when several inputs move together.

For an original per-unit example, use delta 0.60, gamma 0.04, vega 0.12 per volatility percentage point and theta -0.03 per day. An underlying rise of 2, a volatility fall of one percentage point and one elapsed day produce 1.20 + 0.08 - 0.12 - 0.03 = 1.13 of approximate premium change. This is a change in value; add it to the starting premium only if the question asks for the estimated new premium.

Near expiry, around strikes or exercise boundaries, or under a large joint shock, full repricing can be more reliable. A delta hedge alone does not control volatility exposure. A vega hedge does not necessarily control skew or term-structure exposure. Use the risk-factor mapping lesson when combining those drivers across instruments, and keep the valuation assumptions explicit when comparing the approximation with a repriced result.

Does gamma change sign when the underlying move changes direction?

The gamma contribution to the second-order price approximation contains the squared price change. For a fixed positive gamma it is positive for both up and down moves. The delta contribution changes sign with the move. A short position reverses its position gamma, so distinguish direction of movement from direction of the trade.

Is delta exactly the probability of finishing in the money?

It is fundamentally a price sensitivity. Under particular models it has relationships to probability-like quantities, but those relationships depend on the model, numeraire and instrument. Use the definition needed by the question and avoid substituting an informal probability interpretation for a hedge calculation.

Try a changed assumption

The underlying falls by 2 instead of rising. With delta 0.60 and gamma 0.04, what is the approximate option-price change?

Show answer and reasoning

-1.20 + 0.08 = -1.12, holding other inputs fixed. The delta contribution changes sign; the positive gamma contribution does not.

Original learning check. No email or account needed.

Does delta equal the probability an option expires in the money?

Do not treat them as interchangeable. Delta is a price sensitivity. Some models produce related quantities, but exercise probability and hedge ratio have different definitions.

Sources and scope

Links checked 2026-10-10. GARP establishes exam scope; the technical references support the methods identified below. These are selected study notes and original examples, not complete curriculum coverage or official exam questions.

Prepared by MiloPrep editorial team. MiloPrep is independent of GARP. References do not imply endorsement or independent review of these notes.