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Option Greeks for the FRM® Exam Part I: delta, gamma, vega, theta and rho

Option Greeks measure sensitivity to price, volatility, time and rates. Worked examples explain delta hedging, rebalancing and the gamma and vega risks that remain.

FRM® Exam Part I guideBy High Performer CertsPublished 8 min read

Key facts

  • Delta measures local price sensitivity; gamma measures how delta changes with the underlying price.
  • Vega measures volatility sensitivity, theta the effect of time passing and rho interest-rate sensitivity.
  • Delta neutrality offsets current directional exposure but does not by itself remove gamma or vega risk.
  • Contract multipliers, position signs and percentage-point conventions must be included in Greek calculations.
  • A gamma-neutral portfolio can still have vega exposure, particularly across different maturities.

Delta measures an option’s local price sensitivity to the underlying; gamma measures how that delta changes as the underlying price moves. Vega measures sensitivity to volatility, theta to time passing and rho to interest rates. Delta hedging offsets current directional exposure. It does not by itself remove curvature or volatility risk, so a delta-neutral position can still lose money. (ASX, option Greeks (secondary source)) (BCBS, MAR40.77–40.80)

This guide supports study for the FRM® Exam Part I, run by the Global Association of Risk Professionals (GARP®). (GARP, FRM Exam information) Our questions complement GARP’s own study materials; they do not replace them. The course page gives the wider context. All numerical examples below use our own illustrative inputs, not market quotations.

What do the five option Greeks measure?

Let VV be option value per underlying share, SS the share price, σ\sigma annualised volatility, rr the risk-free rate and tt elapsed time. Each partial derivative holds the other inputs constant. The table describes long, standard European equity options in the Black–Scholes model before expiry, with positive volatility and a fixed dividend yield. Short positions reverse the sensitivities. (Back, Liu and Loewenstein, option hedging (secondary source)) (ASX, option Greeks (secondary source))

Option Greeks: definitions and signs
GreekSensitivityLong call / long put
Deltaδ=∂V/∂S\delta=\partial V/\partial S: underlying pricePositive / negative
GammaΓ=∂2V/∂S2\Gamma=\partial^2 V/\partial S^2: delta’s response to pricePositive / positive
Vegaν=∂V/∂σ\nu=\partial V/\partial\sigma: volatilityPositive / positive
ThetaΘ=∂V/∂t\Theta=\partial V/\partial t: time passingOften negative; not a universal sign rule
Rhoρ=∂V/∂r\rho=\partial V/\partial r: interest ratePositive / negative

Gamma measures curvature, not another independent market input. A share itself has delta one and gamma zero. Buying shares therefore changes portfolio delta without changing gamma. (OIC, gamma (secondary source))

Which units belong in a Greek calculation?

Mathematical vega and rho above use volatility and rates as decimals. Market displays often quote the change per percentage point instead. Theta may be quoted per day rather than per year. Match the sensitivity to the shock’s units before multiplying. (ASX, option Greeks (secondary source)) (OIC, theta (secondary source))

Suppose a call has vega £0.14 per volatility percentage point. A rise from 24% to 26.5% is 2.5 percentage points, or 0.025 in decimal units. The equivalent unscaled vega is £14:

ΔVvol≈0.14×2.5=14×0.025=£0.35\Delta V_{\mathrm{vol}}\approx0.14\times2.5=14\times0.025=\pounds0.35

Separately, daily theta of −£0.035 gives approximately −£0.07 over two days. Rho of £0.09 per rate percentage point gives +£0.036 for a 0.40-percentage-point rate increase. These are isolated effects, with other inputs fixed. Constant theta over several days is only an approximation: time decay changes as expiry approaches. (OIC, theta (secondary source))

How do you calculate a delta hedge?

Add signed position sensitivities, including the contract multiplier. For signed contract count nn, shares per contract mm and a holding of hh shares, delta neutrality requires: (ASX, option Greeks (secondary source)) (Back, Liu and Loewenstein, option hedging (secondary source))

δportfolio=nmδ+h=0⇒h=−nmδ\delta_{\mathrm{portfolio}}=nm\delta+h=0\quad\Rightarrow\quad h=-nm\delta

Assume 32 short call contracts, each covering 100 shares, with call delta 0.57. The multiplier is an assumption for this example, not a rule for every option contract.

δoptions=−32×100×0.57=−1,824\delta_{\mathrm{options}}=-32\times100\times0.57=-1{,}824

Buy 1,824 shares to offset that exposure. With other inputs fixed, a rising share price raises call premiums, so the short-call position loses value while the shares gain. A long call instead requires a short-share hedge. A long put has negative delta, so its delta hedge requires buying shares. These directions follow from the same signed equation.

Why does delta hedging require rebalancing?

Delta changes with the underlying price, time and volatility. Gamma estimates the price-driven change in delta, holding the other inputs fixed. (Back, Liu and Loewenstein, option hedging (secondary source)) (ASX, option Greeks (secondary source)) Suppose our call’s gamma is 0.026 per £1 and the share price rises £1.50:

δnew≈0.57+0.026×1.50=0.609\delta_{\mathrm{new}}\approx0.57+0.026\times1.50=0.609

The new hedge is approximately 32 × 100 × 0.609 = 1,948.8 shares. Starting from 1,824 shares, buy another 124.8 shares in the idealised calculation. Whole-share trading requires rounding and leaves a small residual delta. Recalculate actual model sensitivities before trading; gamma itself need not remain constant. (OIC, gamma (secondary source))

With other inputs fixed, rebalancing a short-gamma position’s delta hedge means buying shares after price rises and selling after falls. Large absolute gamma makes the hedge change quickly. Gamma can be particularly high near the strike shortly before expiry, making a fixed share holding less effective between adjustments. (Back, Liu and Loewenstein, option hedging (secondary source)) (OIC, gamma (secondary source))

What loss can gamma leave after delta hedging?

For a small price move, the second-order Taylor approximation adds curvature to the delta estimate. Holding time, volatility and rates fixed gives: (Back, Liu and Loewenstein, option hedging (secondary source))

ΔV≈δΔS+12Γ(ΔS)2\Delta V\approx\delta\Delta S+\tfrac12\Gamma(\Delta S)^2

Our call’s estimated increase per share is 0.57 × 1.50 + ½ × 0.026 × 1.50² = £0.88425. The short options lose £2,829.60, while the original share hedge gains £2,736. The £93.60 shortfall is the gamma effect:

Γportfolio=−32×100×0.026=−83.2\Gamma_{\mathrm{portfolio}}=-32\times100\times0.026=-83.2ΔΠgamma≈12(−83.2)(1.50)2=−£93.60\Delta\Pi_{\mathrm{gamma}}\approx\tfrac12(-83.2)(1.50)^2=-\pounds93.60

A £1.50 fall produces the same negative curvature term because the move is squared. This is an isolated gamma calculation, not total trading profit: theta, financing, dividends and trading costs also matter. Rebalancing after the move does not reverse the loss already incurred. (Back, Liu and Loewenstein, option hedging (secondary source))

Can you hedge gamma and vega with shares?

Shares have no option gamma or vega, so those exposures require another instrument, typically options. Delta alone does not describe option risk; the Basel delta-plus framework explicitly treats gamma and vega separately. (Back, Liu and Loewenstein, option hedging (secondary source)) (BCBS, MAR40.77–40.80)

Return to the initial snapshot. Assume a second call on the same underlying, with a different maturity, has delta 0.43, gamma 0.052, vega £0.22 per percentage point and the same 100-share multiplier. To offset gamma of −83.2, buy:

N=83.2100×0.052=16 contractsN=\frac{83.2}{100\times0.052}=16\text{ contracts}

The options’ combined delta is −1,824 + 16 × 100 × 0.43 = −1,136. Hold 1,136 shares to make the combined position delta-neutral too. This calculation first offsets gamma, then adjusts delta with shares. (Back, Liu and Loewenstein, option hedging (secondary source))

Gamma neutrality need not mean vega neutrality. Using the original call’s vega of £0.14, the initial short options have vega −£448 per percentage point. After buying the second calls:

νportfolio,pp=−448+16×100×0.22=−£96\nu_{\mathrm{portfolio,pp}}=-448+16\times100\times0.22=-\pounds96

A parallel 2.5-percentage-point rise in both implied volatilities still produces an estimated £240 loss. The gamma hedge reduces this particular volatility exposure without eliminating it. This estimate uses the same percentage-point shock for both options; it does not measure every possible volatility scenario.

There is a useful model-specific link. Dividing the Black–Scholes formulas for vega and gamma gives the following relationship, with remaining maturity τ\tau in years and vega measured per unit of decimal volatility: (Back, Liu and Loewenstein, option hedging (secondary source))

ν=ΓS2στ\nu=\Gamma S^2\sigma\tau

Under that model, options on the same underlying with the same maturity and a common volatility input share this proportionality factor. With a common volatility input, different maturities give different factors, so neutralising gamma need not neutralise vega. For vega per percentage point, divide the right-hand side by 100.

What should you check before trusting the estimate?

Check position signs, contract size and shock units first. Then ask which inputs moved and whether the approximation remains local. Larger moves and time between rebalances introduce hedge error; more trading also incurs costs. Greeks describe sensitivities under a model, not a complete loss distribution. (Back, Liu and Loewenstein, option hedging (secondary source)) The value at risk guide explains how linear approximations differ from scenario-based portfolio revaluation.

Frequently asked questions

What is the difference between delta and gamma?

Delta measures option value’s local sensitivity to the underlying price. Gamma measures how delta changes as that price moves.

How do you delta hedge a short call position?

Buy shares equal to the number of short contracts multiplied by shares per contract and call delta. In our example, 32 × 100 × 0.57 gives 1,824 shares.

Why must a delta hedge be rebalanced?

Delta changes with the underlying price, time and volatility. A share position that offsets delta now may no longer offset it after those inputs change.

Does delta-neutral mean risk-free?

No. Gamma, volatility changes and time passing can still change value. Financing and trading costs also affect the result.

How do you use vega quoted per percentage point?

Multiply it by the volatility change in percentage points. Vega of £0.14 and a rise from 24% to 26.5% give an estimated £0.35 increase per underlying share.

Does hedging gamma also hedge vega?

Not necessarily. Our second-option hedge cancels gamma but leaves vega of −£96 per percentage point, so a parallel volatility increase can still cause a loss.

Sources

  1. Australian Securities Exchange (market education, secondary source): Option pricing in detail — Topics 3–4: the Greeks, position delta and equivalent stock positions. Accessed 29 October 2026.
  2. Kerry Back, Hong Liu and Mark Loewenstein (academic textbook, secondary source): Pricing and Hedging Derivative Securities — Chapter 9: Greeks, discrete delta hedging and delta-gamma hedging. Accessed 29 October 2026.
  3. Options Industry Council, OCC (market education, secondary source): Gamma — delta changes, long and short positions, moneyness and expiry. Accessed 29 October 2026.
  4. Options Industry Council, OCC (market education, secondary source): Theta — daily sensitivity and non-linear time decay. Accessed 29 October 2026.
  5. Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework MAR40.77–40.80 — delta-plus treatment of option gamma and vega risk. Accessed 29 October 2026.
  6. Global Association of Risk Professionals (GARP): FRM Exam information: format and exam topics. Accessed 26 October 2026.

High Performer Certs is an independent study platform. It is not affiliated with, endorsed by, or sponsored by CFA Institute or the Global Association of Risk Professionals (GARP). We are not a CFA Institute Prep Provider and not a GARP Exam Preparation Provider. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of our products and services. GARP does not endorse, promote, review or warrant the accuracy of our products or services. All questions and explanations are our own original material and contain no CFA Institute or GARP exam, curriculum or practice questions. CFA®, Chartered Financial Analyst®, CIPM® and GIPS® are registered trademarks owned by CFA Institute. FRM® and GARP® are trademarks owned by the Global Association of Risk Professionals.

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