FRM Exam Part I · Option Sensitivity Measures: The "Greeks"

Greeks of Portfolios and the Taylor Approximation for Option Price Changes

Updated 11 October 2026 · Fact-checked

Portfolio Greeks are the position-weighted sums of each option's Greek, with short positions counted as negative. The Taylor approximation then estimates the value change as ΔV ≈ δΔS + ½ΓΔS² + ΘΔt. Add the terms using one consistent time unit, and include vega and rho only if volatility or rates also move.

Understand Greeks of Portfolios and Taylor Approximation

An option's value depends on the underlying price, time, volatility and interest rates. The Greeks measure how sensitive the value is to each input. Delta is the change in value per unit change in the underlying. Gamma is the change in delta per unit change in the underlying. Theta is the change in value as time passes.

Greeks add up across a portfolio. If you hold n_i units of option i, the portfolio Greek is the sum of n_i times that option's Greek. Long positions have positive n_i and short positions have negative n_i. This works because value is additive. It applies to delta, gamma, theta, vega and rho. Include the underlying itself, which has delta 1 and gamma 0, and futures, which have delta 1 per unit in the futures price.

The Taylor approximation turns Greeks into a P&L estimate. Value is a curved function of the underlying price. Delta is the slope, so δΔS is the straight-line estimate. Gamma captures the curvature, so ½Γ(ΔS)² corrects it. Theta adds the effect of time passing. Because (ΔS)² is always positive, a positive gamma adds value for a move in either direction. A negative gamma loses value for a move in either direction.

For a delta-neutral portfolio, δ = 0 and the delta term drops out. The change in value is then roughly ½Γ(ΔS)² + ΘΔt. A long-gamma portfolio usually has negative theta. It profits from big moves and pays for that through time decay. A short-gamma portfolio is the reverse.

The formulas change slightly by underlying. For an index or a currency, treat the underlying as paying a continuous yield q. For an index, q is the dividend yield. For a currency, q is the foreign risk-free rate. Delta of a European call becomes e^(−qT) N(d1). For options on futures (Black's model), the futures price is the underlying and the discount factor e^(−rT) plays the role of e^(−qT). A futures price has no income, so it behaves like an asset with q = r. Always read which price moves in the question: spot, index or futures.

Key formulas to remember

Portfolio Greek: Portfolio Greek = Σ n_i × Greek_i; n_i is positive for long and negative for short. Multiply by the contract size when the Greek is quoted per unit of the underlying.
Delta-gamma-theta approximation: ΔV ≈ δ ΔS + ½ Γ (ΔS)² + Θ Δt; Use the portfolio δ, Γ and Θ. Θ and Δt must use the same time unit, for example Θ per year with Δt in years.
Full Taylor with vega and rho: ΔV ≈ δ ΔS + ½ Γ (ΔS)² + Θ Δt + ν Δσ + ρ Δr; Add the vega and rho terms only if volatility or the interest rate changes. Δσ and Δr must be in the units that vega and rho use.
Delta-neutral portfolio: ΔV ≈ ½ Γ (ΔS)² + Θ Δt when δ = 0; Delta-neutral protects only against small moves. Gamma and theta still drive the result.
Gamma-neutralising trade: n_T = − Γ_P ÷ Γ_T; n_T is the number of traded options to add. Then change the underlying position by − (δ_P + n_T × δ_T) to restore delta neutrality.
Options on an index or currency: Call delta = e^(−qT) N(d1); Put delta = e^(−qT) [N(d1) − 1]; Gamma = e^(−qT) N′(d1) ÷ (S σ √T); q is the index dividend yield, or the foreign risk-free rate for a currency option. Call and put have the same gamma for the same strike and maturity.
Options on futures (Black's model): Call delta = e^(−rT) N(d1); Put delta = e^(−rT) [N(d1) − 1]; Gamma = e^(−rT) N′(d1) ÷ (F σ √T); Same as the yield formulas with q = r and S replaced by the futures price F. Delta is relative to the futures price.
Black-Scholes-Merton differential equation (no income): Θ + r S δ + ½ σ² S² Γ = r Π; Π is the portfolio value. For a delta-neutral portfolio, Θ + ½σ²S²Γ = rΠ, so theta and gamma have opposite signs when Π is small.

How to solve Greeks of Portfolios and Taylor Approximation questions

Use this order for any question on portfolio Greeks or the Taylor approximation.

1List each position with its signed quantity: positive if long, negative if short. Note the contract size (for example 100 shares per contract) and whether the Greeks are per option or per contract.
2Compute the portfolio delta, gamma and theta as Σ n_i × Greek_i. Include any position in the underlying (delta 1, gamma 0, theta 0).
3Identify the price change ΔS and the time step Δt. Convert Δt into the same unit as theta (days to years by dividing by 365, or by 252 if the question says trading days).
4Decide whether volatility or rates also change. Add vega × Δσ or rho × Δr only if they do.
5Plug in: delta term = δΔS, gamma term = ½Γ(ΔS)², theta term = ΘΔt. Square ΔS before multiplying by ½Γ.
6Add the terms with their signs. For a gamma-neutral or delta-neutral hedge, solve for the trade first: n_T = −Γ_P ÷ Γ_T, then fix delta with the underlying.
7For index, currency or futures options, check which delta formula applies (q or r). Sanity-check the sign: a short-gamma portfolio should lose value on a large move.

Quickest way: Three-term check

When to use it: Use when the question gives you Greeks and a price move and asks for the approximate change in value. Most FRM questions work this way.

Sum delta, gamma and theta in one pass with signs: write each as units × Greek.
Compute the three terms in order: δΔS, then ½Γ(ΔS)², then ΘΔt.
Check the units before adding: theta per year needs Δt in years, so a day is 1 ÷ 365.
If the portfolio is delta-neutral, skip the delta term and look at gamma and theta only.
Match your result to the options. A wrong sign on a short position, or a missing ½, is the usual trap.

Common mistakes in Greeks of Portfolios and Taylor Approximation

Treating short positions as positive when summing Greeks.
Greeks are usually quoted for a long option, so students copy the quoted number.
Fix: Write the signed quantity first. Short 1,000 puts means n = −1,000. Multiply that by the put's delta, gamma and theta.
Leaving out the ½ or not squaring ΔS in the gamma term.
Students recall 'gamma times the move' and mix the formula with delta's.
Fix: Write ½ × Γ × (ΔS)² every time. If ΔS = 2, the squared term is 4, not 2.
Mixing time units: theta per year with Δt in days.
Many sources quote theta per day, but the question may give it per year.
Fix: Read the unit of theta. Convert Δt to match: one day is 1/365 of a year unless the question uses 252 trading days.
Using the plain delta N(d1) for an index, currency or futures option.
Students memorise the non-dividend stock formula only.
Fix: Include the factor e^(−qT) for index and currency options. For options on futures, use e^(−rT). The put delta is the call delta minus the same factor, not minus 1.
Believing a delta-neutral portfolio has no risk.
Delta-neutral is described as hedged, so students forget the gamma term.
Fix: Delta-neutral removes only first-order risk. A large move still produces ½Γ(ΔS)², and time still produces ΘΔt.
Adding the vega term when volatility is unchanged, or ignoring it when volatility moves.
The three-term formula is memorised as complete.
Fix: Read the question for a change in implied volatility or rates. If given, add ν Δσ or ρ Δr, with Δσ in the same units as vega.

Worked examples

Example 1

A trader is long 2,000 calls and short 1,000 puts on the same stock. Each call has delta 0.60, gamma 0.05 and theta −6.0 per year. Each put has delta −0.40, gamma 0.04 and theta −3.0 per year. The stock rises by $2 in one day (Δt = 1/365 year). Estimate the change in portfolio value using delta, gamma and theta only.

Show the solution

Signed quantities: calls +2,000, puts −1,000.
Portfolio delta = 2,000 × 0.60 + (−1,000) × (−0.40) = 1,200 + 400 = 1,600.
Portfolio gamma = 2,000 × 0.05 + (−1,000) × 0.04 = 100 − 40 = 60.
Portfolio theta = 2,000 × (−6.0) + (−1,000) × (−3.0) = −12,000 + 3,000 = −9,000 per year.
Delta term = 1,600 × 2 = 3,200.
Gamma term = ½ × 60 × 2² = ½ × 60 × 4 = 120.
Theta term = −9,000 × (1/365) = −24.66.
Total = 3,200 + 120 − 24.66 = 3,295.34.

Answer: The portfolio gains about $3,295.34.

Example 2

A dealer's option book is delta-neutral with a portfolio gamma of −3,000. A traded option has delta 0.50 and gamma 0.60. What trades make the book gamma-neutral and delta-neutral?

Show the solution

Gamma-neutralising quantity: n_T = −Γ_P ÷ Γ_T = −(−3,000) ÷ 0.60 = +5,000. Buy 5,000 of the traded option.
New portfolio delta = 0 + 5,000 × 0.50 = +2,500.
To restore delta neutrality, take a position in the underlying with delta −2,500. The underlying has delta 1 per unit, so sell 2,500 units.
Check gamma: −3,000 + 5,000 × 0.60 = 0. Check delta: 2,500 − 2,500 = 0.

Answer: Buy 5,000 of the traded option and sell 2,500 units of the underlying.

Exam tips

Read the units of theta first. Many wrong answers come from a per-year theta used with a one-day step, or the reverse.
For a delta-neutral portfolio, the delta term is zero. Look for gamma and theta only, and remember that negative gamma loses on a large move in either direction.
Index, currency and futures options often hide a changed delta formula. Look for the dividend yield, the foreign rate, or the words 'option on futures' before you use N(d1).
Do the gamma hedge before the delta hedge. Trading the option changes delta, so fix delta last with the underlying.
With a financial calculator or spreadsheet, store the portfolio delta, gamma and theta as three separate numbers. That makes it easy to recheck each term.

Practice questions from Option Sensitivity Measures: The "Greeks"

Greeks of Portfolios and Taylor Approximation: frequently asked questions

How do I combine the Greeks of an options portfolio?

Multiply each option's Greek by the signed number of options held, then add. Long positions count as positive and short positions as negative. Add the underlying too, with delta 1 and gamma 0.

Why is the gamma term ½Γ(ΔS)² and not ΓΔS?

It comes from the second-order Taylor expansion. The first-order term δΔS is the slope. The next term is half the second derivative times the squared move. The ½ is part of the Taylor series and must always be included.

Is the delta-gamma approximation exact?

No. It is an approximation that works well for small moves. For large moves, higher-order terms and changes in volatility matter, so full revaluation is more accurate.

How are the Greeks of options on futures or currencies different?

The formulas use an adjusted discount factor. For index and currency options it is e^(−qT), where q is the dividend yield or foreign interest rate. For options on futures it is e^(−rT), so the structure is the same as the stock formula with q = r and F in place of S.

What does a delta-neutral portfolio still lose or gain from?

It still changes in value because of gamma, theta, vega and rho. For a delta-neutral book the main terms are ½Γ(ΔS)² for price moves and ΘΔt for time passing.