Option Greeks Explained: FRM Part 1 Notes with Worked Examples [2026]

Option greeks explained in one idea: every option price is pushed around by five forces, and each Greek measures the push from one of them. Delta measures sensitivity to the underlying price, gamma measures how fast delta itself changes, vega measures sensitivity to implied volatility, theta measures the cost of time passing, and rho measures sensitivity to interest rates. FRM Part 1 does not ask for essays on these. It asks for signs, sizes and hedges: the delta of a put, the gamma of a short position, the vega profit when volatility jumps, and the share purchase that makes a book delta neutral. This note gives each Greek a definition, a sign convention, a worked calculation and the behavior examiners probe, then closes with a summary table, the traps and a one-page revision box.

Short on revision time? Map out your remaining FRM topics with our free study timetable generator. This note sits in a series of three calculation-heavy FRM Part 1 notes with duration and convexity and credit risk expected loss.

Table of Contents

• Option greeks explained: what each Greek measures
• Delta and the delta-neutral hedge
• Gamma: why the hedge keeps moving
• Vega, theta and rho with worked examples
• Option greeks explained: the summary table
• How the Greeks change with moneyness and time
• Option greeks explained: exam traps
• One-page revision: option greeks explained
• Frequently asked questions
• Wrapping up

Option greeks explained: what each Greek measures

The option Greeks measure how an option price reacts to the five forces that move it. Delta tracks the underlying price, gamma tracks delta itself, vega tracks implied volatility, theta tracks time passing, and rho tracks interest rates. FRM questions test the signs, the sizes and the hedges built from all five.

Formally, each Greek is a partial derivative of the option price: delta = change in option price per unit change in the underlying, vega = change per point of implied volatility, and so on. Two conventions matter before any calculation. First, the Greeks in this note are stated per option on one share; exchange-traded contracts often cover 100 shares, and the exam will say so when it matters. Second, signs flip with the position: a Greek that is positive for a long option is negative for a short option, and puts carry the opposite delta and rho signs to calls.

Delta and the delta-neutral hedge

Delta is the workhorse Greek. A call delta sits between 0 and +1: roughly +0.50 at the money, near +1 deep in the money, near 0 deep out of the money. A put delta sits between -1 and 0, roughly -0.50 at the money. Delta also gives the hedge ratio directly: it is the number of shares that replicates the option exposure over a small price move.

Worked example, part 1. A market maker is short 10,000 call options (one share each) with delta 0.55 per option. The position delta is -10,000 x 0.55 = -5,500, meaning the book behaves like a short position in 5,500 shares. To neutralize it, the trader buys 5,500 shares. The combined book now has delta zero: small moves in the stock change the option liability and the share holding by offsetting amounts.

Two cautions examiners build into questions. Delta is a local measure: it holds for small moves and short horizons, which is why gamma exists. And delta is not exactly the probability that the option finishes in the money. In the Black-Scholes-Merton model delta equals N(d1); the risk-neutral probability of exercise is N(d2). The two are close, and loose market talk treats them as the same, but a precise exam question can separate them.

Position deltas add up across a portfolio, and portfolio Greeks feed straight into market risk measurement: a desk that knows its delta, gamma and vega can estimate losses under price and volatility shocks, the same shock logic used in our FRM value at risk cheat sheet.

Gamma: why the hedge keeps moving

Gamma is the change in delta for a one-unit change in the underlying price. Long options, calls and puts alike, have positive gamma: delta moves in the holder favor as the price moves. Short options have negative gamma, and that is where hedging gets expensive.

Worked example, part 2. Continue the book above: delta neutral after buying 5,500 shares. The calls have gamma 0.06. The stock then rises by USD 3. Each call delta rises by gamma x price change = 0.06 x 3 = 0.18, from 0.55 to 0.73. The short option position delta becomes -10,000 x 0.73 = -7,300, while the shares still contribute +5,500. Net book delta = -7,300 + 5,500 = -1,800. The trader must buy 1,800 more shares to restore neutrality.

Notice the pattern: the short-gamma trader buys after a rise and sells after a fall, systematically buying high and selling low. That rebalancing cost is the price of being short gamma, and it grows with the size of the price move. A long-gamma book has the mirror experience: delta moves in its favor, and hedging locks in gains. Gamma is largest for at-the-money options close to expiry, which is why short-dated ATM books need the most frequent rebalancing.

Vega, theta and rho with worked examples

Vega is the change in option value for a one percentage point change in implied volatility. Long options have positive vega: they gain when volatility rises. Worked example: a trader is long 5,000 options with vega 0.12 each. Implied volatility rises by 3 points. Gain = 5,000 x 0.12 x 3 = USD 1,800. Vega is largest for at-the-money options and for longer maturities, because both have the most value riding on future volatility.

Theta is the change in option value as one day passes, other things equal. Long options usually have negative theta: time decay works against the holder. Worked example: a long position of 1,000 options with theta -0.045 per option loses 1,000 x 0.045 = USD 45 per day to decay alone. Short positions collect that decay, which is the income side of option selling, earned in exchange for carrying gamma and vega risk.

Rho is the change in option value for a one percentage point change in the risk-free rate. Calls have positive rho, puts have negative rho, and both are small for short maturities. Worked example: a call with rho 0.05 gains about USD 0.05 per option if rates rise by 1%. Rho rarely decides an FRM calculation question, but its signs appear in theory questions, and it matters more for long-dated options such as LEAPS.

Option greeks explained: the summary table

Greek	Measures sensitivity to	Long call	Long put	Short position	Peaks when
Delta	Underlying price (per USD 1)	0 to +1 (ATM about +0.50)	-1 to 0 (ATM about -0.50)	Opposite sign to the long	Approaches 1 or -1 deep in the money
Gamma	Change in delta (per USD 1)	Positive	Positive	Negative	At the money, near expiry
Vega	Implied volatility (per 1 point)	Positive	Positive	Negative	At the money, longer maturity
Theta	Time passing (per day)	Usually negative	Usually negative	Positive	Most negative at the money, near expiry
Rho	Interest rate (per 1 point)	Positive	Negative	Opposite sign to the long	Longer maturities

How the Greeks change with moneyness and time

Moneyness moves the Greeks in patterns the exam revisits constantly. As a call moves deeper in the money, delta climbs toward 1 and the option behaves more like the stock itself; gamma and vega fade because the outcome is nearly settled. Deep out of the money, delta, gamma and vega all shrink toward zero. The action concentrates at the money, where the outcome is most uncertain: gamma is highest there, vega is highest there, and theta bites hardest there.

Time reshapes the picture. As expiry approaches, at-the-money gamma spikes, so delta becomes unstable and hedges need constant attention. Theta decay accelerates for at-the-money options in the final weeks. Vega falls as expiry nears because less time remains for volatility to matter. Delta of an at-the-money option snaps toward 0 or 1 as expiry arrives, since the uncertainty resolves. A favorite exam framing: compare two otherwise identical options, one with a week to expiry and one with a year. The week option has higher gamma and more negative theta (for ATM); the year option has higher vega and higher rho.

Option greeks explained: exam traps

• Put delta sign. A long put delta sits between -1 and 0. If a question gives a put delta of 0.40, it usually means -0.40; hedging a long put means buying shares, not selling them.
• Gamma is not call-only. Long puts have positive gamma too. Gamma depends on long versus short, not on call versus put.
• Vega units. Vega is quoted per one percentage point of implied volatility (20% to 21%), not per 1% relative change (20% to 20.2%).
• Theta units. Theta is usually per day in trading platforms and per year in some textbook formulas. Check the unit before multiplying by a position size.
• Delta is not the exercise probability. Delta = N(d1) in Black-Scholes-Merton; the risk-neutral probability of finishing in the money is N(d2).
• Contract multiplier. Options on 100 shares multiply every Greek by 100. Hedge share counts built from per-share deltas must use total share exposure.
• The underlying has Greeks too. A share has delta 1 and zero gamma, vega and theta. Futures are close to delta 1 per unit, which is why they make clean hedging instruments.
• Delta neutral is not risk free. A delta-neutral book still carries gamma, vega and theta exposure. Large moves and volatility shifts will still move its value.

One-page revision: option greeks explained

Greek	Formula idea	One-line memory hook
Delta	dV/dS	Shares needed to mimic the option, right now
Gamma	dDelta/dS = d2V/dS2	How fast the hedge ratio runs away from you
Vega	dV/d(sigma), per 1 point	Long options love rising volatility
Theta	dV/dt, per day	The rent a long option pays for time
Rho	dV/dr, per 1 point	Calls gain, puts lose, when rates rise

• Hedge worked example: short 10,000 calls, delta 0.55, hedge = buy 5,500 shares.
• After a USD 3 rise with gamma 0.06, delta becomes 0.73 and the re-hedge buys 1,800 more.
• Long gamma: delta moves in your favor. Short gamma: you buy high and sell low when rebalancing.
• ATM options carry the most gamma, vega and theta pain; time decay accelerates into expiry.

Frequently asked questions

Which Greek matters most once a book is delta hedged?

Gamma, because it decides how quickly the delta hedge goes stale, and vega, because a volatility shift reprices the options without any move in the underlying. A delta-hedged short option book is mostly a short gamma and short vega book, earning theta in exchange.

Why is gamma highest at the money?

At the money, a small price move changes the character of the option the most: it tips the option toward clearly in or clearly out of the money, so delta swings fastest there. Deep in or out of the money, the outcome is nearly decided and delta barely responds to small moves, so gamma is near zero.

Is theta always negative for a long option?

Almost always for long calls and for typical long puts, because the holder pays for time and time only runs out. A rare exception is a deep in-the-money European put, where the interest earned on the strike price received at exercise can outweigh the remaining time value, giving positive theta. Exam questions normally assume the standard negative case unless they specify otherwise.

What does a vega of 0.12 actually tell a trader?

That the option gains about USD 0.12 if implied volatility rises by one percentage point, and loses about USD 0.12 if it falls by one point, per option on one share. Multiply by position size and contract multiplier to get the book exposure, as in the USD 1,800 worked example above.

How is delta used to build a hedge with futures?

The same way as with shares: total position delta divided by the delta of one futures contract gives the number of contracts, with the opposite sign. Stock index futures have a delta close to their contract value in share equivalents, so a book that is long the equivalent of 50,000 shares of exposure is hedged by shorting futures covering 50,000 shares.

Option greeks explained: wrapping up

The exam routine is: fix the signs first (call or put, long or short), take delta as the hedge ratio, use gamma to update delta after a price move, and scale vega and theta by position size and time. Practice the chain on the summary numbers until the re-hedge calculation is automatic. Then move to the sister notes: duration and convexity for interest rate sensitivity and credit risk expected loss for PD, LGD and EAD. Every note in the set is collected on the FRM notes hub.

Found this useful? Share it with a fellow FRM candidate on WhatsApp or Telegram: “The Greeks stop being confusing once you hedge a book step by step. This note runs delta, gamma, vega, theta and rho with real numbers.”