CFA Level I Exam · Pricing and Valuation of Options

Option Greeks and Delta Hedging Explained

Updated 7 October 2026 · Fact-checked

Option Greeks measure how an option's price changes when one input changes: delta for the underlying price, gamma for delta itself, vega for volatility, theta for time and rho for the interest rate. To delta hedge, hold an underlying position that offsets the option's delta, then rebalance because gamma makes delta change.

Understand Option Greeks and Delta Hedging

An option's price depends on the underlying price, volatility, time to expiry and the interest rate. A Greek tells you the sensitivity of the option price to one of these, holding the others fixed. Think of each Greek as a slope.

Delta is the change in option price for a one-unit change in the underlying price. A call has delta between 0 and 1. A put has delta between -1 and 0. Gamma is the change in delta for a one-unit change in the underlying price. It is the curvature of the option price. Gamma is largest for at-the-money options close to expiry and is positive for long calls and long puts.

Vega is the change in option price for a change in volatility. Long calls and long puts both have positive vega. Theta is the change in option price as time passes. For long options it is usually negative, which is time decay. Rho is the change in option price for a change in the risk-free rate. A long call has positive rho. A long put has negative rho.

Delta hedging makes a portfolio's total delta zero, so small moves in the underlying leave its value almost unchanged. If you are short calls, you buy shares. If you are long calls, you short shares. The hedge only works for small moves, because delta changes as the price moves.

That change is gamma risk. A delta-neutral portfolio with large gamma still gains or loses on big price moves. A long-gamma position gains from large moves in either direction. A short-gamma position loses. You must rebalance the hedge as delta drifts, and rebalancing costs money.

Key formulas to remember

Delta: Delta = ΔOption price ÷ ΔUnderlying price; Call delta is between 0 and 1. Put delta is between -1 and 0.
Put delta from call delta: Put delta = Call delta − 1; Holds for European options on an underlying with no dividends. The call and put must share strike and expiry.
Gamma: Gamma = ΔDelta ÷ ΔUnderlying price; Highest at the money and near expiry. Same for a call and put with the same terms.
Price change estimate: ΔOption ≈ Delta × ΔS + ½ × Gamma × (ΔS)²; Delta gives the first-order estimate. The gamma term corrects for curvature.
Portfolio delta: Portfolio delta = Σ (number of units × delta of each unit); Shares have delta of 1 each. Short positions carry a negative sign.
Delta hedge size: Options needed = Shares held ÷ Option delta; Assumes each option covers one share. Sell calls against long shares; buy shares against short calls.
Vega, theta and rho: Vega = ΔOption ÷ ΔVolatility; Theta = ΔOption ÷ ΔTime; Rho = ΔOption ÷ ΔRate; Long options: vega positive, theta usually negative. Rho is positive for calls and negative for puts.

How to solve Option Greeks and Delta Hedging questions

Use this order for any Greek or delta hedging question.

1Identify the position: long or short, call or put, and how many options and shares.
2Write the sign of each position. Short positions flip the sign of delta, gamma, vega and theta.
3Name the Greek the question asks about and match it to the input that changes: price, delta, volatility, time or rate.
4For a hedge, compute portfolio delta as units × delta for every position, shares included.
5Add the opposite position in the underlying so total delta is zero. Divide by option delta if you are sizing an options hedge.
6If the underlying moves, update delta with delta + gamma × ΔS, then recompute portfolio delta to find the rebalancing trade.
7Check the sign and size: a call delta must stay between 0 and 1, and a delta-neutral book can still lose through gamma.

Quickest way: Sign table and delta arithmetic

When to use it: Use for any three-option MCQ asking which Greek, which direction or how many units to trade.

Memorise for long options: delta call +, put −; gamma +; vega +; theta −; rho call +, put −.
For a short option, flip every sign.
For hedge size, compute delta × quantity for each position and trade the opposite amount in the underlying.
If delta is hedged but gamma is non-zero, large moves still matter: long gamma gains, short gamma loses.
Eliminate options that break a sign rule or invert the hedge ratio (multiplying instead of dividing by delta).

Common mistakes in Option Greeks and Delta Hedging

Believing a delta-neutral portfolio has no risk
Delta neutral sounds like risk free.
Fix: Delta neutral protects only against small price moves. Gamma, vega, theta and rho still affect value, and delta changes as the price moves.
Mixing up delta and gamma
Both relate to price moves in the underlying.
Fix: Delta is the slope of option price. Gamma is how fast delta changes, the curvature. Delta is about price, gamma is about delta.
Giving a put a positive delta
Students remember that call delta is positive and apply it to both.
Fix: A long put loses value when the underlying rises, so its delta is between -1 and 0. Use put delta = call delta − 1 only for European options with the same strike and expiry on an underlying with no dividends.
Multiplying instead of dividing when sizing a hedge
The ratio is read the wrong way round.
Fix: Options needed = shares ÷ delta. With delta below 1, you need more options than shares. Check that the answer is larger than the share count.
Ignoring the position sign for short options
Greeks are usually quoted for a long option.
Fix: A short call has negative delta, negative gamma, negative vega and positive theta. Apply the sign before you add up the portfolio.
Assuming theta is always a loss
Time decay is taught for long options only.
Fix: Time decay hurts the option holder and helps the option writer. Say whose position you are assessing.

Worked examples

Example 1

An investor holds 10,000 shares and wants to be delta neutral by selling call options. Each call is on one share and has a delta of 0.40. How many calls should the investor sell? A) 4,000 B) 10,000 C) 25,000

Show the solution

Delta of the shares = 10,000 × 1 = 10,000.
Each short call contributes -0.40 of delta per option.
Set total delta to zero: 10,000 − 0.40 × N = 0.
N = 10,000 ÷ 0.40 = 25,000.
Check: 4,000 would be multiplying by 0.40, and 10,000 would ignore delta.

Answer: C) 25,000 calls

Example 2

A dealer is short 1,000 calls, each on one share, with delta 0.55 and gamma 0.04. The dealer holds 550 shares to be delta neutral. The share price rises by 2. To restore delta neutrality, how many shares should the dealer trade? A) Sell 550 B) Buy 80 C) Buy 630

Show the solution

New call delta ≈ 0.55 + 0.04 × 2 = 0.63. This is an approximation, because it uses gamma as a constant and updates delta linearly.
Short call delta = -1,000 × 0.63 = -630.
Share delta = +550. Net delta = 550 − 630 = -80.
The dealer is short delta, so buy 80 shares to bring net delta to zero.
Check the P&L: the calls lose about 1,000 × (0.55 × 2 + ½ × 0.04 × 4) = 1,000 × (1.10 + 0.08) = 1,180, and the shares gain 550 × 2 = 1,100. Net P&L = -80. This equals the gamma term, ½ × 1,000 × 0.04 × 2² = 80, which is the cost of short gamma. It is a separate quantity from the net delta of -80 used for the rebalancing trade.

Answer: B) Buy 80 shares

Exam tips

Memorise the sign table for long calls and long puts. Many questions are pure sign checks.
Read whether the position is long or short before reading the Greek. Short positions flip the sign.
For hedge-size questions, estimate first: delta below 1 means more options than shares, so reject any smaller answer.
Questions on gamma risk usually test that a delta-neutral position still loses or gains on large moves, and that gamma is highest at the money near expiry.
You do not need a calculator for most of these items. Save the BA II Plus for time value questions and use mental arithmetic here.

Practice questions from Pricing and Valuation of Options

Option Greeks and Delta Hedging in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Option Greeks and Delta Hedging: frequently asked questions

What is the difference between delta and gamma?

Delta is the change in option price for a one-unit move in the underlying. Gamma is the change in delta for that same move. Delta is the slope, and gamma is the curvature.

How do you delta hedge an option position?

Add up the delta of every position, then trade the underlying in the opposite direction to bring the total to zero. For example, if you are short calls with a total delta of -630, buy 630 shares. Rebalance as delta changes.

Why is gamma risk a problem in a delta-neutral portfolio?

Delta neutrality holds only for small price moves. When the underlying moves a lot, delta changes by gamma times the move, so the hedge becomes wrong. A short-gamma position loses on large moves in either direction.

Is theta positive or negative?

For a long option, theta is usually negative because the option loses time value as expiry nears. For an option writer, the same decay works in their favour.