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

Option Greeks and Delta Explained for FRM Part I

Updated 11 October 2026 · Fact-checked

Delta is the change in an option's value for a small change in the underlying price. In Black-Scholes-Merton with a non-dividend stock, a European call has delta N(d1) and a put has N(d1) − 1. Calls range from 0 to 1, puts from −1 to 0. A long forward on one unit has delta 1.

Understand Option Greeks: Overview and Delta

The Greeks are sensitivities of an option's value to its risk factors. Each Greek is a partial derivative of the option price: delta for the underlying price, gamma for delta itself, vega for volatility, theta for time and rho for the interest rate. Traders use them to measure and hedge risk without repricing every position.

Delta (Δ) is the rate of change of the option value with respect to the underlying price: Δ = ∂V ÷ ∂S. If a call has delta 0.60 and the stock rises by $1, the call gains about $0.60. It is the slope of the option value curve at the current price. The approximation is good for small moves and worsens for large ones, because the curve bends. That bending is gamma.

For a European call on a stock paying no dividends, Δ = N(d1). N(·) is the standard normal cumulative distribution, so delta lies between 0 and 1. A deep in-the-money call has delta near 1. A deep out-of-the-money call has delta near 0. An at-the-money call has delta a little above 0.5. For the put, Δ = N(d1) − 1, which lies between −1 and 0. A put loses value when the stock rises.

The underlying asset has delta 1 per unit: a $1 rise in the price changes its value by $1. A short position has delta −1 per unit. A forward on a non-dividend stock has delta 1 per unit, because its value is S − K·e^(−rT). A futures contract has delta e^(rT) with respect to the spot price, because F = S·e^(rT). Futures delta with respect to the futures price is 1. With a continuous yield q on the underlying, the call delta becomes e^(−qT)·N(d1) and the put delta becomes e^(−qT)·[N(d1) − 1].

Portfolio delta is the sum of position quantity × delta for each position. A portfolio with delta zero is delta-neutral: small price moves leave its value roughly unchanged. Delta is also the hedge ratio: to hedge a short call you hold delta shares of the underlying.

Key formulas to remember

Definition of delta: Δ = ∂V ÷ ∂S; Sensitivity of option or portfolio value to the underlying price.
d1 (no dividends): d1 = [ln(S ÷ K) + (r + σ²÷2)·T] ÷ (σ·√T); r is the continuously compounded risk-free rate; σ is annual volatility; T is in years.
d2: d2 = d1 − σ·√T; Used for the price, not for delta.
European call delta: Δcall = N(d1); Between 0 and 1 for a non-dividend stock.
European put delta: Δput = N(d1) − 1 = −N(−d1); Between −1 and 0.
Delta with continuous yield q: Δcall = e^(−qT)·N(d1); Δput = e^(−qT)·[N(d1) − 1]; d1 uses (r − q + σ²÷2). Applies to indices and, with q = foreign rate, to currencies.
Delta of forward and futures: Forward on non-dividend stock: Δ = 1 per unit; futures: Δ = e^(rT) with respect to spot; Delta with respect to the futures price itself is 1.
Portfolio delta: ΔP = Σ (nᵢ × Δᵢ); Use negative quantity for short positions.
Delta-neutral hedge: Shares to hold = −(portfolio delta); Short 1 call with delta 0.6 needs 0.6 shares long per option unit.
Put-call delta relation: Δcall − Δput = 1 (no dividends); Follows from put-call parity.

How to solve Option Greeks: Overview and Delta questions

Use this method for any delta question, whether it asks for a value, a hedge or a portfolio exposure.

1Identify the instrument: call, put, stock, forward or futures, and whether you are long or short.
2Check the underlying: does it pay a dividend or yield q, or is it a currency or index?
3If the question gives N(d1) or d1, use it. If not, compute d1 from S, K, r, σ, T (and q).
4Apply the right rule: call N(d1), put N(d1) − 1, adjusted by e^(−qT) if there is a yield.
5Multiply by position size and sign (short = negative) and by the contract multiplier.
6Sum across positions for portfolio delta.
7For a hedge, take the opposite of portfolio delta in the underlying, then sanity-check the sign and size.

Quickest way: Sanity-check delta by moneyness

When to use it: When options list four candidate values and you need to eliminate wrong ones fast.

Call delta must be between 0 and 1; put delta between −1 and 0. Reject anything outside.
Put delta = call delta − 1 (no yield). Use this instead of recomputing.
At-the-money: call delta slightly above 0.5; deep ITM near 1 (call) or −1 (put).
Short positions flip the sign. Apply this last.
On the calculator, N(x) comes from a normal table; use symmetry N(−x) = 1 − N(x).

Common mistakes in Option Greeks: Overview and Delta

Giving a put a positive delta.
You remember N(d1) and forget the −1.
Fix: Put delta = N(d1) − 1. It is always negative for a long put.
Treating delta as the probability of finishing in the money.
N(d1) looks like a probability.
Fix: The risk-neutral probability of exercise is N(d2) for a call. Delta is a hedge ratio, not that probability.
Forgetting the sign for short positions.
You compute option delta and stop.
Fix: Multiply by −1 for each short position before summing.
Ignoring e^(−qT) for dividend-paying indices or currencies.
You memorise the no-dividend formula only.
Fix: Check the underlying. Use e^(−qT)·N(d1) for calls with yield q.
Using delta of a forward as e^(rT) or futures as 1 without checking the reference variable.
Delta depends on whether you differentiate by spot or by futures price.
Fix: Futures delta is e^(rT) with respect to spot and 1 with respect to the futures price. Read what the question states.
Applying delta to large price moves as if exact.
Delta looks like a fixed number.
Fix: Delta is a first-order estimate. For big moves add gamma or reprice.

Worked examples

Example 1

A European call on a non-dividend stock has N(d1) = 0.64. A trader is short 10,000 calls (one share each). How many shares should the trader hold to be delta-neutral?

Show the solution

Call delta = N(d1) = 0.64.
Position delta = −10,000 × 0.64 = −6,400.
To neutralise, add +6,400 delta from the stock, which has delta 1 per share.

Answer: Buy 6,400 shares.

Example 2

A portfolio holds long 2,000 European puts with N(d1) = 0.55 each, and short 1,000 European calls with N(d1) = 0.55 each, all on the same non-dividend stock (one share per option). What is the portfolio delta in shares?

Show the solution

Put delta = 0.55 − 1 = −0.45.
Long puts: 2,000 × (−0.45) = −900.
Call delta = 0.55. Short calls: −1,000 × 0.55 = −550.
Portfolio delta = −900 + (−550) = −1,450.

Answer: −1,450 shares. The portfolio loses about 1,450 per $1 rise in the stock.

Exam tips

Memorise the sign and range of call and put delta; many questions are solved by elimination.
Read the underlying: dividend yield or foreign rate changes the formula.
Watch for per-share versus per-contract quantities and the multiplier.
Know the difference between delta with respect to spot and futures price.
Be ready to link delta to hedging: hedge shares equal minus portfolio delta.

Practice questions from Option Sensitivity Measures: The "Greeks"

Option Greeks: Overview and Delta 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: Overview and Delta: frequently asked questions

What is the delta of a call and a put option?

For a European option on a non-dividend stock, call delta is N(d1) and put delta is N(d1) − 1. Calls range from 0 to 1 and puts from −1 to 0. Dividends or yields scale both by e^(−qT).

How do I calculate delta in Black-Scholes?

Compute d1 = [ln(S÷K) + (r + σ²÷2)T] ÷ (σ√T), then look up N(d1) in a normal table. That is the call delta. Subtract 1 for the put.

Is delta the probability that an option expires in the money?

No. Delta is the hedge ratio. Under risk-neutral valuation, the probability a call finishes in the money is N(d2), which is close to delta but not equal.

What is the delta of a forward or futures contract?

A long forward on a non-dividend stock has delta 1 per unit with respect to spot. A futures contract has delta e^(rT) with respect to spot, and 1 with respect to the futures price.