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

Delta Hedging and Delta-Neutral Portfolios Explained

Updated 11 October 2026 · Fact-checked

Delta hedging removes a portfolio's first-order exposure to small moves in the underlying price. You find the portfolio's total delta, then trade the underlying (delta of one share is 1) in the opposite amount so total delta is zero. Because delta changes, you must rebalance as price and time move.

Understand Delta Hedging and Delta-Neutral Portfolios

Delta is the rate of change of an option's value with respect to the underlying price: Δ = ∂V ÷ ∂S. A call has delta between 0 and 1. A put has delta between -1 and 0. One share of the underlying has delta 1. A short share has delta -1.

A delta-neutral position has total delta equal to zero. For small price moves, gains on one part offset losses on the other. Say you sell 1,000 calls, each with delta 0.6. You are short 600 shares' worth of exposure, so you buy 600 shares. Your total delta is -600 + 600 = 0.

Delta is only a local measure. It is the slope of the option value curve at one price. For large moves the curve bends, and the hedge leaves an error. That curvature is gamma. A delta-neutral position with nonzero gamma still gains or loses on big moves.

Delta also changes as time passes and as volatility or rates change, even if the price stays put. So a hedge that is neutral today is not neutral tomorrow. Dynamic hedging means rebalancing regularly to bring delta back to zero. Static hedging sets up a hedge once and leaves it, for example by buying an offsetting option. Dynamic hedging has transaction costs, and the cost rises with gamma and with how often you trade.

A short option position hedged by delta tends to need buying the underlying when the price rises and selling when it falls. That is buying high and selling low, and it is the source of the rebalancing cost. A long option position hedged by delta does the opposite: it sells high and buys low, which offsets time decay.

Key formulas to remember

Delta of an option position: Position delta = N × Δ_option (N is signed: positive long, negative short); Use the contract multiplier if options cover many shares, such as 100 per contract.
Delta-neutral condition: Σ (Nᵢ × Δᵢ) = 0; Sum across all options and the underlying. The underlying has Δ = 1 per unit.
Shares needed to hedge: Shares to trade = -(Total option position delta); A positive option delta means sell shares short. A negative option delta means buy shares.
Black-Scholes-Merton delta (no dividend): Call Δ = N(d₁); Put Δ = N(d₁) - 1; With a continuous dividend yield q, call Δ = e^(-qT) N(d₁) and put Δ = e^(-qT)[N(d₁) - 1].
Delta of a forward-like exposure: Call Δ - Put Δ = 1 (same strike, same maturity, no dividends); This follows from put-call parity.
Delta-gamma approximation of value change: ΔV ≈ Δ × ΔS + ½ × Γ × (ΔS)²; For a delta-neutral position, the first term is zero and gamma drives the P&L.
Gamma to neutralize with another option: N_T = -Γ_P ÷ Γ_T; Trading option T changes delta, so you then re-hedge delta with the underlying.

How to solve Delta Hedging and Delta-Neutral Portfolios questions

Use this method for any question on building or maintaining a delta hedge.

1List every position with its signed quantity: long is positive, short is negative. Note the contract size (for example 100 shares per option).
2Find or compute each delta. If given d₁, use N(d₁) for a call and N(d₁) - 1 for a put, adjusting for any dividend yield.
3Multiply quantity × delta × contract size for each position, then add them to get total delta in share terms.
4Set the hedge trade equal to minus the total delta. Positive total delta means sell the underlying. Negative means buy.
5If the question includes a rebalance, recompute delta at the new price or time. The change in required shares is the new hedge minus the old hedge.
6Check direction: for a short option hedge, a price rise raises the delta, so you buy more shares.
7If gamma is given, use it to estimate the P&L or to find the option position that makes gamma zero, then re-hedge delta.

Quickest way: Total delta in shares, then flip the sign

When to use it: Use this for any simple number-of-shares question with given deltas.

Multiply contracts × multiplier × delta for each option and add with signs.
Flip the sign of the total. That is your trade in the underlying.
For a rebalance, subtract: new total delta minus old total delta, then flip the sign to find the additional trade.
Sanity check: a short call gives long shares. A long put gives long shares. A long call or short put gives short shares.

Common mistakes in Delta Hedging and Delta-Neutral Portfolios

Hedging in the wrong direction, such as buying shares to hedge a long call.
Students match the option's direction to the hedge direction instead of offsetting total delta.
Fix: Compute the position delta with its sign first. Then trade the opposite amount in the underlying.
Forgetting the contract multiplier.
The question gives contracts and per-share delta, and students use the number of contracts as the number of shares.
Fix: Always write contracts × shares per contract × delta.
Using N(d₁) as the put delta.
Students remember the call formula only.
Fix: Put delta is N(d₁) - 1, which is negative. With a dividend yield, include e^(-qT).
Believing delta-neutral means risk-free.
The word neutral suggests no risk.
Fix: Delta-neutral protects only against small price moves. Gamma, vega, theta and transaction costs remain.
Assuming the hedge stays valid as time passes.
Delta looks like a fixed number.
Fix: Delta moves with price, time and volatility. Rebalancing is needed, and the question usually asks for the extra shares to trade, not the full hedge.
Mixing up static and dynamic hedging.
Both are described as keeping risk low.
Fix: Static: set once, typically with an offsetting instrument. Dynamic: adjust repeatedly and pay costs each time.

Worked examples

Example 1

A bank has sold 2,000 European call options on a non-dividend stock. Each option covers 100 shares. The call delta is 0.55. How many shares must the bank buy or sell to be delta-neutral?

Show the solution

Position: short 2,000 calls, so quantity = -2,000.
Shares covered = 2,000 × 100 = 200,000.
Position delta = -200,000 × 0.55 = -110,000 shares.
Hedge trade = -(-110,000) = +110,000.

Answer: Buy 110,000 shares.

Example 2

A trader is short 10,000 calls on one share each, with delta 0.50, and holds the hedge of 5,000 shares. The stock rises and the call delta becomes 0.62. What trade restores delta neutrality?

Show the solution

New option position delta = -10,000 × 0.62 = -6,200.
Required shares held = +6,200.
Current shares held = 5,000.
Trade = 6,200 - 5,000 = 1,200 shares to buy.

Answer: Buy 1,200 more shares, which means buying after the price rise.

Exam tips

Most numerical questions come down to total delta × quantity × multiplier. Do this arithmetic carefully and check the sign.
Know the delta ranges: call 0 to 1, put -1 to 0, stock 1. Use them to eliminate wrong answer choices fast.
Conceptual questions test that delta-neutral still has gamma and vega risk. Large moves produce losses for a short-gamma position.
Expect a question on cost: frequent rebalancing of a short option position means buying after rises and selling after falls.
Read whether the question asks for the full hedge or the rebalancing trade. They give different numbers.

Practice questions from Option Sensitivity Measures: The "Greeks"

Delta Hedging and Delta-Neutral Portfolios in other exams

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

Delta Hedging and Delta-Neutral Portfolios: frequently asked questions

How do I calculate the number of shares for a delta hedge?

Multiply the number of options by the shares per option and by the delta, keeping the sign for long or short. Then trade the opposite amount in the underlying. Short 1,000 calls with delta 0.6 needs 600 shares bought.

What is the difference between static and dynamic hedging?

A static hedge is set up once and left, often using an offsetting option. A dynamic hedge is adjusted as delta changes, which means repeated trades and transaction costs. Delta hedging is usually dynamic.

Why does a delta-neutral portfolio still lose money?

Delta only captures the first-order effect of small price moves. Large moves hit gamma, and changes in volatility hit vega. Time decay and trading costs also affect the result.

Does delta change with time even if the stock price stays the same?

Yes. As expiry approaches, delta of an in-the-money option moves toward 1 for a call, and of an out-of-the-money option toward 0. So you still need to rebalance.