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CFA Level II Exam · Valuation of Contingent Claims

Option Greeks and Delta Hedging for CFA Level II

Updated 7 October 2026 · Fact-checked

Option Greeks measure how an option's price responds to changes in the underlying price (delta, gamma), time (theta), volatility (vega) and interest rates (rho). To delta hedge, hold an underlying position whose delta offsets the option position's delta. Then rebalance as delta changes, and use gamma to judge how often.

Understand Option Greeks and Delta Hedging

An option price depends on several inputs. The Greeks tell you how much the price changes when one input moves and the others stay fixed. You need them to understand risk and to build hedges.

Delta is the change in option price for a 1-unit change in the underlying price. A call delta lies between 0 and 1. A put delta lies between -1 and 0. Deep in-the-money calls approach 1, deep out-of-the-money calls approach 0. For a European option on an asset with no dividends, put delta = call delta - 1.

Gamma is the change in delta for a 1-unit change in the underlying price. Delta is a straight-line estimate, but the option price is curved. Gamma measures that curve. Gamma is positive for long calls and long puts, and highest for at-the-money options close to expiry. A call and a put with the same strike and expiry have the same gamma. The stock itself has zero gamma.

Theta is the change in option value as time passes. For most long options it is negative: the option loses time value. Vega is the change in option value for a 1-percentage-point change in volatility. It is positive for long calls and puts, and largest for at-the-money options with longer time to expiry. Rho is the change in value for a change in the risk-free rate. It is usually positive for calls and negative for puts.

Delta hedging makes a portfolio insensitive to small moves in the underlying. If you are short calls, you buy shares so that gains on the shares offset losses on the calls. The hedge works only for small moves, because delta changes as the price moves. That change is gamma. A delta-neutral position with negative gamma loses money on large moves in either direction. One with positive gamma gains. To neutralise gamma, you add another option, because the underlying cannot change gamma. Then you re-fix delta using the underlying.

Key formulas to remember

Delta
Delta = Δ option price ÷ Δ underlying price
Call: 0 to 1. Put: -1 to 0. For a European option on a non-dividend asset, put delta = call delta - 1.
Delta approximation of price change
ΔOption ≈ Delta × ΔS
Accurate only for small moves in the underlying.
Delta-gamma approximation
ΔOption ≈ Delta × ΔS + ½ × Gamma × (ΔS)²
The gamma term is always positive for a long option, whichever way S moves.
New delta after a move
New delta ≈ Old delta + Gamma × ΔS
Use this to find how many shares to trade when you rebalance.
Delta hedge of an option position
Shares to hold = -(Number of options × Delta × Contract size)
Long calls need short shares. Short calls need long shares. Long puts need long shares.
Hedging a stock position with options
Options to trade = -(Number of shares ÷ Option delta ÷ Contract size)
To hedge long shares with calls, sell calls. Divide by delta, not multiply.
Gamma hedge
Hedge options = -(Portfolio gamma ÷ Gamma of hedge option)
Positive result means buy. After this, portfolio delta changes, so reset delta using the underlying.
Greek signs for long positions
Theta usually < 0; Vega > 0; Call rho > 0; Put rho < 0
Deep in-the-money European puts can have positive theta. Treat the signs as typical, not universal.

How to solve Option Greeks and Delta Hedging questions

Use this order for any Greeks or delta-hedging question in an item set.

  1. 1Read the vignette and list the position: long or short, calls or puts, number of options, contract size, and any shares held.
  2. 2Pull each Greek from the exhibit. Check whether it is per option or for the whole position, and whether the underlying move is 1 unit or another size.
  3. 3Fix signs first. Short positions flip the sign of every Greek. A short call has negative delta and negative gamma.
  4. 4Compute position delta: options × contract size × delta, with the right sign. Add the delta of any shares held (+1 per long share).
  5. 5Set the hedge: trade the underlying so that total delta is zero. Shares needed = -position delta.
  6. 6For a price-change question, use delta × ΔS, then add ½ × gamma × (ΔS)² if gamma is given and the move is not small.
  7. 7For a gamma hedge, trade a second option to bring total gamma to zero, then recompute delta and fix it with the underlying.
  8. 8Check the answer: does the direction make sense, and does the hedged position have the exposure the question describes (for example, a loss on large moves when gamma is negative)?

Quickest way: Sign-and-scale shortcut

When to use it: Use it when the question gives delta and asks for shares to buy or sell, or asks about direction of profit or loss.

  1. Multiply options × contract size × delta to get position delta in shares.
  2. Give it a minus sign if you are short the option; keep plus if long.
  3. The hedge is the opposite sign of that number.
  4. For gamma: long option means gains from large moves; short option means losses. Delta-neutral does not remove this.
  5. If the next price is given, new delta = old delta + gamma × ΔS, then trade the difference.

Common mistakes in Option Greeks and Delta Hedging

  • Multiplying instead of dividing when hedging shares with options.

    Students mix up the two hedges: options hedged with shares, and shares hedged with options.

    Fix: Ask which side you hold. Hedging options with shares: multiply by delta. Hedging shares with options: divide by delta.

  • Using a positive delta for a put.

    Students remember call deltas and forget that a put loses value when the underlying rises.

    Fix: Put delta is negative. Check with put delta = call delta - 1 for a non-dividend European option.

  • Thinking gamma can be hedged with the underlying.

    Delta hedging is taught first, so students assume shares fix every risk.

    Fix: The underlying has delta 1 and gamma 0. Only another option changes gamma.

  • Ignoring the sign of gamma on short positions.

    Greeks in exhibits are often given for one long option.

    Fix: Flip signs for short positions. A delta-neutral short-gamma position loses on big moves either way.

  • Confusing delta and gamma.

    Both relate to the underlying price.

    Fix: Delta is the change in option price. Gamma is the change in delta. Gamma is the curvature.

  • Saying a delta hedge stays valid after the price moves.

    Students treat the hedge as set once.

    Fix: Delta changes with price, time and volatility. A hedge needs rebalancing, more often when gamma is large.

Worked examples

Example 1

A dealer is short 10,000 call options on a share, each on one share. The share trades at 50. Call delta is 0.60 and gamma is 0.04 per option. (1) How many shares should the dealer hold to be delta neutral? (2) If the share rises by 1, what is the approximate gain or loss on the combined position using the delta-gamma approximation? (3) After the rise, how many shares should the dealer hold?

Show the solution
  1. Position delta of the short calls = -10,000 × 0.60 = -6,000 shares.
  2. (1) To offset, buy 6,000 shares.
  3. (2) Change in each call ≈ 0.60 × 1 + ½ × 0.04 × 1² = 0.62. Loss on 10,000 short calls = 10,000 × 0.62 = 6,200.
  4. Gain on 6,000 shares = 6,000 × 1 = 6,000.
  5. Net = 6,000 - 6,200 = -200.
  6. (3) New delta ≈ 0.60 + 0.04 × 1 = 0.64. Required shares = 10,000 × 0.64 = 6,400. The dealer already holds 6,000, so buys 400 more.

Answer: (1) Buy 6,000 shares. (2) Net loss of about 200, because the short call position has negative gamma. (3) Hold 6,400 shares, so buy 400 more.

Example 2

A portfolio is short 1,000 options of type A, each with delta 0.50 and gamma 0.05. A second option, B, has delta 0.40 and gamma 0.02. (1) How many B options make the portfolio gamma neutral? (2) What is the portfolio delta after that trade, and what trade in the underlying makes it delta neutral? (3) Which exposure remains?

Show the solution
  1. Portfolio gamma = -1,000 × 0.05 = -50. Portfolio delta = -1,000 × 0.50 = -500.
  2. (1) B options needed = -(-50) ÷ 0.02 = 2,500. Buy 2,500 B options. Gamma = -50 + 2,500 × 0.02 = 0.
  3. (2) Delta after the trade = -500 + 2,500 × 0.40 = -500 + 1,000 = +500.
  4. To offset +500, sell 500 units of the underlying (underlying delta is 1 per unit). Gamma stays 0 because the underlying has zero gamma.
  5. (3) The position is delta and gamma neutral, but it is still exposed to time decay (theta), volatility (vega) and rates (rho).

Answer: (1) Buy 2,500 B options. (2) Delta is +500; sell 500 units of the underlying. (3) Theta, vega and rho exposures remain.

Exam tips

  • Write the sign of each Greek next to the position before computing anything. Most lost marks come from sign errors on short positions.
  • Check the contract size and whether a Greek is quoted per option or per position.
  • Questions about a hedged position's profit usually test gamma: delta-neutral, short gamma means loss on large moves.
  • If the item set asks how often to rebalance, higher gamma means delta changes faster, so rebalance more often.
  • Eliminate options by sign and direction first. Often two of the three answers have the wrong direction of trade.

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 an option's price for a 1-unit change in the underlying. Gamma is the change in delta for that same move. Delta is the slope and gamma is the curvature of the price curve.

How do you delta hedge an option position?

Compute the position delta as options × contract size × option delta, with the correct sign. Then trade the underlying in the opposite amount so the total delta is zero. Rebalance as delta changes.

Why does a delta hedge fail for large price moves?

Delta is only a straight-line estimate and changes as the price moves. The change is gamma. A delta-neutral position with negative gamma loses on large moves, and one with positive gamma gains.

Can I hedge gamma using shares?

No. The underlying has a constant delta of 1 and zero gamma. You need another option to change the position's gamma, then use the underlying to reset delta.