A bank sells call options on a million shares. If the share price rises, the bank loses money on every option it wrote. It could wait and hope. Or it can buy just enough of the shares that any loss on the options is matched by a gain on the shares. How many shares is “just enough”? The option’s delta tells you.
Delta and its relatives, the Greeks, are how option traders measure and manage the different ways an option’s value moves.
This topic covers delta and what it means, delta hedges for written call and put options, currency delta hedges, using delta to size an options hedge for a company, and the Greeks (delta, gamma, vega, rho and theta). The previous topic covered real options and applications of Black-Scholes. This is the last topic in the course. Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on delta hedging.
In 30 seconds
Delta = change in option value ÷ change in the underlying price. It lies between 0 and +1 for a call and 0 and −1 for a put; deep in the money is near 1 (or −1), at the money about 0.5 (or −0.5), deep out of the money near 0. From Black-Scholes, delta of a call = N(d1) and the put delta uses N(−d1). A delta hedge makes a position delta neutral.
Writer of calls: hold quantity × delta of the underlying. Writer of puts: be short that amount (borrow shares, or sell currency forward). A company hedging with bought options: amount to cover ÷ delta gives the quantity of options needed for a complete hedge, which is costly. The Greeks: delta (value vs price), gamma (delta vs price), vega (value vs volatility), rho (value vs interest rate), theta (value vs time).
Delta changes as the price moves, so the hedge must be rebalanced.
Key terms in plain English
| Term | What it means |
|---|---|
| Delta | How much an option’s value moves for a one-unit move in the underlying |
| Delta neutral | A position whose total delta is zero, so small price moves cause no gain or loss |
| Gamma | How much delta changes when the underlying price changes |
| Vega | The change in option value for a 1% change in volatility |
| Rho | The change in option value for a change in interest rates |
| Theta | The change in option value as time to expiry passes |
Syllabus check
This topic is in syllabus area E (treasury and advanced risk management). It covers:
- explaining and constructing delta hedges;
- explaining the other option sensitivities (the Greeks).
CAMA 2020 and risk management
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Risk should be managed prudently for the benefit of the company and its members |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors of financial institutions should set and monitor limits for option risks |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
Delta defined
Options are a zero-sum game: to the extent the holder is protected, the writer is exposed. Banks that write many options can hedge their exposure with a delta hedge. The delta of an option is the change in its price relative to the change in the price of the underlying: if the share rises by $1, a call rises (and a put falls) by no more than $1. So delta is in the range −1 to +1.
- A deeply in-the-money call has a delta close to +1; a deeply in-the-money put close to −1.
- A deeply out-of-the-money option has a delta close to 0.
- An at-the-money option has a delta near +0.5 (call) or −0.5 (put).
Deltas are often shown as percentages of the shares in the contract: a call with a delta of 0.25 behaves like 25 of 100 shares for small price moves. Delta can also be read as the approximate probability that the option will expire in the money: about 100% deep in the money, 50% at the money, near 0 deep out of the money.
Delta hedge for written calls
A bank that writes calls loses if the share price rises. It hedges by holding the underlying. A portfolio is delta neutral when its delta is zero: the loss on the options is matched by the gain on the shares. Number of shares to hold = quantity of underlying in the options × delta. The delta of a call is N(d1) from the Black-Scholes model.
ICAN’s example: call options on 1,000,000 shares with a delta of 0.45 are hedged by holding 1,000,000 × 0.45 = 450,000 shares. A Nigerian bank writing currency calls on €200m with delta (N(d1)) of 0.5326 holds €200m × 0.5326 = €106.52m.
Delta hedge for written puts
A bank that writes puts loses if the price falls, so it must be short the underlying: borrow the shares, or sell the currency forward for the same date as the option expiry. ICAN’s example: put options on 1,000,000 shares with a delta of −0.27 are hedged by borrowing 1,000,000 × 0.27 = 270,000 shares, with the return date matched to expiry. The put delta is N(−d1) (as ICAN states it, taken as a magnitude).
A bank writing currency puts on €50m with delta 0.4674 sells forward €50m × 0.4674 = €23.37m.
Note that N(−d1) = 1 − N(d1): 1 − 0.5326 = 0.4674, the same delta magnitude as in the example above.
Delta and a company’s own options hedge
A company that buys options as a hedge can use delta to find how many options give a complete hedge against small price moves. Options needed (in underlying units) = amount to hedge ÷ delta. ICAN’s examples: a company needing €100m in three months, with at-the-money calls of delta 0.5326 and contracts of €100,000, needs €100m ÷ 0.5326 = €187.76m of calls, which is 1,878 contracts.
A company receiving €20m, with at-the-money puts of delta 0.4764, needs €20m ÷ 0.4764 = €41.98m, which is 420 contracts. This eliminates the risk entirely but is very expensive because of the premiums; in practice the company would buy about €100m ÷ €100,000 = 1,000 call contracts (or 200 put contracts for €20m).
The Greeks
Option dealers monitor ratios named after Greek letters (vega is not actually a Greek letter).
| Greek | Measures the change in | Due to a change in | Notes |
|---|---|---|---|
| Delta (Δ) | Option price | Underlying price | Rises with the share price for a call; also the chance of ending in the money |
| Gamma (Γ) | Delta | Underlying price | Low when the outcome is certain; high when uncertain (near the money) |
| Vega (v) | Option price | Volatility (per 1%) | Important in volatile markets; higher volatility raises vega |
| Rho (ρ) | Option price | Interest rate | Calls rise and puts fall when rates rise; the least used |
| Theta (Θ) | Option price | Time to expiry | Value lost per day; a theta of −0.012 means 1.2 cents a day; higher near expiry |
Because gamma is the rate at which delta changes, a delta hedge must be rebalanced as the price moves: if the delta of a call rises from 0.45 to 0.50, the writer must hold more shares.
The rules in more depth
Why delta is between −1 and +1
An option’s value cannot move by more than the underlying, nor in the opposite direction for a long call.
Why the hedge needs rebalancing
Delta changes with the price (gamma), with time (theta) and with volatility.
Why a complete hedge with bought options is costly
You must buy more option contracts than the amount to hedge, and each carries a premium.
Why a put writer shorts the underlying
A fall in the price creates the loss on the written put, so a short position gains.
Effect on analysis
Compute delta (N(d1) or N(−d1)), multiply by the quantity, state long or short and say that the hedge is for small moves and needs updating.
Worked examples
Example 1: a delta move. A call has a delta of 0.5. The share rises by ₦2.
- Change in option value ≈ delta × change in share price.
- = 0.5 × 2.
- = ₦1.00.
Example 2: a call writer’s hedge. A bank has written calls on 500,000 shares. The delta is 0.6.
- Shares to hold = 500,000 × 0.6.
- = 300,000 shares.
- This is delta neutral for small price moves.
Example 3: rebalancing. The delta rises from 0.60 to 0.70.
- New shares needed.
500,000 × 0.70 = 350,000. - Already held = 300,000.
- Buy 50,000 more shares.
Example 4: put delta. N(d1) = 0.5326.
- Put delta magnitude.
N(−d1) = 1 − 0.5326. - = 0.4674.
- A put writer on €50m sells forward €50m × 0.4674 = €23.37m.
How to score full marks
- State whether the option is a call or a put and who writes it.
- Calculate delta from N(d1) or N(−d1) if needed.
- Multiply by the quantity of underlying.
- Say long or short, and the dates.
- Mention rebalancing and gamma.
- Compare with buying options (cost).
Common mistakes in the exam
- Hedging a written call by selling the underlying.
- Using delta as the number of options when it is the number of shares.
- Forgetting the opposite sign for puts.
- Not stating that the hedge is for small changes.
- Dividing by delta when you should multiply.
- Mixing up gamma and vega.
Quick revision
- Delta: call 0 to +1, put 0 to −1; call delta N(d1).
- Call writer holds quantity × delta of the underlying; put writer is short.
- Options to buy for a complete hedge: amount ÷ delta.
- Gamma: delta vs price; vega: value vs volatility; rho: value vs rate; theta: value vs time.
- Rebalance as delta changes.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. Delta of an option measures the change in:
- Option price per unit change in the underlying
- Volatility
- Time
- Interest rate
Show the answer
A. The first of the Greeks.
Question 2. A deep in-the-money call has a delta close to:
- +1
- 0
- −1
- +0.5
Show the answer
A. It moves almost like the share.
Question 3. An at-the-money call has a delta of about:
- 0.5
- 0
- 1
- −0.5
Show the answer
A. About a 50% chance of ending in the money.
Question 4. A writer of calls should hedge by:
- Holding the underlying
- Selling the underlying
- Buying puts only
- Doing nothing
Show the answer
A. Quantity × delta.
Question 5. A writer of puts should hedge by being:
- Short the underlying
- Long the underlying
- Long volatility
- Neutral
Show the answer
A. Borrow the shares or sell forward.
Question 6. Calls on 1,000,000 shares with delta 0.45 need a hedge of:
- 450,000 shares
- 1,000,000 shares
- 550,000 shares
- 45,000 shares
Show the answer
A. Quantity × delta.
Question 7. Gamma measures the change in:
- Delta for a change in the underlying
- Price for time
- Price for volatility
- Price for rates
Show the answer
A. It shows how fast delta moves.
Question 8. Vega measures sensitivity to:
- Volatility
- Time
- Interest rates
- Share price
Show the answer
A. Per 1% change.
Question 9. Theta measures sensitivity to:
- Time to expiry
- Volatility
- Strike price
- Dividends
Show the answer
A. Value lost each day.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) Define delta and state its range for calls and puts. (4 marks) (b) Explain the meaning of a delta-neutral position. (3 marks) (c) Explain how delta indicates the probability of expiring in the money. (3 marks)
Question 12 (20 marks). A bank has written call options on 2,000,000 shares with a delta of 0.45 and put options on 1,000,000 shares with a delta of 0.27 (magnitude). (a) State and calculate the delta hedge for the calls. (4 marks) (b) State and calculate the delta hedge for the puts. (4 marks) (c) The call delta rises to 0.50. State the adjustment needed.
(3 marks) (d) A company needs €100m in three months and uses at-the-money calls with delta 0.5326 and contracts of €100,000. Calculate the contracts for a complete hedge and comment on cost. (5 marks) (e) Explain gamma and theta. (4 marks)
Question 13 (10 marks). (a) A Nigerian bank writes currency calls on €200m, delta 0.5326. Calculate the hedge. (3 marks) (b) A bank writes currency puts on €50m, delta 0.4674. State and calculate the hedge. (3 marks) (c) State the Greek that measures sensitivity to volatility and to interest rates. (4 marks)
Model answers
Show the model answer to Question 11
(a) Delta (4 marks). Delta is the change in the option price divided by the change in the underlying price. (2) For calls it lies between 0 and +1; for puts, between 0 and −1. (2)
(b) Delta neutral (3 marks). A position whose total delta is zero. (1.5) Small price moves cause no gain or loss because the loss on the options is offset by a gain on the hedge. (1.5)
(c) Probability (3 marks). A deep in-the-money option has a delta near 1 (or −1) and is almost certain to expire in the money. (1.5) At the money delta is about 0.5, a 50% chance; deep out of the money near 0. (1.5)
Show the model answer to Question 12
(a) Calls (4 marks).
- A writer of calls loses if the price rises, so it holds the shares. (1)
- 2,000,000 × 0.45 = 900,000 shares. (3)
(b) Puts (4 marks).
- A writer of puts loses if the price falls, so it is short: borrow shares, returning them at expiry. (1)
- 1,000,000 × 0.27 = 270,000 shares. (3)
(c) Adjustment (3 marks).
- New holding.
2,000,000 × 0.50 = 1,000,000. (1.5) - Buy a further 100,000 shares. (1.5)
(d) Complete hedge (5 marks).
- €100m ÷ 0.5326 = €187.76m. (2)
- Contracts = 187.76m ÷ 100,000 = 1,878. (1.5)
- This is costly because of the premiums; in practice 1,000 contracts would be bought. (1.5)
(e) Greeks (4 marks). Gamma is the change in delta for a change in the underlying price. It is high when it is uncertain whether the option will finish in the money. (2) Theta is the change in option value as time passes, a loss per day that is larger nearer expiry. (2)
Show the model answer to Question 13
(a) Calls (3 marks). The bank holds euros: €200m × 0.5326 = €106.52m. (3)
(b) Puts (3 marks). The bank is short euros: it sells forward €50m × 0.4674 = €23.37m, with settlement on the option expiry date. (3)
(c) Greeks (4 marks). Vega measures sensitivity to volatility, per 1%. (2) Rho measures sensitivity to interest rates. (2)
Pathfinder on Delta, delta hedges for written calls and puts, and the option Greeks
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on delta hedging. Work the call, put and currency examples and the rebalancing example.
Revise first: Delta hedge for written calls
Questions ask what each Greek measures. See the table in the section on the Greeks.
Revise first: The Greeks
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
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