Economic Modelling · Black-Scholes derivative-pricing model

Option Greeks and Delta Hedging Explained

Q: What is the difference between delta hedging and gamma hedging?

Delta hedging holds shares so that portfolio delta is zero, which protects against small share price moves. Gamma hedging uses other options to make portfolio gamma zero, which protects against larger moves and reduces rebalancing. Shares cannot change gamma, so options are required.

Q: How do I calculate the delta of a European call option?

Compute d1 = [ln(S ÷ K) + (r + σ²/2)T] ÷ (σ√T). The call delta is Φ(d1), the standard normal distribution function at d1. For a put, delta is Φ(d1) − 1.

Q: Why is theta usually negative for a long option?

A long option loses time value as expiry gets closer, because there is less time for the share price to move favourably. Theta is the rate of that loss. Some deep in-the-money European puts can have positive theta, so the rule is not universal.

Q: Which Greeks are the same for calls and puts?

Gamma and vega are the same for a European call and put with the same strike, expiry and underlying. Delta, theta and rho differ. The difference in delta is exactly 1 for a non-dividend-paying share, from put-call parity.

Q: Why does a delta-hedged portfolio still make losses?

The hedge is exact only for tiny moves and continuous rebalancing. Large share price moves, changes in volatility, passage of time and transaction costs all create profit or loss. Short-gamma positions lose when the share price moves sharply.

Updated 11 October 2026 · Fact-checked

The Greeks measure how an option's price changes when one input moves: delta (share price), gamma (delta itself), vega (volatility), theta (time) and rho (interest rate). To delta hedge, hold −Δ shares per option written, so small share price moves leave the portfolio value roughly unchanged. Rebalance as delta changes.

Understand The Greeks and Hedging

An option price depends on the share price, volatility, time to expiry and the interest rate. The Greeks are the partial derivatives of the option price with respect to each of these. They tell you how exposed a position is to each input.

Delta (Δ) is the change in option value for a small change in share price. A European call has delta between 0 and 1. A put has delta between −1 and 0. Gamma (Γ) is the rate of change of delta with the share price. It is the same for a call and a put with the same strike and expiry, and it is always positive for a long option.

Vega is the change in value for a change in volatility. Theta (Θ) is the change in value as time passes. Rho is the change in value for a change in the risk-free rate. A long option has positive vega. Its theta is usually negative, meaning it loses time value. A call has positive rho and a put has negative rho.

Delta hedging removes the first-order exposure to the share price. If you write one call with delta Δ, you hold Δ shares. A small rise in the share price gains on the shares what it loses on the call. The hedge only works for small moves, because delta changes as the share price changes. That is the role of gamma. A portfolio with large gamma needs frequent rebalancing. Gamma hedging adds other options so that portfolio gamma is close to zero. Vega hedging works the same way for volatility.

The Greeks are linked through the Black-Scholes PDE. For a delta-hedged position, theta and gamma largely offset each other. A long-gamma position pays for its gains through time decay. A short-gamma position, like a writer of options, earns theta but loses when the share price moves a lot. Assumptions behind all this: constant volatility and rate, continuous trading, no transaction costs. In practice hedging is discrete and costly.

Key rules to remember

d1 and d2: d1 = [ln(S ÷ K) + (r + σ²/2)T] ÷ (σ√T); d2 = d1 − σ√T; Non-dividend-paying share, constant r and σ, continuously compounded r. S is the share price, K the strike, T the time to expiry in years.
Black-Scholes prices: Call = S·Φ(d1) − K·e^(−rT)·Φ(d2); Put = K·e^(−rT)·Φ(−d2) − S·Φ(−d1); Φ is the standard normal distribution function. Put-call parity: C − P = S − K·e^(−rT).
Delta: Δ(call) = Φ(d1); Δ(put) = Φ(d1) − 1; Call delta is in (0, 1). Put delta is in (−1, 0). Put delta equals call delta minus 1.
Gamma: Γ = φ(d1) ÷ (S·σ·√T); φ is the standard normal density, φ(x) = e^(−x²/2) ÷ √(2π). Same for a call and a put.
Vega: Vega = S·φ(d1)·√T; Same for a call and a put. This is per unit change in σ (so divide by 100 for a one percentage point change).
Theta: Θ(call) = −S·φ(d1)·σ ÷ (2√T) − r·K·e^(−rT)·Φ(d2); Θ(put) = −S·φ(d1)·σ ÷ (2√T) + r·K·e^(−rT)·Φ(−d2); Per year. Divide by 365 for an approximate per-day figure.
Rho: ρ(call) = K·T·e^(−rT)·Φ(d2); ρ(put) = −K·T·e^(−rT)·Φ(−d2); Per unit change in r.
Black-Scholes PDE in Greeks: Θ + r·S·Δ + ½·σ²·S²·Γ = r·V; Holds for the value V of any derivative on a non-dividend-paying share. It links theta and gamma.
Delta-gamma approximation: ΔV ≈ Δ·δS + ½·Γ·(δS)²; Use for the change in option value for a share price change δS. Add Vega·δσ and Θ·δt if those change.
Delta hedge: Shares held = −(number of options held × Δ); For a written call, hold +Δ shares per option. Portfolio delta is the sum of position deltas. Gamma-neutral hedging: choose option quantities so that total Γ = 0.

How to solve The Greeks and Hedging questions

Use this order for any Greeks or hedging question. State assumptions first: Black-Scholes, constant σ and r, no dividends unless told.

1List S, K, r, σ and T. Check that r and σ are annual and continuously compounded, and T is in years.
2Compute d1, then d2 = d1 − σ√T. Keep four decimal places.
3Read Φ(d1) and Φ(d2) from the tables. Use Φ(−x) = 1 − Φ(x) for negative arguments.
4Write the required Greek using the correct formula for call or put. Compute φ(d1) if gamma, vega or theta is needed.
5For a portfolio, multiply each Greek by the signed quantity (long positive, short negative) and add them up.
6For a delta hedge, set the share holding to −(portfolio delta). For a gamma hedge, first choose the option quantity that makes gamma zero, then re-hedge delta with shares, since shares have zero gamma.
7Estimate the effect of a move with ΔV ≈ Δ·δS + ½Γ(δS)². Interpret the sign and say what rebalancing is needed.
8State the result with units (rupees, shares, per unit σ) and mention limits: discrete hedging, costs and constant volatility.

Quickest way: Shortcut using bounds, symmetry and signs

When to use it: Use for MCQs and for checking written answers when time is short.

Get put delta from call delta: Δ(put) = Δ(call) − 1. Do not recompute.
Remember that gamma and vega are identical for a call and a put. Compute once.
Check signs. Long option: gamma > 0, vega > 0. Call: delta > 0, rho > 0. Put: delta < 0, rho < 0. Theta is usually negative for a long option.
Sanity-check delta. At the money (K near S), call delta is roughly 0.5 to 0.65 depending on r, σ and T. Deep in the money tends to 1, deep out of the money to 0.
For hedge size, multiply: shares = options × Δ. Do the rounding only at the end.
For short-time estimates, use ΔV ≈ Δ·δS + ½Γ(δS)² and drop the gamma term only if δS is tiny.

Common mistakes in The Greeks and Hedging

Using the call delta formula Φ(d1) for a put.
The call formula is learned first and the put version is forgotten under time pressure.
Fix: Write put delta as Φ(d1) − 1, which is negative. Check that your put delta is between −1 and 0.
Getting the hedge direction wrong, such as buying shares when the position is a long call.
Students mix up hedging the option with replicating the option.
Fix: Work out portfolio delta first, then hold the opposite amount in shares. Long call has positive delta, so short Δ shares. Writer of a call buys Δ shares.
Saying a delta-hedged portfolio is risk-free.
Delta neutrality removes only first-order share price risk.
Fix: Mention gamma, vega, theta and discrete rebalancing. The hedge is exact only for infinitesimal moves with continuous trading.
Treating gamma hedging as the same as delta hedging, or hedging gamma with shares.
Both are described as neutralising a Greek.
Fix: Delta hedging uses the underlying share. Gamma hedging needs other options because a share has zero gamma. After adding options, re-adjust shares to restore delta neutrality.
Ignoring units: using T in months, or quoting vega per unit when the question asks per 1% change.
Formulas are applied mechanically without checking inputs.
Fix: Convert T to years. Divide vega by 100 for a one percentage point change in σ, and say which you are giving.
Dropping the sign for short positions when adding portfolio Greeks.
Quantities are entered as positive numbers.
Fix: Use negative quantities for written options. A short call has negative delta, negative gamma and negative vega.

Worked examples

Example 1

A non-dividend-paying share has price ₹100. A European call and a European put each have strike ₹100 and expire in 1 year. The risk-free rate is 5% per year (continuously compounded) and volatility is 20% per year. Φ(0.35) = 0.6368 and Φ(0.15) = 0.5596. Calculate (a) the call delta, (b) the put delta, (c) the gamma and (d) the vega, and (e) the call price. Use φ(0.35) = 0.3752 and e^(−0.05) = 0.9512.

Show the solution

d1 = [ln(100 ÷ 100) + (0.05 + 0.02) × 1] ÷ (0.2 × 1) = 0.07 ÷ 0.2 = 0.35.
d2 = 0.35 − 0.2 = 0.15.
(a) Call delta = Φ(d1) = 0.6368.
(b) Put delta = Φ(d1) − 1 = 0.6368 − 1 = −0.3632.
(c) Gamma = φ(d1) ÷ (S·σ·√T) = 0.3752 ÷ (100 × 0.2 × 1) = 0.3752 ÷ 20 = 0.01876.
(d) Vega = S·φ(d1)·√T = 100 × 0.3752 × 1 = 37.52 per unit of σ, which is about 0.375 per one percentage point.
(e) Call = 100 × 0.6368 − 100 × 0.9512 × 0.5596 = 63.68 − 53.23 = 10.45.

Answer: Call delta 0.6368, put delta −0.3632, gamma 0.01876 (same for both), vega 37.52 per unit of σ, and call price about ₹10.45.

Example 2

Using the data in the previous example, a bank has written 10,000 of these calls (delta 0.6368, gamma 0.01876 per call). (a) How many shares should it hold to be delta neutral? (b) The share price then rises by ₹1 immediately. Use the delta-gamma approximation to estimate the change in the value of the hedged portfolio. (c) How many shares must it hold afterwards to be delta neutral again?

Show the solution

(a) Portfolio is short 10,000 calls. Option delta = −10,000 × 0.6368 = −6,368. Hold +6,368 shares.
(b) Change in value of one call ≈ Δ·δS + ½Γ(δS)² = 0.6368 × 1 + 0.5 × 0.01876 × 1 = 0.6368 + 0.00938 = 0.64618.
Loss on 10,000 written calls ≈ 10,000 × 0.64618 = ₹6,461.80.
Gain on 6,368 shares = 6,368 × ₹1 = ₹6,368.
Net change ≈ 6,368 − 6,461.80 = −₹93.80, about −₹94. It equals −½ × (total gamma) × (δS)², which is the cost of being short gamma.
(c) New delta per call ≈ 0.6368 + 0.01876 × 1 = 0.6556. Required shares ≈ 10,000 × 0.6556 = 6,556.
Extra shares to buy ≈ 6,556 − 6,368 = 188 (using the unrounded gamma, 10,000 × 0.01876 = 187.6).

Answer: Hold 6,368 shares. The hedged portfolio loses about ₹94 for a ₹1 rise because it is short gamma. Buy about 188 more shares to restore delta neutrality.

Exam tips

Know both call and put forms of delta, theta and rho, and remember that gamma and vega are the same for both.
In written questions, always say what delta hedging does not remove: gamma, vega, theta and rate risk, plus the effect of discrete rebalancing and transaction costs.
For a portfolio question, build a table of positions with signed quantities and add each Greek column. This avoids sign errors and earns method marks.
Explain gamma in words: it shows how fast the hedge goes out of date. Short gamma means losses when the share price moves sharply either way.
In computer-based questions in R or Excel, code d1, d2, then each Greek as a separate cell or line. Check with put-call parity and a finite-difference bump of S to test your delta.

Practice questions from Black-Scholes derivative-pricing model

The Greeks and Hedging in other exams

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

The Greeks and Hedging: frequently asked questions

What is the difference between delta hedging and gamma hedging?

Delta hedging holds shares so that portfolio delta is zero, which protects against small share price moves. Gamma hedging uses other options to make portfolio gamma zero, which protects against larger moves and reduces rebalancing. Shares cannot change gamma, so options are required.

How do I calculate the delta of a European call option?

Compute d1 = [ln(S ÷ K) + (r + σ²/2)T] ÷ (σ√T). The call delta is Φ(d1), the standard normal distribution function at d1. For a put, delta is Φ(d1) − 1.

Why is theta usually negative for a long option?

A long option loses time value as expiry gets closer, because there is less time for the share price to move favourably. Theta is the rate of that loss. Some deep in-the-money European puts can have positive theta, so the rule is not universal.

Which Greeks are the same for calls and puts?

Gamma and vega are the same for a European call and put with the same strike, expiry and underlying. Delta, theta and rho differ. The difference in delta is exactly 1 for a non-dividend-paying share, from put-call parity.

Why does a delta-hedged portfolio still make losses?

The hedge is exact only for tiny moves and continuous rebalancing. Large share price moves, changes in volatility, passage of time and transaction costs all create profit or loss. Short-gamma positions lose when the share price moves sharply.