Strategic Financial Management · Options

Option Greeks: Delta, Gamma, Theta, Vega and Rho

Updated 11 October 2026 · Fact-checked

Option Greeks measure how an option's price changes when one input changes. Delta is the change for a ₹1 move in the underlying, gamma is the change in delta, theta is time decay, vega is volatility sensitivity and rho is interest rate sensitivity. Multiply the Greek by the change in the input to estimate the price change.

Understand Option Greeks and Sensitivity

An option's price depends on the underlying price, strike price, time to expiry, volatility, interest rate and expected dividends. The Greeks isolate the effect of each input, holding the others constant.

Delta is the change in option price for a ₹1 change in the underlying price. A call has delta between 0 and 1. A put has delta between -1 and 0. 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 of about 0.5.

Gamma is the change in delta for a ₹1 change in the underlying. It is highest for at-the-money options close to expiry. It is the same for a call and a put with the same strike and expiry. A high gamma means your hedge goes out of date quickly.

Theta is the change in option value as one day passes. For a long option it is usually negative, because time value decays. Vega is the change in option value for a 1 percentage point change in volatility. It is positive for long calls and puts. Rho is the change in option value for a 1 percentage point change in the risk-free rate. It is usually positive for calls and negative for puts.

Delta hedging uses delta to remove price risk. If you are short one call with delta 0.6, you buy 0.6 units of the underlying per option. The position then has near-zero delta for small price moves. As price changes, delta changes, so you must rebalance. Gamma tells you how fast.

Key rules to remember

Delta: Δ = change in option price ÷ change in underlying price; Call: 0 to 1. Put: -1 to 0. In Black-Scholes, call delta = N(d1) and put delta = N(d1) - 1.
Put-call delta link: Δ(put) = Δ(call) - 1; Applies to European options on a non-dividend-paying asset with the same strike and expiry.
Gamma: Γ = change in delta ÷ change in underlying price; Same for call and put with the same strike and expiry. Always positive for a long option.
Price change estimate: ΔC ≈ Δ × ΔS + ½ × Γ × (ΔS)²; Delta alone is a first-order estimate. Gamma corrects it for larger moves.
Delta-neutral hedge: Units of underlying to hold = - (option delta × number of options); Short call: buy the underlying. Long put: buy the underlying, as put delta is negative.
Theta, vega, rho: Θ = change in value per day; ν = change per 1% change in volatility; ρ = change per 1% change in rate; Check the unit given in the question: per day or per year, per 1% or per 1 unit.
Direction of effects on a call / put: Call: higher S ↑, higher σ ↑, longer time usually ↑, higher r ↑, higher strike ↓; Put: higher S ↓, higher σ ↑, longer time usually ↑, higher r ↓, higher strike ↑. Longer time to expiry usually raises the value of both calls and puts. For European options it can lower the value of deep in-the-money puts, or of calls when dividends are large.

How to solve Option Greeks and Sensitivity questions

Use this method for any Greek-based question, whether it asks for a price estimate, a hedge or an interpretation.

1Identify the option type (call or put), position (long or short) and number of options and lot size.
2Write down the Greeks given and their units, such as per ₹1, per day or per 1%.
3Identify which input changes and by how much.
4Multiply each Greek by the matching change. Add gamma's second-order term only if gamma is given and the move is large.
5Apply the sign for your position. A short position reverses the sign of the Greek.
6For a hedge, compute net delta of the position and take the opposite position in the underlying.
7State the new option value or hedge size and add one line of interpretation.

Quickest way: Greek times change, then flip for short

When to use it: Use it for 2-mark MCQs and for the first part of a numerical.

Pull out the Greek and the change in the input.
Multiply them to get the change in value per option.
Multiply by lot size and number of options.
Reverse the sign if the position is short.
For hedging, hold underlying equal to net delta times quantity in the opposite direction.

Common mistakes in Option Greeks and Sensitivity

Treating delta as a probability of finishing in the money without qualification.
Call delta N(d1) looks like a probability, and it is often described loosely that way.
Fix: Treat delta as the price sensitivity first. Use it as a rough guide to the chance of expiring in the money only if the question says so.
Using the wrong hedge direction for puts.
Put delta is negative, and students ignore the sign.
Fix: A long put has negative delta, so you buy the underlying to neutralise it. A short put has positive delta, so you sell the underlying.
Forgetting to multiply by the lot size.
The Greek is per unit of the underlying, but trades are in lots.
Fix: Multiply the per-unit Greek by the number of units in the position before stating the rupee impact or hedge.
Saying theta is always negative.
Long options lose time value, so this becomes a blanket rule.
Fix: Say theta is usually negative for a long option. It is positive for the seller. For a long option, theta can occasionally be positive in special cases, such as a deep in-the-money European put, or a deep in-the-money European call on an asset with high dividends.
Ignoring the units of vega and rho.
Questions give vega per 1% change but students use 0.01 or 1 incorrectly.
Fix: Read the unit in the question. If vega is ₹2 per 1% and volatility rises 3 points, the change is ₹6.
Believing a delta hedge stays perfect.
Delta looks like a fixed number.
Fix: Delta changes as the price moves, which is gamma. Mention rebalancing in your answer.

Worked examples

Example 1

A trader has sold 2,000 call options on a share. Each call has delta 0.60. How many shares should the trader buy to be delta-neutral? If the share price rises by ₹1, estimate the gain or loss on the option position and on the hedge.

Show the solution

Net delta of the short call position = -0.60 × 2,000 = -1,200.
To neutralise, buy 1,200 shares.
Option position: the calls rise by about ₹0.60 each, so the trader loses 0.60 × 2,000 = ₹1,200.
Hedge: 1,200 shares gain ₹1 each = ₹1,200.
Net change ≈ ₹1,200 - ₹1,200 = ₹0 for a small move.

Answer: Buy 1,200 shares. The ₹1,200 loss on the calls is offset by the ₹1,200 gain on the shares for a small price move.

Example 2

A call option on a share is priced at ₹40. Delta is 0.55, gamma is 0.04, and vega is ₹0.30 per 1% change in volatility. The share price rises by ₹2. Estimate (a) the new call price after the ₹2 rise using delta and gamma, and (b) the price after volatility also falls by 4 points.

Show the solution

Delta effect = 0.55 × 2 = ₹1.10.
Gamma effect = ½ × 0.04 × 2² = ½ × 0.04 × 4 = ₹0.08.
(a) Price after the share move = 40 + 1.10 + 0.08 = ₹41.18.
Vega effect = 0.30 × (-4) = -₹1.20.
(b) Price after both changes = 41.18 - 1.20 = ₹39.98.

Answer: (a) The estimated call price is ₹41.18 after the share price rise. (b) It is ₹39.98 after volatility also falls by 4 points.

Exam tips

In MCQs, learn the ranges: call delta 0 to 1, put delta -1 to 0, gamma positive for long options.
Always state the sign of the position. Short option means the opposite sign of the Greek.
Show the delta-neutral calculation as net delta, then the number of shares to buy or sell.
If a numerical gives gamma, add the ½ × Γ × (ΔS)² term. If not, use delta alone and say it is an approximation.
Link the Greeks to the Black-Scholes inputs when asked for factors affecting option price.

Practice questions from Options

Option Greeks and Sensitivity 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 Sensitivity: frequently asked questions

What is the delta of an at-the-money call?

It is roughly 0.5. A deep in-the-money call is close to 1 and a deep out-of-the-money call is close to 0. The exact value depends on time, volatility and rates.

How do I calculate the delta of an option in the exam?

If Black-Scholes is used, compute d1 and read N(d1) from the normal table for a call. For a put, delta is N(d1) - 1. If delta is given, use it directly.

Why is gamma important for delta hedging?

Gamma shows how fast delta changes when the price moves. A high gamma means the hedge becomes inaccurate quickly, so you rebalance more often.

What factors affect an option's price?

They are the underlying price, strike price, time to expiry, volatility, risk-free rate and expected dividends. Higher volatility raises the value of both calls and puts. Longer time to expiry usually does too, but for European options it can lower the value of deep in-the-money puts, or of calls when dividends are large.