NISM-Series-X-A: Investment Adviser (Level 1) · Understanding Derivatives

Option Pricing and the Greeks: Delta, Gamma, Theta, Vega

Updated 11 October 2026 · Fact-checked

Option premium = intrinsic value + time value. Six factors drive it: spot price, strike price, time to expiry, volatility, interest rate and dividends. The greeks measure sensitivity: delta to spot price, gamma to delta, theta to time, vega to volatility, rho to interest rate. Solve questions by identifying the factor that changes and its direction.

Understand Option Pricing and the Greeks

An option's price is called the premium. The buyer pays it to the seller (writer) for the right, not the obligation, to buy (call) or sell (put) at the strike price. The premium is not random. It has two parts.

Intrinsic value is the amount the option would be worth if exercised right now. For a call it is the higher of (spot − strike) and zero. For a put it is the higher of (strike − spot) and zero. An option with intrinsic value is in the money (ITM). If spot equals strike it is at the money (ATM). Otherwise it is out of the money (OTM) and its intrinsic value is zero.

Time value is the rest of the premium: premium − intrinsic value. It is what buyers pay for the chance that the option becomes more valuable before expiry. It is highest for ATM options and shrinks to zero at expiry. An option's premium can never be negative, and time value is never negative for the buyer.

Six factors affect premium. Higher spot raises a call and lowers a put. Higher strike lowers a call and raises a put. More time to expiry raises both. Higher volatility raises both, because bigger swings make a payoff more likely. Higher interest rates slightly raise calls and lower puts. Expected dividends lower calls and raise puts, because the share price falls when it goes ex-dividend.

The greeks measure how premium reacts to each factor, holding the others constant. Delta is for spot price, gamma for delta itself, theta for time, vega for volatility and rho for interest rate. The Black-Scholes model is the best-known formula for pricing European options. It uses the six inputs above and assumes volatility is constant, markets have no frictions and returns are lognormal. You do not need to calculate with it. You need to know its inputs and assumptions.

Key formulas to remember

Premium: Option premium = Intrinsic value + Time value; Time value = premium − intrinsic value. It is zero at expiry.
Intrinsic value of a call: Max(Spot − Strike, 0); ITM when spot is above strike.
Intrinsic value of a put: Max(Strike − Spot, 0); ITM when spot is below strike.
Delta: Delta = Change in premium ÷ Change in spot price; Call delta lies between 0 and +1. Put delta lies between −1 and 0. ATM call delta is roughly 0.5.
Gamma: Gamma = Change in delta ÷ Change in spot price; Highest for ATM options near expiry. Same sign for calls and puts (positive for buyers).
Theta: Theta = Change in premium ÷ Change in time; Negative for option buyers, positive for sellers. Time decay speeds up near expiry for ATM options.
Vega: Vega = Change in premium ÷ Change in volatility; Positive for buyers of both calls and puts. Highest for ATM options with longer time to expiry.
Rho: Rho = Change in premium ÷ Change in interest rate; Positive for calls, negative for puts. Usually the least important greek.

How to solve Option Pricing and the Greeks questions

Most questions ask you to compute a value or to predict the direction of a change. Use the same routine each time.

1Note whether the option is a call or a put, and whether you are the buyer or the seller.
2Compare spot with strike to decide ITM, ATM or OTM.
3If asked for intrinsic value, apply Max(Spot − Strike, 0) for a call or Max(Strike − Spot, 0) for a put.
4If asked for time value, subtract intrinsic value from the premium.
5If asked about a change, identify the single factor that moves and hold all others constant.
6Match the factor to its greek: spot to delta, delta change to gamma, time to theta, volatility to vega, interest rate to rho.
7Apply the sign: calls have positive delta, puts negative; buyers have negative theta and positive vega.
8Check the answer against the limits, such as delta range and premium never below intrinsic value.

Quickest way: Sign and greek lookup

When to use it: Use it for direction-of-change and which-greek MCQs, where the time limit matters most.

Name the factor that changes.
Recall the greek for that factor.
Recall the sign for your position: long call delta +, long put delta −, long option theta −, long option vega +.
Estimate: premium change ≈ greek × size of change in the factor.
Eliminate options that break the limits, for example call delta above 1 or negative time value.

Common mistakes in Option Pricing and the Greeks

Treating the whole premium as intrinsic value.
Students forget that an option has time value as well.
Fix: Always compute intrinsic value from spot and strike first, then subtract it from the premium to get time value.
Giving a put a positive delta.
Students remember delta as 'between 0 and 1' without the call/put split.
Fix: Put delta is between −1 and 0 because a put gains when spot falls.
Saying higher volatility lowers a put premium.
Students link a put with falling prices and assume risk hurts it.
Fix: Higher volatility raises the premium of both calls and puts. Vega is positive for buyers of both.
Saying theta is positive for the option buyer.
Students confuse time passing with value growing.
Fix: Time decay hurts the buyer and helps the seller. Buyer's theta is negative.
Confusing gamma with delta.
Both relate to spot price movement.
Fix: Delta is the change in premium per unit change in spot. Gamma is the change in delta per unit change in spot.
Applying dividend and interest effects the wrong way round.
Students memorise the effect on calls only.
Fix: Higher interest rates: call up, put down. Higher expected dividends: call down, put up.

Worked examples

Example 1

A call option on a share has a strike of ₹500. The share trades at ₹540 and the call premium is ₹52. Find the intrinsic value and time value.

Show the solution

The option is a call, and spot ₹540 is above strike ₹500, so it is in the money.
Intrinsic value = Max(540 − 500, 0) = ₹40.
Time value = premium − intrinsic value = 52 − 40 = ₹12.

Answer: Intrinsic value is ₹40 and time value is ₹12.

Example 2

A call has a delta of 0.6 and a gamma of 0.04. The share price rises by ₹2. What is the approximate new delta and the approximate change in premium using the starting delta?

Show the solution

Change in premium ≈ delta × change in spot = 0.6 × 2 = ₹1.20.
Change in delta = gamma × change in spot = 0.04 × 2 = 0.08.
New delta = 0.6 + 0.08 = 0.68.

Answer: Premium rises by about ₹1.20 and the new delta is about 0.68.

Exam tips

Learn the sign table by heart: call delta +, put delta −, buyer theta −, buyer vega +, call rho +, put rho −.
In calculation questions, compute intrinsic value first. The time value follows by subtraction.
Read carefully whether the question is from the buyer's or the seller's side. The theta sign flips.
Expect questions on which option has the highest gamma, theta or vega. The answer is usually the ATM option.
For Black-Scholes, know the inputs and that it prices European options. Do not spend time on the formula itself.

Practice questions from Understanding Derivatives

Option Pricing and the Greeks in other exams

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

Option Pricing and the Greeks: frequently asked questions

What is the difference between intrinsic value and time value of an option?

Intrinsic value is the gain if you exercised the option right now, and it is zero for OTM options. Time value is the premium minus intrinsic value. It reflects the chance of further gain before expiry and falls to zero on expiry.

Which factors affect option premium?

Spot price, strike price, time to expiry, volatility, interest rate and expected dividends. Volatility and time to expiry raise the premium of both calls and puts. Spot, strike, interest and dividends affect calls and puts in opposite directions.

Why is theta negative for option buyers?

Time value shrinks as expiry nears, so the premium falls even if nothing else changes. The buyer loses this value and the seller gains it. That is why theta is negative for buyers and positive for sellers.

Do I need to calculate Black-Scholes for the NISM exam?

Questions are generally conceptual. Know that it prices European options and uses spot, strike, time, volatility and the interest rate as inputs, with dividends in the adjusted version. Also know its main assumptions, such as constant volatility.