Every option chain you see on a trading screen shows prices, implied volatility and Greeks. Behind most of those numbers sits a formula published in 1973 by Fischer Black and Myron Scholes, with key work by Robert Merton: the Black-Scholes model. It gives a fair price for a European option, one that can only be exercised at expiry, which is how all index and stock options in India work.
You do not need the mathematics to trade options. But knowing what goes into the model, and what it gets wrong, explains why options cost what they cost and why they move the way they do.
The five inputs
| Input | Meaning | Effect on a call | Effect on a put |
|---|---|---|---|
| Underlying price (S) | Where NIFTY or the stock trades now | Higher S, higher call | Higher S, lower put |
| Strike (K) | The option's fixed price | Higher K, lower call | Higher K, higher put |
| Time to expiry (T) | In years: 7 days is 7/365 | More time, higher | More time, usually higher |
| Interest rate (r) | A risk-free rate such as a treasury bill yield | Higher r, slightly higher | Higher r, slightly lower |
| Volatility (σ) | Expected yearly swing, as a % | Higher σ, higher | Higher σ, higher |
For stocks, expected dividends before expiry are a sixth input; they lower calls and raise puts. Notice that the model has no input for direction. It does not ask whether you think the market will rise. That is a key insight: option prices reflect how much the market might move, not which way.
The formula, without fear
Call = S × N(d1) − K × e^(−rT) × N(d2)
Put = K × e^(−rT) × N(−d2) − S × N(−d1)
d1 = [ln(S/K) + (r + σ²/2) × T] ÷ (σ × √T); d2 = d1 − σ × √T
N( ) is the standard normal probability: N(0) is 0.5, larger inputs give numbers closer to 1. Read the call formula as "what you expect to receive (the shares, weighted by N(d1)) minus what you expect to pay (the strike, discounted, weighted by N(d2))". N(d1) is the call's delta, and N(d2) is roughly the model's probability that the call ends in the money.
A worked NIFTY example
NIFTY at 25,000, strike 25,000, 7 days to expiry, interest 6.5% a year, implied volatility 13%.
- T = 7 ÷ 365 = 0.0192, so √T = 0.1385.
- σ × √T = 0.13 × 0.1385 = 0.0180.
- d1 = [0 + (0.065 + 0.0085) × 0.0192] ÷ 0.0180 ≈ 0.08; d2 ≈ 0.06.
- N(d1) ≈ 0.531 and N(d2) ≈ 0.524.
- Call ≈ 25,000 × 0.531 − 24,969 × 0.524 ≈ ₹195.
- Put, from put-call parity: 195 − 25,000 + 24,969 ≈ ₹165.
So the model says the 25,000 straddle is worth about ₹360. If the market is charging ₹360, its implied volatility is 13%. If the market charges ₹450, it is pricing more movement than 13%.
The ATM shortcut and the expected move
For an at-the-money option the formula simplifies to a rule traders use in their heads:
ATM straddle ≈ 0.8 × S × σ × √T, and ATM call ≈ ATM put ≈ 0.4 × S × σ × √T
Check: 0.8 × 25,000 × 0.13 × 0.1385 ≈ 360, the same as the full formula. This shortcut is why the at-the-money straddle is used as the expected move: it is a direct translation of implied volatility into points. Here is NIFTY's expected move worked out from its live straddle:
The range shown is the straddle price added to and subtracted from NIFTY. Using the shortcut in reverse, you can turn that range back into an implied volatility: divide the straddle by 0.8 × NIFTY × √T.
The shortcut also shows two important facts. Doubling volatility doubles the ATM price. And because of the square root, an option with four times as long to expiry costs only about twice as much, which is why time decay speeds up near expiry (see the theta guide).
Implied volatility: running the model backwards
Of the five inputs, four are visible: price, strike, time and the interest rate. Volatility is the only unknown. Instead of guessing it, traders take the option's market price and solve for the volatility that makes the formula match. That number is implied volatility (IV), the market's estimate of future movement. The implied volatility guide explains how to use it.
Every trading platform does this calculation for each strike, every few seconds. When you see an IV column on an option chain, it is the Black-Scholes model run backwards on that strike's price. The Greeks shown next to it (delta, gamma, theta, vega) come from the same formula with that IV plugged in, which is why they can differ slightly between platforms that use different interest rates or time conventions; the Greeks guide explains each one.
IV is more useful than the premium for comparing options, because it removes the effect of strike and time. Whether today's IV is high or low compared with its own past is shown by IV rank:
Look at where each index's IV sits within its past range. A high rank means options are expensive by their own history; a low rank means they are cheap. Neither says which way the price will go.
What the model assumes, and where it fails
- Constant volatility. The model uses one volatility for every strike. In reality, out-of-the-money puts on NIFTY usually trade at a higher IV than calls, because crashes are feared more than rallies. This pattern is the volatility skew.
- No jumps. Prices are assumed to move smoothly. Gaps on results, budget day or global shocks are more frequent than the model expects, so far out-of-the-money options are often worth more than it says.
- Normal returns. Real returns have "fat tails": very large moves happen more often than a bell curve predicts.
- Frictionless trading. No brokerage, taxes, spreads or margins, and continuous hedging. None of these hold in practice.
Traders know all this. They keep the model because it gives a common language: every option, whatever its strike and expiry, can be described by one number, its IV, and the Greeks that come out of the same formula.
Common mistakes
- Treating the model price as the "true" price. It is only as good as the volatility you put in.
- Using historical volatility as the input without thought. Past movement and expected movement often differ, especially before events; see historical vs implied volatility.
- Measuring time in trading days in one place and calendar days in another. Be consistent.
- Trusting model Greeks blindly near expiry. Gamma and theta become extreme and change by the minute.