On Wednesday 10 Jun 2026 the June Nifty future settled at 23,240.10. The 23,250 call, 9.9 points above it, settled at Rs 321.65 a unit with 20 days to run. I can rebuild that premium from five numbers: the future, the strike, the days left, the interest rate and one volatility. Four of them are on the screen. The fifth is the one people argue about, and the formula cannot find it alone. Nobody at the exchange computes a premium. The market trades it, and the formula then runs backwards, from the price to the volatility. This lesson builds the formula from its smallest piece, a tree with a single step, and then shows where it holds and where it bends. Every figure comes from NSE end of day files, dated and not live.

3By the end you can

  1. Build a call price from a one step tree and say why the odds come from the price, not from a forecast.
  2. Name the five inputs of Black 76, rebuild a settle price from them and say which input is solved, not seen.
  3. Explain why one volatility fails across strikes, and why a gap to the formula is not proof of a mispricing.
A radar sweep in teal over a polar grid dotted with lit points

The idea, plainly

An option price is an average of payoffs. Start with the smallest market there is. A toy future stands at a hundred. In one step it ends at a hundred and ten or at ninety. A call struck at a hundred pays ten in the first case and nothing in the second.

One step is the whole idea: set the odds so the average is today's price

A toy tree with one step: the future stands at a hundred and ends at a hundred and ten or ninety with even odds, a call struck at a hundred pays ten or nothing, and its value today is fiveDiagram with 4 boxes. Future today: 100 to Up to 110 (half); Future today: 100 to Down to 90 (half); Up to 110 to Call today = half of 10, plus half of 0 = 5; Down to 90 to Call today = half of 10, plus half of 0 = 5.halfhalfFuture today: 100call struck at 100Up to 110call pays 10Down to 90call pays 0Call today = half of 10, plushalf of 0 = 5even odds, interest set tozero
A toy future at a hundred that ends at a hundred and ten or at ninety, a call struck at a hundred, interest set to zero. The even odds are the ones that keep the average ending price at a hundred. Illustration with round numbers, no market data. Source: RupeeCase illustration with round numbers, not market data. As of 17 Oct 2026.

Call = e−rT x [F N(d1)
− K N(d2)]
d1 = [ln(F/K) + s*s*T/2]
/ (s * sqrt(T))
d2 = d1 − s * sqrt(T)

Read 4 more paragraphs, 302 words

What is that call worth today? Pick the odds so that the average ending price equals today's price. A future costs nothing to hold, so its expected change is zero, and here that means even odds. The call is worth half of ten, which is five. No view on direction went in. The odds came from the price, not from a forecast. With interest in the picture, the answer is discounted back by a small factor.

Now add steps. Cut the life of the option into two steps, then ten, then hundreds. Each step still goes up a little or down a little, and the odds are still set to keep the average on the future. The paths fan out into a bell shaped spread of ending prices. That is the model: the index at expiry is spread around the forward, and the width of the spread is the volatility. It is the standard way to explain the formula, and it has a name, the binomial tree.

When the steps get small enough the tree adds up to a closed form. For index options priced off the future it is called Black 76.

F is the future, K the strike, T the years left, r the rate and s the volatility. N turns a number into a chance between 0 and 1. Read d2 as how many volatility widths the forward sits from the strike. N(d2) is then the model's chance of finishing above the strike, and N(d1) is the call's delta before the small discount, which module F21 uses. Black Scholes is the same idea started from the spot with a dividend yield. I use the future because it already carries the interest and the dividends. Nifty options are European and cash settled, so there is no early exercise to price.

The five dials

Each input pushes the price one way. A higher future lifts a call and lowers a put. A longer time or a higher volatility lifts both, because both add room for the index to move. The rate works two ways. With the spot held still, a higher rate lifts the future, so the call gains a little and the put loses a little. With the future held still, as in Black 76, the rate only discounts, and both dip a little. Over a few weeks either effect is small, which is why the rate gets the least attention. A higher strike lowers a call and lifts a put. Module F15 shows the same effects on real prices, and the Greeks measure each slope.

So how is an option price calculated?

In three moves. First list the payoffs the option can have at expiry. Second, choose odds for those payoffs so that the average ending price equals the forward. Third, average the payoffs and discount them. Black 76 does the three moves in one line. Everything that is not a tree is bookkeeping. The short version is on the question page on how an option price is calculated.

Worked example: rebuilding one settle price, 10 Jun 2026

Take the 23,250 call of the June chain. Here are its five inputs and where each comes from.

The RupeeCase terminal on sample data: Market cap bands: how does the app call a stock large or small?
Watch in the terminal: how does the app call a stock large or small? A short tutorial on sample data.
InputValueSource
Future23,240.10Settlement
Strike23,250Contract
Time left0.0548 yearsDays over 365
Rate5.11%T bill proxy
Volatility15.09%Solved

The tree overshoots, then undershoots, then settles on the formula

Binomial tree price less the Black 76 price for the Nifty 23250 call, from 1 step to 250 steps: the error swings above and below zero and shrinks as the steps grow1 step 82.7; 2 steps −34.7; 5 steps 16.6; 10 steps −6.9; 25 steps 3.3; 50 steps −1.1; 100 steps −0.5; 250 steps −0.1.Tree price less Black 76 price, rupees a unit1 step+82.72 steps−34.75 steps+16.610 steps−6.925 steps+3.350 steps−1.1100 steps−0.4250 steps−0.1
Model, not market data. Figures are rupees per unit. The tree uses the future, strike, days, rate and the call's own volatility from the 10 Jun 2026 chain, with equal steps over the 20 days. A coarse tree can be far off. A fine one is within paise of Black 76. Source: RupeeCase binomial tree calculation, inputs from the NSE end of day chain. As of 10 Jun 2026.
OptionSettleFormulaSettle less formula
22,600 put112.8597.7215.13
23,250 call321.65320.461.19
23,900 call89.45100.14−10.69

One volatility does not fit every strike: the strikes below the index carry more

Implied volatility of Nifty out of the money options against strike on 10 Jun 2026: higher for strikes below the future and lower above, not a flat line30 Jun 2026 expiry: from 22.0% at 20,950 to 18.4% at 25,550, high 22.0% at 20,950, low 14.3% at 23,950.Implied volatility, % a year14%16%18%20%22%21,00022,00023,00024,00025,000Strike priceForward 23,240ATM 15.0%
Implied volatility of the out of the money option at each strike, from settle prices on the 10 Jun 2026 June chain. A flat line would be the formula's own assumption. The volatility at the money, nearest the June future, is 15.03% a year. Source: NSE end of day data, computed by RupeeCase. As of 10 Jun 2026.

Before expiry the formula's value sits above the expiry line, and the gap is time value

Profit and loss on one lot of the Nifty 23250 call bought on 10 Jun 2026, with Black 76 values on that day and with 10 days left: both curves sit above the straight expiry linePayoff of the position at expiry and before expiry. Breakeven at 23,572. Maximum profit in the range ₹53,980, maximum loss ₹20,907; at least one end of the range is still sloping, so the true extreme lies outside it. Spot 23,215.Profit or loss, ₹ for the whole positionAt expiryOn the example date10 days to expiry−₹20,000₹0₹20,000₹40,000₹60,00022,50023,00023,50024,000Nifty at expirySpot 23,215BE 23,572Max loss ₹20,907No cap
Profit and loss on one lot of the 23,250 call bought at its 10 Jun 2026 settle price, with Black 76 values on the example date and with 10 days left, using the call's own volatility. The formula gives the curves, the contract gives the straight line. Source: NSE end of day data, computed by RupeeCase. As of 10 Jun 2026.
Read 8 more paragraphs, 421 words

Black 76 returns Rs 321.72 against a settle price of Rs 321.65. The few paise of difference are rounding in the volatility. Do not read that as a test of the formula. I made it match: the volatility was solved from that very price. This is the step the screen hides. The premium and the volatility are one fact in two units, and traders quote whichever helps. The volatility is also what the 23,250 call and the 23,250 put share, a point module I05 comes back to. The call's delta came to 0.50.

Does the tree agree with the formula? I priced the same call on a tree with the same volatility and different step counts.

That is why I trust the formula as a shortcut, and why I do not trust it as the truth. It is a tidy way to do a tree sum that never ends. A trading screen runs it thousands of times a second, which is the other reason the closed form wins: nobody waits for a tree of hundreds of steps for each strike on the chain.

The same arithmetic also runs backwards. Hand the formula a premium and ask which volatility returns it, and you have the implied volatility on every option chain. Module I13 shows how that solve is done.

Now the assumption that bends. The formula takes one volatility for the whole life of the option and for every strike. The market does not. I priced three options with the one at the money volatility of 15.03%, read at the strike nearest the June future because this is a monthly series, and set each against its settle price. Each option's own volatility, solved from its settle price, was 15.98% for the put, 15.09% for the middle call and 14.35% for the call above.

Near the money one volatility does well. Far below the index the put settles Rs 15.13 above the formula, and the call above the index settles Rs 10.69 below it. The market charges a higher volatility for the strike below and a lower one for the strike above. Plot the volatility of every strike and you get the smile, taken apart in module I17.

One more view helps. The same inputs give a value at every index level, not only at expiry.

The Greeks calculator below opens on this chain. Move the volatility by a single point and watch the price move with it. Then compare the rupees with the Greeks of module I10.

The common mistake

The belief: The formula gives the fair price, so an option trading above it is overpriced. The formula gives no price until someone picks a volatility, and the usual source of that volatility is the market. Put the at the money volatility into the 22,600 put and the formula lands Rs 15.13 under its settle price. Put in that strike's own volatility and the gap closes. So "overpriced" only says the strike's volatility differs from the one chosen. It does not say which is right, and it says nothing about where Nifty goes next. Compare volatilities, not rupees, and then ask why they differ.

Key takeaways

  • An option price is a discounted average payoff, with odds set by today's price.
  • A tree with many small steps turns into the Black 76 formula.
  • Five inputs set the price. Volatility is the one the market does not quote.
  • Nifty options are European and cash settled, so no early exercise is priced.
  • One volatility fits near the money and misses the wings.
  • A gap to the formula measures the volatility chosen, not a market error.

The term this section turns on: Black 76

The term this section turns on: Black 76The term this section turns on: Black 76. Black 76: The closed form option price when the underlying is a future or forward. It takes the future, strike, time, rate and volatility.Black 76The closed form option price when theunderlying is a future or forward. It takesthe future, strike, time, rate and volatility.
From the module glossary; every term is in the Learn glossary too. Source: Module 14.9, From a binomial tree to Black Scholes and Black 76.

Before you trade. Derivatives trading carries a high risk of loss. SEBI's study "Profitability of Individual Traders in the Equity Derivatives Segment (FY25 to FY26)" (August 2026) found that 87.7% of individual traders in equity F&O made net losses in FY26, with aggregate net losses of Rs 91,685 crore. Read the SEBI study