🧮 How options are priced: Black-Scholes, binomial trees and put-call parity

🎯 In 30 seconds: An option's price depends on six things: the underlying price, the strike, time to expiry, interest rates, dividends and volatility. Five of those you can look up. Volatility you cannot, which is why traders talk about implied volatility. The Black-Scholes formula turns the six inputs into a price for a European option, and every option traded on NSE and BSE is European style. NSE itself uses Black-Scholes to set the starting price of new option contracts. Put-call parity ties calls and puts together so tightly that one tells you the other.

The six inputs

Every pricing model, from a spreadsheet to what large trading firms run, starts from the same list. Here is how each input moves the price of a call and a put, with everything else held still.

The Greeks you met in The Greeks are these sensitivities with numbers attached: delta for the underlying price, theta for time, vega for volatility, rho for interest rates.

Volatility is the odd one out. The others are known today. Future volatility is not. So traders turn the question around: given the market price of an option, what volatility would the model need to produce it? That number is implied volatility. India VIX, which NSE computes from Nifty option prices, is a summary of it for the index. See Implied volatility and India VIX.

A shop-floor way to think about it

Imagine a mango wholesaler in Ratnagiri who sells buyers the right, not the duty, to buy a crate at ₹2,000 in three months. What should he charge for that right? It depends on today's price (if mangoes already sell at ₹2,200, the right is clearly worth something), on how long the buyer has, on how wildly mango prices swing in a season, and on the interest the buyer could earn on ₹2,000 while waiting. Option pricing models are a careful way of putting those same questions into one number.

Value today vs value at expiry

At expiry an option is worth only its intrinsic value: the hockey stick in every payoff chart. Before expiry it is worth more, because there is still a chance of a bigger payoff. The model curve sits above the hockey stick, and the gap is time value.

The Black-Scholes formula

In 1973, Fischer Black and Myron Scholes published a formula for the price of a European option, one that can only be exercised at expiry. Robert Merton derived it another way. Scholes and Merton received the 1997 economics Nobel prize for it; Black had died in 1995.

This matters more in India than in the US. US stock options are mostly American style and can be exercised early, so the formula is only an approximation there. Indian index options and stock options are European style (NSE contract specifications, October 2026), so Black-Scholes fits their structure directly. NSE states that the base price of a newly introduced options contract is its theoretical Black-Scholes value.

For a European call on a share that pays no dividend before expiry:

C = S x N(d1) - K x e^(-rT) x N(d2)

d1 = [ ln(S/K) + (r + σ²/2) x T ] / (σ x √T)
d2 = d1 - σ x √T

And for the matching put:

P = K x e^(-rT) x N(-d2) - S x N(-d1)

What the symbols mean, in plain words:

• S is the share or index price, K the strike, T the time to expiry in years, r the risk-free interest rate per year (continuously compounded), σ (sigma) the volatility per year.
• ln is the natural logarithm, e is about 2.718, and e^(-rT) is the discount factor that turns money at expiry into money today.
• N(x) is the standard normal cumulative distribution: the chance that a bell-curve variable is below x. In a spreadsheet it is NORM.S.DIST(x, TRUE).
• Loosely, N(d2) is the model's probability that the call finishes in the money, and N(d1) is the call's delta.

Worked example in rupees

Hypothetical inputs: Sharma Foods Ltd at S = ₹1,000, strike K = ₹1,000, T = 0.25 years (about three months, roughly the furthest monthly stock option usually listed), r = 6.5%, σ = 25%, no dividend before expiry. The 6.5% is a made-up rate for the example. For real work, use a current short-term government or money-market rate and note the date; RBI and FBIL publish these.

1. ln(S/K) = ln(1) = 0.
2. (r + σ²/2) x T = (0.065 + 0.03125) x 0.25 = 0.0240625.
3. σ x √T = 0.25 x 0.5 = 0.125.
4. d1 = 0.0240625 / 0.125 = 0.1925. d2 = 0.1925 - 0.125 = 0.0675.
5. N(d1) = 0.5763, N(d2) = 0.5269, N(-d1) = 0.4237, N(-d2) = 0.4731.
6. K x e^(-rT) = 1,000 x e^(-0.01625) = ₹983.88.
7. Call = 1,000 x 0.5763 - 983.88 x 0.5269 = ₹57.91.
8. Put = 983.88 x 0.4731 - 1,000 x 0.4237 = ₹41.79.

For a lot of, say, 500 shares (a made-up lot), the call would cost about ₹28,955 and the put about ₹20,895.

What the formula assumes, and where India differs

• Constant volatility. The model uses one σ for the whole life of the option. Real volatility changes, especially around results, RBI policy days and elections. That is why implied volatility differs by strike and expiry.
• Smooth prices. The model assumes prices move continuously. Indian markets gap at 9:15 IST after overnight news, and individual stocks can hit price bands. Gaps make big moves more common than a bell curve says.
• No dividends. For a stock with a dividend before expiry, subtract the present value of the dividend from S (or use the version of the formula with a dividend yield q). NSE does not adjust strikes for ordinary dividends below 2% of the share price, so the option price itself has to reflect them. Dividends at or above 2% lead to a strike adjustment instead.
• Free borrowing and no costs. The model ignores brokerage, STT, the bid-ask spread and margin. Real prices sit inside a band around the model value, not on it.

The model is not wrong because of these. It is a reference point. Traders use it to translate prices into implied volatility, so that options with different strikes and expiries can be compared on one scale.

The binomial idea in one step

Black-Scholes can look like magic. A binomial tree shows where it comes from. Imagine Sharma Foods can do only one of two things over the next three months: go up 10% to ₹1,100 or down 10% to ₹900.

Step 1: build a copy of the call. Buy 0.5 shares and borrow ₹442.75. In three months you owe ₹442.75 x 1.0164 = ₹450. If the share is at ₹1,100, your half share is worth ₹550, you repay ₹450, and you have ₹100. If it is at ₹900, your half share is worth ₹450, you repay ₹450, and you have ₹0. That is exactly what the call pays.

Step 2: price the copy. The copy costs 0.5 x ₹1,000 - ₹442.75 = ₹57.25. Since the copy and the call pay the same in every case, they must cost the same. If the call cost more, you could sell it, build the copy, and keep the difference with no risk. If it cost less, you would do the reverse. That is the no-arbitrage argument behind all option pricing.

Step 3: see the shortcut. The same answer comes from the formula in the figure, using the weight p = (1.0164 - 0.90) / (1.10 - 0.90) = 0.582. That p is called a risk-neutral probability. It is not anyone's forecast. It is the weight that makes the share earn exactly the risk-free rate, and it gives the same price as the copy.

Real models use hundreds of small steps instead of one big one. As the steps get smaller, the binomial price gets closer to the Black-Scholes price. Binomial trees can also handle early exercise, which matters for American options in the US, but not for European options in India.

The one-step price, ₹57.25, happens to be close to the Black-Scholes ₹57.91 here. That is partly luck: a single 10% step and 25% annual volatility are different assumptions. The method is the point, not the number.

Put-call parity

For European options with the same strike and expiry, on an underlying that pays no dividend before expiry:

C - P = S - K x e^(-rT)

Why it must be true: a long call plus a short put pays exactly (price at expiry - strike), whatever happens. Above the strike the call pays S - K and the put expires. Below it the call expires and you pay K - S on the put. That is the same as owning the underlying and owing K at expiry. Two positions with identical payoffs must cost the same today.

Check it with the worked example.

On the Nifty. In the 14-day chain used in Vertical spreads, the 25,000 call is 286 and the put 224, a difference of 62 points. With r = 6.5% and no dividends, 25,000 - 25,000 x e^(-0.065 x 14/365) = 62.25. It matches because the chain came from the same model.

In real markets, the Nifty's constituents pay dividends and the cash index cannot be traded as one instrument. Traders therefore check parity against the Nifty future for the same expiry: C - P = (F - K) x e^(-rT), where F is the futures price. If the call-minus-put difference is far from what this gives, either a price is stale or you are looking at bids and asks rather than mid prices.

What a violation would look like. Suppose the Sharma Foods 1,000 call showed ₹57.91 but the put ₹36. Then C - P = ₹21.91, about ₹5.79 more than parity allows. A trader could sell the call, buy the put and buy the shares, locking in about ₹5.79 per share at expiry. In practice such gaps close fast, and after STT, brokerage, the bid-ask spread on three instruments and margin, small gaps are not free money. For you, parity is a sanity check, not a strategy.

Where the market's numbers come from

• NSE's option chain shows implied volatility for each strike, worked out from traded prices.
• India VIX is computed by NSE from the order book of Nifty options in the near and next expiries, using a method adapted from the CBOE VIX, and expressed as an annualised percentage (NSE India VIX methodology).
• The interest rate used by NSE for India VIX is a MIBOR rate for the relevant tenor, according to its methodology note. For your own work, any reasonable short-term rate is fine, as long as you note it.

Try it

1. Build the Black-Scholes call and put in a spreadsheet as above and check you get ₹57.91 and ₹41.79.
2. Pick a near-month Nifty strike close to the current level from NSE's option chain. Enter the current Nifty, the strike, days to expiry divided by 365, a short-term rate, and the implied volatility shown on the chain. How close is your model price to the last traded price?
3. Check put-call parity for that strike using the Nifty future of the same expiry instead of the index. Use mid prices.
4. Change only the volatility input until your model price matches the market price. That number is your own implied volatility estimate. Compare it with the chain's figure and with India VIX.

Sources

• NSE: Equity derivatives on individual securities (European style, base price from the Black-Scholes model)
• NSE: Corporate action adjustments in equity derivatives (2% dividend threshold)
• NSE: India VIX computation methodology
• NobelPrize.org: The Sveriges Riksbank Prize in Economic Sciences 1997, press release