This guide explains the inputs, the formula, a worked example on Nifty, put-call parity, which interest rate to use in India, and the limits of the model.
The inputs
| Input | What it is | Where to find it |
|---|---|---|
| Underlying price (S) | The current price of the index or stock | Live market data |
| Strike (K) | The option's strike price | The option chain |
| Time to expiry (T) | Years until expiry; 12 days is 12 ÷ 365 | The contract's expiry date |
| Interest rate (r) | A short-term risk-free rate | Treasury bill yields |
| Volatility (σ) | Expected annual volatility of the underlying | Implied volatility, or historical volatility as a starting point |
The formula
For a call:
C = S × N(d1) − K × e^(−rT) × N(d2)
For a put:
P = K × e^(−rT) × N(−d2) − S × N(−d1)
where:
- d1 = [ln(S ÷ K) + (r + σ² ÷ 2) × T] ÷ (σ × √T)
- d2 = d1 − σ × √T
- N(x) is the standard normal cumulative distribution function.
You do not need to calculate this by hand; the calculator does it. But knowing the structure helps: N(d1) is the call's delta, and N(d2) is the model's probability that the call expires in the money.
A worked example on Nifty
Inputs: Nifty at 25,000, strike 25,000, 12 days to expiry, interest rate 6.5% and volatility 14%.
- Call value: about ₹280.48
- Put value: about ₹227.11
- Call delta: about 0.54; put delta: about −0.46
- Call theta: about −₹13.08 a day; vega: about ₹18.00 per point of volatility
With a lot of 65, the call is worth about ₹18,231 per lot in the model.
Put-call parity
For European options, calls and puts at the same strike and expiry are linked:
C − P = S − K × e^(−rT)
In the example, 280.48 − 227.11 = 53.37, and 25,000 − 25,000 × e^(−0.065 × 12 ÷ 365) is also about 53.37. If market prices break this relationship by more than costs, there may be an arbitrage, though in practice such gaps are small and quickly closed.
Which interest rate to use in India
The model needs a risk-free rate for the option's horizon. Common choices are the yield on government Treasury bills with a similar maturity, or a short-term money market rate. For options expiring within weeks, the choice makes little difference: rho, the sensitivity to rates, is small for short-dated options.
Using futures instead of spot
Index options settle against the index, but traders often price them against the futures price for the same expiry instead of the spot index. The futures price already includes the cost of carry and expected dividends, which the basic Black–Scholes formula leaves out. The version of the model that uses the futures price is known as Black-76.
What the calculator is useful for
- Fair value check. Compare a market price with the model value at your volatility estimate.
- Greeks. See how an option responds to moves, time and volatility. See the options Greeks.
- Scenario analysis. Change the underlying price, days to expiry or volatility to see what an option could be worth later.
- Implied volatility. Work backwards from a market price to the volatility it implies. See the implied volatility calculator.
- Strategy payoffs. Price each leg to see a multi-leg position today and at expiry in the option profit calculator.
Limits of the model
- Constant volatility. The model assumes one volatility for all strikes and over the option's life. Real markets show a volatility smile or skew, with out-of-the-money puts on Indian indices usually priced at higher IV than calls.
- Smooth prices. The model assumes prices move continuously. Real markets gap, especially around events.
- No dividends in the basic version. Use futures prices or adjust for dividends.
- European exercise only. Index and stock options in India are European-style, so this is not a problem here, though stock options are physically settled.
- No costs or liquidity. The model gives a theoretical price, not the price you can trade at.
The model's value lies less in its precision than in its structure: it shows how each input affects an option's price.
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Frequently asked questions
What is the Black–Scholes model used for?
To calculate the theoretical value of a European option and its Greeks from the underlying price, strike, time, interest rate and volatility.
Does Black–Scholes work for Nifty options?
Yes, as a reference model. Nifty options are European-style. Real prices differ because each strike trades at its own implied volatility.
What volatility should I enter?
To value an option, use your estimate of future volatility. To understand a market price, enter the price and solve for implied volatility instead.
Why does the model price differ from the market price?
Because the market's implied volatility differs from the volatility you entered, and because of skew, supply and demand, dividends and liquidity.
Sources
- Black, F. and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654
- Merton, R. C. (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4(1), 141–183
- Black, F. (1976). The Pricing of Commodity Contracts. Journal of Financial Economics, 3(1–2), 167–179
- Hull, J. C. Options, Futures, and Other Derivatives. Pearson