Expected range calculator

See the full expiry distribution an implied volatility implies, and the model probability of finishing beyond any strike you choose.

Quick answer: The expected range calculator converts an implied volatility into the full set of expiry bands — one, two and three standard deviations — and computes the model probability that the underlying finishes beyond any strike you nominate.

Expected Range Calculator

The probabilities come from a lognormal model. Real index returns have fatter tails, so the outer bands understate the true risk of an extreme move.

Figures are per unit of the underlying and exclude brokerage, STT, exchange charges, stamp duty and GST.

How this calculator works

The bands, and what they are worth

One standard deviation captures about 68.3% of a normal distribution, two about 95.4%, three about 99.7%. Those numbers are properties of the normal curve, not of the market. Real index returns are fat-tailed: three-standard-deviation days that the normal model says should happen once in several years happen several times a decade. The 1σ band is a well-calibrated first approximation. The 3σ band is a polite fiction.

The probability of finishing beyond a strike

Under the lognormal model, the probability that the underlying expires above a strike K is N(d₂), where d₂ is the second Black–Scholes argument. This is nearly, but not exactly, the option's delta — delta is N(d₁), which is slightly larger. The difference is the reason a 0.30-delta option has a somewhat lower than 30% chance of expiring in the money, and the reason 'delta is the probability of finishing ITM' is a useful lie rather than a fact.

Why the distribution is not symmetric

Prices are bounded below by zero and unbounded above, so the terminal price distribution is lognormal rather than normal: it is right-skewed in price space even though it is symmetric in log space. The upside band is therefore slightly wider than the downside one in absolute points. Separately and much more importantly, the equity-index volatility skew means the market charges more for the downside strike than this single-volatility model does — so the true risk-neutral probability of a large fall exceeds what this calculator reports.

Probability of touching is not probability of expiring

This calculator reports the probability of EXPIRING beyond a strike. The probability of the underlying TOUCHING that strike at any point before expiry is roughly twice as large for an at-the-money-ish strike. A stop-loss cares about touching. An option's payoff cares about expiring. Conflating them is how people conclude a short strangle is safer than it is.

Probability of expiring beyond a strike (lognormal)

P(S_T above K) = N(d₂), d₂ = [ ln(S ÷ K) + (r − σ² ÷ 2)·T ] ÷ (σ·√T)

This is the risk-neutral probability, not the real-world one — it is the probability under the measure the option market prices in, which embeds the risk premium. It is close to, but slightly less than, the call's delta N(d₁).

P(S_T above K)Risk-neutral probability the underlying expires above the strike.
N(·)The cumulative standard normal distribution function.
d₂The second Black–Scholes argument. d₁ = d₂ + σ√T is the one that gives delta.
SSpot price today.
KThe strike being tested.
σImplied volatility, annualised, as a decimal.
TTime to expiry in years = calendar days ÷ 365.
rRisk-free rate, as a decimal. 6.5% is used as an Indian proxy.

Using it, step by step

Enter spot, the at-the-money implied volatility for the expiry in question, and calendar days.
Read the 1σ, 2σ and 3σ bands. Treat the 1σ band as informative and the 3σ band as an understatement of tail risk.
Enter a strike to test. The calculator gives the model probability of expiring above and below it.
Compare that probability with the strike's delta on your broker's chain. Delta will be slightly larger, and the gap is not an error in either number.
Remember that the probability of TOUCHING the strike before expiry is roughly double the probability of expiring beyond it.

Worked example

NIFTY

NIFTY at 24,000, 30 days, implied volatility 13%. The one-standard-deviation move is 24,000 × 0.13 × √(30/365) ≈ 894 points, so the 1σ band runs from roughly 23,106 to 24,894 and captures about 68% of the model distribution. Now test the 25,000 strike: d₂ works out to −0.97, so N(d₂) ≈ 16.6% is the model probability of expiring above it. Note that the same strike's delta is 0.175 — larger, as it always is, because delta is N(d₁) and the probability is N(d₂). That 16.6% looks like comfortable odds for a call seller. The catch is not in the probability, which is about right; it is in the 16.6% of outcomes, which are unbounded for a naked short, and in the fact that the probability was computed with a single volatility on a chain whose skew says the market disagrees with that single volatility at every strike but one.

Assumptions and limitations

What this calculator assumes. Every number it produces is only as good as the assumptions below, and each of them is wrong to some degree in a real market. The output is a model's opinion, not a measurement.

Returns are lognormal with constant volatility. The market disagrees: the volatility skew means each strike prices its own implied volatility, and using one figure across all strikes misprices the wings systematically.
The probabilities are RISK-NEUTRAL, not real-world. They are the probabilities implied by option prices, which already embed a risk premium. The real-world probability of a large fall is lower than the risk-neutral one; the real-world probability of a large rise is higher.
Fat tails are ignored entirely. The 2σ and 3σ probabilities understate the true frequency of extreme moves, often by a large multiple.
The result is the probability of EXPIRING beyond the strike, not of touching it at any time. For a near-the-money strike the touch probability is roughly twice as large.
Dividend yield is taken as zero, correct for NIFTY and BANKNIFTY and wrong for a single stock with an ex-dividend date before expiry.

Common mistakes

Reading the 3σ band as a worst case. Real markets deliver moves beyond it far more often than the normal model allows.
Confusing the probability of expiring beyond a strike with the probability of touching it. Short-option positions are frequently sized as though these were the same number.
Treating delta as exactly the probability of finishing in the money. Delta is N(d₁); the probability is N(d₂), which is smaller. The gap widens with volatility and time.
Using one at-the-money implied volatility to price probabilities in the wings, where the chain's own skew says the volatility is different.
Interpreting risk-neutral probabilities as real-world forecasts. They are the probabilities the market's prices imply, and they are deliberately shifted by the risk premium.

Frequently asked questions

What is the expected range of an option?

It is the set of price bands that an implied volatility implies at expiry — one, two and three standard deviations from the current spot, covering roughly 68%, 95% and 99.7% of a lognormal distribution.

How do I calculate the probability of an option expiring in the money?

Under Black–Scholes it is N(d₂) for a call finishing above the strike, where d₂ = [ln(S/K) + (r − σ²/2)T] / (σ√T). This calculator computes it for any strike you enter.

Is delta the probability of expiring in the money?

Not exactly. Delta is N(d₁) and the probability is N(d₂), which is always smaller. Delta is a good approximation and a bad identity; the gap grows with volatility and with time to expiry.

What is the probability of touching a strike?

For a strike near the money it is roughly twice the probability of expiring beyond it, because a path can touch the strike and come back. A stop-loss cares about touching; an option's payoff cares about expiring.

Why are these probabilities called risk-neutral?

Because they are derived from option prices, which embed the market's risk premium. They are not forecasts of what will happen; they are the probabilities under which today's option prices are fair. The real-world probability of a crash is lower than the risk-neutral one.

Why is the upside band wider than the downside band?

Because the price distribution is lognormal: prices cannot fall below zero but can rise without limit, so the distribution is right-skewed in price space. In log space it is perfectly symmetric.

Does this account for the volatility skew?

No. It uses a single implied volatility across all strikes, which is exactly the assumption the skew disproves. The true risk-neutral probability of a large fall is higher than this calculator reports, and the option chain prices it that way.

How reliable is the 3-sigma band?

Not very. The 99.7% coverage is a property of the normal distribution. Real index returns are fat-tailed, and moves beyond three standard deviations arrive several times a decade rather than once in several years.

Voice search & related questions

What is the probability my option expires worthless?

For a call, it is one minus N(d₂), the model probability of finishing above the strike. Enter your strike above to compute it — but remember these are risk-neutral probabilities and they ignore the volatility skew.

How likely is NIFTY to move more than 1000 points this month?

With NIFTY at 24,000 and implied volatility at 13%, a 30-day one-standard-deviation move is about 1,012 points, so the model puts the odds of exceeding it at roughly one in three. Fat tails make the true odds somewhat higher.

Is a 16 delta option safe to sell?

A 0.16-delta option has roughly a 16% model probability of finishing in the money. Nothing about that number is safe: the 16% of outcomes are unbounded for a naked short option, and the probability was computed from a model whose tails are too thin.

Last reviewed 10 July 2026. Educational content only — not investment advice.