The Butterfly: A Bet That Nothing Happens, Priced Precisely

The sharpest way an option position can say "here, and nowhere else"

A long call butterfly is three strikes and four contracts. You buy one call at a lower strike, sell two calls at a middle strike, and buy one call at a higher strike, with the two outer strikes placed at equal distances from the middle one. The whole thing costs money to put on, and that cost is the most you can lose. At expiry the position is worth the most if the underlying finishes exactly at the middle strike, and it decays away to nothing in both directions, reaching zero at each outer strike.

What makes it distinctive is not the shape but the resolution of the statement it makes. A vertical spread says the underlying will finish above or below a level. A straddle or a strangle says it will move a lot, without saying which way. An iron condor says it will stay inside a wide corridor. The butterfly says it will finish near a point, and it charges very little for that opinion precisely because the opinion is so unlikely to be right. Everything else about the structure follows from those two sentences.

That is why the butterfly is a useful thing to compute even for someone who has no intention of ever putting one on. It is the clearest available demonstration that in a market where options are priced off a distribution, a high reward-to-risk ratio is not a discovery. It is a receipt. The ratio is high because the event is rare, and the two are tied together by the pricing, not by anything the structure does. Read the ratio alone and it looks like a bargain. Read the probability alone and it looks hopeless. Read them together and you get the truth, which is that the structure is priced.

The mechanics of strikes, premiums and expiry that the rest of this page assumes are covered in our framework for thinking about options, which is the parent page for everything here. If terms like the middle strike or the wing are unfamiliar, what a strike price actually is is the shorter starting point. This page assumes both and goes straight to the arithmetic.

The structure, stated in full so it can be checked

Everything on this page comes from one seeded model. The model is stated in full below so that the arithmetic can be reproduced or disputed, and so that it is obvious which numbers are computed and which are assumptions. The index level, the volatility and the contract multiplier are illustrative. They are not a quotation from any market on any date, and nothing here should be read as a description of what any index has done or will do.

Two entries in that table deserve a warning. The contract multiplier is the one most likely to be quoted back as fact from an article, and it should not be. The minimum contract value for index derivatives was moved up to the Rs 15 lakh to Rs 20 lakh band, lot sizes were raised sharply in November 2024 to fit inside it, and they were then revised back down from January 2026 as index levels rose. Any specific lot number printed anywhere has a shelf life. Our page on how lot sizes are set and revised explains the mechanism, which is the durable part.

The flat volatility assumption is the other one. Real option chains are not flat across strikes, and a butterfly is unusually sensitive to the shape because it is long the two outer strikes and short the middle one twice. A skew changes the debit, and therefore changes every ratio computed below. That is a genuine limitation and it is stated here rather than buried, because the point of the exercise is the relationship between the reward and the probability, and that relationship survives the simplification even though the individual numbers would move.

The payoff, computed rather than sketched

Pricing the three legs at the stated volatility gives 594.84 points for the 23,700 call, 409.92 for the 24,000 call and 265.76 for the 24,300 call. The structure costs the lower call plus the upper call less twice the middle call, which is 40.76 points, or Rs 2,649.28 at the illustrative multiplier. That is the debit, and because every leg is defined and the wings are equal, it is also the entire downside. There is no scenario in this structure where more than the debit is lost, and no scenario where a margin call arrives to ask for more.

The maximum profit sits at the middle strike and is the wing width less the debit: 300 less 40.76, or 259.24 points, which is Rs 16,850.72 illustrative. The reward-to-risk ratio is therefore 6.36 to one. The two breakevens are the lower strike plus the debit and the upper strike less the debit, which puts them at 23,740.76 and 24,259.24. Those are 1.08 percent below and 1.08 percent above the starting level, and the band between them is 518.48 points wide, or 2.16 percent of the index.

Read that last number carefully, because it is the whole page in one figure. For this structure to produce anything at all, an index has to spend thirty days moving and then finish inside a window 2.16 percent wide. For it to produce anything close to the headline 6.36, the index has to finish within a few points of one specific strike. The payoff diagram makes the peak look like a destination. It is a single point on a continuum, and the structure is worth its maximum at exactly one index level and less at every other level in the universe.

It is worth being precise about how much less. The payoff inside the band is a triangle, not a plateau. Halfway out from the strike toward either breakeven, at 129.62 points away, the position returns half of its maximum. At the breakevens it returns nothing. Under the stated distribution, the model puts 23.49 percent of outcomes inside the band, but only 11.88 percent above half the maximum and only 2.30 percent within twenty five points of the strike. The average result across all the outcomes that finish inside the band is 130.24 points, or Rs 8,465.86 illustrative, which is 3.20 times the debit rather than 6.36 times it. The headline ratio is a boundary condition, not a typical outcome, and the distinction is the difference between a structure that looks generous and one that is merely priced.

The ratio and the probability are one fact, not two

Here is the arithmetic almost everyone does, at least silently. The structure pays 6.36 times what it risks. The model says it lands 23.49 percent of the time. Multiply the reward by the probability of winning, subtract the loss weighted by the probability of losing, and the answer is positive 0.73 per rupee at risk. On that reasoning the butterfly is one of the most attractive things available anywhere in a listed market, and the only remaining question is how many to put on.

The correct answer, computed on exactly the same distribution that produced the prices, is 0.0054 per rupee. Not 0.73. The expectation of the structure before any charge is 0.22 points, which works out at Rs 14.19 per contract illustrative, and that residue is simply the interest on the debit over thirty days. A Monte Carlo run of 400,000 paths on the same measure returns 0.35 points with a standard error of 0.13, which is the same number inside sampling noise. There is no edge in the structure and there was never going to be one, because the structure was priced off the distribution being used to evaluate it.

The naive calculation goes wrong in two places at once. It treats the maximum profit as the payoff whenever the index lands inside the band, when in fact the payoff inside the band is a triangle whose average is half its peak. And it treats the maximum loss as the payoff whenever the index lands outside, when 3.56 percent of outcomes land inside the wings but outside the band and lose only part of the debit. Correct both and the two sides cancel to nothing, which is what a priced instrument is supposed to do.

The tidiest way to see that the pricing, not the shape, is doing the work is to build the same structure at seven different widths and look at what changes. A 100 point butterfly costs 4.57 points, pays 20.89 times its cost, and lands inside a band 0.80 percent wide 8.8 percent of the time. A 600 point butterfly costs 158.34 points, pays 2.79 times its cost, and lands inside a band 3.68 percent wide 38.9 percent of the time. The ratio moves by a factor of seven and a half across that range. The expectation moves from Rs 1.59 to Rs 55.13 illustrative, every one of which is the interest on a different debit, and every one of which is a rounding error against maximum profits running from Rs 6,203 to Rs 28,708 illustrative.

None of this means the outcome is fixed. It means the model has no opinion, which is a different and more useful statement. Everything that could make the structure worth anything has to come from a disagreement with the price, and for a butterfly that disagreement has a specific shape: it is a view that realised volatility will come in below what the options were priced at. Rerun the terminal distribution at different realised volatilities while leaving the entry price alone and the expectation moves from Rs 1,637 illustrative at 7.5 percent realised, through zero at 12.5 percent where implied and realised agree, to a loss of Rs 868 illustrative at 19 percent.

So the honest description of a butterfly is not that it is a bet on a price. It is a bet on a volatility, expressed with a payoff that also requires the price to cooperate. That double requirement is what separates it from a straddle, which needs only the volatility view to be right. Whether implied volatility is high or low relative to its own history is a separate measurement with its own pitfalls, and our page on implied volatility rank and percentile works through why that measurement is harder than the two numbers make it look, while what implied volatility actually is covers the underlying idea.

The pinning question, and what this model cannot answer

Butterflies are frequently justified by an argument that goes roughly as follows. Enormous open interest builds up at round strikes. Dealers who are short those options hedge their positions dynamically, and the hedging flow buys weakness and sells strength near the strike. The result is that the underlying gets drawn toward the heavily traded strike as expiry approaches, which is exactly what a butterfly needs. If that is true, the probability of landing on the peak is higher than a plain distribution suggests, and everything computed in the section above is too pessimistic.

The honest position on this page is that the model above cannot test that claim, and that saying so is worth more than producing a number that looks like support. The simulation contains a price process and nothing else. It has no order book, no open interest, no dealer inventory and no hedging rule. Feeding it a distribution of open interest across strikes would change nothing at all, because there is no channel through which that information could reach the price. A model that cannot represent a mechanism cannot provide evidence about it, and a computation that ignores the mechanism and reports a probability anyway is not neutral. It is an assertion wearing a decimal point.

What can be done is to show what the claim would have to be worth to matter. Insert a restoring pull toward the middle strike by hand, ramping up over the final third of the life, and vary its strength. Every point on the resulting curve is an assumption, and the parameter has no measurement behind it. The exercise is not evidence that pinning exists. It is a sensitivity, showing how much pinning would be needed before the conclusion changed.

With no pull at all the model puts 23.4 percent of outcomes inside the band, which reproduces the closed-form answer and confirms the simulation is doing what it should. At a pull strength of 0.5 that becomes 24.9 percent, at 1.0 it is 26.7 percent, and at the largest strength tested, 4.0, it reaches 38.4 percent. That last figure sounds like it settles the argument in favour of the structure until you look at the other line on the same chart. The chance of finishing within fifty points of the strike, which is where the payoff is actually large, moves only from 4.6 percent to 7.7 percent even under the most aggressive assumption. A pull that concentrates the distribution enough to matter for the band barely concentrates it at all where the peak is.

There is a deeper reason the simulation cannot help here, and it is worth stating because it applies to a lot of modelling that looks more rigorous than it is. A pull toward the strike and a lower realised volatility deform the terminal distribution in the same direction. Both narrow it. A model of this kind fitted to terminal prices could not tell the two apart, so even a good fit would not identify which mechanism produced it. Distinguishing them needs intraday order flow and dealer positioning data, which is a different exercise on a different dataset, and it is not the exercise this page ran.

Two further things are worth saying plainly. The classic evidence for pinning in the academic literature concerns single stocks, where a physical delivery obligation and identifiable dealer inventory give the effect a concrete channel. Index options in India are cash settled, which removes the delivery obligation that gives the single-stock version of the effect one of its channels, and this page has not verified any measurement of pinning in Indian index options from a primary source. And even if the effect were real and were measured, it would be a property of specific strikes on specific expiries, not a general licence to assume the peak is more reachable than the price of the structure implies. Where open interest concentration is a legitimate and readable signal is a separate topic, covered in what open interest does and does not tell you and in how to read an option chain.

Nearly worthless, and then suddenly worth everything

The expiry diagram is a picture of one instant. For the whole month before that instant the position behaves very differently, and the difference governs whether the structure can be exited at all. A butterfly is long two options and short two options, so its value at any moment is a small difference between large numbers. Early in its life that difference barely moves, because time value is draining out of all four legs at similar rates and the short middle legs offset the long wings almost exactly.

Mark the structure every day with the index held exactly at the middle strike, which is the most favourable possible path, and the numbers are stark. At thirty days the position is worth its debit, by construction. At twenty one days it is worth 48.71 points, a gain of 7.95 against a maximum of 259.24, which is 3.1 percent of what is available. At fourteen days, the halfway point, it is worth 59.43, a gain of 18.67 points or 7.2 percent of the maximum. At seven days it has reached 16.1 percent. At three days, 30.5 percent. At one day, 52.9 percent. The remaining 47 percent arrives in the final session, when the last of the middle legs' time value disappears and the wing intrinsic is all that is left.

Put the other way round: 83.9 percent of the entire gain arrives in the last seven days, and 69.5 percent in the last three. And that is on the perfect path, with the index sitting motionless on the strike the whole time. On any other path the numbers are worse.

This has a practical consequence that the payoff diagram cannot show. A butterfly held at a profit is not a position that can be banked early. At the halfway mark, on the best possible path, there is 18.67 points of profit on the screen against a Rs 2,649.28 illustrative debit. Closing there costs four more chargeable orders and four more crossings of the bid-ask spread, which as the next section computes runs to more than the profit on the screen. The structure has to be carried into the last few days to be worth anything, which means carrying it through exactly the period when the underlying has the least time left to come back if it moves away.

It also means the position is nearly insensitive to being right early and highly sensitive to being right at the end. An index that sits on the strike for twenty five days and then moves 400 points in the final three produces the maximum loss, and the twenty five correct days contribute nothing. The reverse is equally true. The behaviour of options in the last few sessions before settlement is its own subject, and what actually happens on expiry day covers the mechanics that this convergence assumes.

Four legs of charges, and the tax that is largest exactly at the peak

Cost models for options usually describe a round trip on a single instrument. A butterfly is four contracts, every statutory levy is computed on the leg it applies to rather than on the net debit, and the debit is tiny compared with the premium the four legs move between them. That mismatch is the whole of this section, and it is more extreme for a butterfly than for any other common structure.

The rates used are the verified ones. Securities transaction tax on the sale of an option is 0.15 percent of the premium and falls on the seller, a rate set from 1 April 2026 by section 159 of the Finance Act 2026 amending serial 4 of the section 98 table of the Finance (No. 2) Act 2004. Securities transaction tax on an exercised option is 0.15 percent of intrinsic value and falls on the purchaser. Stamp duty on options is 0.003 percent on the buy side only, under Article 56A(d) of Schedule I to the Indian Stamp Act 1899. Goods and services tax at 18 percent applies to brokerage and to the exchange transaction charge, and not to the transaction tax or the stamp duty, which pass through under the pure agent route. The exchange transaction charge is modelled at Rs 3,250 per crore of premium turnover on index options, the rate notified by an exchange notice of 27 September 2024 with effect from 1 October 2024. Brokerage is a commercial charge rather than a statutory one, and the model assumes an illustrative flat Rs 20 per executed order.

One rate is missing rather than guessed. The regulator's turnover fee is verified at Rs 10 per crore for the cash segment under regulation 41(1) of the SEBI (Stock Brokers) Regulations, 2026, but no primary source was available for the corresponding derivatives-segment figure, so it is left out of the model entirely. Every cost figure below is therefore an understatement rather than an overstatement, which is the direction an unverified omission should err in. The sibling pages on the iron condor and the calendar spread reached the same conclusion on the same evidence and omit it too.

Apply that stack to the four legs and the first number is the striking one. The four opening premiums sum to 1,680.44 points, which at the illustrative multiplier is Rs 1,09,228.70 of premium turnover, against a debit of Rs 2,649.28. Turnover is 41.2 times the net premium, which for this structure is also the entire amount at risk. The iron condor page computes the same ratio, turnover against net premium, at 4.59 times for its four legs, so the butterfly is roughly nine times worse on this measure, and the reason is entirely the deep leg. Buying a call 300 points in the money means paying 594.84 points of premium for something whose contribution to the debit is almost cancelled by the legs around it. The exchange charge and the transaction tax are computed on that 594.84, not on the 40.76 the position actually costs.

The totals are smaller than the multiple suggests, and the composition is the useful part. Opening the position costs Rs 217.90: transaction tax Rs 79.93, brokerage Rs 80.00, the exchange charge Rs 35.50, tax on brokerage and the exchange charge Rs 20.79, and stamp duty Rs 1.68. Left to settle rather than closed, one further charge arrives, and it is the one worth the section title. At settlement the long 23,700 call is 300 points in the money and is exercised, and securities transaction tax on an exercised option falls on the purchaser, which is you. That is Rs 29.25, taking the total to Rs 247.15. Closed out with four more orders instead, the round trip is Rs 349.03.

Look at where that exercise charge is largest. At the peak, with the index at 24,000, the long lower call is 300 points in the money and the charge is Rs 29.25. It is not a coincidence that the best outcome carries a charge the worst outcome does not: below 23,700 nothing is in the money and nothing is exercised, so the maximum loss case pays no exercise charge at all. Further up it gets worse in a way the payoff diagram cannot show, because above the upper strike both bought legs finish in the money. At 24,300 the exercise charge is Rs 58.50, at 25,000 it is Rs 195.00 and at 25,500 it is Rs 292.50, while the payoff sits flat at the maximum loss of Rs 2,649.28 across all three. The general mechanic, that the leg whose function is to cap the loss is the one that bills you at settlement, is worked through in more depth on the iron condor page. What is specific to the butterfly is that the charge is also present, and unavoidable, at the single best outcome the structure has.

Then there is the cost that never appears on a contract note. Four legs means the bid-ask spread is crossed four times on entry, and four more times if the position is closed rather than left to settle. At an illustrative 1.5 points of spread per leg, entry alone costs 6 points, or Rs 390 illustrative, which is well over the entire statutory stack. That figure is an assumption offered as a sensitivity rather than a measurement: at 0.5 points per leg the cost is Rs 130 and at 3 points it is Rs 780. The calendar spread page reached the same verdict from a two-leg structure, that the spread crossed on the illiquid leg exceeded the whole tax stack, and the butterfly inherits the problem twice over because its deep in-the-money leg is the least actively traded of the three strikes.

Adding the modelled stack to an illustrative 1.5 points per leg gives an effective entry cost of 50.11 points against a quoted 40.76. What that does is instructive precisely because it is uneven. The breakevens barely move, from 23,740.76 and 24,259.24 to 23,750.56 and 24,249.89, narrowing the band by 19.15 points out of 518.48 and cutting the probability of landing inside it from 23.49 percent to 22.65 percent. The peak falls from Rs 16,850.72 to Rs 16,213.57 illustrative, a reduction of under four percent. But the reward-to-risk ratio falls from 6.36 to 4.98, because costs are charged against the debit and the debit is the small number. A fifth of the headline ratio is gone, and none of it is visible on the payoff diagram.

Two sensitivities are worth recording because they change the answer more than anything in the options themselves. The whole cost profile turns on brokerage, which has nothing to do with options: at Rs 10 per order the settlement stack is Rs 199.95, at Rs 20 it is Rs 247.15 and at Rs 40 it is Rs 341.55. And if the two middle contracts are sent as a single order rather than two, entry costs Rs 194.30 instead of Rs 217.90. Finally, the honest summary of the whole cost section: on this model, all-in costs of Rs 637.15 illustrative require realised volatility to come in at 10.01 percent against an implied 12.50 percent simply to break even, a gap of 2.49 volatility points that has to be forecast correctly before the structure returns anything at all.

The iron butterfly is the same shape, assembled from different parts

The variant most people meet first is the iron butterfly, and the relationship between the two is exact rather than approximate. Instead of four calls, the iron butterfly sells a call and a put at the middle strike and buys a put at the lower strike and a call at the upper one. It is still four contracts across three strikes, and it still produces a tent with its peak at the middle strike and zero value at the outer ones. The visible difference is the cash flow: the call butterfly is paid for on entry, and the iron butterfly is received on entry.

On the same strikes and the same volatility the iron butterfly brings in a credit of 257.64 points, or Rs 16,746.82 illustrative, against a maximum loss of 42.36 points, or Rs 2,753.18. Its breakevens are 23,742.36 and 24,257.64, a band of 515.29 points or 2.15 percent, and the model puts 23.35 percent of outcomes inside it. Those figures are almost identical to the call butterfly's, and they are not close by accident. Put-call parity forces the credit to equal the wing width discounted to the present, less the call butterfly's debit. The residual between them, 1.60 points, is exactly the interest on 300 points for thirty days at the stated rate. Anyone who finds a larger gap than that between the two structures has found a quoting artefact or a mispricing, not a better structure.

Where they genuinely differ is in what the charge stack sees, and the difference runs the other way from the intuition that a credit structure must be more expensive. The call butterfly moves Rs 1,09,228.70 of premium turnover; the iron butterfly moves Rs 73,208.56, a third less, because it uses only at-the-money and out-of-the-money options and never touches the 594.84 point deep leg. Entry charges are Rs 190.79 against Rs 217.90. And at the peak, where the index finishes exactly at the middle strike, every leg of the iron butterfly is at or out of the money, so nothing is exercised and the exercise charge is zero rather than Rs 29.25. On this model the iron butterfly is the cheaper way to buy the same payoff, by Rs 56.36 illustrative on entry and settlement combined.

Two practical caveats belong with that. The iron butterfly has two sold legs, so it attracts an initial margin requirement that the fully paid call butterfly does not, and the capital tied up is a real cost even though it is not a charge. And selling an at-the-money put alongside an at-the-money call means the position is short two of the most liquid contracts on the board, which usually helps on the spread but concentrates the assignment question at one strike.

How the structure fails

Defined risk is not the same as low risk, and a structure whose worst case is small can still be a reliable way to lose money if the worst case arrives often enough. The failure modes below are not hypothetical: every one of them is a number from the model above, restated as the thing that goes wrong.

The last row is the one that matters most and it is the one a payoff diagram is least able to convey. A structure that risks a small sum feels like it can be run repeatedly without consequence, and it is that feeling, rather than any single position, that does the damage. The regulator's own measurement of the outcome across the whole retail derivatives population is the relevant frame: about 93% of individual traders in equity derivatives made net losses over FY22 to FY24, with aggregate net losses exceeding Rs 1.8 lakh crore (SEBI, September 2024).

What the computation is actually for

It would be easy to read this page as an argument against butterflies. It is not. It is an argument against reading a reward-to-risk ratio as though it were free information, and the butterfly is simply the clearest place to see why, because its profitable region comes to a point rather than a plateau, which pushes the ratio up and the probability down by the same amount and in the same breath.

The transferable habit is the one this page performed rather than described. Whenever a structure is quoted with a ratio, compute the probability attached to it under a stated distribution, and then compute the expectation. If the expectation comes out at zero, the structure is priced and the ratio told you nothing. If it comes out positive, the next question is which assumption is carrying the result, and for a butterfly the answer is always the volatility assumption. That sequence takes a few lines of arithmetic and it converts a persuasive number into a testable one. Our options payoff calculator will draw the shape for any combination of legs, but the shape is the easy part, and the probability and the expectation are the parts a payoff drawing will never supply.

The second habit is to compute the charge stack per leg before deciding whether a structure is available at all. Multi-leg positions at retail size are not made unviable by the tax on them; on this model the statutory levies on entry come to Rs 117.11, and Rs 146.36 including the tax on exercise. They are made difficult by the per-order component and by the spread, both of which scale with the number of legs and neither of which appears in a payoff diagram. Any structure whose case rests on a ratio computed before costs has not been evaluated, and the more legs it has, the further from evaluated it is.

What remains after all of that is a legitimate and quite narrow use: a butterfly is the cheapest way to express a precise view about where something will settle, and its price is an honest quotation of how unlikely that view is. Reading the quotation correctly is the skill. Building the arithmetic that reads it, rather than accepting the diagram that hides it, is the habit that separates a considered position from a hopeful one, and if that distinction is the interesting part of this page rather than the tedious part, that is the method we teach.

Every figure comes from one deterministic model, seeded so that it reproduces identically on each run. The four legs are priced with Black-Scholes on an illustrative index at 24,000, strikes 300 points apart, thirty days to expiry, a flat implied volatility of 12.5 percent and an interest rate of 6.5 percent. Probabilities and expectations are computed under the same lognormal distribution that produced those prices, in closed form where a closed form exists and by Monte Carlo across 400,000 paths where it does not, with the two cross-checked against each other and the difference reported. The pinning section adds a restoring pull toward the middle strike whose strength is an assumed parameter, stated as such and never presented as a measurement. Charges are computed leg by leg from the primary instruments cited in the text; brokerage, the bid-ask spread and the contract multiplier are illustrative assumptions and are labelled wherever they appear.

All results are illustrative and simulated. They are not a track record, not a forecast, and not an indication of what any structure would produce in a live account. The purpose is to demonstrate the relationship between a reward to risk ratio, the probability attached to it and the expectation that follows, which is a property of how options are priced rather than of any particular market.

• The iron condor, and eight chargeable events against one credit →
• The calendar spread and the decay differential →
• Vertical spreads, the simplest defined-risk structure →
• Straddles and strangles: paying for movement →
• The framework for thinking about options →