The Volatility Skew: Why Puts Cost More Than Calls

One volatility for every strike is a modelling convenience

The closed-form option pricing model that every screen and every calculator runs on takes a single number for volatility and applies it to every contract on the underlying. That is not a small assumption tucked into a footnote. It is the load-bearing one. The model works by assuming the underlying moves in continuous lognormal steps whose size is governed by one constant, and once that is granted, a unique fair price falls out for every strike and every expiry at once.

The convenience is enormous and the fiction is obvious the moment you look at a live chain. Because the model maps one volatility to one price, it can be run backwards: take the traded price, hold everything else fixed, and solve for the volatility that would have produced it. That solved number is implied volatility, and the mechanics of backing it out of a premium are set out in our guide to what implied volatility actually measures. The important consequence for this page is what happens when you do that inversion for every strike on the same expiry and write the answers down side by side.

They do not agree. Not slightly, not within a rounding tolerance, but by a margin large enough that the far strikes and the near strikes look like they belong to different instruments. If the model were describing the world, the inversion would return the same number everywhere, because there is only one underlying and one thirty-day horizon. Instead the answers trace a curve, and the curve has a stable, repeatable, direction-specific shape.

There are two ways to read that. The first is that the market is wrong and the model is right, which would make the curve an inefficiency waiting to be traded away. The second is that the model is wrong and the market is right, which would make the curve a correction the market applies to a known modelling defect. The second reading is the one that survives contact with the evidence, and the rest of this page is an attempt to demonstrate it rather than assert it. The demonstration matters because the two readings lead to opposite conclusions about what to do with a rich-looking put.

The practical stake is larger than a debate about model assumptions. If volatility differs by strike, then a trader with a view about direction is unavoidably also holding a view about the shape of the curve, whether or not the view was intended. Any structure that touches more than one strike prices off more than one point on that curve, and the arithmetic of the position depends on all of them. That is the part most retail treatments skip, and it is the part with rupees attached.

The curve the market actually quotes

Every number on this page comes from one stated configuration, so it can be reproduced or disputed. The underlying is an illustrative index at 25,000, which is a round number chosen for arithmetic and not a quote for any real instrument. Expiry is thirty calendar days away. The financing rate is 6.5 percent continuously compounded and there is no dividend adjustment, which puts the forward at 25,133.92 rather than at spot. Volatility at the forward is 14.0 percent.

The curve itself is a stated parametric form. Writing m for the natural logarithm of the strike divided by the forward, implied volatility is 14.0 percent, less 0.55 times m, plus 1.40 times m squared. The linear coefficient is the slope, which is what makes the curve lean, and the quadratic coefficient is the curvature, which is what lifts both wings away from the middle. Those three coefficients are illustrative and representative of an index shape rather than a fit to any particular day's chain. Nothing about the argument depends on their exact values, and the section after next derives a curve of the same shape from a return distribution without using them at all.

Two properties of that shape are worth naming because they get conflated. The lean is the skew: the whole curve tilts, so the put side sits above the call side. The lift is the smile: both wings curl upward away from the low point, which is why the far call strikes stop falling and eventually turn back up. Equity index curves are dominated by the lean, which is why the word skew, or sometimes smirk, is the one that gets used. Currency curves are closer to symmetric and earn the word smile. Reading the shape off a live chain is a mechanical exercise covered in our guide to how an option chain is laid out, which treats the curve as one of the columns you learn to read; this page is about what generates it and what it costs.

Tabulated, the curve looks like this. The premium columns are what the pricing model returns when each strike is fed its own volatility from the curve rather than a shared one.

Read down the volatility column and the size of the effect becomes hard to dismiss. The lowest strike in the table carries 21.80 percent and the highest carries 10.18 percent. That is not a bid-ask artefact or a stale quote. It is a difference of more than a factor of two in the single most important input to the price, on contracts that expire on the same afternoon.

The same distance, two different prices

The cleanest way to size the asymmetry is to hold distance constant and let everything else follow. Take a strike five percent below spot and a strike five percent above it. Same underlying, same expiry, same distance to travel before either contract is worth anything at expiry. The put carries 17.56 percent and the call carries 11.87 percent, and the premiums that fall out are Rs 79.93 against Rs 41.64 per unit of index. The put costs 1.92 times the call for the same distance of movement.

On one illustrative contract that is Rs 5,195.45 against Rs 2,707.60, a difference of Rs 2,489.85. The contract size used for that conversion is 65 units, chosen only so that contract value lands at Rs 16.25 lakh, inside the Rs 15 lakh to Rs 20 lakh minimum contract value band the exchanges now work to. That figure is illustrative and nothing more. Exchange lot sizes are revised periodically to hold contract value inside the band, having been raised in November 2024 when the index lot moved from 25 to 75 and revised downwards again from January 2026 as index levels climbed, so anyone reproducing this arithmetic should substitute the size in force on the day rather than treat this one as current.

It is worth separating how much of that put premium is the curve and how much is simply the level. Priced at the volatility that sits at the forward, with no skew at all, the same put would cost Rs 34.90. The curve adds Rs 45.03, which is 56.3 percent of the quoted premium. More than half of what the buyer of that put is paying is the shape of the curve rather than the general level of volatility, which is a useful thing to know before treating a rich put as evidence that volatility is high.

Here is the complication the tidy version of this story leaves out. At strikes close to spot the put is not the more expensive contract. Two forces act in opposite directions. The skew makes the put dearer, and the forward, which sits above spot because of financing, makes the call the closer of the two strikes in the terms the model actually works in. Close to the money the second force wins. At two percent from spot the put comes out at Rs 192.57 against Rs 226.42 for the call, so the call is the pricier contract despite carrying 2.26 volatility points less. The crossover on this curve is at about 2.93 percent from spot. Beyond it the skew dominates and the gap widens fast: at ten percent out the put is Rs 22.69 and the call is Rs 0.22.

That refines rather than contradicts the general point made in our comparison of calls against puts as instruments, which isolates the skew by holding the forward at spot and concludes that the put is the richer contract. It is, once you are far enough out. Inside about three percent, on this curve, measured from spot, it is not, and a reader who has been told only the headline will misread the near-the-money chain. Publishing the distance at which the statement becomes true is more useful than repeating the statement.

Where the shape comes from

Everything so far describes the curve. None of it explains why the curve should exist, and an explanation matters, because the two readings offered at the start of this page lead to opposite conclusions. If the skew is a demand artefact then it is something to be traded against. If it is what the return distribution implies then harvesting it means selling something for what it is worth and calling the proceeds an edge.

The test is straightforward to construct. Generate two return series. Give them exactly the same volatility, so that the single input the pricing model cares about is identical. Make one symmetric, which is what the model assumes, and give the other the negative skew and fatter left tail that equity indices actually display. Then price options off each series by simulation, invert the resulting prices back to implied volatility strike by strike, and look at the two curves.

The parameters are these. Both series are simulated to a thirty-day horizon at an annualised volatility of 16.0 percent, over 60,00,000 paths, with the same random draws used for the diffusion component of each so the comparison is like for like. The symmetric series is plain lognormal. The asymmetric series adds occasional downward jumps, at a rate of two a year, averaging 4.5 percent in logarithmic terms with a dispersion of 5.5 percent, and its continuous component is reduced to 12.45 percent so that total variance comes out identical rather than merely similar. That last step is what makes the comparison honest: the jumps are not extra volatility bolted on, they are a reallocation of the same volatility into a different shape.

The two distributions that come out look like this.

Now price options off each of them and invert. The symmetric series is the control, and it is the reason the result can be trusted. If the machinery were manufacturing shape out of simulation noise or an inversion bug, the symmetric series would show it, because a plain lognormal series must return a flat line at the volatility it was given. It returns 15.996 percent at its lowest strike and 16.016 percent at its highest against an input of 16.0 percent, a maximum error of 0.016 of a volatility point across the whole ladder, with the largest simulation standard error at any strike being 0.0158 of a point. The machinery is clean.

That is the payoff of the page. Nothing in the second simulation contains an opinion about crash insurance, a demand imbalance, or a hedging flow. It contains a return distribution with a realistic negative skew and nothing else, and the options priced off it come back with a downward sloping implied volatility curve that runs from 22.45 percent at the lowest strike to 14.09 percent at its trough before curling back up. The market is not adding a skew to a symmetric world. The pricing model is subtracting one from an asymmetric world, and the curve is what the market puts back.

The comparison against the quoted curve produces the inconvenient result, and it is worth keeping. The distribution generates a gap of 3.57 volatility points between the five percent put and the five percent call. The quoted curve shows 5.69. The distribution accounts for 62.8 percent of what the market charges, and the remaining portion is not a failure of the model. It is the premium the market adds on top of the raw distribution for bearing a left tail that cannot be hedged away in the moment it matters. Sellers of downside protection are compensated for more than the statistical frequency of a crash, because the frequency is not the whole of the risk: the correlation of a crash with everything else the seller owns is part of it too.

Which means the honest verdict sits between the two readings offered earlier. Most of the skew is the distribution. Some of it is a premium on top. Neither part is free money, and the part that is a premium is compensation for a risk that shows up all at once.

Anything that spans two strikes is a position on the curve

The consequence of a strike-dependent volatility is that a structure with legs at different strikes is priced off different points on a curve, and pricing it off one point instead gets a systematically wrong answer. This is not a subtle effect visible only to a desk. It is large, it is signed, and it is computable in advance.

Take four ordinary two-leg structures and price each of them twice: once with a single flat volatility, the way a calculator fed the at-the-money number would, and once with each leg taking its own volatility off the curve. Everything else is held identical.

Work through the first one leg by leg, because the direction of the error is not the one most people guess. The structure buys the 24,000 put and sells the 23,000 put. Priced flat, the long leg costs 60.86 and the short leg brings in 4.55, for a net debit of 56.31 points. Priced off the curve, the long leg carries 16.84 percent volatility and costs 106.67, while the short leg carries 19.98 percent and brings in 35.88. The sold leg gets the bigger lift in volatility, 5.98 points against 2.84 for the bought leg, so the naive expectation is that the skew makes the spread cheaper.

It does the opposite. The net debit rises from 56.31 to 70.79 points, an increase of 14.48 points or 25.7 percent. The reason is that sensitivity to volatility is not constant across strikes: the far strike has much less of it, so a larger move in its volatility still produces a smaller move in its price. The bought leg gains 45.81 points of premium and the sold leg only 31.33. Intuition about which leg the skew favours is unreliable, and the arithmetic is not.

Everything downstream of the debit moves with it. Maximum profit on the structure, which is the distance between the strikes less what was paid, falls from 943.69 points to 929.21. The breakeven at expiry moves from 23,943.69 to 23,929.21. A trader who sized the position off the flat number planned around a cost that was 25.7 percent too low.

The last row is the one that should end the argument. A risk reversal, which buys the five percent call and sells the five percent put, comes out of the flat model as a debit of 37.93 points. Priced off the curve it is a credit of 38.29 points. The single volatility does not merely get the size wrong, it gets the direction of the cash flow wrong, which means a trader relying on it would be expecting to pay for a structure that in fact pays them, and would have no idea why. That is the purest available demonstration that a risk reversal is a position on the shape of the curve wearing a directional costume.

It is worth putting the size of this error next to the cost item retail traders actually watch. On one illustrative contract, entering and exiting the put spread costs Rs 115.68 across four chargeable events, which works out at 1.78 index points on the breakeven. The charge stack is built from securities transaction tax at 0.15 percent of premium on every sale of an option, stamp duty at 0.003 percent on the buy side, an exchange transaction charge on premium turnover, goods and services tax at 18 percent on brokerage and the exchange charge, and an illustrative flat brokerage. The mispricing introduced by using one volatility instead of the curve is Rs 941.20 on the same contract, which is 8.1 times the entire round trip of charges. The item nobody checks is worth several times the item everybody argues about, and it is available for free before the order is placed.

None of the four structures above is presented as something to trade. The relevant context is that 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), and a page that explains how these instruments are priced is not a page that suggests using them. The mechanics of what a two-leg debit structure does at expiry are covered separately in our treatment of how a vertical spread behaves, which works the payoff, the breakevens and the charge stack in full; the point being added here is only that the entry price itself depends on a curve most descriptions of these structures never mention.

The curve moves, and it moves in two ways at once

A curve that sat still would be a nuisance to model and nothing more. What makes it consequential is that it moves, and that it moves along two independent axes. The level of the whole curve rises and falls, which is the thing most traders already watch. The slope of the curve steepens and flattens, which is the thing most traders do not watch, and which can move a position on its own with the level unchanged.

Model three regimes on the same strike ladder. A calm regime with volatility at the forward of 11.0 percent and a shallow slope, the base regime used throughout this page, and a stressed regime with volatility at the forward of 22.0 percent and a much steeper one. Spot is held at 25,000 and time to expiry is held at thirty days in all three, so nothing moves except the curve itself.

This is the distinction a single headline volatility number cannot carry. India VIX and its relatives summarise the level of the curve into one figure, and our page on what that index measures and what it cannot works through how the number is constructed. It is a reading of height. It says nothing about lean, and two chains with the same headline volatility can price a five percent put twenty percent apart if their slopes differ.

What that does to positions already held is the table below. The marks are recomputed with spot and time frozen, so every change is attributable to the curve and to nothing else. The last column is the important one: it isolates the shape by holding volatility at the forward fixed at the base level and steepening only the slope and curvature, which separates the two axes that the middle columns confound.

Two rows deserve reading twice. The straddle at the money is the position most people think of as the pure volatility trade, and on level it is: its mark rises by 465.89 points when the curve lifts into stress. On shape alone, with the level held fixed, it moves by only 12.20 points. It is almost blind to the lean, which is exactly what makes it a clean expression of level and a useless one for anything else.

The risk reversal is the mirror image. Its mark falls by 58.14 points on shape alone, which is 152 percent of where it started, and it does so with the level of volatility completely unchanged. A trader holding that structure who watches only a headline volatility figure has no instrument capable of seeing what is happening to the position. The steepening that hurts it is invisible in the number being monitored.

The general rule falls out of the table without needing to be asserted. A steepening curve marks up anything long the put wing and marks down anything short it, and the effect grows with distance from spot because the far strikes are where the slope has the most room to work. A flattening curve does the reverse. Neither movement requires the index to go anywhere.

The other axis, in one paragraph

Strike is not the only dimension along which volatility varies. Hold the strike fixed and walk out through the expiries and the number changes again, which is the term structure. It matters here for one specific reason: the steepness of the skew is itself a function of maturity, so a curve read on a weekly expiry and a curve read on a quarterly one are not the same object and should not be compared as though they were.

Priced off the same simulated return process used earlier, at the same distance of five percent either side of spot, the gap between the put and the call is 8.33 volatility points at seven days, 3.57 at thirty, 2.20 at sixty and 1.60 at ninety. The mechanism is visible in the return distribution itself, whose negative skewness weakens from 1.77 at seven days to 0.49 at ninety. A jump is a large share of what can happen in a week and a small share of what can happen in a quarter, because ordinary diffusion has three months to accumulate and only five sessions to do so in the first case. Near-dated curves are therefore steep and volatile, and far-dated curves are shallow and slow.

The trading consequences of holding two expiries at once are a subject in their own right and are worked through with computed numbers in our page on what a calendar structure actually is, which owns that ground and prices the decay differential day by day. The only claim being made here is the narrower one: when comparing skews, compare like maturities, because a steep weekly curve and a shallow quarterly one may be describing the identical market.

What the curve is actually for

The temptation, having established that the curve exists and that it is expensive, is to treat it as a signal. It is not one, and the page would be doing damage if it left that door open. The curve prices the cost of insurance at each strike. It contains no information about which way the index will go, and a chain can carry a violently steep put skew through a month in which the index does nothing but rise. Anything that converts the shape into a directional forecast has stopped describing the market and started predicting it.

What the curve is for is arithmetic. Three things follow from it that are usable immediately and require no view of any kind. The first is that a premium cannot be judged rich or cheap against the at-the-money volatility, because a large part of what a far strike costs is the shape rather than the level. On the illustrative curve here, 56.3 percent of the five percent put's premium is the shape. The second is that any structure spanning two strikes must be priced leg by leg off its own points on the curve, because the alternative produces errors that on the four structures modelled above ran from 20.8 percent to more than a hundred percent, and in one case reversed the sign of the cash flow. The third is that a position can move materially with the index perfectly still, because the curve has a shape axis as well as a level axis and a headline volatility figure reports only the second.

None of that is difficult. It is a matter of reading a column that is already on the screen, one row at a time, instead of collapsing it into an average. The reason it gets skipped is not that the mathematics is hard, because it is not, but that the flat-volatility habit is invisible: nothing on a calculator announces that it has just applied a single number to a chain that quotes a different one at every strike. The defence is procedural rather than technical, which is to price every leg at its own strike's volatility and to look at the curve before looking at the premium.

That habit, of taking apart the thing everyone treats as one number and insisting on seeing its parts, is most of what separates a trader who understands an instrument from one who has memorised its payoff diagram. It is also, unglamorously, most of what a serious curriculum in derivatives spends its time on, and if the arithmetic on this page was the interesting part rather than the tedious part, that is the method we teach.

Every figure comes from one deterministic model, seeded at 20260815 so that it reproduces identically on each run, written in Python with numpy as the only dependency. Option prices use the standard closed-form model on an illustrative index at 25,000, thirty days to expiry, a financing rate of 6.5 percent continuously compounded and no dividend adjustment. Implied volatilities are inverted from prices by bisection to a tolerance of ten decimal places. The quoted curve is the stated quadratic in log-moneyness given in the second section, with coefficients that are illustrative and representative rather than fitted to any chain.

The two-series demonstration simulates 60,00,000 terminal values for each series with antithetic sampling on the diffusion component and common random numbers across the two, prices every strike by averaging discounted payoffs, and inverts the results back to implied volatility. The asymmetric series is a jump process whose continuous volatility is reduced so that total variance matches the symmetric series exactly rather than approximately, and both are simulated under a drift that makes the discounted underlying a martingale, so neither series is given an advantage. The symmetric series is retained as a positive control and returns its input volatility to within 0.016 of a volatility point at every strike, with a maximum simulation standard error of 0.0158 of a point.

Charge rates are taken from the site's verified statutory research. Securities transaction tax on the sale of an option is 0.15 percent of premium, set by section 159 of the Finance Act 2026 with effect from 1 April 2026. Stamp duty on an option is 0.003 percent on the buy side, under Schedule I Article 56A(d) of the Indian Stamp Act 1899. The exchange transaction charge is modelled at Rs 3,250 per crore of premium turnover, the rate notified for index options by exchange notice 20240927-37 with effect from 1 October 2024. Goods and services tax at 18 percent is applied to brokerage and to the exchange charge and not to the transaction tax or the stamp duty. Brokerage is an illustrative flat Rs 20 per order, illustrative because naming a rate card would mean naming a firm. The regulator's turnover fee is verified at Rs 10 per crore for the cash segment under regulation 41(1) of the 2026 broker regulations and no derivatives figure is asserted here; at the cash rate it would add a few paise across the four legs, which changes no conclusion.

All results are illustrative and simulated. They are not a track record, they are not a forecast, and they are not an indication of what any structure would produce in a live account. Lot sizes, strike intervals, charge rates and volatility levels all change; the contract size used here is illustrative and was chosen only so that contract value sits inside the current minimum band. Nothing on this page is advice to enter, avoid or adjust any position.

• What implied volatility actually measures →
• Reading an option chain column by column →
• The term structure, and structures that hold two expiries →
• How a vertical spread behaves at expiry →