Wednesday 10 Jun 2026. Three Nifty expiries were trading: the 16 Jun weekly with 6 days left, the 30 Jun monthly with 20 and the 28 Jul monthly with 48. The at the money volatilities were 14.61, 15.03 and 14.77 percent. That is three numbers. The chain holds more than a hundred strikes for each expiry, and every one has its own implied volatility. A volatility surface is the rule that joins them, in strike and in time, without leaving a price that someone could exploit. This module reads the pieces: the conditions a surface must meet, SVI and SABR for one expiry, local and stochastic volatility for the dynamics, and what a fit can and cannot tell you. Every figure comes from NSE end of day files, dated, so nothing here is live and nothing is a view on the next move.
3By the end you can
- State the butterfly and calendar conditions and test them on a dated Nifty chain.
- Say what the parameters of SVI and SABR do, and why a close fit does not make a parameter a market fact.
- Say what local and stochastic volatility add to a fitted surface, and where each one is weak.
The idea, plainly
What is a volatility surface?
An implied volatility belongs to a strike and an expiry. Put the strikes on one axis and the expiries on the other and you have a surface. Models work with two transforms of it. Log moneyness is k, the log of the strike over the forward. Total variance is w, the implied volatility squared times the time to expiry. Total variance adds up over time, which makes it the natural quantity for a term structure, and k removes the drift of the forward, which makes slices of different expiries comparable. The quotes are a set of points. The NSE option chain shows an implied volatility for each strike, and that is a set of points, not a surface. A surface is something you build.
What must a surface satisfy?
Two conditions keep a surface free of static arbitrage, which means no set of trades that costs nothing today and cannot lose at expiry.
- Across strikes, the butterfly condition. The price of a call must be convex in the strike. A butterfly, long one call at each wing and short two in the middle, can never cost less than nothing. To first order the butterfly price is the market's implied density at that strike, discounted, times the square of the wing width.
- Across time, the calendar condition. At the same log moneyness, total variance must not fall as the expiry gets later. A longer option cannot hold less variance than a shorter one.
The maths: the two conditions in symbols
Butterfly: C(K − h)
− 2 C(K) + C(K + h)
is at least zero
Calendar: w(k, T2) is at
least w(k, T1) for T2 > T1
In the limit of a narrow wing the butterfly is the second derivative of the call price in the strike, which is the risk neutral density times the discount factor. A negative second derivative is a negative probability. A model built so that both conditions hold by construction is arbitrage free. A fit to quotes is only arbitrage free if someone checks.
3 terms this section turns on
What does SVI fit?
SVI draws one expiry's total variance with five numbers: a, b, rho, m and sigma. The curve is a hyperbola in k with straight wings. The level is a. The slope of the wings is set by b. The tilt is rho, which rotates the curve so one wing rises faster, and for equity indices it is usually the left wing. The vertex is moved sideways by m, and sigma rounds it. Straight wings are a standard choice because, far from the money, total variance can grow at most linearly in log strike, with a slope of at most 2 (Lee's moment formula). A handful of constraints keeps the lowest point of the curve positive. A check of the fitted curve for the butterfly condition is a separate step.
The maths: raw SVI
w(k) = a + b x (rho x (k − m)
+ sqrt((k − m)^2 + sigma^2))
The constraints are b of at least zero, rho strictly between minus one and one, sigma above zero, and a + b x sigma x sqrt(1 − rho^2) of at least zero. The implied volatility at k is the square root of w over T. Fitting a surface joins expiries by tying the parameters across T, so the calendar condition holds across the whole grid and not slice by slice.
What does SABR fit?
SABR describes the forward as a random walk whose volatility is itself random. It has four parameters: alpha, the level of volatility; beta, which sets how the volatility scales with the level of the forward; rho, the correlation of the forward and its volatility; and nu, the volatility of volatility. Beta is usually fixed, not fitted, and the other three are fitted expiry by expiry. A closed form gives the implied volatility at each strike, so the fit is fast. With beta of one, the case I use here, the skew at the money is rho times nu over two for each unit of log strike, and the curvature comes from nu. The closed form is an expansion. It is weakest for long expiries and far strikes, and in the left wing it can imply a negative density.
The maths: SABR with beta of one
sigma(K) = alpha x (z / x(z))
x {1 + [rho nu alpha / 4
+ (2 − 3 rho^2) nu^2 / 24] T}
z = (nu / alpha) x ln(F / K)
The function x(z) is the log of (sqrt(1 − 2 rho z + z^2) + z − rho) over (1 − rho). Near the money z over x(z) is one less rho z over two, so the slope against ln K is rho nu over two, as in the text. The bracket with T is a small correction that grows with expiry.
What is local volatility?
Local volatility turns the whole surface into one diffusion in which the volatility depends only on the index level and the date. Dupire's formula reads that function straight off the surface, and by construction the model reprices every vanilla option on it. At the forward, and when the smile is flat, the local variance is the forward variance between two expiries. That is the same quantity as the forward volatility module. Away from the forward it also depends on the slope and the curvature of the smile, which is why noisy prices ruin it: the formula takes second derivatives in strike.
The maths: Dupire in total variance
local variance =
(dw / dT) divided by
[1 − (k / w) w_k
+ (1/4)(−1/4 − 1/w + k^2 / w^2) w_k^2
+ (1/2) w_kk]
Here w_k and w_kk are the first and second derivatives of w in k. At k of zero with a flat smile the bracket is one and the local variance is the time derivative of total variance, which two expiries estimate as a difference.
What does stochastic volatility add?
In a stochastic volatility model the variance moves on its own. One popular version has five parameters: the starting variance, a long run variance, a speed of reversion to it, the volatility of variance and a correlation with the index. The correlation tilts the smile and the volatility of variance bends it. These models fit longer expiries reasonably. They struggle with the steep skew of the shortest expiries, which is why practitioners add jumps or other features. The worked example below shows that steepness on one chain.
From noisy quotes to a surface and then to a model
Which model for which job?
| Job | What I reach for | What to watch |
|---|---|---|
| Mark one expiry | SVI or SABR | Parameters that mean nothing alone |
| Interpolate strikes | A fitted slice, checked | Butterfly condition off the quotes |
| Price a path dependent option | Local or stochastic volatility | The smile's future shape |
| Hedge a book | Whichever matches the dynamics seen | The module on skew dynamics |
Worked example: 10 Jun 2026
Do the quotes meet the conditions?
Take the 30 Jun expiry and price a butterfly with wings of 200 points at eight strikes, using puts for the centres below the forward and calls for those above it. Every price is positive, and they trace the shape of a distribution, highest near the money, 22.50 at the 23,400 strike, and thin in the wings. At the 25,000 strike the butterfly costs 0.80.
At 200 points the butterflies are all positive
Total variance rises with the expiry, as it must
Read 2 more paragraphs, 143 words
Narrow the wings and the answer changes. The 23,550, 23,600 and 23,650 calls settled at 186.65, 171.00 and 152.70. A butterfly of 50 point wings on them is worth −2.65, below zero. At the 25,000 strike a butterfly of 100 point wings is −0.30. These are not free trades. A settlement price is not a quote anyone offered, and the tick is coarse against a small butterfly. That is why a raw chain cannot be fed into a formula that takes second derivatives, and why a fitted slice is a repair of the quotes and not a measurement.
Across time, total variance for the three expiries was 3.51, 12.38 and 28.68 in points squared times years, with times to expiry of 0.0164, 0.0548 and 0.1315 years. It rises with the expiry, as the calendar condition needs.
How does the skew change with the expiry?
The shortest expiry has the steepest left side
Read the put wing five percent below the forward against the at the money volatility. For the 6 day expiry the gap was 5.02 points, for the 20 day expiry 2.27 and for the 48 day expiry 1.81. The shorter the expiry, the steeper the tilt per unit of log moneyness, which is the pattern SVI and SABR must reproduce expiry by expiry and a one factor stochastic volatility model has trouble with.
The chart also shows the two nearer expiries curling up on the right, a call wing. That matters below: it is why the SVI fit for the 30 Jun expiry wants its vertex far to the right.
The wing premium shrinks as the expiry lengthens
What would SABR say, and what would SVI say?
For SABR with beta of one the slope at the money is rho times nu over two. Take the 22,550 put at 16.13 percent and the 23,950 call at 14.34 percent, three percent either side of the forward. The log distance between the strikes is 0.0602, so the slope is −0.297 of volatility for each unit of log strike, which is −0.30 vol points for each one percent move of the strike. Twice the slope gives rho times nu, about −0.59. The data fix the product. They do not say whether rho is near minus one and nu small, or rho is near minus a half and nu larger. The curvature, and other expiries, decide that.
| Parameter | 30 Jun | 28 Jul |
|---|---|---|
| a | −0.0226 | 0.0022 |
| b | 2.018 | 0.011 |
| rho | 0.98 | −0.43 |
| m | 0.353 | 0.039 |
| sigma | 0.062 | 0.032 |
Read 2 more paragraphs, 230 words
I fitted SVI to the traded out of the money strikes within eight percent of the forward for the 30 Jun and 28 Jul expiries, 74 and 75 strikes, with the fitting routine of the pricing engine behind this site's tools. The root mean square error was 0.10 and 0.15 vol points. Both fits pass the butterfly check on the range scanned. The parameters tell a different story.
Both curves track the quotes to within 0.15 vol points, and yet b differs by a factor of about 190, and rho changes sign. The 30 Jun smile has a rising right wing, so the fit moves its low point out to m of 0.353, outside the window it was fitted on, and pins the right wing slope, b times one plus rho, at 4.0, the fit's upper bound. That is twice the limit of 2 that Lee's moment formula sets for the slope of total variance far from the money, so this curve may only be read inside the window it was fitted on. Refit with that stricter limit and b halves to 1.018, at the same error of 0.10 vol points. Five parameters can draw nearly the same curve in many ways inside a narrow window. I read the fit as an interpolator. I do not read the parameters as the market's view of the skew.
What local volatility is implied between the monthlies?
Between the 30 Jun and 28 Jul expiries the forward variance is 212.5 points squared a year, a forward volatility of 14.58 percent, below both at the money volatilities, 15.03 and 14.77. To first order that is the local volatility at the forward for those weeks. Away from the forward the formula also needs the slope and curvature of a smooth fitted surface, which is why the sequence is fit, check, then differentiate. The skew dynamics module asks what the smile does when the index moves, which is where local and stochastic volatility part ways.
Two terms to hold on to from 16.15
The common mistake
The claim: If a model fits the quotes to a tenth of a vol point, its parameters are the market's view of the skew.
They are not. The SVI fits above are close to the quotes, yet the tilt flips sign between two neighbouring expiries and one vertex lands far outside the strikes that were fitted. A fit is evidence that the family can draw the curve. It is not evidence that the parameters mean anything alone, and it says nothing about next day's curve. I check the fit against the butterfly and calendar conditions, compare parameters across expiries, and keep the parameters for the interpolation they were fitted to do.
Key takeaways
- Total variance, sigma squared times T, is the quantity a surface joins.
- No arbitrage means convex call prices in strike and rising total variance in time.
- Raw settlement prices can break the butterfly condition at narrow wings.
- SVI fitted two expiries closely, but the parameters were far apart.
- SABR fixes the skew product rho times nu, not the two alone.
- Fit is not dynamics: local volatility fits the surface and misses how it moves.
Where this section sits in module 16.15
Before you trade. Derivatives trading carries a high risk of loss. SEBI's study "Profitability of Individual Traders in the Equity Derivatives Segment (FY25 to FY26)" (August 2026) found that 87.7% of individual traders in equity F&O made net losses in FY26, with aggregate net losses of Rs 91,685 crore. Read the SEBI study