Implied volatility and the smile

Every chapter so far has put volatility in and taken a price out. This one does it the other way round, which is how the market actually works.

Implied volatility, defined precisely

Implied volatility is the volatility that makes the formula reproduce the market price. Four inputs are observable; the fifth is solved for.

The inversion is always well-defined for an admissible price, and chapter 10 supplies the reason: vega is strictly positive, so the price is strictly increasing in $\sigma$. One price, one volatility, no ambiguity.

There is no closed form for it, so it is found numerically — bisection or Newton's method on the pricing formula. That is a detail, except for one consequence that matters: a price outside chapter 1's bounds has no implied volatility at all. The solver does not return a bad answer; it returns nothing, because no $\sigma$ can produce that price. Keep that in mind for the worked problem.

And what it is not. Derman and Kani are explicit: implied volatility is the constant volatility "that makes the BS formula value equal to the market price," and "in essence, $\sigma_{\text{imp}}$ is a means of quoting prices." It is a change of units, not a measurement of the future. Quoting an option at 14% rather than ₹450.57 is like quoting a bond at a yield rather than a price — more comparable across contracts, and no more informative.

The problem: they do not agree

If the model were right, this chapter would be two paragraphs long. Derman and Kani state the implication exactly:

The Black-Scholes model assumes that the index level executes a random walk with a constant volatility. If the Black-Scholes model is correct, then the index distribution at any options expiration is log-normal, and all options on the index must have the same implied volatility.

They do not have the same implied volatility, and have not for forty years:

But, ever since the '87 crash, the market's implied Black-Scholes volatilities for index options have shown a negative relationship between implied volatilities and strike prices — out-of-the-money puts trade at higher implied volatilities than out-of-the-money calls.

Their evidence is specific and dated. The downward slope of implied volatility against strike for 44-day S&P 500 options on 5 May 1993; and the same structure in 47-day European-style March options on 31 January 1994, across strikes from 90% to 105% of spot, with call prices used above spot and put prices below.

And the verdict on persistence: "Though the exact shape and magnitude vary from day-to-day, the asymmetry persists and belies the BS theory."

Two shapes, two names. The downward slope across strikes is the skew. The variation of at-the-money implied volatility with expiry — which the same source documents rising with time to expiration — is the term structure. Collectively, the smile.

What the skew actually says

This is where care is needed, because the skew gets over-read constantly.

What it says about the distribution. Derman and Kani put it precisely: by varying volatility with strike, "traders are implicitly attributing a unique non-lognormal distribution to the index." Charging more for downside strikes means pricing a fatter left tail than lognormal — a larger chance of a big fall than the model's distribution allows.

That is a real and defensible claim. Equity indices do fall faster than they rise; the 1987 crash is the event that taught the market so, and the skew dates from it.

What it does not say. Chapter 3's result forbids reading the skew as a probability forecast. The prices contain a risk-neutral distribution, which differs from any real-world distribution by the premium investors demand for bearing that risk. Downside options are insurance, and insurance trades above expected loss.

So a steep skew is some mixture of two things — a belief about crash probability, and the price of wanting protection — and no amount of arithmetic separates them. A commentary that reads "the options market is pricing a 15% chance of a crash" has silently assumed the risk premium is zero.

The Behavioural finance subject supplies the demand side. Investors' documented preference for lottery-like payoffs explains persistent demand at far strikes; the desire not to lose explains persistent demand for downside protection. Both push implied volatilities up at the wings, and neither is a forecast.

Why the industry kept a model it knows is wrong

A reasonable reader objects: if the constant-volatility assumption is false, why is the formula still used to quote prices?

Because it works as a language. A price of ₹450.57 means nothing without knowing the index, strike, expiry and rate. "14% volatility" is comparable across every contract on the board, which is what a quoting convention is for.

Because the alternatives cost more than they fix. Chapter 7's argument, in Derman and Kani's words: stochastic volatility and jump models lose the preference-free property, since there are no securities with which to hedge volatility or jump risk directly, and they introduce "several additional parameters whose values must be estimated," which "often makes confident option pricing difficult."

And because the inconsistency is contained. The same source draws the line: "It may be convenient to keep quoting options prices in terms of Black-Scholes-equivalent volatilities, but it is probably incorrect to calculate options prices using the BS formula." Quoting with it is fine. Pricing a new contract with a single volatility is not. That distinction is the professional practice, and the implied-tree method their paper introduces is one way of respecting it.

Computing a smile from a chain

NSE publishes a worked example chain in its volatility index white paper, and inverting it is the exercise this chapter exists for. The chain gives best bid and ask by strike, a near-month future at 5,129, an at-the-money strike $K_0$ of 5,100, 0.02466 years to expiry and a 3.90% rate.

Taking the midpoint of each spread and solving for the volatility that reproduces it, using the forward form so the futures price does the work:

Strike	$K/F$	Out-of-money option	Mid price	Implied volatility
3,800	0.741	put	₹0.45	69.7%
4,000	0.780	put	₹0.77	62.3%
4,200	0.819	put	₹1.10	53.3%
4,400	0.858	put	₹1.95	45.6%
4,600	0.897	put	₹4.47	39.0%
4,800	0.936	put	₹13.30	33.8%
5,000	0.975	put	₹40.45	28.6%
5,100	0.994	put	₹74.45	27.6%
5,100	0.994	call	₹144.75	40.5%
5,300	1.033	call	₹79.05	45.7%
5,400	1.053	call	₹34.88	38.5%
5,600	1.092	call	₹3.62	30.9%
5,700	1.111	call	₹1.82	32.2%

Implied volatilities computed from the quoted mid prices in the source's example chain, not published by NSE.

The put wing is a textbook skew — monotone, steep, 27.6% at the money rising to 69.7% at three-quarters of the forward. That is the shape Derman and Kani describe.

And then the chain fails its own consistency tests, which is the more valuable half of the exercise.

Working the problem

What can be concluded about the market, and what only about the data?

Test 1 — the same-strike test, which parity forces. A call and a put on the same strike and expiry must imply the same volatility. Chapter 2's parity fixes the difference between their prices exactly; so if one price is consistent with a volatility, the other is consistent with the same one. There is no freedom.

At the 5,100 strike, the put implies 27.6% and the call 40.5%. They cannot both be right, and the gap is not small. This is chapter 2's ₹41.33 parity failure, converted into volatility units where it is easier to see.

Test 2 — the bounds. Chapter 1's floor for a put, in forward form, is $e^{-rT}(K-F)$. For the 5,500 put that is ₹370.64 — and the quoted mid is ₹368.00, which is ₹2.64 below a limit that holds on arithmetic. There is no implied volatility for that quote, which is why the row is absent from the table above. The bid, at ₹340, is ₹30.64 below the floor.

Test 3 — the spread filter, which the source itself applies. NSE discards any strike whose spread exceeds 30% of the mid price. On this chain the 3,900 put's spread is 78% of its mid, the 4,300 put's 35%, the 4,100 put's 32%. Those implied volatilities sit inside a bid-ask band wide enough to swallow the answer — the 3,900 put at ₹0.35 bid and ₹0.80 ask implies 62.6% at the bid and 68.2% at the ask, a range of nearly six volatility points for a single strike. The mid of 65.8% is a convention, not a measurement.

So the honest conclusions divide cleanly.

About the market: the put wing's shape is real and matches what the literature documents — downside strikes trade at higher implied volatilities, and the slope is steep. That is a genuine structural feature of index options, in India as elsewhere.

About the data: nothing. The chain fails the same-strike test by 13 volatility points, violates a no-arbitrage bound at one strike, and carries spreads up to 78% of mid at others. It is a hypothetical example, as the white paper says it is, published to demonstrate a calculation rather than to be internally coherent.

And the transferable method is the point. Those three tests — parity at a strike, the arbitrage bounds, the spread as a fraction of mid — take about a minute and should precede any conclusion drawn from any chain. Chapters 1, 2 and 10 are not preliminaries to this chapter; they are its instruments.

One last limit, stated plainly. Even a clean chain with a clean skew does not tell you whether options are expensive. Comparing today's implied volatility with its own history tells you where the quote sits in its range; comparing implied with subsequently realised volatility tells you, after the fact, whether it was too high. Neither is a trading signal, and the Technical analysis subject's conclusion about rules found in past data applies with full force here.

The point

Implied volatility is the volatility that makes the formula match the price — a unique number, since vega is positive, and a quoting convention rather than a measurement. Options on the same underlying imply different volatilities, with downside strikes dearer, and have done since 1987; the source states that under Black-Scholes they all would have to agree, so the skew means the market is pricing a non-lognormal distribution with a fatter left tail. That shape is part distribution and part the price of protection, and nothing separates them, so a skew is not a crash forecast. The industry keeps the formula as a language while declining to price with a single volatility. And before reading anything from a chain, test it: calls and puts at one strike must imply the same volatility, prices must clear the arbitrage bounds, and a quote whose spread is most of its value carries no information at all.