Vega, and volatility as a traded quantity

Chapter 9 established what a delta-hedged option owner is really trading: realised volatility against implied. This chapter prices the second half of that.

Vega, and an awkward fact about it

Vega is the sensitivity of the price to volatility.

$\mathcal{V} = \frac{\partial C}{\partial \sigma} = S\,n(d_1)\sqrt{T}$

Quoted conventionally per one percentage point of volatility, which means dividing that expression by 100.

The awkward fact first. Chapter 7 listed constant, known volatility as the model's first assumption. Vega differentiates the price with respect to that constant. Strictly, it is the derivative of the formula with respect to a parameter the formula says cannot change — which is not a coherent risk measure inside the model's own terms.

What it actually measures is a quoting sensitivity. Derman and Kani describe the market's practice exactly: traders quote a call's price "in terms of whatever constant local volatility $\sigma_{\text{imp}}$ makes the BS formula value equal to the market price," and they are blunt that "in essence, $\sigma_{\text{imp}}$ is a means of quoting prices."

So vega answers: if the number people quote this contract in changes by one point, what happens to the price? That is a genuinely useful question, and it is not the same as a model-consistent measure of volatility risk. Chapter 11 is what happens when you take the inconsistency seriously.

Vega across strikes

Index 24,000, thirty days, 14% volatility, 6.5% interest.

Strike	Price	Vega per vol point	Vega as % of price
22,000	₹2,120.58	₹1.86	0.09%
23,000	₹1,176.02	₹13.14	1.12%
24,000	₹450.57	₹27.13	6.02%
25,000	₹101.39	₹18.90	18.64%
26,000	₹12.22	₹5.04	41.26%

In absolute terms vega peaks at the money, like gamma. A deep in-the-money option is nearly a forward contract and barely cares about volatility; a far out-of-the-money option is too cheap for a vol point to be worth much in rupees.

In proportional terms the ranking inverts completely. The 26,000 call moves 41% of its own value per vol point; the 22,000 call moves 0.09%. Far out-of-the-money options are almost pure volatility positions, which is why their prices look so unstable and why chapter 11's smile is measured there.

The full range makes it vivid — the thirty-day 26,000 call across volatilities:

Volatility	26,000 call
10%	₹1.03
14%	₹12.22
20%	₹64.63
30%	₹227.76

A factor of 220 from a factor of 3 in volatility. Nothing else about the contract changed.

Vega across maturities

Time left	Price	Vega per vol point	Vega as % of price
7 days	₹200.85	₹13.22	6.58%
30 days	₹450.57	₹27.13	6.02%
90 days	₹868.45	₹45.90	5.29%
180 days	₹1,353.41	₹62.67	4.63%
365 days	₹2,189.47	₹83.01	3.79%

Vega grows with the square root of time — the $\sqrt{T}$ in the formula — so long-dated options carry far more of it in rupees.

Which sets up the contrast with chapter 9 cleanly:

	Peaks where	Grows with
Gamma	at the money, near expiry	$1/\sqrt{T}$
Vega	at the money, far from expiry	$\sqrt{T}$

Short-dated options are gamma instruments; long-dated options are vega instruments. A view that the index will move a lot this week and a view that volatility will be higher this year are different views and are expressed with different contracts. Treating a weekly option as a way to be long volatility is a category error — it is a way to be long one week's realised movement.

The near-linearity worth knowing

At-the-money option prices are almost exactly proportional to volatility, which makes mental arithmetic possible.

Volatility	30-day ATM call	Linear estimate from vega at 14%	Error
8%	₹288.86	₹287.80	+₹1.06
12%	₹396.39	₹396.31	+₹0.08
14%	₹450.57	₹450.57	—
16%	₹504.89	₹504.83	+₹0.06
20%	₹613.77	₹613.35	+₹0.42
30%	₹886.56	₹884.64	+₹1.92
50%	₹1,432.54	₹1,427.23	+₹5.31

The linear estimate is within a rupee across a range from 8% to 30%. So for an at-the-money option, vega is not a local approximation that decays — it is nearly the whole relationship, and "the price moves ₹27 per vol point" is a usable rule rather than a first-order guess.

This is the formal version of chapter 4's observation that trebling volatility roughly trebled the price. The reason is that $n(d_1)$ barely changes when $d_1$ is near zero, so the price is close to $0.3989 \, S \sigma \sqrt{T}$ plus the financing term.

It fails away from the money, as the 26,000 call's table shows. Those prices are convex in volatility, not linear — the sensitivity itself rises sharply with volatility, which is why a far strike can multiply many times over on a volatility shock.

Measuring implied volatility properly

If every option implied a different volatility, which is the market's volatility? NSE's answer is a published method worth reading as an engineering document.

India VIX is computed "using the order book of the underlying index options and is denoted as an annualised percentage", depicting expected volatility over the next 30 calendar days — bracketing that period with near-month and next-month NIFTY expiries, and rolling to the next and far month with three trading days left.

Five design choices tell you what the problem is.

It uses out-of-the-money options, not one at-the-money option. Calls above the at-the-money strike $K_0$, puts below it. The index is built out of a strip precisely because a single strike's implied volatility is not representative — which is chapter 11's subject, admitted in the construction of the measure.

It takes the forward from the futures market. Where the CBOE infers the forward from the strike with the smallest call-minus-put difference, NSE uses "the latest available traded price of the NIFTY futures of the respective expiry month," because the futures market is actively traded and liquid. That is chapter 2's parity used as an input rather than a test.

It uses MIBOR as the risk-free rate — the 30-day or 90-day rate as appropriate, which is a concrete answer to chapter 2's question of which rate.

It measures time in minutes, not days: minutes to midnight of the current day, plus minutes on expiry day to the 3:30 p.m. close, plus whole days between. For a seven-day option, a day's rounding is a material error.

It discards bad quotes. A strike whose spread exceeds 30% of the mid price is "considered as not appropriate", and its mid is replaced using a cubic spline. An option chain's far strikes are often quoted too wide to carry information, which is the practical limit on everything in this chapter and the next.

Working the problem

A seven-day at-the-money call at ₹412.63 on 30% implied volatility before a scheduled event. A day later: event passed, index unchanged, volatility back to 14%.

Step 1 — reprice. Six days left, 14% volatility, index still 24,000:

$C = ₹184.89$

A loss of ₹227.74, or 55.2% of the position, with the index exactly where it started.

Step 2 — decompose it. Split the change into its two causes by moving one variable at a time:

Cause	Effect
One day's decay, volatility still 30%	−₹31.62
Volatility 30% → 14%, six days left	−₹196.13
Total	−₹227.75

Decay accounts for about an eighth of the loss and the volatility collapse for the rest — roughly 16 vol points at a vega of about ₹13 a point.

Step 3 — the breakeven move.

Index move	New price	Change
0	₹184.89	−55.2%
+200	₹308.27	−25.3%
+338	₹412.63	0%
+400	₹463.17	+12.2%
+600	₹640.41	+55.2%
+800	₹830.64	+101.3%

The index had to rise 338 points — about 1.41% — for the holder merely to break even. Being right about direction was not enough; they had to be right about direction by more than the volatility collapse cost them.

What this position actually was. The buyer thought they were taking a view on the event. They were taking two views at once:

That the index would move, which is what they intended.

That it would move more than 30% annualised volatility implied — about 1.6% a day over the week — which they probably never considered and almost certainly did not believe.

The second view is the one that lost. Implied volatility was high because an event was scheduled; the market had already priced the uncertainty the buyer was paying to be exposed to. Once the uncertainty resolves, the price of uncertainty falls whatever the resolution was. The Derivatives chapter on what an option price is made of names this as a classic way to be right about direction and lose money; this chapter supplies the arithmetic.

And the general rule it gives you. Before buying an option into a known event, convert its implied volatility into the move it implies over your horizon, and ask whether you expect more than that. If you do not, the option is the wrong instrument however confident you are about direction — chapter 3's conclusion, arrived at from the volatility side.

One honest limit on all of this. Vega tells you what happens if implied volatility moves. It does not tell you whether it will, and Derman and Kani's warning applies: once volatility is allowed to be random, there is no security with which to hedge it directly, so option valuation "is in general no longer preference-free." Volatility risk is the one exposure in this subject that the replication argument cannot neutralise — which is exactly why it is priced, and why the price contains fear as well as forecast.

The point

Vega is $S\,n(d_1)\sqrt{T}$, the price's sensitivity to the one input nobody can observe and the model says is constant — so it measures a quoting convention rather than a model-consistent risk. It peaks at the money in rupees and far from the money in proportional terms, where options behave as almost pure volatility positions; it grows with $\sqrt{T}$, making long-dated options the vega instruments and short-dated ones the gamma instruments. At-the-money prices are near-perfectly linear in volatility, so "₹27 a vol point" is a rule rather than an approximation. Because no single strike's implied volatility is representative, India VIX is built from a strip of out-of-the-money quotes, with the forward taken from futures, MIBOR as the rate, time measured in minutes, and wide quotes discarded. And buying an option before a known event is buying a volatility level as well as a direction.