Variance Swaps and Volatility Desks: Replication, the 1/K² Weighting, Convexity, and the 2008 Lessons

A variance swap is a forward contract that pays the difference between the variance a stock or index actually realizes over a set period and a fixed variance strike agreed at the start. Dealers can hedge it, in theory almost perfectly, with a static portfolio of out-of-the-money options whose quantities are weighted by one over the strike squared (1/K²), plus a simple futures hedge.

In short: a variance swap gives exposure to realized volatility without the stock-price dependence that comes with a delta-hedged option. Its fair strike is set by the cost of the replicating option strip, the same calculation that underlies the VIX index. Because the payoff is in variance (volatility squared), a variance swap is convex in volatility: the buyer gains more from a volatility spike than it loses from an equal drop, so sellers face uncapped losses. The replication assumes the underlying moves without jumps and that options exist at every strike. In 2008 both assumptions failed, single-stock variance swap trading largely evaporated, and caps at 2.5 times the strike became standard even on indices.

This article builds on implied volatility explained and on the gamma and theta trade-off covered in delta hedging and gamma scalping. It describes how dealers and volatility desks work. It is not a recommendation.

A common misconception: “an option is a pure volatility bet”

An option’s value depends on volatility, but its volatility exposure changes with the stock price. A delta-hedged at-the-money option has a lot of vega and gamma; if the stock drifts far from the strike, both fade, and the position stops making or losing money from volatility at all. In their 1999 Goldman Sachs Quantitative Strategies research note, “More Than You Ever Wanted to Know About Volatility Swaps,” Kresimir Demeterfi, Emanuel Derman, Michael Kamal and Joseph Zou called options “impure” volatility vehicles for this reason, and showed how to build a position whose sensitivity to variance does not depend on where the stock goes. That position is the variance swap.

The contract

The payoff of a variance swap at expiry, per Demeterfi and coauthors, is:

Payoff = (realized variance - K_var) x N_var

Realized variance is the annualized variance of daily log returns over the life of the swap, usually computed with a zero mean: 252 / n x the sum of (ln(S_i / S_(i-1)))² over n observations, quoted in “variance points” (volatility in percent, squared). A realized volatility of 20% is 400 variance points.
K_var is the variance strike. By convention it is quoted as a volatility: a “20 strike” means K_var = 20² = 400.
N_var is the variance notional, in dollars per variance point.

Because traders think in volatility, notional is usually quoted as vega notional: the dollar gain per volatility point for a small move around the strike. The conversion is N_var = N_vega / (2 x K). A $100,000 vega notional swap struck at 20 has a variance notional of $100,000 / 40 = $2,500 per variance point.

Demeterfi and coauthors list what a term sheet must specify: the price source and observation frequency (for example, daily S&P 500 closes), the annualization factor, and whether the mean return is subtracted. The zero-mean convention is preferred because it matches what the option strip replicates.

Where the market came from

Peter Carr and Roger Lee (“Volatility Derivatives,” Annual Review of Financial Economics, 2009) date the take-off of index variance swaps to 1998, a year of very high implied volatility. Hedge funds found it attractive to sell realized variance at rates well above statistical forecasts. Banks bought it, knowing from practitioner research that most of the market risk could be hedged by selling a strip of options and delta hedging them. In Carr and Lee’s words, the banks were “buying realized variance at a low rate through variance swaps and simultaneously selling it at a high rate through delta-hedged options,” while the funds paid for a payoff they lacked the infrastructure to build themselves.

Why a 1/K² strip of options replicates variance

The intuition

A delta-hedged option earns, over each small interval, roughly half its dollar gamma times the squared return, minus time decay. If you held a portfolio whose dollar gamma (gamma times the stock price squared) was the same at every stock price, its hedged P&L would accumulate realized variance at a constant rate wherever the stock went.

The gamma of an option is concentrated near its strike, and the dollar gamma of an option struck at K scales roughly with K². So to make dollar gamma flat across stock prices, hold options in quantities proportional to 1/K²: more options at low strikes, fewer at high strikes. Demeterfi and coauthors show this graphically: a portfolio with equal weights has dollar gamma that rises with the stock price, a portfolio with weights proportional to 1/K has it rising less, and only 1/K² gives a flat profile.

Strike as % of spot	50%	75%	100%	125%	150%
Relative quantity (1/K², spot strike = 1.00)	4.00	1.78	1.00	0.64	0.44

The formal result

The strip is a way of building a log contract, a payoff equal to the logarithm of the final price. Demeterfi and coauthors show that if the stock moves continuously, realized variance equals 2/T times the gains from a dynamic position holding a constant dollar amount of stock, minus 2/T times a static short position in the log contract. The log contract, in turn, can be built from a forward plus out-of-the-money puts below a reference level and calls above it, each in quantity proportional to 1/K². The fair variance strike is the cost of that portfolio, grossed up by interest:

K_var = (2/T) x [ rT - (S0 e^(rT) / S* - 1) - ln(S* / S0) ] + e^(rT) x (2/T) x [ sum over puts and calls of (option price / K²) x strike spacing ]

where S* is the boundary between puts and calls (usually near the forward). No volatility model appears anywhere. The only inputs are option prices.

The VIX is the same calculation

Cboe’s VIX methodology document describes the 2003 redesign, done with Goldman Sachs, as one that “estimates expected volatility by aggregating the weighted prices of S&P 500 Index puts and calls” over a wide range of strikes, supplying “a script for replicating volatility exposure with a portfolio of SPX options.” Its generalized formula sums each out-of-the-money option’s midpoint price times its strike spacing divided by K², multiplies by 2/T and e^(RT), and subtracts a correction for the gap between the forward and the first strike below it. The VIX is 100 times the square root of that 30-day variance. In other words, the VIX squared is a discretized 30-day S&P 500 variance swap rate. The details are in the VIX in depth.

Worked example 1: pricing a variance swap with a skew

Demeterfi and coauthors give a three-month example. Index at 100, interest rate 5%, no dividends, strikes from 50 to 150 every 5 points, at-the-money implied volatility of 20%, and a skew that adds 1 volatility point for every 5-point drop in strike. Selected rows of their Table 1:

Option	Strike	Implied vol	Weight	Option value	Contribution
Put	75	25%	72.26	0.0276	1.99
Put	85	23%	56.23	0.2854	16.05
Put	95	21%	45.00	1.6747	75.36
Put	100	20%	20.98	3.3537	70.36
Call	100	20%	19.63	4.5790	89.87
Call	105	19%	36.83	2.2581	83.16
Call	115	17%	30.69	0.2578	7.91
Call	125	15%	25.98	0.0057	0.15
Total cost of the strip					419.87

The weights fall with strike roughly as 1/K², while the far wings contribute little to cost despite their large weights because the options are nearly worthless. Plugging the strip cost into the formula gives a fair variance of (20.467)², about 418.9 variance points.

We recomputed the example in Python. Using the piecewise-linear weighting in the paper’s Appendix A, we get a strip cost of 419.68 and a fair strike of 20.462. Using the VIX-style 1/K² sum with 5-point spacing instead, we get 20.467, matching the paper. With 1-point spacing the VIX-style estimate falls to 20.063 (about 402.5 variance points), close to the paper’s statement that with a continuum of strikes this skew converges to a fair variance of “about 402,” against exactly 400 with no skew. Two lessons:

Skew raises the variance strike above at-the-money volatility, because the expensive downside puts carry the largest weights.
Discreteness matters. Coarse strike spacing overstates the cost by more than the skew itself does here, which is why desks use every listed strike.

Worked example 2: why variance is convex in volatility

Take a $100,000 vega notional variance swap struck at 20 (variance notional $2,500 per point), and compare it with a volatility swap of the same vega notional, whose payoff is simply $100,000 x (realized vol - 20). The capped column applies the standard 2.5x cap, so realized volatility is capped at 50 for payoff purposes:

Realized volatility	Variance swap payoff	Volatility swap payoff	Capped variance swap payoff
10	-$750,000	-$1,000,000	-$750,000
15	-$437,500	-$500,000	-$437,500
20	$0	$0	$0
25	+$562,500	+$500,000	+$562,500
30	+$1,250,000	+$1,000,000	+$1,250,000
40	+$3,000,000	+$2,000,000	+$3,000,000
60	+$8,000,000	+$4,000,000	+$5,250,000
80	+$15,000,000	+$6,000,000	+$5,250,000

Near the strike, the two payoffs agree, which is the point of the vega-notional convention. Away from it, the variance swap always pays more: the difference is (realized vol - strike)² / (2 x strike) per dollar of vega notional, which Demeterfi and coauthors call the convexity bias. The buyer of variance is long that convexity. The seller is short it, with a loss that grows with the square of realized volatility. At 80 realized, an uncapped seller loses $15 million on a position that would have made at most $1 million if volatility had collapsed to zero.

Two consequences follow:

The fair volatility swap strike is below the square root of the variance strike. Since a variance swap always outperforms the matching volatility swap, the volatility swap must be struck lower to avoid an arbitrage. How much lower depends on how volatile volatility is. As an illustration only (our assumption, not the paper’s), if realized variance were lognormally distributed with a mean of 400, the expected realized volatility would be 19.78 with a log standard deviation of 0.3, 19.38 with 0.5, and 18.46 with 0.8, against a naive 20. Demeterfi and coauthors note that volatility swaps, unlike variance swaps, cannot be replicated statically; they must be hedged dynamically with variance, and their value depends on the volatility of volatility.
Variance accrues in jumps. One day’s contribution to a 63-day swap is 252 / 63 x (daily return in %)². A 1% day adds 4 variance points; a single 10% day adds 400, as much as a whole quarter at 20 volatility.

How jumps break replication

The replication argument requires continuous prices. Demeterfi and coauthors work out exactly what a jump does. For a short variance swap hedged with the replicating strategy, a single jump of size J (J = 0.10 for a 10% fall, negative for a rise) produces a hedging error of:

P&L from the jump = (2/T) x [ -J - ln(1 - J) ] - J² / T

The squared term cancels between the swap and the hedge, so the leading error is cubic in the jump size, about (2/3) x J³ / T. Down jumps produce a gain for the hedged short-variance position, and up jumps a loss. Our computation from that formula, which reproduces the figures in their Table 5 (per $1 of variance notional, in variance points):

Single jump	Three-month swap	One-year swap
15% down	+101.5	+25.4
10% down	+28.8	+7.2
5% down	+3.5	+0.9
5% up	-3.2	-0.8
10% up	-24.8	-6.2
15% up	-81.0	-20.2

That table assumes perfect replication with every strike. The second failure is the strike range. Demeterfi and coauthors show that with flat 25% implied volatility, a strip of strikes from 75% to 125% of spot prices a one-year variance swap at (23.0)², not (25.0)², because once the stock leaves the strike range the strip’s gamma disappears and it stops accruing variance. A large move that pushes the underlying beyond the listed strikes does exactly that, and it does it at the moment variance is accruing fastest.

What 2008 did to variance swaps

Carr and Lee, writing in 2009, described the effect bluntly: “Historically profitable strategies based on the naked selling of realized variance suffered huge losses in the final quarter of 2008.” They go on: “Dealers learned the hard way that the standard theory for pricing and hedging variance swaps is not nearly as model-free as previously regarded.” Their diagnosis matches the two failure modes above. Sharp moves exposed books to cubed and higher-order returns, and the inability to hold deep out-of-the-money options meant that “as the underlying index or stock moved away from its initial level, dealers found themselves exposed to much more vega than a complete hedging strategy would permit.”

The damage was worst for single stocks, because their options are less liquid and their extreme moves are larger. Carr and Lee: “the market for single-name variance swaps has evaporated in 2009.” Ian Martin (“Simple Variance Swaps,” NBER Working Paper 16884, 2011) opens with the same observation, that the large price jumps of 2008 and 2009 “caused the single-name variance swap market to dry up completely,” and proposes an alternative contract that can be priced and hedged even with jumps.

The contractual response was the cap. Carr and Lee explain that because variance swaps use log returns, a single-name payoff becomes “literally infinite if the underlying ever closes at zero,” so single-name swaps had long capped realized volatility at 2.5 times the strike. After 2008, that convention became standard for index variance swaps as well. A cap turns a variance swap into a variance swap plus a short call on variance struck at the cap, which complicates pricing and hedging. In the table above, the 2.5x cap limits the long’s payoff, and the short’s loss, to $5.25 million.

How a volatility desk runs the book

Putting the pieces together, here is the workflow the literature describes for a dealer that quotes variance:

Quote from the strip. The bid and offer start from the cost of the replicating options across the full listed strike range, adjusted for the skew, the discreteness of strikes, and the cost of wings that cannot be bought.
Hedge statically, then dynamically. Buy or sell the strip (or the nearest liquid approximation), and run the small dynamic futures hedge the formula requires. What remains is exposure to jumps, to the strike range, and to the gap between listed and theoretical options.
Warehouse what cannot be hedged. Wing risk and jump risk are priced into the spread or capped by contract.
Net the flows. Index variance and single-stock variance can offset in a correlation book: the gap between index variance and the variance of its components is the basis of dispersion and correlation trading.

The economic backdrop is the variance risk premium: on average, implied variance has exceeded subsequently realized variance, so sellers of variance have been paid for bearing crash risk. Peter Carr and Liuren Wu (“Variance Risk Premiums,” Review of Financial Studies, 2009) measured it using synthetic variance swap rates built from option prices, the same replication shown above. The premium is the subject of the variance risk premium. The convexity table shows why it exists: the seller carries a loss distribution with a long right tail.

What can go wrong

Uncapped convexity. A short variance position loses with the square of realized volatility. A few violent days can outweigh years of premium.
Jumps. Replication captures realized variance only if prices move continuously. The cubic error is small for small jumps and large for large ones, and it differs by direction.
Strike range and liquidity. The strip needs deep out-of-the-money options. When the underlying moves outside the listed range, the hedge stops working, as dealers found in 2008.
Single-name default. With log returns, a stock that goes to zero produces unlimited realized variance, which is why single-name swaps are capped.
Cap and contract mechanics. Caps, market disruption clauses, and how missed fixings are handled all change the payoff. A capped swap is a different instrument from the textbook one.
Volatility swaps need a vol-of-vol view. They cannot be replicated statically, so their pricing depends on a model of how volatility itself moves.

The short version

A variance swap pays realized variance minus a fixed strike, and a dealer can replicate it with out-of-the-money options weighted by 1/K² plus a futures hedge, so its fair strike comes straight from option prices. That same calculation, discretized, is the VIX. In the Demeterfi, Derman, Kamal and Zou example, a three-month index with 20% at-the-money volatility and a downside skew has a fair variance strike of about 20.5 volatility with 5-point strikes. Variance is convex in volatility, so buyers gain more from spikes than they lose from lulls, and sellers carry a long tail of losses. The replication assumes no jumps and a full strike range; 2008 broke both, the single-name market evaporated, and 2.5x caps became the norm. Running a variance book is professional work that depends on dealer infrastructure, option access across all strikes, and capital to absorb the tails.