Variance Swap

Also known as: Var swap, Vol swap (cousin)

A pure bet on how much a market moves — direction irrelevant. Volatility as a tradable asset.

3 min read · 535 words · Updated

1 · SnapshotThe one idea to remember
Key intuition: a variance swap is a weather derivative for markets — it pays on the size of the storm, not the direction of the wind.
2 · BeginnerWhat is it, really?

Most instruments pay off based on where a price goes. A variance swap pays off based on how much it moved along the way — up or down, doesn't matter.

The two sides agree on a "strike" level of volatility, say 20%. At the end, the actual (realised) volatility of the stock's daily moves is measured. Came out at 30% — a turbulent period? The buyer of variance collects. A calm 12%? The buyer pays.

This turns turbulence itself into an asset. Investors buy variance as crash insurance (markets get wild when they fall), and sellers harvest the premium that insurance buyers persistently overpay — most of the time.

Long variance pays off convexly in realised volatility — gains accelerate as vol rises.
StrikeVariance legVol-linearRealised volatilityPayoff (long variance)

Point at a line to pick it out from the others.

Settling on how much it moved, not where it ended
Variance buyerlong volatilityVariance sellershort volatility1No premium — a strike isagreed2The daily returns arerecorded3Realised minus strike, ifpositive4Strike minus realised, ifit was quiet

only if a condition is metnot a payment

One payment, at the end, on a number that is computed rather than quoted.

At the start

  1. Variance buyer → Variance seller The strike is the level of variance at which the contract is fair. Nothing changes hands but collateral.

Every day until expiry

  1. Variance seller → Variance buyer Realised variance accumulates from the actual closing prices. Neither party can influence it and neither pays anything yet.

At expiry, once only

  1. Variance seller → Variance buyer Paid on the variance notional. Because the payoff is in variance rather than volatility, a large move is worth disproportionately more — the convexity that makes the seller's loss open-ended.
  2. Variance buyer → Variance seller The seller's gain is capped at the strike itself: variance cannot go below zero.
Asset class
Equity derivatives (volatility)
Instrument type
Swap on realised variance
Traded
OTC
Typical users
Vol traders, hedge funds, structurers

Which risks decide the outcome

Not how risky this is, and not a rating — there is deliberately no total. It says which of five failure modes drives what happens here, in the same order on all 129 products so they can be compared. This publication's own reading; see the notice below.

  • Marketdecides it
  • Creditbarely applies
  • Liquiditybarely applies
  • Fundingdecides it
  • Operationalmatters

What decides it here. The seller's loss grows with the square of the move, so a single violent week is a funding event before it is a valuation one.

What the five mean, and which one decides where →

3 · IntermediateHow it works in practice

The contract

Payoff at expiry, with vega notional expressed as variance notional \(N_{var}\):

$$ \text{Payoff} = N_{var}\,\big(\sigma^2_{\text{realised}} - K^2_{var}\big), \qquad \sigma^2_{\text{realised}} = \frac{252}{n}\sum_{i=1}^{n} \Big(\ln \tfrac{S_i}{S_{i-1}}\Big)^2 $$
What the symbols mean
  • Nthe normal distribution, or a count
  • rthe interest rate, per year
  • sigmavolatility, the standard deviation of returns
  • Kthe strike: the price written into the contract
  • nhow many periods, or how many things
  • Sthe price of the underlying today

Note it settles on variance (vol squared): a move from 20 → 30 vol pays more than 20 → 10 costs. That convexity is why the market quotes variance, which dealers can replicate exactly, rather than volatility, which they can't.

Quoting in vega

Traders think in "vega notional" — P&L per volatility point near the strike: \(N_{vega} = 2 K_{var} N_{var}\). A trade of "100k vega at 20 strike" makes ≈ $100k per vol point of realised above 20 (more, due to convexity).

Uses and abuses

  • Hedging: long variance offsets equity drawdowns (vol spikes when markets crash — strong negative correlation).
  • Carry harvesting: implied variance usually exceeds subsequent realised — selling variance collects this "variance risk premium", with occasional violent losses (short variance in 2008 or the 2018 "Volmageddon" was ruinous).
  • Dispersion: selling index variance vs. buying single-name variance trades correlation.
Worked example: long $100k vega at strike 20. Realised comes in at 26 → payoff ≈ $100k × (26² − 20²)/(2×20) = $100k × 6.9 = $690k — versus $600k a linear "vol swap" would have paid. Convexity worked for you.
4 · AdvancedPricing & valuation

Replication: the log contract

The theoretical heart: realised variance can be replicated model-free by delta-hedging a portfolio of options whose payoff is \(-\tfrac{2}{T}\ln(S_T/S_0)\). By the Carr–Madan expansion, that log payoff decomposes into a strip of out-of-the-money options weighted by \(1/K^2\):

$$ K^2_{var} \;=\; \frac{2 e^{rT}}{T}\left[\int_0^{F_0}\frac{P(K)}{K^2}\,dK + \int_{F_0}^{\infty}\frac{C(K)}{K^2}\,dK\right] $$
What the symbols mean
  • Kthe strike: the price written into the contract
  • rthe interest rate, per year
  • Tmaturity, in years
  • Fthe forward or futures price
  • Pa price, or a present value
  • Cthe price of a call option

The fair strike is thus readable off the entire volatility smile — no model of dynamics required, only continuous paths. This same formula (discretised) is how the VIX is computed: the VIX is essentially a 30-day variance-swap strike.

Where replication breaks

  • Jumps: a single crash makes discrete realised variance exceed what the hedge captures; dealers cap payoffs (e.g. at 2.5× strike) to bound this.
  • Wings: the \(1/K^2\) weighting demands deep OTM puts that may not exist or trade at extreme spreads; the missing wing is a model reserve.

P&L accrual

A seasoned variance swap decomposes into realised-so-far plus implied-remaining, weighted by elapsed time \(t\):

$$ K^2_{t,T} = \frac{t}{T}\,\sigma^2_{\text{real},[0,t]} + \frac{T-t}{T}\,K^2_{\text{imp},[t,T]} $$
What the symbols mean
  • Kthe strike: the price written into the contract
  • ta point in time
  • Tmaturity, in years
  • sigmavolatility, the standard deviation of returns

The formulas above are standard textbook formulations, simplified for teaching. They explain the mechanism — they are not a valuation tool, and they will not reproduce a dealer’s price.

5 · Desk notesHow practitioners think about it
Practitioner note: variance is the only volatility instrument with an exact static replication — everything else (vol swaps, VIX futures, options on vol) carries model risk on top.

Now say it back

Close the page and give Variance Swap in four sentences. It takes a minute and it is the only way to find out whether reading it was enough.

  1. Who wants what — two parties wanted opposite things badly enough to write it down.
  2. What the contract obliges, and when — not the payoff; the obligation.
  3. Where the money comes from — name the source, or you have described a hope.
  4. What makes it lose — the ordinary way, not the dramatic one.

Do it with a clock → · why these four

Put Variance Swap beside any other instrument →

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