Variance Swap Replication: The Log Contract and the Var Strike, From Options

If you searched for variance swap replication or the variance swap strike, this is the working version: what the instrument is, why a static option strip replicates it, what the strike-vs-ATM spread means, and where to get fair strikes computed per expiry without building the strip yourself.

Variance swap, defined in one paragraph: a variance swap is a forward contract on realized variance - at expiry it pays (realized variance minus the strike) times the variance notional. Quoted in volatility units for convenience, it settles on variance, the squared quantity, which makes its payoff convex in vol terms: a long position earns more per vol point in a blowup than it loses in a calm-down.

Why variance, not vol

Variance is the quantity option P&L is actually written in. The delta-hedged option's income statement - derived in the gamma scalping guide - is ½ΓS² times a variance spread, and variance adds across days and across independent sources the way vol does not. The variance swap packages that quantity directly: no re-hedging, no path dependence, one number at settlement. That is why var swaps (and their strip replication) are the professional benchmark for measuring the variance risk premium, the legs of clean dispersion trades, and the definition of VIX itself.

The replication, in four steps

1. Realized variance is the P&L of a hedged log contract. Apply Ito to log(S): the payoff log(S_T/S₀), delta-hedged, accrues exactly minus one-half of realized variance. So a short log contract, hedged, earns one-half of realized variance - the thing we want to price, up to the factor of two the strike formula restores.
2. Any smooth payoff decomposes into options. A static portfolio of calls and puts across strikes can replicate any terminal payoff; for the log contract the required density works out to 1/K² per strike - more weight to low strikes, less to high.
3. Therefore the fair var strike is an integral over the smile. Buy the strip of OTM puts and calls weighted by 1/K², delta-hedge the package, and it pays realized variance. Its cost today, annualised, is the fair strike: K²_var = (2e^rT/T) ∫ Q(K)/K² dK, with Q(K) the OTM option price at strike K.
4. VIX is this formula. The VIX methodology is the discretised strip on SPX at 30 days - VIX is not "average implied vol"; it is a variance-swap strike quoted in vol units. Every property of the strip (wing sensitivity above all) is inherited by VIX, which is worth remembering when reading VIX term-structure signals.

The convexity spread, with live numbers

Because the strip integrates the whole smile, the fair var strike always sits above ATM implied vol - the wings are extra variance the ATM quote ignores. The gap is a direct read on smile curvature. SPY on 2026-08-03, from the platform's per-expiry fair values:

Two readings worth internalising. First, the fair strike curve is nearly flat (about 17 across tenors) while ATM IV rises with tenor - the short end's smile is much more curved, so the wings contribute more variance there; the spread narrowing from +4.9 to +1.4 is the term structure of tail pricing, cousin to the tail convexity series. Second, an honest warning from the same dataset: at very short tenors the strip degenerates - the same computation at 2 days printed a 39.26 fair vol against 9.95 ATM, because sparse, wide-quoted wings dominate 1/K² weighting as T shrinks. Fair strikes under roughly two weeks should be treated as noise unless the wing quotes are carefully filtered; this is a property of the mathematics, not a data defect, and it is why VIX uses interpolation between two expiries rather than one short strip.

Variance swaps vs vol swaps

A volatility swap pays realized vol (not variance) against a vol strike. Because variance is vol squared and squaring is convex, the var swap strike always exceeds the fair vol swap strike - the difference is a convexity premium the var buyer pays for the quadratic payoff. Practical consequences: var swaps are replicable (hence liquid to price) while vol swaps are not statically replicable and embed a vol-of-vol view; and P&L on var swaps accelerates in blowups, which is exactly why single-name variance books were capped (payout ceilings, typically 2.5x the strike) after uncapped structures produced outsized losses in historical crash episodes. Index var with caps is the surviving standard.

Fair strikes as an API field

The advanced-volatility endpoint computes the strip from the fitted, arbitrage-checked SVI surface - not raw sparse quotes - and returns fair variance, fair vol and the convexity adjustment per listed expiry, alongside the SVI parameters and greeks surfaces:

import requests

r = requests.get("https://lab.flashalpha.com/v1/adv_volatility/SPY",
                 headers={"X-Api-Key": KEY}).json()

for row in r["variance_swap_fair_values"]:
    print(row["expiry"], row["days_to_expiry"],
          f"fair_vol {row['fair_vol']}",
          f"atm_iv {row['atm_iv']}",
          f"convexity {row['convexity_adjustment']}")

Fitting the surface first matters: the strip's wing sensitivity is precisely where raw-quote replication breaks, and the SVI fit with liquidity filtering and arbitrage checks is what makes short-and-mid tenor strikes usable. Background on the surface machinery: the advanced volatility API guide and the variance swap concept page. The endpoint is Alpha tier, and its historical mirror replays fair strikes point-in-time for backtesting variance-premium strategies (SPY from 2017-01-03).

Frequently asked questions

What is a variance swap in simple terms?

A bet on how much a stock or index actually moves, settled in cash: you agree a strike (say 17 vol, i.e. 289 variance points), and at expiry you receive the difference between realized variance and that strike, times your notional. More movement than agreed, the buyer collects; less, the seller collects. No options to manage, no hedging - the instrument does it all at settlement.

How is VIX related to variance swaps?

VIX is a variance swap strike: the CBOE methodology computes the 30-day fair variance strike on SPX from the 1/K²-weighted strip of OTM options and quotes its square root. Reading VIX as "expected vol" is close but subtly wrong - it includes the smile's wings, which is why VIX sits above SPX ATM implied vol in normal markets.

Why is the variance swap strike higher than ATM implied volatility?

Because the replicating strip owns the whole smile, wings included, while ATM IV prices only the center. Skewed, curved smiles add variance the ATM quote does not see - SPY's 30-day gap was 4.9 vol points on 2026-08-03. The spread widens when tails are bid and compresses when the smile flattens, making it a clean curvature gauge.

Can I replicate a variance swap myself with listed options?

In principle yes - buy the OTM strip weighted 1/K², delta-hedge daily, and you hold synthetic variance. In practice discrete strikes, truncated wings, and spread costs across many small option positions make retail-scale replication expensive; the strip is best used as the pricing benchmark (what fair variance costs) rather than the implementation.

The variance swap is where volatility trading becomes exact: a replicable strike, a path-free payoff, and a strike-vs-ATM spread that quantifies the smile in one number. The strip mathematics explains VIX, prices the tails, and - served per expiry from the fitted surface on the Alpha tier - turns variance-premium monitoring into a loop over an API response. Related: variance vs volatility risk premium, gamma scalping (the path-dependent way to trade the same spread), and the volatility surface guide.