Variance Swaps and the VIX

Delta-hedging an option gives volatility exposure with baggage: the dollar-gamma weighting means when the vol arrives changes your P&L (the delta-hedging lesson's caveat). Variance swaps remove the baggage — pure, clean exposure to realized variance — and the machinery behind them also explains what the VIX literally is.

The contract

A variance swap pays, at expiry,

$$\text{Notional} \times \left(\sigma^2_{\text{realized}} - K_{\text{var}}\right)$$

where realized variance is computed from daily log returns over the life, and $K_{\text{var}}$ (the strike, quoted in vol-squared units) is set so the swap costs nothing upfront. No delta, no gamma path-dependence in the payoff definition: realize more variance than the strike and you collect, full stop. Market convention quotes notional in vega terms (P&L per vol point), with the conversion $\text{vega notional} \approx 2K \times \text{variance notional}$ — and the payoff's convexity in vol (linear in variance) means a long variance swap gains more from a vol spike than it loses from an equal drop: bought convexity, priced into the strike.

The replication idea

Why can dealers quote this? Because variance is replicable: a portfolio of options across all strikes, weighted by $1/K^2$, plus delta-hedging in the underlying, produces realized variance as its P&L. The sketch (interview-depth, not desk-depth): a single delta-hedged option accrues variance weighted by its dollar gamma, which is concentrated near its strike; summing options across strikes with $1/K^2$ weights makes the total dollar gamma flat in spot — variance is captured equally wherever the underlying wanders. The exotic-sounding "log contract" is just the name of that aggregate payoff. Consequence with teeth: the variance strike is an integral over the entire smile — OTM puts (trading at high implied vols) enter the basket, which is why variance strikes sit above ATM implied vol. Skew isn't decoration; it's in the price.

The VIX is this formula

The VIX is (essentially) the 30-day variance-swap strike on the S&P 500, computed from the $1/K^2$-weighted strip of listed SPX options and quoted in vol points. That's the honest one-sentence answer to "what is the VIX?" — not "the fear gauge," but the market-clearing price of 30-day S&P variance. Follow-on facts worth having: VIX futures and ETPs trade the forward strike, whose term structure is usually in contango — the source of the notorious decay in short-vol ETPs, and of the 2018 "Volmageddon" blowup when the curve snapped. Mentioning that episode, briefly, dates the risk properly.

The vol risk premium, again

Across decades, variance strikes have averaged above subsequently-realized variance — the volatility risk premium from the smile lesson, now directly harvestable: systematically short variance swaps collects it, with the exact tail profile this curriculum keeps flagging (steady income, catastrophic left tail; the Greeks lesson's steamroller with cleaner accounting). The risk-management lesson's VaR-gaming critique applies verbatim.

The interview version

"You think realized vol will beat implied over the next month. Best expression?" — Rank the options: a delta-hedged straddle (path-dependent, gamma concentrated near strike), versus a variance swap (clean, but you pay the skew premium in the strike); state the trade-off and pick per your view's shape. "Why is the variance strike above ATM vol?" — the $1/K^2$ strip includes the smile's expensive wings. "What's the VIX?" — the paragraph above, in two sentences. This lesson is level-5 material precisely because it composes five earlier ones — smile, Greeks, hedging P&L, risk premia, tail risk — into one instrument.