Options are the most common way to trade volatility, but they come with a problem: as the underlying moves, the delta changes, and the options position needs to be re-hedged continuously. This means a pure volatility view expressed through options is obscured by the P&L generated by delta hedging. A variance swap solves this by offering a direct, path-independent payoff tied solely to the difference between realised and implied volatility.
What Is a Variance Swap?
A variance swap is an OTC contract between two counterparties where one party agrees to pay the other the difference between realised variance and a pre-agreed strike (implied variance), multiplied by a notional amount. Variance is volatility squared: if realised volatility over the period is 20%, realised variance is 400 (20²). If the variance strike was set at 324 (18²), the payoff per unit of variance notional is 400 − 324 = 76.
The payoff formula is:
Payoff = Variance Notional × (Realised Variance − Strike Variance)
Where realised variance is typically calculated as the annualised sum of squared daily log-returns of the underlying over the life of the trade, multiplied by a convention factor (often 252 for trading days per year).
Variance Notional vs Vega Notional
Variance swaps are often quoted and sized in terms of vega notional rather than variance notional. Vega notional is the approximate P&L for a 1 volatility point move in realised vol around the strike. The relationship between the two is:
Variance Notional = Vega Notional / (2 × Strike Vol)
So a variance swap with a vega notional of £100,000 and a strike of 20% would have a variance notional of £100,000 / (2 × 20) = £2,500. This means a 1 vol point move from 20% to 21% generates a payoff of approximately £100,000 — useful for intuition, though the actual payoff is non-linear in vol because payoff is linear in variance, not vol.
This non-linearity is important: the P&L is convex in volatility. A buyer of variance does better than linear as volatility moves further from the strike in either direction, because each additional vol point adds a larger amount to the realised variance. This is the key difference from a simple volatility swap (which would be linear in vol).
Fair Value of a Variance Swap
The strike of a variance swap is set so that the trade has zero initial value. In the absence of jumps and assuming a complete options market, the fair variance strike can be replicated by a static portfolio of vanilla options across all strikes — a weighted sum of call and put prices where the weights are inversely proportional to the square of the strike price. This result, due to Carr and Madan, means the fair variance strike can be read directly off the options skew.
In practice, the variance strike will be above the at-the-money implied volatility because the options skew (puts trading at higher implied vol than calls) adds positive weight to the wings. The variance strike for major equity indices like the FTSE 100 or S&P 500 typically trades above the ATM vol for a given tenor by 1–3 volatility points, depending on the skew.
Gamma P&L and the Relationship to Delta Hedging
There is a deep connection between variance swaps and the P&L from delta-hedging a vanilla option. When you continuously delta-hedge an option, the cumulative P&L is:
Delta-hedge P&L ≈ ½ × Gamma × S² × (σ²_realised − σ²_implied) × dt
Integrated over the life of the option, this is proportional to the difference between realised and implied variance — exactly the payoff of a variance swap. This means a variance swap is economically equivalent to a continuously delta-hedged option portfolio, where the hedging P&L accumulates over time.
In practice, options desks that are short variance (because they have sold options and are now long gamma) will accumulate positive P&L if markets are quiet. The risk is a vol spike — markets moving sharply — which generates large negative gamma P&L and is precisely the scenario where a long variance swap (bought variance) profits most.
How Variance Swaps Differ from Options
Several features distinguish variance swaps from vanilla options:
- No delta hedging required. The payoff depends only on the path of daily returns, not the spot level. There is no directional delta exposure to manage.
- Path independence (of the payoff, not the risk). The variance swap pays based on the total realised variance over the period, regardless of whether the moves were large early or late in the trade.
- Jump risk is amplified. Variance is squared volatility, so a single large daily move (a 10% gap down, for example) contributes 100 squared = 10,000 units to the realised variance, far more than a series of smaller moves. Sellers of variance are acutely exposed to gap risk.
- No vega decay. Unlike options, whose vega profile changes as the underlying moves and time passes, a variance swap's sensitivity to vol is more stable — it accumulates linearly with time remaining.
Uses in Practice
Variance swaps are used by hedge funds to take pure views on the level of implied vs realised vol without taking on directional risk. A fund that believes the equity market will be quieter than options prices imply can sell variance, collecting the variance strike and paying realised variance at the end. Banks use variance swaps to hedge or monetise the residual variance risk in their options books. Index variance swaps — on the S&P 500 (VIX-linked), Eurostoxx 50, or FTSE 100 — are the most liquid; single-stock variance swaps exist but are less common.