---
description: "Variance swaps are derivative contracts whose payoff is proportional to the difference between realized variance and a preset variance strike, allowing pure volatility bets without the need for continuous Delta-hedging."
tags: [volatility, variance-swap, realized-variance, delta-neutral, derivatives]
---

# Volatility Trading with Variance Swaps

**Section**: 7.6 | **Asset Class**: Volatility | **Type**: Volatility / Derivatives

## Overview
Variance swaps allow traders to bet on future realized variance without the operational burden of continuous Delta-hedging required by options-based volatility strategies. The payoff at maturity is proportional to the difference between the realized variance of the underlying and a preset variance strike K, scaled by a variance notional N.

## Construction / Mechanics

**Payoff at maturity T** (for the long side):
```
P(T) = N × (v(T) - K)                                               (434)
```

**Realized variance** over the period:
```
v(T) = (F/T) Σ_{t=1}^T R(t)²                                        (435)
```

**Log-return** at time t:
```
R(t) = ln[S(t) / S(t-1)]                                             (436)
```

where:
- t = 0, 1, ..., T: sample points (e.g., trading days)
- S(t): underlying price at time t
- F: annualization factor (F = 252 for daily data)
- N: variance notional (preset, in dollar terms)
- K: variance strike (preset at contract inception; determines the fair value entry price)
- Note: the mean return is NOT subtracted in v(T), so the denominator is T (not T-1)

**Long variance swap**: benefits when realized variance v(T) > K (realized vol was higher than expected).
**Short variance swap**: benefits when realized variance v(T) < K (realized vol was lower than expected).

**Fair value of K**: at inception, K is set such that the swap has zero initial value; it equals the market's expected future variance (often proxied by the square of the VIX for equity index variance swaps).

## Payoff / Return Profile
- **Long variance swap** profits when realized variance exceeds the variance strike K: v(T) > K.
  - Typical use: hedge against a volatility spike; gains are convex in the volatility move (since variance is the square of vol).
- **Short variance swap** profits when realized variance is below K: v(T) < K.
  - Typical use: harvest the volatility risk premium (implied vol > realized vol on average); earns N × (K - v(T)) in normal markets.
- The payoff is in units of variance (not volatility); a move from 10% to 20% vol generates 4× the payoff of a move from 10% to 15% vol.

## Key Parameters / Signals
- K: variance strike (the break-even point)
- N: variance notional (dollar value per unit of variance)
- v(T): realized variance (the key realized outcome)
- F: annualization factor (252 for daily, 52 for weekly, etc.)
- T: number of observation periods
- v(T) - K: the net P&L per unit of notional

## Variations
- **Volatility swap**: payoff based on sqrt(v(T)) - K_vol (realized volatility minus vol strike); less convex than variance swap; harder to replicate.
- **Conditional variance swap**: accumulates variance only on days when the underlying is within a specified range.
- **Corridor variance swap**: similar, accumulates variance only when the underlying is above (or below) a barrier.

## Notes
- The key advantage over options-based volatility strategies: no Delta-hedging required, so the trader takes pure variance exposure with no directional risk (assuming no P&L drift from Delta).
- The convexity of the variance payoff (variance = vol²) means long variance swaps gain more from large vol spikes than short options positions gain from vol increases.
- Short variance swaps capture the volatility risk premium (K > expected realized variance) but have unlimited downside: if realized variance is very high (e.g., a market crash), N × (v(T) - K) can be very large.
- Variance swaps on equity indexes are liquid OTC instruments; single-stock variance swaps are less common.
- The mean is not subtracted in v(T) (Eq. 435), which is standard market convention; subtracting the mean would change the denominator to T-1.