Variance Swaps: How to Trade Realized Volatility with Fair Variance Strike

Published: June 25, 2026
Ryan O'Connell, CFA, FRM
Article by Ryan O'Connell, CFA, FRM
Table of Contents

    Variance swaps and volatility swaps are powerful derivatives that let traders take pure exposure to realized volatility without any directional market risk. Unlike options, which mix volatility exposure with delta, these instruments isolate volatility as a tradable asset. This guide covers how variance and volatility swaps work, how to calculate fair variance strike using option replication, and the connection to VIX methodology.

    What Is a Variance Swap?

    A variance swap is an over-the-counter (OTC) derivative contract where two parties agree to exchange realized variance for a fixed variance rate (the variance strike) at maturity.

    Key Concept

    In a variance swap, the variance receiver (long variance) profits when realized variance exceeds the variance strike. The variance payer (short variance) profits when realized variance comes in below the strike. This provides pure volatility exposure with no directional market risk.

    The payoff at maturity for the variance receiver is:

    Variance Swap Payoff
    Payoff = Lvar × (V – VK)
    Where Lvar is variance notional, V is realized variance, and VK is the variance strike

    Variance notional is often quoted in terms of vega notional for easier interpretation. The relationship is:

    Vega Notional Conversion
    Lvar = Lvol / (2 × σK)
    Where Lvol is vega notional and σK is the volatility strike (square root of VK)

    Vega notional is quoted as dollars per volatility point (per 1%). For example, $100,000 vega notional at a 20% volatility strike means Lvar = $100,000 / (2 × 20) = $2,500 variance notional per variance point. If realized volatility is 25% (realized variance 6.25%) versus the 4% strike variance, the payout to the variance receiver is $2,500 × (6.25 – 4.00) = $5,625. Alternatively, using vega notional directly: the 5-point move (20% to 25%) produces approximately 5 × $100,000 = $500,000 — but this linear approximation only holds for small moves near the strike.

    Variance swaps are the institutional standard for trading volatility because variance can be replicated model-free from option prices. For more on implied volatility as an input to variance swap pricing, see our dedicated article. You can also use our Implied Volatility Calculator to extract IV from option prices.

    What Is a Volatility Swap?

    A volatility swap is similar to a variance swap but pays out based on realized volatility rather than variance. This makes the P&L more intuitive since it’s linear in volatility points.

    Key Concept

    The payoff of a volatility swap is Lvol × (σ – σK), where σ is realized volatility. Unlike variance swaps, volatility swaps require a convexity correction because the square root function is concave (Jensen’s inequality).

    While volatility swaps are easier to understand conceptually, they are harder to replicate and price. The convexity correction means the fair volatility strike is not simply the square root of fair variance — it’s slightly lower due to the curvature of the square root function.

    Realized Variance Calculation

    Realized variance in a variance swap is calculated from daily log returns during the observation period. The standard formula used in swap contracts is:

    Realized Variance
    V = (252 / (n – 1)) × Σ[ln(Si+1 / Si)]2
    Where n is the number of observations, Si is the i-th price, and 252 annualizes daily observations

    Key conventions in variance swap contracts:

    • Mean assumed zero — daily log returns are squared directly without subtracting the mean return
    • Annualization factor — typically 252 for equity indices (trading days per year)
    • Divisor — contract-specific; may be n-1, n-2, or n depending on the swap terms
    5-Day Realized Variance Example
    Day Price Log Return (Log Return)2
    0 100.00 — —
    1 101.20 0.01192 0.000142
    2 99.80 -0.01392 0.000194
    3 100.50 0.00699 0.000049
    4 102.00 0.01483 0.000220
    5 101.50 -0.00491 0.000024

    Sum of squared returns = 0.000629. With 5 daily returns: V = (252/5) × 0.000629 = 0.0317 (annualized variance). Realized volatility = √0.0317 = 17.8%.

    Valuing a Variance Swap

    The key insight that makes variance swaps valuable is that expected variance can be calculated model-free from option prices alone. This is because variance can be replicated using a static portfolio of out-of-the-money puts and calls.

    The fair variance strike is determined by Hull equation (26.6):

    Expected Variance (Hull 26.6)
    E[V] = (2/T) × ln(F0/S*) – (2/T) × (F0/S* – 1) + (2/T) × [Σ (ΔKi/Ki2) × erT × Q(Ki)]
    Where F0 is forward price, S* is split strike, T is time to maturity, and Q(Ki) is OTM option price at strike Ki

    The function Q(K) represents:

    • Put price for strikes below S* (the split strike)
    • Call price for strikes above S*
    • Average of put and call at the split strike S* itself

    The split strike S* is typically the highest strike price below the forward price F0. This approach uses out-of-the-money options because they’re more liquid than in-the-money options.

    Pro Tip

    The “model-free” nature of variance swap pricing assumes European options, no-arbitrage option prices, and a sufficiently wide range of liquid strikes. In practice, limited strike availability and bid-ask spreads introduce pricing uncertainty, especially in the wings.

    The value of an existing variance swap where the variance receiver will receive realized variance V and pay fixed variance VK is:

    Variance Swap Value
    Value = Lvar × (E[V] – VK) × e-rT
    Present value of expected variance difference times notional

    Understanding how option prices determine fair variance helps explain the volatility smile and skew phenomenon — the shape of implied volatility across strikes directly affects variance swap pricing.

    Variance Swap Example

    Let’s work through a variance swap valuation adapted from Hull Example 26.4.

    3-Month S&P 500 Variance Swap

    Setup:

    • Index level: 1,020
    • Risk-free rate: 4%
    • Dividend yield: 1%
    • Time to maturity: 3 months (T = 0.25)
    • Variance notional: $100 million
    • Fixed variance strike: VK = 0.045 (equivalent to 21.2% volatility)

    Option Data (9 strikes):

    Strike Implied Vol Q(K) Price
    800 29% $2.22 (put)
    850 28% $5.22 (put)
    900 27% $11.05 (put)
    950 26% $21.27 (put)
    1,000 25% $51.21 (avg)
    1,050 24% $38.94 (call)
    1,100 23% $20.69 (call)
    1,150 22% $9.44 (call)
    1,200 21% $3.57 (call)

    Calculation:

    1. Forward price: F0 = 1,020 × e(0.04-0.01)×0.25 = 1,027.68
    2. Split strike: S* = 1,000 (highest strike ≤ F0)
    3. Weighted sum: Σ(ΔK/K2) × erT × Q(K) = 0.008139 (with ΔK = 50 for all strikes)
    4. Correction term: (2/0.25) × [ln(1027.68/1000) – (1027.68/1000 – 1)] = 8 × (0.0273 – 0.0277) = -0.0032
    5. Fair expected variance: E[V] = -0.0032 + 8 × 0.008139 = 0.0619 ≈ 0.062
    6. Fair volatility: √0.062 = 24.9%

    Swap Value:

    $100M × (0.062 – 0.045) × e-0.04×0.25 = $100M × 0.017 × 0.99 = $1.69 million

    The variance receiver profits because expected variance (6.2%) exceeds the strike variance (4.5%).

    The VIX Index

    The VIX volatility index uses a variance-swap-style methodology to calculate expected 30-day S&P 500 volatility from option prices.

    Key Concept

    VIX uses equation (26.10) — a simplified version of the variance swap replication formula — to calculate 30-day expected variance from SPX option prices, then takes the square root to express the result in volatility points. A VIX of 20 means the market expects annualized S&P 500 volatility of approximately 20%.

    The VIX calculation process:

    1. Select near-term and next-term SPX/SPXW option expirations bracketing 30 days
    2. Calculate expected variance for each expiration using the option strip formula
    3. Interpolate to get 30-day expected variance
    4. Annualize (multiply by 365/30) and take the square root

    VIX is called the “fear index” because it spikes during market stress — option prices rise when investors pay up for downside protection, increasing expected variance. The variance-swap-style methodology ensures VIX reflects actual option market prices rather than model assumptions.

    Important Distinction

    While VIX uses variance-swap methodology, it’s not a direct volatility swap fair strike. VIX includes CBOE-specific strike selection rules, uses both SPX and SPXW options, and has its own correction terms. Trading VIX futures or options involves additional basis risks beyond simple variance swap exposure.

    Valuing a Volatility Swap

    Volatility swaps are harder to value than variance swaps because expected volatility is not simply the square root of expected variance. Due to Jensen’s inequality, E[√V] < √E[V] — the expected value of a concave function is less than the function of the expected value.

    Hull provides a convexity correction formula (equation 26.9):

    Expected Volatility (Convexity Correction)
    E[σ] ≈ √E[V] × (1 – var(V) / (8 × E[V]2))
    Where var(V) is the variance of realized variance (volatility-of-volatility effect)

    The correction term depends on var(V) — how uncertain we are about what realized variance will be. This introduces model risk: we need to estimate variance-of-variance from historical data or a stochastic volatility model like Heston.

    Volatility Swap Valuation

    Using the same setup as the variance swap example:

    • E[V] = 0.0621 (from variance swap calculation, before rounding)
    • var(V) = 0.0001 (estimated variance of realized variance; std dev = 0.01)
    • Volatility strike: σK = 23%
    • Vega notional: $100 million (per vol point)

    Convexity Correction:

    E[σ] = √0.0621 × (1 – 0.0001 / (8 × 0.06212))

    E[σ] = 0.2492 × (1 – 0.0001 / 0.0308) = 0.2492 × 0.9968 = 24.84%

    Swap Value:

    $100M × (24.84 – 23.00) / 100 × e-0.04×0.25 = $100M × 0.0184 × 0.99 = $1.82 million

    Note that E[σ] = 24.84% is slightly less than √E[V] = 24.9%. The convexity correction reduces expected volatility, which affects volatility swap pricing.

    Variance Swaps vs Volatility Swaps

    Both instruments provide volatility exposure, but with important differences:

    Variance Swap

    • Payoff linear in variance, convex in volatility
    • Model-free replication from option prices
    • Amplified gains from vol spikes (variance = vol2)
    • Institutional standard; more liquid
    • Notional quoted in variance or vega terms

    Volatility Swap

    • Payoff linear in volatility, easier to interpret
    • Requires convexity correction (model-dependent)
    • P&L proportional to vol difference
    • Less liquid; fewer dealers
    • Notional in vega terms only

    The key tradeoff: variance swaps are easier to price and hedge (model-free), but their payoff is convex in volatility, meaning large vol moves produce outsized P&L. Volatility swaps have more intuitive P&L but require model-dependent pricing.

    How to Calculate Fair Variance Strike

    Here’s a practical approach to calculating fair variance strike from option prices:

    1. Gather option data: Collect European option prices across a range of strikes. Use OTM puts below the forward price and OTM calls above it.
    2. Identify split strike: Set S* as the highest strike price at or below the forward price F0. At this strike, use the average of put and call prices.
    3. Compute weighted sum: For each strike Ki, calculate (ΔKi / Ki2) × erT × Q(Ki), where ΔKi is the strike interval.
    4. Apply correction: Calculate (2/T) × [ln(F0/S*) – (F0/S* – 1)] and add to 2/T times the weighted sum.
    5. Result: The output is E[V] — the fair variance strike. Take the square root for the implied fair volatility.
    Pro Tip

    The 1/K2 weighting means low-strike options contribute more per dollar of premium. This reflects the fact that downside puts are more valuable for variance replication, which is why the volatility skew directly affects variance swap pricing.

    Explore the Options Greeks Course

    Common Mistakes

    Practitioners frequently make these errors when working with variance and volatility swaps:

    1. Confusing variance with volatility units — A 20% volatility strike means 0.04 (4%) variance, not 0.20 variance. Many P&L misunderstandings stem from this conversion error.

    2. Forgetting convexity correction for volatility swaps — Using √E[V] as expected volatility systematically overestimates fair volatility strike, leading to mispriced trades.

    3. Using inconsistent realized variance conventions — Different contracts use different divisors (n, n-1, n-2), annualization factors, and return conventions. Always verify the exact contract terms.

    4. Ignoring option liquidity at extreme strikes — The 1/K2 weighting amplifies the impact of low-strike puts. Sparse or wide bid-ask spreads in the wings create pricing uncertainty.

    5. Comparing variance and volatility swap P&L directly — A $100M vega notional variance swap has different P&L sensitivity than a $100M vega notional volatility swap due to the nonlinear relationship between variance and volatility.

    6. Confusing decimal notation with percentage notation — Is the strike “20” or “0.20”? Contract terms matter: “20 vol strike” usually means 20% (σK = 0.20), but variance swaps quoted in “variance points” might state the strike as 400 (for 20% vol). Always verify the contract’s notational convention.

    Limitations of Variance Swaps

    Despite their advantages, variance swaps have important limitations:

    Key Limitation

    Variance swap replication requires liquid options across a wide strike range. When wings are illiquid or unavailable, the “model-free” fair value becomes uncertain. This is especially problematic for single-stock variance swaps where option liquidity is limited.

    Discrete monitoring — Contracts monitor prices daily (or at other intervals), while the replication formula assumes continuous observation. This creates small basis risk between theoretical and realized variance.

    Jump risk — The standard replication assumes continuous diffusion. Large price jumps (earnings announcements, market crashes) affect realized variance differently than continuous volatility.

    Convexity risk for vol swaps — Volatility swap pricing requires estimating variance-of-variance, which introduces model risk. Different vol-of-vol assumptions produce different fair strikes.

    P&L asymmetry — Because variance is the square of volatility, variance swap P&L is highly asymmetric. A 10-point increase in realized vol from 20% to 30% produces (9% – 4%) = 5% variance gain, while a 10-point increase from 30% to 40% produces (16% – 9%) = 7% variance gain.

    Frequently Asked Questions

    A variance swap pays the difference between realized variance and fixed variance strike, while a volatility swap pays the difference in volatility terms. The key distinction is that variance swaps are model-free — they can be replicated using a static portfolio of options without any volatility model. Volatility swaps require a convexity correction because E[√V] ≠ √E[V] due to Jensen’s inequality. Variance swaps are the institutional standard because of their easier pricing and hedging.

    The fair variance strike is calculated using a replicating portfolio of out-of-the-money puts and calls across a range of strikes. Each option’s price is weighted by 1/K2, where K is the strike price. This approach, based on Hull equation (26.6), determines the risk-neutral expected variance. OTM puts are used below the forward price and OTM calls above it. The weighted sum, plus a correction term involving the forward price and split strike, gives E[V] — the fair variance strike at which the swap has zero initial value.

    The VIX uses variance swap methodology because variance can be calculated model-free from observable option prices. The CBOE computes 30-day expected variance using SPX option prices from two expirations bracketing 30 days, interpolates to get the 30-day variance estimate, then converts to volatility by taking the square root. This approach ensures VIX reflects actual market prices for volatility insurance rather than any specific model’s assumptions. However, VIX has CBOE-specific rules and is not a pure variance swap fair strike.

    Variance swaps can hedge volatility exposure but are not a perfect vega hedge. A long variance swap position gains when realized volatility exceeds expectations, which can offset losses from short vega positions in an options book. However, variance swaps hedge realized volatility, not implied volatility — the two can diverge, especially in trending markets. Additionally, variance swap P&L is driven by daily realized moves, while options mark-to-market depends on changing implied volatility and skew. Traders use variance swaps as part of a broader hedging toolkit, not as an exact offset.

    Variance can be replicated using a static portfolio of European options because the log contract payoff (which delivers realized variance) can be decomposed mathematically into a strip of puts and calls with 1/K2 weights. This decomposition is exact under standard no-arbitrage assumptions. Volatility involves taking a square root of variance, which is a nonlinear (concave) function. The concavity means expected volatility depends on the distribution of variance outcomes, not just expected variance. This introduces model dependence that doesn’t exist for variance.

    Vega notional is a more intuitive way to express variance swap size by stating how much the swap gains or loses per 1 point change in volatility (at the strike level). If a swap has $100,000 vega notional at a 20% volatility strike, it means a 1% increase in realized volatility (from 20% to 21%) produces approximately $100,000 profit for the variance receiver. The conversion to variance notional is Lvar = Lvol / (2σK). Vega notional makes it easier to compare positions across different strike levels.

    Institutional traders use variance swaps for several purposes: (1) expressing pure volatility views without directional market exposure, (2) hedging options book vega to reduce volatility risk, (3) relative value trades comparing implied variance (from options) to realized variance, (4) dispersion trades comparing index variance to weighted component variances (exploiting correlation assumptions), and (5) tail risk hedging, since long variance positions benefit disproportionately from volatility spikes. The variance risk premium — the tendency for implied variance to exceed realized variance — also attracts systematic short-variance strategies.

    Disclaimer

    This article is for educational and informational purposes only and does not constitute investment advice. Variance and volatility swap trading involves significant risk and is suitable only for sophisticated institutional investors. The examples shown use simplified assumptions and may not reflect actual market conditions. Always consult with qualified professionals before trading derivatives.