{
  "id": "d512b1b1-5da8-5b5e-9e84-9f8b70cb5634",
  "slug": "volatility-swap",
  "term": "Volatility Swap",
  "aliases": [],
  "category": "Derivatives & Options",
  "category_slug": "derivatives-options",
  "difficulty": "advanced",
  "definition": "A volatility swap is an over-the-counter derivative contract in which counterparties exchange the realized volatility of an underlying asset over a specified period against a fixed volatility strike, with payoff proportional to the difference between realized and strike volatility multiplied by a notional vega amount. It is the volatility analog of the variance swap — providing direct, clean exposure to the level of volatility rather than its square.",
  "key_takeaways": [
    "Payoff = Notional Vega × (Realized Volatility - Vol Strike), where realized vol is typically the annualized standard deviation of daily log returns.",
    "Unlike variance swaps, volatility swaps cannot be replicated by a static portfolio of options — they require dynamic replication, making them more difficult and expensive to hedge.",
    "Volatility swaps trade at a discount to the square root of the variance swap strike, by an amount related to the convexity adjustment (Jensen's inequality).",
    "Volatility swaps provide more intuitive P&L attribution than variance swaps for investors who think in volatility rather than variance terms.",
    "The primary users are hedge funds expressing pure volatility views and institutions hedging volatility-linked liabilities (e.g., variable annuity guarantees)."
  ],
  "detailed_explanation": "Volatility swaps and variance swaps are the two primary instruments for trading 'pure' volatility — exposure to the level of an underlying's return volatility without directional (delta) or gamma exposure. While variance swaps trade realized variance (the square of volatility) against a fixed strike, volatility swaps trade realized volatility directly against a fixed strike. This seemingly minor distinction has profound implications for replication, pricing, and risk management.\n\nThe key difference between volatility and variance swaps lies in replicability. A variance swap can be replicated by a static portfolio of options: buy options at every strike, weighted by 1/K², and continuously rebalance the delta-hedge. This static replication makes variance swaps relatively straightforward to price and hedge using observed option prices. A volatility swap, by contrast, cannot be replicated statically — because volatility is a non-linear function of variance (vol = √variance), replicating the square root requires a dynamic strategy. This means volatility swaps are inherently more model-dependent and more expensive to hedge than variance swaps, explaining why the market is less liquid in volatility swaps than variance swaps.\n\nThe pricing relationship between variance and volatility swaps is governed by Jensen's inequality. Because volatility is a concave function of variance (√x is concave), the expected volatility is strictly less than the square root of expected variance: E[√Variance] < √E[Variance]. The difference between the square root of the variance swap strike and the volatility swap strike is the 'convexity adjustment' — the market discount applied to volatility swaps relative to the naively computed level. In a stochastic volatility world, this convexity adjustment depends on the vol-of-vol parameter: higher vol-of-vol increases the difference between var swap and vol swap strikes, as the concavity of the square root function generates a larger Jensen's inequality effect.\n\nFrom a practical risk management perspective, volatility swaps are preferred by investors who think in vol-of-vol space and want P&L that scales linearly with the realized volatility level rather than quadratically with variance. For example, a pension fund that wants to hedge the volatility component of its liability-driven investment (LDI) program — where the liability's duration sensitivity itself varies with interest rate volatility — may prefer a volatility swap because its payoff profile matches the exposure more intuitively than a variance swap. Similarly, variable annuity writers who sell minimum guaranteed return products are implicitly short volatility, and they may use volatility swaps to hedge this exposure in a size that is directly proportional to the volatility level of the underlying funds.\n\nThe settlement of a volatility swap follows the same structure as a variance swap: at maturity, the realized volatility is computed as the annualized standard deviation of daily log returns, √(252/n × Σ[ln(Sᵢ/Sᵢ₋₁)]²), and compared to the vol strike. The payoff flows from the losing party to the winning party, without any exchange of principal. Realized volatility calculations must specify exactly how to handle non-trading days, corporate actions, and discrete price disruptions — all of which are specified in the ISDA volatility swap confirmation.",
  "example": "An insurance company has written $500 million of variable annuity contracts with a guaranteed minimum accumulation benefit (GMAB). The benefit pays out if underlying equity fund volatility drives the account value below a floor. The actuarial team estimates that a 5-vol-point increase in S&P 500 realized volatility increases the fair value of the GMAB liability by $12 million. To hedge this exposure, the company enters a 6-month volatility swap as buyer: vol strike = 20%, notional vega = $2.4 million per vol point. If realized vol over 6 months is 25% (a 5-vol-point overshoot), the volatility swap payoff = $2.4M × (25% - 20%) = $2.4M × 5 = $12M — exactly offsetting the increase in GMAB liability. If realized vol is 15%, the company pays $2.4M × (15% - 20%) = -$12M on the swap, but benefits from a $12M decrease in the GMAB liability, netting to zero — a perfect hedge.",
  "formula": "Vol Swap Payoff = N × (σ_realized - K_vol); Vol Strike ≈ √(Variance Swap Strike) - (Convexity Adjustment); Convexity Adjustment = Vol-of-Vol² / (8 × K_vol)",
  "formula_latex": null,
  "interactive_type": "calculator",
  "calculator_id": null,
  "related_terms": [
    "annuity",
    "bull-spread",
    "convexity",
    "convexity-adjustment",
    "credit-support-annex",
    "delta",
    "duration",
    "embedded-derivative",
    "equity",
    "exchange",
    "floor",
    "gamma",
    "interest-rate",
    "jensens-inequality",
    "martingale-measure"
  ],
  "backlinks": [],
  "cross_references": [
    "annuity",
    "convexity",
    "convexity-adjustment",
    "delta",
    "duration",
    "equity",
    "exchange",
    "floor",
    "gamma",
    "interest-rate",
    "jensens-inequality",
    "netting",
    "option",
    "settlement",
    "standard-deviation",
    "swap",
    "variance",
    "variance-swap",
    "vega",
    "volatility"
  ],
  "tags": [
    "level:advanced",
    "cat:derivatives-options"
  ],
  "asset_classes": [
    "derivatives"
  ],
  "regulators": [],
  "see_also": [],
  "sources": [],
  "wordcount": 844,
  "checksum": "481d883439f503bc",
  "version": "2026.05.03",
  "license": "CC-BY-4.0",
  "updated_at": "2026-09-07T02:15:24+00:00",
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