Volatility Swap Pricing & Convexity Drag Lab

Price institutional volatility swaps and quantify Jensen's inequality convexity drag ($K_{\text{vol}} < \sqrt{K_{\text{var}}}$). Model Brockhaus-Long analytical Taylor approximations and linear vs. convex payoffs.

Market Vol-of-Vol Presets:
Fair Vol Strike (K_vol)

19.32%

Annualized Vol
Convexity Drag

0.68 pts (68 bps)

96.6% of √K_var
Mark-to-Market (MtM)

+$142.5k

Blended Accrual
Variance Notional

$2,588 / vol²

N_vol / (2·K_vol)
Position Vega

$80,245 / pt

Forward Sensitivity
Vol Notional

$100,000

Per 1.0 Vol Point
Swap Terms & Dynamics
%
Implied volatility equivalent of fair variance.
%
Governs variance of realized variance and Jensen's gap.
%
%
Observed volatility across elapsed days.
$
Dollar P&L per 1.0 vol point realized move.
Convexity Drag Math

Brockhaus-Long Second-Order Expansion:

K_vol ≈ √K_var − Var(σ_R²) / [8 (K_var)^(3/2)]

Vega Notional Matching:

N_var = N_vol / [2 · K_vol]
Terminal Payoff Comparison: Volatility Swap vs. Vega-Matched Variance Swap
At Expiration T
Realized Vol ($\sigma_R$) Vol Move ($\sigma_R - K_{\text{vol}}$) Vol Swap P&L Variance Swap P&L Convexity Difference Regime Characteristic
Sensitivity: Variance Strike vs. Vol-of-Vol on Convexity Drag (pts)
Higher volatility of volatility ($\xi$) and lower variance strike amplify the percentage convexity penalty.
Sensitivity: Accrued Realized Vol vs. Days Elapsed on MtM Value ($k)
As observation days elapse, the mark-to-market value locks into realized history, damping remaining forward vega.

Quantitative Derivatives Guide: The Anatomy of Volatility Swaps

1. Jensen's Inequality & The Convexity Gap

In quantitative finance, the market trades two fundamental pure-volatility derivatives: variance swaps, which settle on annualized realized variance $\sigma_R^2$, and volatility swaps, which settle on annualized realized standard deviation $\sigma_R = \sqrt{\sigma_R^2}$.

Because the square root function $f(x) = \sqrt{x}$ has a strictly negative second derivative ($f''(x) = -\frac{1}{4}x^{-3/2} < 0$), it is strictly concave. By Jensen's Inequality, the expectation of a concave function is strictly less than the function evaluated at the expectation:

E[√(σ_R²)] < √(E[σ_R²]) ⟹ K_vol < √(K_var)

The difference $\sqrt{K_{\text{var}}} - K_{\text{vol}}$ is known as the convexity drag (or Jensen's gap). It represents the value of the positive convexity inherent in holding variance rather than volatility.

2. Analytical Pricing & Dynamic Hedging Reality

While variance swaps can be replicated statically using a portfolio of European options weighted by $1/K^2$ (synthesizing the log contract), volatility swaps cannot be replicated statically.

  • Brockhaus-Long Taylor Approximation: Taking a 2nd-order expansion around $E[\sigma_R^2]$ yields: $$K_{\text{vol}} \approx \sqrt{K_{\text{var}}} - \frac{\text{Var}(\sigma_R^2)}{8 K_{\text{var}}^{3/2}}$$ The drag increases quadratically with the volatility of variance ($\xi^2$) and time to maturity $T$.
  • Dynamic Rebalancing: Because $d\sigma_R = \frac{1}{2\sigma_R} d(\sigma_R^2)$, the vega of a volatility swap changes with the level of volatility. If realized volatility surges, a volatility swap's dollar sensitivity declines relative to a variance swap.
  • The Relative Value Trade: Hedge funds frequently trade the spread between volatility swaps and variance swaps to harvest the volatility-of-volatility risk premium without directional equity exposure.
Volatility Swap Mastery Quiz

1. Why is the fair volatility strike $K_{\text{vol}}$ always strictly less than $\sqrt{K_{\text{var}}}$?

2. How does an increase in the volatility of volatility ($\xi$) affect the convexity drag?

3. How is the vega-matched variance swap notional $N_{\text{var}}$ related to the volatility notional $N_{\text{vol}}$?

4. Why are volatility swaps harder to hedge for market makers than variance swaps?

Frequently Asked Questions

A volatility swap is a forward contract whose payoff is strictly linear in annualized realized volatility: Payoff = N_vol * (sigma_R - K_vol). In contrast, a variance swap pays on realized variance: Payoff = N_var * (sigma_R^2 - K_var). This creates a parabolic payoff curve for variance swaps, generating substantial positive convexity during high-volatility spikes.

By Jensen's Inequality, because the square root function is strictly concave (its second derivative is negative), the expected value of the square root of variance is strictly less than the square root of the expected variance: E[sqrt(sigma_R^2)] < sqrt(E[sigma_R^2]). Consequently, K_vol is always lower than sqrt(K_var), with the difference representing the convexity drag.

Brockhaus and Long (2000) derived an analytical second-order Taylor expansion: Convexity Drag = sqrt(K_var) - K_vol approx Var(sigma_R^2) / (8 * K_var^(3/2)). For stochastic volatility processes, this gap is directly proportional to the volatility of variance squared (xi^2) and the time to maturity T.

Variance swaps synthesize the log contract ln(S_T/F), which has a constant second derivative (gamma proportional to 1/S^2) that can be matched statically by an option strip. Volatility swaps require taking the square root of integrated variance; this nonlinearity produces non-zero gamma that shifts with market volatility, demanding continuous dynamic delta and vega rebalancing.

To align the first-order vega sensitivity near the at-the-money strike, the equivalent variance notional is calculated as N_var = N_vol / (2 * K_vol). Because d(sigma^2)/d(sigma) = 2 * sigma, scaling the notional by 2 * K_vol ensures that a 1-point move in volatility produces identical initial dollar P&L.
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