Variance Swaps

A delta-hedged option is a bet on realized variance, but a messy one, the P&L is weighted by a random, path-dependent dollar gamma. A variance swap is the clean version: a forward contract on realized variance itself, with constant dollar exposure to variance regardless of path. Its beauty is that it can be replicated statically by a fixed portfolio of options weighted $1/K^2$, a model-free result that is also the mathematical backbone of the VIX.

The payoff

A variance swap pays, at expiry, the difference between realized variance and a fixed strike, scaled by a variance notional:

$\text{Payoff} = N_{\text{var}}\big(\sigma_R^2 - K_{\text{var}}^2\big), \qquad \sigma_R^2 = \frac{252}{n}\sum_{i=1}^{n}\ln^2\!\Big(\frac{S_i}{S_{i-1}}\Big).$

The strike $K_{\text{var}}$ is quoted in vol points (so $K_{\text{var}}^2$ is the fair variance). Because the payoff is linear in variance, its exposure, the "variance vega", is constant, unlike an option whose gamma drifts with spot. It is the purest vehicle for a view on realized vs implied volatility.

The log-contract derivation

The key identity connects realized variance to a tradeable payoff. Under the risk-neutral diffusion $dS/S = r\,dt + \sigma_t\,dW$, apply Itô to $\ln S$:

$d\ln S_t = \Big(r - \tfrac12\sigma_t^2\Big)dt + \sigma_t\,dW_t.$

Subtract this from $dS/S = r\,dt + \sigma_t\,dW_t$, the two $\sigma\,dW$ terms cancel exactly:

$\frac{dS_t}{S_t} - d\ln S_t = \tfrac12\sigma_t^2\,dt.$

Integrate over $[0,T]$ and multiply by 2:

$\int_0^T \sigma_t^2\,dt = 2\left[\int_0^T \frac{dS_t}{S_t} - \ln\frac{S_T}{S_0}\right].$

The left side is the (continuous) realized variance. The right side is a recipe: $\int dS/S$ is the P&L of a continuously rebalanced position holding $1/S_t$ shares (a self-financing, zero-cost dynamic strategy whose risk-neutral expectation is $rT$), and $-\ln(S_T/S_0)$ is a static payoff, the log contract. So

$\mathbb{E}^{\mathbb{Q}}\!\left[\int_0^T\sigma_t^2\,dt\right] = 2\left(rT - \mathbb{E}^{\mathbb{Q}}\!\left[\ln\frac{S_T}{S_0}\right]\right).$

Realized variance is model-free once you can price the log contract, no volatility model needed, only the ability to value $\ln S_T$.

Static replication: the strip of options

The log payoff is not directly traded, but the Carr-Madan spanning formula rebuilds any smooth payoff $f(S_T)$ from a cash amount, a forward, and a continuum of options:

$f(S_T) = f(S_*) + f'(S_*)(S_T - S_*) + \int_0^{S_*}\!\!f''(K)\,(K - S_T)^+\,dK + \int_{S_*}^{\infty}\!\!f''(K)\,(S_T - K)^+\,dK.$

For the log contract $f(S) = -\ln(S/S_0)$ we have $f''(K) = 1/K^2$. Choosing $S_* = F_0$ (the forward), the linear term vanishes in expectation, and the fair variance strike becomes

$\boxed{\,K_{\text{var}}^2 = \frac{2}{T}\left[rT - \Big(\frac{S_0}{F_0}e^{rT}-1\Big) + e^{rT}\!\int_0^{F_0}\!\frac{P(K)}{K^2}\,dK + e^{rT}\!\int_{F_0}^{\infty}\!\frac{C(K)}{K^2}\,dK\right].\,}$

The essential content: fair variance is a $1/K^2$-weighted integral of OTM put and call prices. The $1/K^2$ weight is the signature, it says a variance swap is long a strip of every strike, with far-OTM options (the wings) heavily weighted. This is a static hedge: buy the strip once, hold to expiry, and (with continuous rebalancing of the $1/S$ stock leg) you have replicated realized variance without any model of volatility.

Volatility swap vs variance swap, the convexity

A volatility swap pays $\sigma_R$ (vol, not variance). Since $\sqrt{\cdot}$ is concave, Jensen gives $\mathbb{E}[\sigma_R] \le \sqrt{\mathbb{E}[\sigma_R^2]} = K_{\text{var}}$, so the fair vol strike is below the square root of the fair variance strike. The gap, the convexity adjustment, grows with vol-of-vol, so a variance swap is implicitly long vol-of-vol relative to a vol swap. This is why variance swaps, not vol swaps, admit clean static replication: variance is the "linear" quantity in options.

Worked example

Suppose the fair-variance calculation on an index yields $K_{\text{var}} = 20$ (so $K_{\text{var}}^2 = 400$ variance points), on a variance notional of $10,000 per variance point. If the index then realizes $\sigma_R = 25\%$, the payoff is

$N_{\text{var}}(\sigma_R^2 - K_{\text{var}}^2) = 10{,}000\,(25^2 - 20^2) = 10{,}000\,(625 - 400) = \$2{,}250{,}000.$

Note the payoff's convexity in vol: going from 20 to 25 vol earns $2.25m, but 20 to 15 loses only 10{,}000(225-400) = -\1.75$m. Long variance is long convexity, you win more from a vol spike than you lose from an equal vol drop, which is exactly why sellers demand a premium and fair variance strikes trade above expected realized.

What breaks in practice

• The wings are not traded to infinity. The replication needs a continuum of strikes out to $0$ and $\infty$; real markets have finite strikes, so the strip is truncated. Truncation underestimates fair variance and, in a crash, caps how much the swap can pay, the replication breaks exactly when variance explodes. Post-2008, capped variance swaps (cap at $2.5\times$ strike) became standard.
• Discrete monitoring and jumps. The log-contract identity assumes continuous paths; a jump adds a term $-2\sum[\ln(S_i/S_{i-1}) - \Delta S_i/S_{i-1}]$ so realized variance and the replication diverge, a big down-gap makes the actual realized variance exceed the option strip's hedge.
• Dividends and discreteness. Discrete dividends and daily (vs continuous) sampling introduce corrections the clean formula omits.

In interviews

The derivation is the prize: show $dS/S - d\ln S = \tfrac12\sigma^2 dt$, integrate to get realized variance $= 2[\int dS/S - \ln(S_T/S_0)]$, and conclude that variance replicates via the log contract, which spans into a $1/K^2$-weighted strip of options. Emphasize it is model-free (no vol model, just option prices) and static. Know the payoff $N_{\text{var}}(\sigma_R^2 - K_{\text{var}}^2)$, that variance swaps are long convexity (fair strike above expected realized, and a vol swap is worth less by a convexity adjustment), and the practical failure, wing truncation and jumps break the hedge in a crash. This machinery is the VIX.