Forward Variance Curve

If a 1-year loan costs 4% and a 2-year loan costs 5%, you can back out the implied rate for borrowing specifically in year two — the "forward rate" nobody quoted directly but that's baked into the two spot rates. Forward variance works exactly the same way, except the two things being compared are variance swap strikes for two different maturities instead of interest rates for two different terms.

The formula, defined

Because variance accumulates additively over non-overlapping time periods, the forward variance between two future dates can be extracted from two spot variance swap quotes.

$$\xi(T_1, T_2) = \frac{\sigma_{T_2}^2 T_2 - \sigma_{T_1}^2 T_1}{T_2 - T_1}$$

In words: $\sigma_{T_1}^2$ and $\sigma_{T_2}^2$ are the fair variance swap strikes (in variance terms, i.e. vol squared) quoted today for maturities $T_1$ and $T_2$. Multiplying each by its own maturity gives total variance accumulated from today out to that date. Subtracting the shorter-dated total variance from the longer-dated one isolates the variance expected to accumulate specifically between $T_1$ and $T_2$, and dividing by the length of that window, $T_2 - T_1$, converts it back into an annualized variance rate for that specific future slice of time — the forward variance.

Worked example 1 — extracting a forward slice

A 3-month variance swap trades at a fair strike corresponding to 20% vol, and a 9-month variance swap trades at 24% vol. First, total variances: $\sigma_{T_1}^2 T_1 = 0.20^2 \times 0.25 = 0.01$, and $\sigma_{T_2}^2 T_2 = 0.24^2 \times 0.75 = 0.0432$. The forward variance for the 3-to-9-month window: $\xi = (0.0432 - 0.01)/(0.75 - 0.25) = 0.0332/0.50 = 0.0664$, so the forward vol for that window is $\sqrt{0.0664} \approx$ 25.8% — noticeably higher than either the 3-month (20%) or the 9-month (24%) spot vol, because the market expects most of the extra variance priced into the longer maturity to be concentrated in the later part of the period, not spread evenly.

Worked example 2 — a flat curve gives a flat forward

Now suppose 3-month and 9-month variance swaps both trade at 22% vol. Total variances: $0.22^2 \times 0.25 = 0.0121$ and $0.22^2 \times 0.75 = 0.0363$. Forward variance: $(0.0363 - 0.0121)/0.5 = 0.0484$, and $\sqrt{0.0484} = 22\%$ exactly — when the spot curve is flat, the forward variance for any window equals that same flat number. Useful sanity check: a flat term structure implies the market expects the same variance rate throughout, and forward variance recovers exactly that unchanged rate.

What this means in practice

Traders use forward variance to see where the market expects volatility to be elevated relative to its own term structure, not just in absolute terms — a steep forward segment around a known future date (an election, a central bank meeting) shows the market pricing in extra variance specifically around that event, even if spot vol today looks calm. It's also the fundamental building block of forward-starting variance swaps and the forward variance models used to price them (see Bergomi forward variance models).

Practice

1. A 1-month variance swap trades at 18% vol and a 4-month variance swap trades at 21% vol. Compute the forward variance (as a vol number) for the 1-to-4-month window.
2. If the 3-month and 9-month variance swaps in worked example 1 (20% and 24%) instead both moved up by the same 3 vol points to 23% and 27%, would the forward vol for the 3-to-9-month window change, and roughly by how much relative to the original 25.8%?