From vega notional to variance notional, realised variance from daily returns, the payoff, the cap and a mid-life mark.
A variance swap pays variance notional × (realised volatility² − strike²), with volatility in points, and the variance notional is the vega notional divided by twice the strike. On an illustrative 100,000 vega notional struck at 20 volatility, realised volatility of 23.46 pays 2,500 × (550.4 − 400) = 375,898, about 30,000 more than the 345,974 the vega notional alone would suggest.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Variance swaps are quoted in volatility but settle in variance. The trader agrees a strike in vol points, 20 here, and a vega notional, the approximate gain or loss per vol point near the strike. The payoff is then computed on the square of realised volatility, because variance, not volatility, is what a static portfolio of options replicates. The conversion between the two is the step most people get wrong, and the squaring is where the swap's convexity comes from.
| Term | Value |
|---|---|
| Vega notional | 100,000 |
| Volatility strike, K | 20.0 |
| Variance strike, K² | 400 |
| Variance notional = vega notional ÷ (2 × K) | 2,500 |
| Observation period, daily closes | 63 returns |
| Annualisation factor | 252 |
| Cap on realised volatility, 2.5 × K | 50.0 |
The 2K in the denominator comes from the derivative of variance with respect to volatility: d(σ²)/dσ = 2σ. At the strike, one vol point moves variance by 2 × 20 = 40 variance points, and 40 × 2,500 = 100,000, which is the vega notional.
Realised variance = (252 ÷ n) × Σ ln(Si ÷ Si−1)² × 100²
Payoff = Nvar × (σR² − K²), with Nvar = Nvega ÷ (2K)
In Excel, with closing prices in B2:B65: =SQRT(252/63*SUMPRODUCT(LN(B3:B65/B2:B64)^2))*100 for realised volatility, then =2500*(D2^2-20^2)
Step 1: square and sum the daily log returns. Over the 63 returns the sum of squared log returns is 0.013759. Most term sheets use this zero-mean convention: the average daily return is not subtracted, unlike a sample standard deviation.
Step 2: annualise. 252 ÷ 63 = 4.0, so annualised variance is 4.0 × 0.013759 = 0.05504. Realised volatility is the square root, 23.46 per cent, and in points squared 550.4.
Step 3: pay the difference in variance. 550.4 − 400 = 150.4 variance points, times the variance notional of 2,500: 375,898 to the long.
Step 4: compare with the vega shortcut. 100,000 × (23.46 − 20) = 345,974. The swap pays 29,924 more, which is the convexity the long variance holder owns.
| Realised volatility | Variance swap payoff | Vega notional × (σ − K) | Convexity |
|---|---|---|---|
| 10.0 | -750,000 | -1,000,000 | 250,000 |
| 15.0 | -437,500 | -500,000 | 62,500 |
| 20.0 | 0 | 0 | 0 |
| 25.0 | 562,500 | 500,000 | 62,500 |
| 30.0 | 1,250,000 | 1,000,000 | 250,000 |
| 40.0 | 3,000,000 | 2,000,000 | 1,000,000 |
| 60.0, capped at 50.0 | 5,250,000 | 4,000,000 | 1,250,000 |
Up to the cap, the long always does better than the linear line: it loses less when volatility falls and gains more when it rises. Its worst case is 1,000,000, at zero realised volatility. The short variance seller carries the opposite profile, which is why sellers insist on a cap. Without the cap, 60 realised volatility would pay 8,000,000; with variance capped at 2,500 (50 squared) the long receives 5,250,000. A cap at 2.5 times the strike is the usual convention on single-stock variance swaps and is also seen on index swaps, but check the term sheet.
Variance is additive in time, so a seasoned swap is valued by blending realised variance to date with the fair strike for the remaining period. Suppose that after 21 of the 63 days realised volatility is 26.0 and the market quotes 22.0 for a new swap over the remaining 42 days.
Expected variance = (21 ÷ 63) × 26.0² + (42 ÷ 63) × 22.0² = 225.3 + 322.7 = 548.0
Value = Nvar × (548.0 − 400) × discount factor = 2,500 × 148.0 × 0.9934 = 367,542
Undiscounted the mark is 370,000; the discount factor is e−4.0% × 0.1667 for the 42 trading days left at an illustrative 4.0 per cent rate. Note that the blended volatility, the square root of 548.0, is 23.41: you blend variances by time, never volatilities.
Convert vega notional to variance notional by dividing by twice the strike, settle on the difference of squares, and expect the long to beat the linear approximation at every realised level below the cap: here 375,898 against 345,974. The book builds pricing and hedging from first principles, and the free workbooks for the book include the volatility models to test these inputs. For the inputs themselves, see how to calculate historical volatility from daily prices and what one vol point is worth on an option.
Vega notional is the approximate payoff per volatility point near the strike; variance notional is the amount paid per variance point, which is what the swap actually settles on. Variance notional equals vega notional divided by twice the strike, so 100,000 of vega at a 20 strike is 2,500 of variance notional.
Because its payoff is convex in volatility. Squaring realised volatility means each extra point is worth more than the last: at 30 realised volatility a 20-strike swap on 100,000 vega pays 1,250,000, against 1,000,000 on a linear volatility payoff. The long pays for that convexity through a variance strike set above at-the-money implied volatility.
Weight realised variance to date and the fair variance strike for the remaining period by their share of the observation days, subtract the strike squared, multiply by the variance notional and discount. After 21 of 63 days at 26.0 realised with 22.0 quoted forward, expected variance is 548.0 and the illustrative mark is 367,542.
This article is one calculation from Quantitative Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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