You buy a one-year variance swap with vega notional $100,000 per volatility point struck at 20. Realised volatility over the year is 22. What is the payout, and why is it not $200,000?

You buy a one-year variance swap with vega notional $100,000 per volatility point struck at 20. Realised volatility over the year is 22. What is the payout, and why is it not $200,000?

Approach: Convert the vega notional into a variance notional using the rule that variance notional equals vega notional divided by twice the strike, then settle on the difference of squared volatilities.

$210,000. The variance notional is vega notional divided by 2*K = 100000/40 = $2,500 per variance point, and the payout is 2500*(22^2 - 20^2) = 2500*(484 - 400) = $210,000. The linear guess of $200,000 misses the convexity of the variance swap in volatility: the payout is quadratic, so it pays more than the vega notional on an upmove and loses less on a downmove. A realised 18 would cost 2500*(324 - 400), a loss of $190,000 rather than $200,000. That convexity is worth real money, which is why variance swaps trade above the at-the-money implied volatility, and the gap widens with the steepness of the skew.

Follow-up: Realised volatility comes in at 45 on a crash. What does the payout become, and how would a capped variance swap change it?

Key concepts: variance swap, variance notional, convexity, vega notional.