A desk holds a variance swap struck at 25 volatility on Rs 1 crore vega notional. What does it pay if realised volatility is 35 or 15, and why is a delta-hedged straddle not the same trade?

The swap pays Rs 12 crore at 35 and loses Rs 8 crore at 15. Vega notional of Rs 1 crore at a 25 strike is Rs 2 lakh per variance point, and it pays on realised variance minus 625. Because variance is volatility squared, the payoff is convex: a 10 point rise earns more than a 10 point fall costs. A delta-hedged straddle earns on volatility only near its strike, so its result depends on the path.

The contract pays on variance, but traders think in volatility, so it is sized in vega notionalThe rough profit or loss for a one point move in volatility near the strike. Dividing it by twice the strike gives the variance notional.. Near the strike, one volatility point is worth about 2 x 25 = 50 variance points, so the variance notional is Rs 1 crore divided by 50, which is Rs 2 lakh per variance point. At 35 realised, variance is 1,225 against 625, a gain of 600 points, or Rs 12 crore. At 15 realised, variance is 225, a shortfall of 400 points, a loss of Rs 8 crore.

Think of the area of a square tile. Going from 25 cm to 35 cm adds 600 square centimetres; going from 25 cm to 15 cm removes only 400. Variance is volatility squared in the same way. So a long variance swap gains more from a move up in volatility than it loses from the same move down, and the straight line of Rs 1 crore per point understates both the gain and the cushion. That curvature is why variance swaps usually strike above the at-the-money implied volatility: the buyer is paying for the convexity.

A delta-hedged option earns, each day, half its gammaHow fast an option delta changes as the stock moves. High gamma means rehedging captures more profit from each move. times the squared move, minus what the time decay costs. The straddle's cash gamma is highest at the strike and fades as spot moves away. With three months left and 25 volatility, a straddle whose stock has drifted 15% above the strike earns only about 0.57 of its at-the-money rate from each squared move, and about 0.40 if the stock is 15% below. The variance swap is built to have the same sensitivity at every spot level, so it collects every squared move equally.

Now compare two paths that both realise 35. On the first, the index chops around the strike all quarter; the straddle collects close to the swap's payoff. On the second, the index trends 15% higher in the first month and then swings violently; most of the variance arrives where the straddle has little gamma, and it collects perhaps half. The swap pays on realised variance wherever the stock goes; the straddle pays on realised variance weighted by where the moves happen. On the downside the same weighting cushions the straddle's losses, so the colleague is right that it is cheaper to run and wrong that it is the same trade.

State the limits. The straddle picture assumes continuous hedging at the right implied volatility; discrete hedging adds noise of its own. And the swap's convexity cuts both ways for a seller: a short variance swap has unlimited exposure to a crash, because realised variance has no ceiling. Desks cap variance swaps for exactly that reason.