Agasthya Capital buys a variance swap with vega notional of Rs 10 lakh at a strike of 20. What is the variance notional, what is the P&L if realised volatility is 24, and if it is 16? Why is the payoff not symmetric?

Variance notional is Rs 25,000 per variance point; Agasthya makes Rs 44 lakh at 24 and loses Rs 36 lakh at 16. The vega notional, Rs 10 lakh a point, is only the slope at the strike. The swap pays on the square of volatility, so each point above 20 is worth more than the last and each point below worth less, which is why the payoff is convex and the buyer gains more on a four-point rise than it loses on a four-point fall.

The trade is quoted in volatility because people think in volatility, but it pays in variance because variance is what a portfolio of options can replicate. The variance notional is set so that, near the strike, one point of volatility is worth the vega notional: since 20 squared changes by about 2 times 20, or 40, variance points when volatility moves by one, variance notional is Rs 10 lakh divided by 40, Rs 25,000 per variance point. At the strike the two descriptions agree. Away from it they do not, and that is the rest of the case.

Square, subtract, multiply. At 24, realised variance is 576 against 400, so Agasthya receives 176 times Rs 25,000, Rs 44 lakh; at 16, realised variance is 256 against 400, so it pays 144 times Rs 25,000, Rs 36 lakh. Both outcomes are four volatility points from the strike, and a volatility swapA swap that pays the vega notional times realised volatility less the strike, linear in volatility. Harder for dealers to hedge than a variance swap, so less common. would have paid plus and minus Rs 40 lakh. The variance swap pays Rs 4 lakh more on the way up and Rs 4 lakh less on the way down.

Think of the area of a square plot. Add a metre to each side of a 20 metre plot and the area grows by 41 square metres; take a metre off and it shrinks by 39. Squares grow faster on the way up. The swap pays on the square of volatility, so the buyer is long convexityHere, the curvature of the payoff in volatility: the gain from a rise exceeds the loss from an equal fall.: every extra point of volatility pays more than the one before, and the asymmetry widens the further realised volatility moves. That convexity is not free. Dealers know it, so variance swap strikes sit above the at-the-money implied volatility of the same maturity, often by a point or more, partly because the replicating portfolio includes far out-of-the-money puts, which trade at higher implied volatility, and partly as payment for the convexity itself. A buyer who expects volatility of 22 should not buy at a strike of 22 and expect to break even.

State the limits for the client. The upside in a crash is large: at a realised volatility of 36 the swap pays Rs 224 lakh, and the dealer on the other side knows this, which is why single-stock variance swaps usually carry a cap, commonly two and a half times the strike, and index swaps sometimes do. The realised number depends on the exact return series in the contract, daily closes, with the method of annualising fixed in the confirmation; a missed day or a market closure can move it. And because the swap is a pure bet on realised against implied, a fund that buys it is paid only if volatility actually turns up, not if the market merely becomes nervous; a rise in implied volatility before expiry helps only the mark.