Variance swaps

A variance swap pays the difference between realised variance and a fixed strike: pure volatility, no delta to hedge. Its fair strike needs no model at all. A strip of out-of-the-money options weighted by one over the strike squared replicates it, so the whole smile is priced into one number. Build the strip from a smile, check it against Heston's exact expectation, and see what the swap pays when volatility comes in high or low.

Market smile

At-the-money volatility: 20%

Skew: -0.10

Curvature: 0.04

Spot

Maturity

Rate

Realised volatility at expiry

Strikes in the strip

Heston check

√v0

√θ

Try this

0 of 3 done

Flat smile, fair strike

Load the flat preset and compare the fair volatility strike with the at-the-money volatility.

What to look for: They are almost equal, with the strike a touch higher: the payoff's convexity in volatility is worth a little even when every strike trades at the same volatility.

Skew is expensive

Load the equity-skew preset and look at the contribution chart.

What to look for: The fair strike jumps well above the at-the-money level and the chart piles up on the low strikes. The 1/K² weight makes cheap puts the most important part of the strip.

Trust but verify

Set the Heston √v0 and √θ to 20% and 22% and compare the two numbers in the Heston check.

What to look for: The strip built from Heston prices lands within a fraction of a vol point of the model's closed-form expectation. A model-free formula and a model agree, as they must.

Fair volatility strike

20.84%

√K_var from the strip; at-the-money is 20.00%

Fair variance

0.04344

puts 0.02617, calls 0.01727

Payoff if realised is 22.0%

+0.012

per unit of vega notional

Heston: strip vs exact

21.16% / 21.16%

the model-free strip on Heston prices against θ + (v0 − θ)(1 − e^(−κT))/(κT)

Puts to the left of the forward, calls to the right, each weighted by 1/K². The 1/K² weight is why low strikes matter so much: a steep put skew raises the fair variance well above the at-the-money volatility.

The payoff is linear in variance, so it is convex in volatility: a long swap gains more from a big rise in volatility than it loses from an equal fall. That convexity is why the fair volatility strike sits above the at-the-money volatility even with a flat smile in variance terms.