A Seasoned Volatility Swap

This is very much what's in the Carr-Lee paper "Robust Replication of Volatility Derivatives", but it wasn't so easy to obtain in practice:

• The formulas as written in the paper are not usable as is: they can be simplified (not too difficult, but intimidating at first)
• The numerical integration is not trivial: a simple Gauss-Laguerre is not precise enough (maybe if I had an implementation with more points), a Gauss-Kronrod is not either (maybe if we split it in different regions). Funnily a simple adaptive Simpson works ok (but my boundaries are very basic: 1e-5 to 1e5).

A Seasoned Volatility Swap

This is very much what's in the Carr-Lee paper "Robust Replication of Volatility Derivatives", but it wasn't so easy to obtain in practice:

• The formulas as written in the paper are not usable as is: they can be simplified (not too difficult, but intimidating at first)
• The numerical integration is not trivial: a simple Gauss-Laguerre is not precise enough (maybe if I had an implementation with more points), a Gauss-Kronrod is not either (maybe if we split it in different regions). Funnily a simple adaptive Simpson works ok (but my boundaries are very basic: 1e-5 to 1e5).

A Volatility Swap and a Straddle

A Volatility swap is a forward contract on future realized volatility. The pricing of such a contract used to be particularly challenging, often either using an unprecise popular expansion in the variance, or a model specific way (like Heston or local volatility with Jumps). Carr and Lee have recently proposed a way to price those contracts in a model independent way in their paper "robust replication of volatility derivatives". Here is the difference between the value of a synthetic volatility swap payoff at maturity (a newly issued one, with no accumulated variance) and a straddle.

Those are very close payoffs!

I wonder how good is the discrete Derman approach compared to a standard integration for such a payoff as well as how important is the extrapolation of the implied volatility surface.

The real payoff (very easy to obtain through Carr-Lee Bessel formula):