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list1
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Joined: July 22nd, 2015, 2:12 pm

Re: A conjecture on volatility swap price

December 22nd, 2016, 10:23 pm

The upper half of the second page with 'intuition' looks suspiciously.   The formula
$$ E \frac{1}{T} \int_0^T  \sigma^2_t dt\,=\, \int_0^{\infty} E(  \frac{1}{T} \int_0^T  \sigma^2_t dt\, | S = K ) p ( K ) d K $$
 
where p ( K ) is the spot density at maturity T ... .  should be more accurately explained. It looks like assumed that $$  \sigma^2_t  =   \sigma^2_t ( S ( t ) )$$
and "p ( K ) is the spot density at maturity T" should be interpreted that p ( T ) is the density of the S ( T ) . On the other hand 
$$  \frac{1}{T} \int_0^T  \sigma^2_t ( S ( t )) dt\,$$ depends on S ( * ) on [ 0 , T ]. Though It might be other interpretation that did not clear represented in the slides.
He is using Gyongy's lemma (or was it theorem), also known as Markovian projection in finance, I believe first derived in a finance context by Dupire and/or Derman.
my confusion comes with observation that
$$\sigma^2_t\,=\, \sigma^2_t ( S_t ) $$ 
i.e [$] \int_0^T \sigma^2_t [$] depends on distribution of the  S ( t ) ] for all t on [ 0 , T ] and not [$]S_T[$] as it was stated. Of course one can assume that we talk about rv [$]\frac{1}{T} \int_0^T \sigma^2_t dt[$] which is definitely defined at the moment T. Nevertheless in this case condition in right hand side should be this random variable and not S = K as it was written
 
pausti
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Joined: August 17th, 2016, 1:12 pm

Re: A conjecture on volatility swap price

January 4th, 2017, 10:42 am

In a pure stochastic vol model with zero spot-vol correlation, the vol swap strike is equal to the at-the-money-forward (atmf) vol to a very good approximation. The implied vol smile will be fairly symmetric in such a model, with the atmf close to the lowest part of the smile. Therefore I believe that your formula will over-value the vol swap in this case
 
frolloos
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Re: A conjecture on volatility swap price

January 4th, 2017, 5:26 pm

In a pure stochastic vol model with zero spot-vol correlation, the vol swap strike is equal to the at-the-money-forward (atmf) vol to a very good approximation. The implied vol smile will be fairly symmetric in such a model, with the atmf close to the lowest part of the smile. Therefore I believe that your formula will over-value the vol swap in this case
pausti, you are correct.

For the forum: I checked whether the IVs I generated based on Heston model down to strike 0.5 (with spot = 100) were correct by comparing the variance strike based on the Gatheral et al formula with the variance strike according to the Heston model. It does. Hence I was comfortable with my IVs. Then comparing the volstrike based on the "conjecture" versus the exact Heston volswap strike: the "conjecture" seems to be an upper bound that lies somewhere between the varstrike and  the volswap strike. I don't know why. But volswap is somewhere between the d2=0 approx, and perhaps the integral expression conjecture (no more).
My obsession for the exact volswap value continues, now exploring fractional derivatives.