Good morning,I was trying to check the accuracy of the pricing formula under Heston for variance swap:1) Heston closed-form Option + Variance Swap Replication Formula (c.f. Derman) (integration from K = 50% to 150% step by 1%)2) Variance Swap: hest_varswap = (1 - Exp(-Speed * t)) / (Speed * t) * (spotvar - longvar) + longvar (c.f. Gatheral)I found very different results: the Variance Swap is correct, however, the replication is wrong (SX5E 1y 38 vs. 54). Is there anyone who did the same exercise and found the same results? Is there any explanation or my replication is wrong?Many thanks

I did this kind of analysis in the past and got close results between the two cases.Check your implementation of 1).

thanks for your answer.I was using K = 50% ... 150% and it matches using 0... 200% (around 5vol difference on the wing.Quick question: what do you think about the various approximation for volatility swap?1) ATMF vol2) Brockhaus-long approximation: convexity adjustment3) replication from Carr/Lee - synthetic volatility swap using Bessel fonction

> I was using K = 50% ... 150% and it matches using 0... 200% (around 5vol difference on the wing.I guess the vol of vol is extreme in this evniroment so that the range of replicating strikes should be wide. It also shows that it is virtually impossible to risk-manage var swaps using the replicating portfolio of vanilla options (sure, "theoretical" mark-to-market is easy given a skew parametrization).> Quick question: what do you think about the various approximation for volatility swap?>1) ATMF vol>2) Brockhaus-long approximation: convexity adjustment>3) replication from Carr/Lee - synthetic volatility swap using Bessel fonction 1) Do you mean using a lognormal distribution for the realized variance with a given ATMF volatility? Easiest thing to use, for sure, but you need to come up with an ATMF vol estimate. Not good for short maturities, less than 6 months.2) I tried this under the Heston model and compared to the exact solution and found that this approximation is not that robust especially if the vol of vol is high. Even if you take the correction up to the fourth moments, it is still not acceptable.3) Never tried this one.Btw, for the Heston model there is an exact solution for the volatility swap (up to numerical integration), see Eq (27) and Eq(30) in http://math.ut.ee/~spartak/papers/varsw ... umerically, it is no more complicated than computing vanilla option price under the Heston model

Thanks for your comments.I was using the ATMF vol as an estimate of the expected realised volatility (the ATMF vol being the implied vol backed our from the price of an Heston option with K=F). Unfortunately, it matches the MC price ONLY if the spot/vol correl is 0 and is dependant on the this parameter whereas the volatility swap is not (?). However, it's an easy observable estimate and the variance swap / volatility swap is of a critical interest to me.

You are welcome.I don't quite follow you. By ATMF, do you mean the expected volatility of the spot price or the expected volatility of the realized variance? In the former case, it does depend on the spot-variance correlation parameter, in the latter it does not. But, in my opinion, you should usethe expected volatility of the realized variance in this setting. Note that volatility of the realized variance is a function of vol-of-vol in the first order, while the volatility of the spot is a function of variance in the first order, so that fundamentally they are quite different.

What I'm trying to achieve is to simply price the variance swap - volatility swap spread i.e. sqr(E{V}) - E{sqr(V)}.E{sqr(V)} is my problem, I had several potential estimates:1) ATMF vol, that is to say under Heston the BS Vol of the Heston option with strike struck at the forward => which is strongly dependant on the spot/var correl2) Brockhaus-long approximation => which is NOT dependant on the spot/var correlLooking at the Monte-Carlo simulation, E{sqr(V)} (like E{V}) seems to be independant of the the spot/var correl - is that wrong?

I see. Under Heston and, generally, in any SV model where the spot is a log-normal type process, the distribution of V does not depend on the spot-var correlation parameter (important: V is the instanteneous variance, not the implied volatility squared). Pricing Vol swap under Heston is similar to pricing a VIX futures, you can use the explicit solution for this in the paper I quoted using formulas (56) - (59) with A^F=0 and B^F=1.1) It can work, but you need to use the volatility of V not the implied volatility of the spot. See my previous post.2) As I noticed, for short-term maturities and high vol-of-vol, I found that it is not acceptable compared to the exact solution even if you use the correction up to the fourth moments.