In the Heston model with zero equity drift rate, suppose a positive time to maturity is given, and the variance at infinity is sufficiently larger than the variance at time $0$.1) When the equity and variance correlation is non-negative, is the at-the-money implied volatility of a European option increasing with respect to the volatility of volatility?2) When the equity and variance correlation is negative, as the volatility of volatility increases, does the implied volatility first decrease (it is possible that this interval length is zero depending on the time to maturity) then increase?

You could play around with eqn (3.5) on pg 84 of "Option Valuation under Stochastic Volatility" to find out.That eqn shows that, if [$]\xi[$] is the vol-of-vol, then the first order ([$]O(\xi)[$]) change in [$]V_{imp}[$] is indeed proportional to [$]\rho \xi[$].But I suspect [$]O(\xi^2)[$] effects, which are also given there can easily mess up the sign, yielding the conclusion that there is no monotonicity. That (3.5) series is likely only asymptotic and not convergent, and in any event, there is no error bound.So if you find a likely counter-example using it, then you should re-verify it with a full Heston calculator.

Hi, Alan. I have just edited my question to make it more concrete and precise. I did look at the asymptotics and have considered the order of magnitude effect as you described. I had the exact concern that these asymptotics were for small [$]\xi][$] and there was no error bound, which would not produce the monotonicity for [$]\xi][$] of arbitrary size. I wonder if there is any (even partial) results of monotonicity or interesting counterexamples.

I don't know; I am just suggesting ways to find out. Here is a likely easier method. Assume the time to maturity is very large.Then, the atm [$]V_{imp}[$] is given by (4.5) on pg 189 of you-know-where. From the second line, I would investigate [$]\rho <0[$],but [$]\frac{1}{5} < \rho^2 < \frac{3}{7}[$]. Looks to me that this renders the [$]O(\xi)[$] term negative, but both the [$]O(\xi^2)[$] and [$]O(\xi^3)[$]corrections positive. So, fix [$]\omega=1[$], plot the exact first line vs [$]\xi[$], and adjust [$]\theta[$] to try to achieve a 'yes' to your Q2 for at leastsome combinations of [$](\rho,\theta)[$]. Update:I just tried this. For [$]\rho = 0.5[$], [$]V_{imp}[$] at first goes up, but then turns down with [$]\xi[$]. This resolves your Q1.But for [$]\rho = -0.5[$], [$]V_{imp}[$] seems to always decrease with [$]\xi[$] for all the cases I tried.Unless I missed something, it looks like taking the time to maturity large is either(a) not a helpful attack on the Q2 problem, or(b) is correctly suggesting that [$]V_{imp}[$] is indeed monotone decreasing with [$]\xi[$] for [$]\rho \le 0[$].Good puzzle! (which I leave to you)

Hello lovenatalya, For fixed maturity, one way to potentially prove some results along these lines would be to use some (Hajek's) comparison theorem. The first step would be to see if the total variance increases/decreases (almost surely) with the vol of vol, and then transfer that to the stock price process.This is a rough idea, and have not had time to give it much though yet, but it might be fruitful.Best,

Hello lovenatalya, For fixed maturity, one way to potentially prove some results along these lines would be to use some (Hajek's) comparison theorem. The first step would be to see if the total variance increases/decreases (almost surely) with the vol of vol, and then transfer that to the stock price process.This is a rough idea, and have not had time to give it much though yet, but it might be fruitful.Best,

Hi, Antonio:
I did not see your reply until now, a full year later. Thank you very much for your idea. Would you mind providing some references for Hajek's comparison theorem? Is it related to the maximum principle and the equivalent comparison theorem of the elliptical operator and the parabolic partial differential equation?
Regards

Dear lovenatalya,

I would look at the following paper: 
http://www.sciencedirect.com/science/ar ... 4987902018.
This is not Hajek's original article (B. Hajek, Mean stochastic comparison of diffusions)

but should provide some useful elements. 
I would be more than happy to discuss about this more specifically. It is easier to contact me directly by email though.

Regards,