Well surely if you hold the future, you will make (very close to) 1% tomorrow if the VIX increases by 1%? Of course in fact you will typically make a little less than 1% (if you're long) due to VX futures gradually decaying since demand exceeds supply due to risk averseness -- but that's the same for the ETFs, right? So why is it that gradually rolling over seems to leverage the ETF compared to just rolling at expiry?

But in the case of VIX futures there is no "stock price". The VIX index is only notionally the future's underlying asset. Unlike e.g. stock index futures, where trading an ETF is pretty much trading the index, one can't buy or sell "VIX", other than via the futures themselves (or ETPs trading the futures). Unconstrained by the possibility of arbitraging via an underlying asset, supply and demand alone determine prices. VIX futures alway decline in the long term, because buying it is essentially buying insurance, and selling it is selling insurance, and sellers of insurance demand a premium.

No, TickData is stupid expensive. What hobbyist has $100,000 knocking around for some data? In any case you did not answer the question at all, merely spewed an anti-piracy rant.

Yes. You might say, the "additional cost" is the cost of the call compared to what it would cost if interest and expected dividend yield were exactly equal (or if we forget dividends, then if interest rates were zero). I'm sure that this additional cost is intimately related to delta, but not sure if it is as simple as it being proportional to delta.

"Explain to me how you think that deep ITM call has a cost of carry?" Well, if my proposed simple formula carry = delta * (rates - yield), then the deep ITM call would not have any appreciable cost of carry. I'm still looking for confirmation that this simple formula works though.

@JanStuller I actually don't quite think my question has been answered yet... but as the bounty expires very soon I've decided to award it to you as you went the extra mile trying to improve my understanding. Though other answers were very interesting too. The one thing I don't know for sure from this is: Is the additional cost (or discount if yield > interest) of the call attributable to carry simply equal to delta * (rates - yield), or is there some other formula I need if I want to express it in terms of delta (or moneyness, or other Greeks)?

Interesting... I think most people define ATM (rightly or wrongly) as having strike = underlying (I've sure I've read this definition several times in the past few days). But presumably what you are saying, then, is that ATM is correctly defined as meaning delta = +/- 0.5? And that at delta = +/- 0.5, the option price includes the compounded interest/yield? And that this is why delta is not quite half when strike = underlying. I need to digest this a little and see what drops out!

Or perhaps there isn't a simple formula for this?

Whilst your answer is illuminating (it would be more so if I understood more of it;) , unless I missed it I'm still lacking the one thing I was very specific about wanting... a simple formula that tells me how much that extra penalty is on the call price (or discount on the futures price) in terms of moneyness, delta or whatever. Unless it's staring at me from somewhere in your post and I'm not smart enough to see it, which is possible!

"Cost of carry play no further role here, besides defining the fair future value (or strike)". Yes. I never imagined it to be otherwise. It's the same for the futures... carry isn't some ongoing "fee" for holding the long future, it's a penalty priced into buying it.

@AKdemy I fail to see anything wrong with my point about "If it were immaterial everyone would buy ATM calls, sell ATM puts and sell futures.". This would be free money if options prices didn't similarly punish (as described in my above comment) the holder of a (long) synthetic future. Right? So (if we rule out the free money scenario) the cost of carry is not immaterial to options.

@AKdemy Your point that technically the "cost of carrying" is associated with the holding the physical stock rather than the future is immaterial. The fact remains that futuresPrice = underlyingPrice + R - D (where R and D are interest/expected dividends up until expiry, and that if rates are high (let's ignore dividends for a moment) you are paying a "premium" (not options premium!) for the benefit of not having to tie up money (that could earn interest) in holding the underlying - or equivalently not having to borrow money to hold the underlying.

@JanStuller Absloutely not! Nothing to do with fees. Carry on a long future = interest rate - dividends. Check the current price of any US future. It is higher than spot price of underlying (at the moment, while interest rates are much higher than expected dividends). Suppose you buy a (long) future and hold till expiry. At expiry, the price is obviously guaranteed equal to the price of the underlying. So there is a positive cost of carry (negative profit), since you are buying high (relative to underlying) but selling at parity with the underlying.

Please don't think I'm being off-hand or ungrateful... I think my options understanding may simply be not up to scratch, and I'm trying to get there;)

@AKdemy Well, I see what you mean, but rearranging the formula doesn't give a formula for carry that is in any way illuminating (to me!). Carry in terms of put price and call price doesn't answer my question "how is the cost of carry split between the put and call legs of a synthetic future?". An illuminating formula (to me) would be in terms of moneyness, delta, etc. Maybe that's just me. And assuming Rodrigo is correct then there is a second term (which I may have misunderstood as "interest payed to the buyer on the premium") which doesn't appear in that formula at all.