Don't follow that at all, I'm afraid. I don't even understand what you were rewriting! Can you dumb this down a little? ;) Not in terms of E(payoff at expiry), and not in terms of 1 day's change. Just the percentage cost of carry in terms of option price, underlying price, normalised r (call it R = r * days /360). If that's possible, of course! I just don't see how taking a derivative with respect to time helps anything; surely you'd just have to reintegrate to get a meaningful answer?

I don't think so. That question/discussion is about the relationship between cost of carry, call price, put price and underlying price. Nowhere does it state how to calculate the cost of carry.

And the magnitude of the second term? I'm thinking it's r * (optionPrice / underlyingPrice) ? Where r is normalised from annual.

@JanStuller Nasdaq 100. I might diversify into S&P and maybe some treasuries later.

Continuous rolling futures (I think the maturity is immaterial... but I usually buy quarterlies at just over 3 months to maturity, as my broker doesn't offer anything further dated). As for the option strikes and maturities, I'm currently putting together some backtested simulations to determine what days-to-expiry, strikes, and purchase dates to use (aiming to minimise drawdown while maximising returns, naturally). Right now (until backtesting done) I'm actually shorting selling QYLD (a covered call ETF) rather than buying the puts myself (selling a covered call is the same as buying puts).

Is it because paying a premium is effectively "shorting cash", i.e. lending money?

Could you expand a little on 2? Why are premiums less when interest rates are higher (regardless of whether call or put)? Thanks

I don't see how the "this increase in cost should be immaterial". If it were immaterial everyone would buy ATM calls, sell ATM puts, and sell futures. Net price movement of such a strategy is zero, yet selling futures earns one (interest rate - dividend yield) every year. Which would be free money that I could leverage as high as my broker would let me. Since everyone isn't doing this to get free money (or if they are, they forgot to tell me:), I assume the long ATM call and short ATM put (synthetic long future) cost of carry is definitely "material" (equal and opposite to that of the future).

"Your main concern is not "the cost of carry" anyway". I accept that obviously delta sensitivity is of much greater influence, the question says that "cost of carry" (which I'm defining as interest rate - dividend) is my concern here :) I'm continually holding leveraged, hedged option positions (e.g. holding long N100 future plus regular monthly N100 puts) , and if interest rates are considerably higher than dividends, then the ongoing effect of interest rates is significant. The leveraged (long) future has high cost of carry, and I want to know how much the puts reduce that.

There is something absolutely fundamentally different between options and cash instruments. Cash instruments don't expire!