Stock options are very different from stock shares. I assume you're talking about shares.

Yes, sorry about the imprecision.

If the public companies produced value, say, this year, the value of the companies themselves increased. This means that the worth of the stock in the stock market has increased -- even if no transactions have taken place. That is why stock market is not a zero-sum game.

I didn't claim that owning stock shares produces no value. I claimed that most trades involving stocks (those which neither increase nor decrease the amount of stocks) are zero-sum w.r.t. monetary value.

Consider Alice and Bob who have the same utility function w.r.t. money and the same discounting strategy. Alice sells a share to Bob at price X. Alice is betting that the total discounted utility from the dividends gained by owning the share indefinitely is less than the immediate utility of owning X, while Bob is betting that it is greater than that. Clearly they can't be both right. The gain of one is the loss of the other.