Fully is a hard threshold to meet, but most of the US Equity markets have determinism windows in the ballpark of 1-3us. Some Futures markets have got under the 1us window, though I don’t recall exactly what the threshold minimum is there.
No market is fully deterministic, however there are a lot of correlations in all markets. Take options. The option is a derivative and therefore its price is defined by the underlying stock. This was mathematically proven by Black & Scholes (and they got a Nobel prize for it).
So, when the underlying moves (the stock) the option should move according to the model. But there are time inefficiencies in the market so a HFT can trade against the misplaced option and lock in a bit of profit. Do that a million times a day any you may make a few bucks.
The parent post is talking about the determinism of the order gateway/matching engine/market data publisher, not the price.
Determinism is achieved there by ensuring there is only one path to the order gateway with everyone having cables of equal length, and that the order in which network packets hit the gateway is guaranteed to be the order the matching engine processes the packets in. Likewise you need to ensure everyone sees the market data at the same time (equal length cables, multicast). Most exchanges fail hard at guaranteeing this.
For your option example, I think you paint a somewhat misleading picture of options trading. You can't trivially arbitrage the spot and the option. There are several unknown factors between the two. An options market-maker works by taking on risk, decomposing it based on the sensitivities of the price to the various inputs, spreading it across the option universe and biasing its prices so that trades statistically take them back to zero. It's closer to a dispersion strategy. For some markets, delta (sensitivity to the underlying) risk is bad to take on so you try to hedge out of it fast, or you avoid trading options with too much delta in them. Trading high-delta requires being fast to react on underlying moves. But that's just an extra requirement when operating this kind of strategy and isn't how you make profit. Even taking out mispriced options after a move is usually mostly done as a way to get out of risk for cheap, and what you'd care about most there is modelling the spot-volatility dynamics.