The concept of deriving the financial product price is a logic of cash flow.
The outcomes of go/back cash flow should be identical.
Otherwise, you invent a money machine.
An (equity) option is linked to a specific stock. The price of the option is much less than the price of the underlying stock, which is a major reason for the attractiveness of options. If the price of the stock changes, the price of the option also changes, although by a smaller amount. As the price of a stock goes through its daily ups and downs, the price of an associated option undergoes related fluctuations.
The price of an option can be viewed and followed in much the same way as a stock price. There are numerous online services, including the data feed for your brokerage account, which provide the prices of options.
For a call option, if the stock price goes up, the option price also increases. If the stock price goes down, the price of the call decreases.
For a put option, if the stock price goes down, the option price increases. If the stock price goes up, the price of the put decreases.
This sounds like owning a call option is similar to holding a long position in the stock because you have the potential to make a profit when the stock price goes up. And owning a put option is similar to holding a short position in the stock because you have the potential to make a profit when the stock price goes down. In a rough sense, this analogy is true, but there are some significant differences.