Everytime that the server has a new value for a stock, the app is passed the new data. Finally, the data is displayed to the user.
This leaves a lot out of the story. This is more a use case than a design.
There are several ways to approach this problem. It's your fundamental limitations that really guide the design. I'll be making assumptions about what your limitations are.
Presumably you don't own the server that is the source of stock data. You simply have a rest API you can connect to. Likely you are limited in how much you can download and how often you can connect.
Schedule an Api call every X seconds
This is called polling. It has severe limitations over events but can work well if done well.
1) I'll be downloading much more data than is necessary, since I don't know what stocks have changed their value, I will have to fetch all of them.
That depends entirely on the limitations of the API. If the API really won't let you inquire about a particular stock but insists on sending you the whole ticker then yes this is true.
2) This is not really "reactive programming". I somehow feel that the server should "push" the data to the client and not the other way around.
Polling, done right, is actually what events really are. It would be an awful design if every x seconds every user hit the API and searched through the ticker list for the stock they cared about and checked if the price had moved. That's a lot of work and pointless moving of data.
If the API won't let you do anything more advanced then pull down the whole ticker when you call it I recommend setting up a server of your own. Your one server will hit the API of their server, get the whole ticker, do this every x seconds, but will be the only thing doing this.
Your server will allow users to register as observers of stocks and every time a stock changes your server will loop that stocks registered observers and push a notification out to them.
You'll need some reasonable timeouts and limitations on registering as an observer or you'll be the subject of denial of service attacks.
Done this way notifications can be limited both to stocks of interest, price movements of interest, and with timeouts, to observers still interested.
This would also allow you to push notifications out in any form you wish, UDP, TCP, SMS, email, phone calls, post cards.