And your affection not gone forth, I'll make you
The Queen of Naples.
(1.2.538-540)
These words are said by Ferdinand shortly after he encounters Prospero and Miranda while he is wandering around the island after the shipwreck. He says to this to Miranda just after having met her and this shows that they, or at least he has fallen in love with her. It is evident that she has also fallen in love with Ferdinand by the way that she defends Ferdinand while her father calls him a traitor and compares him to Caliban. Although Ferdinand is the prince now, he says that he will make Miranda the Queen of Naples because he believes his father, King Alonso, has died in the shipwreck on the return voyage from Africa. Therefore, he thinks that he has inherited the kingdom from his father. But, he does not know that his father has actually survived the wreck and is fine. Conversely, Alonso and the royal party believe that Ferdinand has died because nobody has seen him since the shipwreck.
When this is looked at from the bigger picture , it is seen that Prospero, with the help of Ariel his servant, has actually set all of this up as a part of his bigger plan. The storm is caused by the spirit Ariel using his magical powers. Then, the fact that the ship actually wrecks on the island where Prospero and his daughter Miranda are living is no coincidence either. Also, Ferdinand does not just wander around and stumble upon Miranda and Prospero on his own. He is actually led by Ariel who uses song to guide him to Miranda. All of these things are part of Prospero's brilliant plan and they each lead to this quote by Ferdinand speaking to Miranda. At this point, this is exactly what Prospero wants. He wants Ferdinand and Miranda to fall in love with each other and get married so that when he becomes the King of Naples, she will become the Queen of Naples. Prospero may just want the power for his daughter or he may be utilizing Miranda to gain power for himself and get revenge on his brother for usurping his power when he ruled as the duke of Milan. The question is just that: Does Prospero just want to use his daughter Miranda to gain power for himself or does he just want to give Miranda a life of royalty to enjoy for herself?