The author of the Badshah Nama was Abdul Hamid Lahori.
Option 1: Abdul Hamid Lahori is the correct answer. He was the author of the Badshah Nama, a historical chronicle of the Mughal emperor Shah Jahan`s reign.
Option 2: Abul Fazl was not the author of the Badshah Nama. He was a courtier and historian in the court of Emperor Akbar and the author of Akbarnama, a biography of Emperor Akbar.
Option 3: Shah Jahan was not the author of the Badshah Nama. He was the Mughal emperor during whose reign the book was written. Shah Jahan commissioned Abdul Hamid Lahori to write the Badshah Nama to document his achievements and the events of his rule.
Option 4: Sadullah Khan was not the author of the Badshah Nama. There is no known connection between Sadullah Khan and the book in question.
In conclusion, the correct answer is option 1, Abdul Hamid Lahori, as he was the author of the Badshah Nama.