Who is the author of 'Badshah Nama'?

Abdul Hamid Lahori

The correct answer is Option (4) → Abdul Hamid Lahori

A pupil of Abu’l Fazl, Abdul Hamid Lahori is known as the author of the Badshah Nama. Emperor Shah Jahan, hearing of his talents, commissioned him to write a history of his reign modelled on the Akbar Nama. The Badshah Nama is this official history in three volumes (daftars) of ten lunar years each. Lahori wrote the first and second daftars comprising the first two decades of the emperor’s rule (1627-47); these volumes were later revised by Sadullah Khan, Shah Jahan’s wazir. Infirmities of old age prevented Lahori from proceeding with the third decade which was then chronicled by the historian Waris.
The gifting of precious manuscripts was an established diplomatic custom under the Mughals. In emulation of this, the Nawab of Awadh gifted the 'illustrated Badshah Nama' (One containing paintings along with written text) to King George III in 1799. Since then it has been preserved in the English Royal Collections, now at Windsor Castle.