Option 1: Punjab National Bank is not the correct answer. Although it is one of the oldest banks in India, it was established in 1894, while the first commercial bank managed by Indians was established before that.
Option 2: Oudh Commercial Bank is the correct answer. It was the first limited liability bank managed by Indians, established in 1881 in Faizabad, Uttar Pradesh. It was founded by Lala Raja Ram Shukla and Rai Bahadur Lala Kanhaiya Lal, and played a crucial role in providing financial services to Indian merchants and businessmen.
Option 3: Imperial Bank of India is not the correct answer. It was established in 1921, much later than the first commercial bank managed by Indians.
Option 4: Bank of Baroda is also not the correct answer. It was established in 1908, after the establishment of the first commercial bank managed by Indians.
Therefore, the correct answer is option 2, Oudh Commercial Bank. It holds the distinction of being the first commercial bank of limited liability managed by Indians in India.