In India, the first bank of limited liability managed by Indians and founded in 1881 was

Oudh (Awadh) Commercial Bank was established in Faizabad in 1881 and is cited as the first commercial bank in India to have limited liability and be managed by an Indian board. Historical summaries and entries identify Oudh Commercial Bank (founded 1881) as the earliest bank meeting both criteria—limited liability status and Indian management—whereas the other options (Hindustan Commercial Bank, Punjab National Bank, Punjab & Sind Bank) were founded in different years and contexts later or do not match this specific distinction [1]. Therefore the correct answer is Oudh Commercial Bank (option 2).

Sources

1. [1] https://en.wikipedia.org/wiki/Oudh_Commercial_Bank