Stubborn old boy expected to hold rodent (8)
I believe the answer is:
obdurate
'stubborn' is the definition.
(obdurate means stubborn or obstinate)
'old boy expected to hold rodent' is the wordplay.
'old boy' becomes 'OB' (abbreviation for a private school 'old boy').
'expected' becomes 'due' (similar in meaning).
'to hold' is an insertion indicator.
'rodent' becomes 'rat' (rat is a kind of rodent).
'due' enclosing 'rat' is 'durate'.
'ob'+'durate'='OBDURATE'
(Other definitions for obdurate that I've seen before include "Obstinate and unrepentant" , "A doubter being stubborn" , "hard to influence" , "Hard-hearted" , "Implacable" .)