I have been asked to prove mathematically that a binary option close to maturity should be hedged using a call spread with the same maturity.
I understand that far from maturity, one would use delta hedging to sell or purchase the underlying asset. Yet as time to expiry tends to zero the delta profile tends towards a dirac delta function and so renders the hedge impractical. See: delta of a binary option
Other than calculus to derive delta, are there any other rigorous ways to construct hedges of this kind?
As this is a homework question, hints rather than full answers are most welcome.
Thanks in advance,