From a continuous standpoint, I understand why an ATM call has delta = 0.5 and for ITM call, the delta approaches 1 since each move in the underlying corresponds to same unit of value change in call option.

Howwever, if we take a discrete case where call option expires in 1 period. If strike price is 100 and the underlying is at 100. If the underlying moves to 105 in 1 period, then dS = 5 and the change in call option is also 5? So then this ratio of $\frac{dc}{dS} = 1$ not 0.5?


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The theoretical idea of Delta hedging is placed in a setting of infinitesimally small time steps. In such small time steps and if no jumps occur (eg. in the diffusion case) the underlying can not move that much and Delta makes sense (at least theoretically). In continuous time things work out.

As in practice (or in simulations) the Delta hedge can only be done in small discrete time steps (not continuously) and thus only captures a part of the change in value, it is merely an approximation.

In your case the time step is discrete and the move is too large. Thus the Delta Hedge does not work at all.


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