Greeks for Strike :-)

As is known, the option Delta is the change in Premium for a $1 change in the underlying Spot.
Can the Greeks be used also to compute the Premium for $1 below or above the Strike?

 

Impossible to do. Can only be done for $1.

 

Funny: in the past I had tested & verified & confirmed that the premise of Delta is mathematically correct,
but right now when I try to verify it again, it somehow doesn't sum up as it gives a different value than "Premium + Delta = NewPremium_for_$1_change_in_Spot":

Example:
S=50 K=50 DTE=365(t=1.0) IV=10 r=0% q=0% --> Call=1.993881 CallDelta=0.519939
For a $1 increase in Spot (ie. for Spot $51) it gives a Premium = 1.993881 + 0.519939 = 2.513820

Verification via BSM:
S=51 K=50 DTE=365(t=1.0) IV=10 r=0% q=0% --> Call=2.553142

There's a difference of 0.039322.
How come? Shouldn't that be exact? Is there a bug?

 

Clarification: I was meaning premium for a strike above/below $1 off the original strike, keeping everything else the same, incl. spot. If possible w/o using IV, ie. then calc'ing via the Put/Call parity & Greeks...

 

how would you want to calculate the greeks without IV?

 

Dunno, just asking whether possible or not etc...