Is 5/8 and 3/8 sum up to 1 is coincident here or there is reason behind it.
No, it’s not a coincidence but it’s also not a rule.
In that example, the option’s delta is 0.50. When the stock moves up, delta increases a bit (because of gamma), so the option rises by 5/8. When the stock moves down, delta decreases a bit, so it falls by 3/8.
Those two numbers (5/8 and 3/8) add up to 1 simply because they’re centered around a 0.50 delta. Their average is 0.50, and 0.50 × 2 = 1.
If the delta had been 0.30 instead of 0.50, the up and down changes would not add to 1.
So there is a reason behind it but it only works that way when the starting delta is about 0.50.
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What happen if delta was 0.3? avg = 0.3?
If delta is 0.30 and the stock moves up and down by the same amount:
• Up delta will be slightly above 0.30
• Down delta will be slightly below 0.30
• Their average ≈ 0.30
Their sum ≈ 0.60 (which is 2 × 0.30).
So only when delta is 0.50 does the sum become 1.
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