I have noticed in several Option cases, that while option premiums of ATM and ITM options are increasing, certain deep ITM option premiums decresed/drop significantly. Why does this happen

Underlying Price was @ 388 (correcting earlier typo). Up around 4.5/- from previous close. below is the option chain

At the time of viewing details on NSE site

premium for Strike Price of 390 was at 8.75 - up 1.80

premium for Strike Price of 380 was at 12.80 - up 2.70

premium for Strike Price of 370 was at 19.50 - up 4.00

HOWEVER premium for Strike Price of 350 (DEEP ITM) was at 19.5033.00 - DOWN by 13.75 (-13.75 from previous close)

Please explain

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Not just deep OTM but even deep ITM option premiums can fall. This is because the way the deltas behave - it flattens out for strikes in the deep OTM and deep ITM region.

I’d suggest you read this to get a complete idea - http://zerodha.com/varsity/chapter/delta-part-2/

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sorry there was a typo in my query. The underlying was @ 388 (not 308) making SP 350 deep ITM.
As per the link (theory also suggests) deep ITM have a delta of 1 or ~1. So theoretically every point upward movement of underlying will result in a point upward movement in premium of DEEP ITM option.Then in the above case why did it lose value ? (also notice that all other ATM and ITM options gained value as per theory)
Kindly correct my understanding
Expected change in option premium = Option Delta * Points change in underlying

You got to look at the Delta acceleration. In fact it is not just Delta that plays here but also Gamma and Vega. For example the effect of Vega for deep ITM/OTM is minimum. I’d suggest you read the theory in detail as the premium derives its value from all greeks, and not just 1.