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If I long put and short call, assuming share price is going down, does it mean the delta will be further "negative"??? let's say initially -0.5, after price going down, -0.6.

I think this depends.. While the put delta decreases further, the negative delta resulting from the short call increases (becomes less negative). Depending on the gamma of both options, the overall delta-exposure could be de- or increasing. Hope this helps..

It would depend on the strike prices of the two options. Clearly, if both are struck at the same level, delta remains constant at -1.If put is struk at a lower strike than call, then delta would at first increase (less -ve than -1) and then approach -1 as stock price approaches 0. If put is struk at a higher strike, then delta would initially decrease (more -ve than -1) and then increase to -1.