is the explanation of that in a put option, when price goes down, you gain from delta AND volatility going up a valid answer? which would be greater than the loss if the price went up given the change in volatility is lower when it goes down

if i remember well - this can be explained with the put pricing chart, which is positive convexity - price increase more when interest rate decrease comparing with the same level price decrease when interest increase.

(try to draw a long put option chart and draw the convex line above - or check this one https://www.google.co.uk/search?q=long+put+chart&source=lnms&tbm=isch&sa=X&ved=0ahUKEwjlxN6RmYLNAhUnJsAKHffIAJQQ_AUIBygB&biw=1366&bih=646#tbm=isch&q=long+put+option+chart&imgrc=btCEKZUsN5tJTM%3A)

yeah i figured that after I read the solution. Would that mean the volatility explanation is wrong?

hmm i am not quite sure about the volatility part - can we deduct the volatility would increase when the option is in the money and would decrease when option is OTM?

The question is not asking about volatility changes. It’s asking about prices changes. (This is like comparing distance and speed) I think delta increasing is significant and perhaps should be clarified as to how that relates to changes. Since changes in price don’t imply changes in volitliity that could affect if the grader feels you understand the topic.

Delta measures the change in price for a change in the underlying.

The absolute delta of a put increases as the price of the underlying falls.

Thus, if delta measures the option’s sensitivity to changes in the underlying, that sensitivity will increase as the absolute delta increases and the absolute delta will increase as the price of the underlying falls.

If a stock drops from $10 to $5, the put’s delta will have increased in absolute terms along the way, so it will have been more responsive to that price change than if the stock rose from $10 to $15, in which case the absolute delta will have been decreasing along the way.