The following is an interview question.
It is 10 months since you sold a one-year European call option to a customer. You have been delta-hedging your exposure to the written call since it was sold. The option is now well in-the-money, and the delta of your replicating portfolio is correspondingly high (at around 0.90, say). Suppose that you watch the underlying stock price falling gently over the last two months of the life of the option. As the stock price falls over this time period, what happens to the delta of the replicating portfolio? That is, are you buying stocks or selling stocks as you watch the stock price fall? You may have to describe different possible scenarios—be clear on the assumptions you make.
Clearly delta decreases if stock price decreases.
Since we short call option, to delta hedge, we long $\Delta$ shares of stocks. As the new $\Delta$ decreases, we are holding more stocks than necessary. So, we need to sell stocks for our portfolio to remain delta-neutral.
What I do not understand from this question is that what different possible scenarios do I need to consider other than the above? Also, what assumptions do I need here other than the Black-Scholes assumptions?