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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?

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    $\begingroup$ You need to consider the time aspect. You start at .9 delta, but if the option ends up in the money, you will end up with 1 delta. So you need to buy the .1. At the same time it is true that 2 months before expiry, you will sell as the price goes down. So these two effects work in different directions. I assume, since the setting is that you are “well ITM” and the stock only decreases “gently”, that you do not need to consider the scenario where the option actually finishes OTM. $\endgroup$
    – Ivan
    Commented Dec 6, 2019 at 14:17
  • $\begingroup$ @Ivan As the option remains 2 months to expire and it has a high delta $0.9$ now, so we buy stock till delta is $1$? $\endgroup$
    – Idonknow
    Commented Dec 6, 2019 at 14:20

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If the stock prices falls "gently" and the option remains in the money, your delta will converge to 1 and you will have to buy stocks: the gains on your long stock positions will be lower, but the payoff of the option you wrote will be lower too.

If the stock prices falls more sharply and gets closer to the strike, there are higher chances that the option will not be exercised and you will sell stocks.

N.B.: The worst situation for a short vanilla option trader is to be ATM very close to expiry. At that point, you hold roughly 0.5 share per option. At expiry, it is then binary: either your option is OTM ($S < K$ for a call), and you lose roughly $K - S$ per stock bought, or the option is ITM ($S > K$ for a call), but as you have only purchased 50% of the stocks needed you lose $S - K$ per stock you have to buy to deliver the shares to the holder.

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