How a Call Writer’s Delta Hedge Changes as Expiration Nears
Summary
The document explains how a trader hedging a written European call may need to adjust the stock position as the underlying price falls near expiration. The hedge holds shares against the call’s positive delta, so the direction of rebalancing depends on how the option’s moneyness changes as time passes. If the price declines gently and the call remains well in the money, its delta can move closer to one; the trader may need to buy more shares. If the price falls toward the strike, the chance of exercise decreases and delta can fall, leading the trader to sell shares.
The answer also highlights the risk of being near the strike at expiration: delta is around one half, but the option’s payoff becomes binary at expiry, potentially leaving a short-call hedger with too few or too many shares. These are qualitative scenarios rather than a complete pricing analysis. The result depends on the path of the stock relative to the strike and on assumptions about the option model and other market inputs.
Key ideas
- A short call is commonly hedged by holding shares in proportion to its delta.
- If a falling stock remains well above the strike near expiry, call delta may rise toward one, requiring more shares.
- If the stock falls toward the strike, call delta may decline and the hedge may require selling shares.
- A short call near the strike at expiry faces substantial hedge uncertainty because the payoff changes sharply at expiration.
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Full text
# Delta of an option which is approaching expiration when stock price decreases # Delta of an option which is approaching expiration when stock price decreases 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? ## Answer by siou0107 (score 3, accepted) https://quant.stackexchange.com/a/50134 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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