# Confusion Regarding Dynamically Delta Hedging a Short Option

To my understanding, market makers (mm) in the options market dynamically delta-hedge their portfolios by buying/shorting the underlying, thus eliminating directional risk and profiting from providing liquidity. For example, if a mm buys long a 0.5 delta call, they hedge by shorting (0.5 * spot) worth of the underlying. If the underlying moves up and the delta rises to 0.6, they adjust their hedge to be short 0.6 * spot, thus maintaining net-0 delta.

I'm confused about this delta-hedging when writing an option. Say a mm writes an ATM call that has a delta of 0.5. This makes them have a -0.5 delta on their position, and they hedge by buying 0.5 * spot. If the underlying rises by \$1, they are down \$0.5 on their option but gain \$0.5 from stock. In the same scenario, though, let's say the underlying drops \$10, making the call OTM with a new delta of 0.4. The call is worth less than what the mm sold it for, so why would the mm need to readjust their delta hedge? Say they do readjust to long 0.4 * spot of stock, wouldn't this only expose them directionally? If the underlying drops again, they would lose money on their 'hedge', but can't profit more from the call they wrote than what they sold it for. Would the mm take their hedge off the table after criteria are met to avoid this?

I'm assuming I'm fundamentally misunderstanding something about writing options, delta, or how dealers delta hedge. Would greatly appreciate any insight. Sorry for the wordiness and thank you in advance.

• The only difference between delta hedging a short position of an option compared to a long position is the sign of the amounts of stock you're buying/selling. As you said : hedging the long position of the option requires to sell $N(d_1)$ amount of stock. Hedging a short position requires to buy $N(d_1)$ amount of stock. Whatever extra money is needed to finance this needs to be borrowed and, of course, paid back at maturity. Likewise, whatever amount of stock is required to be sold needs to be borrowed, unless you are hedging a covered call/put. Sep 28, 2021 at 10:56