# Barrier shift consideration in delta hedging down and in puts (PDI)?

I have a question regarding the barrier shift used when risk managing a down and in put (PDI). I'm reading Exotic Options Trading by Frans de Weert and he gave this example.

Trader is long one PDI 100/70 (strike 100% spot, down and in barrier 70% spot), striked at spot = 100. Now spot is at 70.1, and the put's absolute delta (called delta from now on) is larger than 1, say 2.5 so the trader needs to buy 2.5 stocks to hedge for 1 PDI. The stock then drops to 69.9 and the PDI turns into a normal ITM put, which has a delta of 1 for simplification. Now the trader has to sell 1.5 stocks for each PDI he is long. And if he is long a lot of PDIs he cannot sell the excess delta at exactly 69.9 but much lower and make a loss.

I understand it up to here. However, the author says that the trader can treat the PDI 100/70 as a PDI 100/67 and risk manage it accordingly to give himself a cushion of 3%. What does this mean ?

As I understand, European PDI with American barrier has a lower absolute delta if the barrier is lower, for the same level of spot, i.e in the example the 100/67 PDI will have a delta of say 2 instead of 2.5 like the 100/70 PDI. If the stock drops right from 70.1 to 66.9, the trader will lose less since he only has to sell 1 stock for each PDI he owns instead of 1.5. However, if the stock stays above 67, the 100/70 PDI that the trader actually owns now has a delta of 1, while since he treats it as a 100/67, he should still hedge it at a delta larger than 1, say 2.6.

This means that if he treats the 100/70 PDI as a 100/67 PDI, if the stock is above 67 and under 70, he is long 1.6 delta ? How is this good hedging ? What good is treating the 100/70 as a 100/67 in this specific case ?

Shadow barriers (the fx name for this strategy) work for market makers as they super-replicate the option in question. The trick here is to realize that the option bought from the client was priced and hedged with the lower barrier in this case. So we bought it for cheaper than fair value and have moved the delta gap below the in barrier of the real option so we never have to worry about it. That is why it is a better hedging strategy.

• Can you explain a bit the "super-replicate" term ? And also how the market makers hedge the delta around the barrier ? So they move it down in this case to avoid a hectic delta. But say you are hedging for a 100/70 PDI, treating it as a 100/67, what happens if the stock crosses from 70.01 to 69.9 ? The 100/70 is supposed to have a delta of close to -1, while the 100/67 delta is even lower. When the stock does cross the barrier at 70, does the trader start selling stocks right away from his 100/67 delta to reach the 100/70 delta and the cheaper pricing makes up for the potential loss ? Jul 7, 2021 at 8:27