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Shadow Barriers for Managing Down-and-In Put Delta Risk

Article Quant Q&A · Author: Bach Pham
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Summary

The document asks how a dealer can manage the discontinuous hedge exposure of a down-and-in put near its activation barrier. In the example, the put’s delta before activation is greater than the delta of the vanilla put it becomes after the barrier is crossed, potentially forcing the holder to sell shares into a falling market. The author asks why risk-managing the position as if its barrier were lower provides a useful cushion, despite creating apparent over-hedging when spot lies between the real and lower barriers.

The answer describes this as a shadow barrier approach used by market makers to super-replicate the client option. The dealer prices and hedges the client trade using the lower barrier, which means the option is bought more cheaply than its fair value under the actual barrier. This shifts the delta gap below the real activation level, reducing concern about a sudden hedge adjustment at that level. The explanation is brief and gives no quantitative validation or discussion of the costs and assumptions behind the pricing and hedge.

Key ideas

  • A down-and-in put can have a sharp delta change when its barrier is crossed.
  • A lower shadow barrier can move the problematic delta gap below the contract’s actual barrier.
  • The described market-making approach prices and hedges the client option using the shadow barrier.
  • The answer characterizes the method as super-replication but provides no quantitative evidence or detailed assumptions.

Tags

OptionsDerivatives pricingRisk managementMarket making
Full text
# Barrier shift consideration in delta hedging down and in puts (PDI)?


# 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 ?

Thanks for your help.

## Answer by river_rat (score 1)

https://quant.stackexchange.com/a/65869

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.