Delta hedging on Barrier/Digital Options

Question

I would like to adress a question I have in mind and I didn't found a clear answer online.

When we deal with Barrier or Digital Options we have a discontinuty in the payoff, so that the derivatives (the Greeks) are very spiky and take large values around the barrier/strike. As a consequence, it would be a very complicated task to hedge this type of options because the trader would have to buy or sell huge amounts of the underlying, risking to have problems with liquidity and suffer some losses.

In practice I know that we have to apply a barrier shift to sligthly change the payoff structure and have more manageable deltas around the barrier/strike.

How does it work? How can we choose this shift? Wouldn't be pricing a different option (with a different price) wrong?

Most importantly, what if the underlying price goes around the shifted barrier? Here, we would have to buy/sell large amounts in order to delta hedge anyways.

Answer 1

You're right that the "real" greeks of a digital option are unwieldy, e.g. delta is zero everywhere except at the barrier where it is an impulse. So sell-side trading desks model/price digital options as tightly struck call/put spreads that will sit and play nicely with the rest of the book.

Here's a simple example: let's say a bank sells a digital call on AAPL that pays \$1000 if the stock is over \$150 at expiry. This could be modeled as being short 100 \$140/\$150 call spreads. So if the stock ends up at \$155, the trader would have hedged against a payout that matches the digital (\$1000 = 100 call spreads * \$10 strike difference). In this case, there is no gap/fixing risk upon expiry.

So, what's the problem with this? In short, the price the trader quotes for the digital is going to be very noncompetitive, as the person asking for the digi could go and just purchase call spreads themselves (though this is not possible for all underlyings), and so the trader will soon be out of a job.

How can he/she show a better price? Basically in one of two ways:

• Using more leverage, meaning that they would model the digi call as a tighter call spread. The tighter the strikes used to model, the more call spreads it would require, and the larger/more discontinuous the greeks would be in the book. E.g. if modeled as 145/150, it would require 200 spreads, 1000 spreads for 149/150 and so on. Still no gap risk.

• Using what you call a barrier shift. Here's where things can get spicy. In short, this is changing where the call spread used to model the digi is centered. Let's say the trader models the \$150 digi call as 500 \$149/\$151 call spreads (so one on either side of the digi strike rather than both at/below in our first example). Then his/her price on the digi is going to certainly be a bit better for the client. But what happens if the stock is at \$150.50 upon the digi expiry? Then the trader would effectively be due to pay out the \$1000 but only have \$750 from their hedges (since they hedged against the call spreads, paying 500 * (150.50-149) = 750), and so would realise a fixing loss upon expiry of \$250.

The trader must assess how much leverage is realistic or practical (a) in the context of the rest of their trading book and greek management and (b) to your point, given the liquidity in the underlying market for delta/vega hedging. At this point, they're likely to still be uncompetitive without taking some (unhedgeable) gap risk via a barrier shift, but they would seek to keep their total digital exposure to a modest amount at any given time.

Answer 2

I nearly agree with @phlsmk's answer, but with some small differences.

First off, the delta of a digital is not "zero everywhere except at the barrier where it is an impulse". This is what it is at $t=T$. before this, it is smoothed out, exactly like a regular option is.

The problem is on what the delta may become. This is not the only place where it happens.

Let's look at an example, just a vanilla call. K=100, r=q=0 (i.e. no drift, fwd is whatever the spot is), $\sigma$=20% (changing the vol. in all these examples will just be like changing the time, so it doesn't really make a difference to the answer). Below, adu is absolute delta/unit, and aguu is absolute gamma/unit/unit.

note how as the time goes to zero, the payoff approaches a cusp. Here are the delta and gamma:

Now, i think it's fairly easy to see that as $t \to T$ this becomes unmanageable to hedge.

One method to deal with this is to break up the option into several, with different strikes spread around the original one. Here's what that looks like (with the $t=1.0$ pv overlayed in orange for comparison):

and here's the difference in price as time goes on:

Okay, so delta and gamma?

So the delta has an extra step in the middle, and the gamma is still spikey, but now each spike is half the size of before. Obviously we can do better still.

Let's do a strip of options spread from 90 to 110.

Here's our new payoff, and the comparison to the original:

And again, delta and gamma:

So, you can see that by spreading out our payoff, we can reduce the potential risk we may face in the future, with only a minimal increase in cost.

This is just for a vanilla option - but a digital is essentially the first derivative of a call wrt. strike, so just swap gamma for delta in the examples above.

Obviously the more you smear out the barriers, the higher the cost will be. This is where risk limits come in. If you want to do one of these trades in small size, there is no need for this, since the maximum risk will be small. If you come to me and ask to do a 100mm digital though, then i could potentially have a very large hedge on my hands, so i will smooth the payout to reduce that risk - this is just another cost of doing a large trade. If you can find someone else who is happier to accept enormous risk, then perhaps they will offer you a lower price.

I disagree that doing this will make your prices non competitive, as it is an extremely standard thing to do -> everyone does it. How much may differ, but as i mentioned already, this is also dependent on size. And as you move to larger size, the market is going to shrink anyway, so you're going to expect costs to increase.

And for the barrier shift - why would the bank ever do this in any direction other than against the client? When done like this, the PnL at expiry is always $\geqslant0$.

eidt: so, i spent a little longer toying around with my spreadsheet for this - you can do even better. Why weight the notionals of the options evenly? If we weight them according to a normal, we can get nicely behaved greeks all the time, even at $t=T$:

where that last one is the delta of your digital. Notice that the two methods are different - the first has a lower level for the whole region, but it appears abruptly. the second has a larger maximum but approaches slowly. Which you decide to go with will depend on how you want to manage things.