Sorry I posted this in the trading forum also but I think this also qualifies as a technical question and I see that this forum gets much higher traffic so here goes.....From what I understand exotics traders utilize a technique known as barrier shifting (or bending) when pricing/hedging intruments with a barrier. For example if they sell a european digital call to a client with a strike of 100 the idea would be to actually charge them for the more expensive digital call with a strike of 98 and then dynamically hedge against the 98 strike over the lifetime of the contract. The idea then being that the bank gets the extra premium for charging at the 98 strike price and any windfall incurred if the contract expires between 98 and 100. This all to cover the risk of the position.My question is what is the advantage of doing this? Why not just charge a premium up front. For instance look at the p/l histogram and charge the client a premium that guarantees making money 95% of the time or whatever your metric is. Why obtain this premium indirectly in the form of barrier shifting? Is there something else that I'm missing here? Thanks for any insights.

Just guessing, but one rationale might be to relate the adjusted price to what ismissing from your valuation model. Say you are using a diffusion model, but no jumps.The (incorrect) hedging rules that follow from this use the ability to trade right as the barrier is hit.To compensate, you estimate that if a jump through the barrier does occur, it might be around 2% (on average).

Also, when you're running a book you're not going to be reviewing each trade whenever you hedge,so you'd like this rule-of-thumb to be applied automatically - entering a barrier shift is one way of doing this.Just overcharging upfront won't do the trick.

Barrier shifting doesn't work for pure digitals : to use your example, you won't be able to replicate your payoff when the spot is at 98 (think in terms of diffusion and dirichlet conditions).

barrier shifting is done for digital because that is the way to hedge in the market. you must be aware of call spreads to hedge a digital right?

I'm with you on that, CS for digitals.But barrier shifting doesn't solve any hedging problem for them ; using that KI digital at 100 :- if you shift to 98, you'll have to hedge your exploding greeks without any sort of protection- if you shift to 102, you won't have that problem but you'll be underpricing severely your option (since PriceShifted(100) < 1)Or am I missing something ?

hi yes you will be underpricing but the idea is that the px is lower because the trader needs that margin to manage the risk in smooth manner if the spot does really go close to the barrier.

i agree wtih upadastra. it is really a PNL cushion to shift the barrier, and you may end up with a windfall gain if you are lucky.what i have done in the past to smooth the greeks for large barrier options is to price and hedge them as a series of barrier options with different barriers. this smooths out the greeks pretty well. for example, if you have a 90 strike barrier you might price it as the sum of 10 barrier options with barriers evenly spaced between 87 to 90.this is the equivalent philosophy to pricing a "european digital" (meaning not a one-touch, only expiry trigger) using a put spread or call spread. a staircase digital payoff looks a lot like a call spread.for small everyday barriers though it is too much of a pain to book so many options in your instrument, so a simple barrier shift is practical.

Thanks for the answers (erstwhile, I read your previous posts about large options and digitals and found them very informative).What do you mean though when you speak of windfall gain ? Are you talking about the gap risk, which if we're still talking of this good old american digital and assuming we're short the option, will benefit us ?I fail to see why you don't CS american digitals too, which would prevent this underpricing problem and solve those greeks management problems at the same time.

Where the windfall gain comes from: let's say you have sold a one-touch digital that pays $1mm if the asset/rate hits 90% of original level within one year, and you priced and hedged it as a 92% barrier instead of a 90% barrier. Let's say the market is reasonably behaved and delta hedging works. If the market hits 91.9%, then you will have liquidated your hedge right around here and you have the $1mm ready to pay out. If the asset/rate bounces back up and you don't need to pay out the $1mm after all, you have a wnidfall of $1mm (plus interest)!If the thing continues on down and hits 90% then you alerady have the cash to pay out and you are fine. You again may have made a little interest on the $1mm waiting for the market to get past 90%.Using a call spread to price american digitals: It makes a lot of sense to hedge with call spreads or put spreads, and you can use this method to estimate the pricing as well. The connection you need is that an american (meaning one touch) digital is roughly equal to two european digitals. Why? Well if the forward curve is flat, then when the market gets to the barrier level the american digital is worth 100 (by definition) and the european is worth around 50 (probably a bit more if the skew is equity-like). Intuitively, the digital it is at the money, so has even odds of being above/below.So you can roughly pretend that the american digital is 200% of a european digital and then use the call/put spread overhedge method for the european digital, but use twice the number of options.A nice exercise is to use a local vol monte carlo (yes i know it is wrong for barrier options) and simulate the *difference* between a one-touch digital and the call spread or put spread. You get a pretty small stdev of results, indicating a stable hedge.One refinement (and there could be many) would be to calculate the theoretical (unadjusted for reality) prices of the pure european and american digitals, and see what the price ratio looks like (maybe it is 180%, not 200% for example).

Hi erstwhileThanks for the example. Can you explain why an American digital is worth two Europeans ?

since you can only trigger your option once, comparing to an european digital which can be trigger only at expiry, it's not rigorously true, but you can view it as if all the odds of hitting the barrier where concentrated on only side. So roughly speaking, it's like you having twice the opportunity to hit the barrier, or in price terms, twice the european barrier. hope this helps...

When you're at barrier - epsilon (knock-up), an american digital is almost worth the discounted rebate since there's a very small probability that the underlying doesn't cross the barrier once.For a european barrier, if you simplify saying (r-q) ~= 0, then you have about 50% chance to be above the barrier at maturity, hence it's worth about 2 times less that the discounted rebate.