Mini Chapter Seventeen
Barrier options
European Knock In / Knock out options:Â
- These options officially fall in the realm of structured products as they can be understood as a combination of at least two different options. As an extension of a vanilla option, the payoff of barrier strategies is additionally a function of attaining a barrier level (either higher or lower vs the strike) that can either knock in or knock out the underlying vanilla option.
- As an example, the risk of a buyer of a vanilla call option at 100 strike that can knock in if at maturity the stock is at 110 can be understood as being long a call option at 100 only if the underlying at maturity is at 110 or higher. In case the option knocks in it would already be worth 10 dollars at 110 (at parity) emulating a long vanilla call payoff thereafter. This discontinuous 10 dollars pay-off as we touch 110, replicates the risk of being long a 10 dollar notional digital at 110. For this structure, 110 is referred to as the barrier.
Graph 64 – Payoff for an Up and In Call
Source: Pandemonium.
- In the same vein from a buyer’s point of view combining long vanilla puts or calls with knock in and knock out (KI and KO hereafter) either above or below the strikes we have these 8 scenarios: Down and In, Down and Out, Up and In and Up and Out – each of these applied to a Put and Call.
- Barrier monitoring – while these are path dependent options where barriers can be monitored periodically ahead of maturity (in case of American options), for our discussion we would stick to European barriers where the monitoring only happens at maturity. Nonetheless for the sake of completion greater the frequency of barrier monitoring for knockout/knock-in options, cheaper/more expensive they would be as the probability of the embedded vanilla option expiring worthless/going live would go higher.
- Placement of barrier versus strike – when barrier is placed close to at the money strike a knockout option is a lot cheaper since the probability of the option expiring worthless increases. Conversely if for a knock-in the barrier is placed close to at the money strike it would be a lot more expensive since the probability of the option knocking in/going live increases.
Play of the greeks
Let’s start with a popular structure with retail / institutional investors of a down and in put (DIP)
- Consider a European structure – long 3m put option on USDJPY at 130 which only knocks in at maturity at 125 with current ATMF at 135. Extending from above this would be a combination of a long 125 put (remember the put at 130 would only knock in once we touch 125) and a long 5 dollar digital at 125.
- We can also think of it as being a long 130 put but having a lower probability of ending in the money due to the scenario of never touching 125. Hence the delta of a 130/125 DIP would be less than the delta of a 130 put when USDJPY is far from the barrier. The price of the DIP then is also lower than the price of a vanilla 130 put. Safe to say that the change in the price of the DIP is also underwhelming as long as we are far from the barrier i.e. delta continues to be small (negative as we are long a put).Â
- As we get close to the barrier (remember we are long a 5$ digital), delta of a long digital would come into play. Imagine a scenario where USDJPY is trading 125.05 just before maturity. A move lower by 0.1 (124.95) will immediately result in the price of the DIP jumping by at least 5$ as we are now long an ITM put. This would imply that delta of the DIP is now -5 (note that max delta of a vanilla put can be -1). Thus we see a massive spike in delta of the put just before the barrier.Â
- Once the barrier is breached delta of this KI option should be the same as the delta of a vanilla 130 put (ITM) and would revert to -1. The rise in delta (deeper in negative zone) effectively means that the long DIP holder would actually be long gamma up until the barrier with delta inceasing as we get closer to the barrier but then has a gap adjustment to the delta of a vanilla option once the barrier is breached.
- Delta Hedging as a result would happen in a positive gamma backdrop as long as we do not breach the barrier but flips to a negative gamma territory just as we cross it. The negative gamma emanating from the barrier breach at maturity would mean a much larger notional swing on the delta (from -5 to -1) and getting rid of the excess (selling the underlying) at lower prices. Therefore, Delta hedging in the run-up to the option knocking in would have net costs associated to account for the in-the-moneyness of the 130 put once the option knocks in.
Graph 65 – Delta Profile of a 130/125 DIP (Close to Maturity)
Source: Pandemonium.
Graph 66 – Gamma profile of a 130/125 DIP
Source: Pandemonium.
- Vega – it should be straightforward that a Knock in option price should be more sensitive to higher implied vols relative to a vanilla option’s price (with the same strike) as higher volatility would mean a higher probability of hitting the barrier and the option coming into existence. Vega of a KI barrier is higher than that for a vanilla option. A Down in Put option holder would also be long the downside skew i.e. expects the implied vol for out of the money (downside) barrier to nudge higher and get knocked in. For Knock out options the reverse holds true where higher vol increases the probability of a live vanilla option knocking out. Hence, Vega for a knockout barrier option is generally lower than that for a vanilla option, in fact it could even be negative for a knockout depending on how close the barrier is to the vanilla option’s strike (and spot).
- Barrier Shift – due to the discontinuous nature of the payoff around the barrier that creates large swings in delta and gamma while hedging even for a long option holder, managing the risk of the structure using a ‘barrier shift’ below the original barrier level i.e. a more out of the money barrier strike to ‘under-hedge’ in the run-down to 125 is generally resorted to. The width of the original and shifted barrier would be a function of a) underlying’s liquidity – the reduction of excess delta as the barrier is crossed would need adequate liquidity i.e. poorer liquidity would mean a wider shift b) the distance between vanilla option strike and original barrier – higher the in-the-moneyness of the vanilla option, larger the digital notional and hence wider the shift c) volatility of the underlying – larger IV at and around the barrier strike would mean a larger probability for the underlying to gap and inflict negative gamma losses.   Â
- Manner of applying the barrier shift – the magnitude of barrier shift assumed in the price of the KI can either be put into effect immediately after trading with the investor/client or can be reached as progressively as we approach the option expiry. Former would be a constant barrier shift applied right at inception while the latter would be a progressive linear or non-linear shift that would eventually get to the shifted barrier at maturity. The order of aggressive pricing of the barrier would flow as below – (non-linear convex shifts would imply a more aggressive pricing versus a linear shift, but a non-linear concave shift would be less aggressive than a linear shift):
     Price non-linear barrier < Price linear (convex) barrier < Price Constant Shift
- Regular vs Reverse barriers options – in our example above of a ‘down and in put’ note that the barrier was at a strike more out of the money versus the original put strike i.e. has the bias of a put. But once knocked in this lends a larger intrinsic value to the option appearing as the payoff of the digital at the time of knock-in (would also apply if it were a knock-out option) which in turn created large gamma and delta swings (generating large gap risks) much harder to manage for both the buyer and seller. These are called reverse barrier options. Contrary to these are options where risk management does not have to deal with such gap risks as the barrier strikes are in the opposite direction of the initial option bias i.e. long 135 call with 125 either a knock-in or a knock-out. These are called regular barrier options that are so out of the money (0 intrinsic value, only time value as notional for the digi) once knocked in or knocked out that gamma/delta swings are far more tame.
- To elaborate let’s consider a down and out call – a 3m ATMF 130 call with knockout at 125. As the underlying’s price goes lower i.e. out of the money, so would the delta and even as it approaches the barrier delta would keep dropping to finally touch 0 as the barrier gets hit. Please note the gamma doesn’t flip in regular knockout options.
- Replicating reverse barrier payoff at maturity with vanilla options
An alternative way of managing risk/replicating cash flows at maturity of European Reverse Barrier options is by trading vanilla options on the underlying. A long position in a 3m ATMF 130 put that knocks in at 125 at maturity can be replicated as:
- Buy 5x 126 puts and Sell 4x 125 puts
- Or Buy 5x 126 to 125 put spread to achieve max payout of 5 dollars and buy x 125 put
Also note this is a more conservative replication for the down in put as it may not pay as much as an activated barrier option (in case the max payout isn’t realised) but also easier to manage in the absence of gap risks. Put spread width is a function of the distance between the barrier and the strike and the leverage one can manage subject to market pricing and liquidity conditions.
In-Out Parity for Barrier Options
For Barrier options having the same underlying strike, maturity and barrier levels the following always holds true:
- A combined long position in an up and out + a long position in an Up and In is equivalent to a long position in the vanilla underlying option.
- One can think of it as one and only one of the up and out vs up and in would be live (existing) for any price of the underlying and hence the combination is equivalent to always being long the underlying option.
- This will also hold true for any combination of calls/puts and up or down barriers.
- Thus price of the KO and KI options combined = price of the vanilla
- Pleasantly enough if you use this parity you can apply your intuition to understand the barrier risks better.
- Px (put @ 130) = Px (DIP @ 130/125) + Px (DOP @ 130/125)
- Px (DOP @ 130/125) = Px (Put Spread 130/125) –Â $5 Digi @ 125
- Equating the two we get: Px (DIP 130/125) = Px (put @125) + $5 Digi @ 125
Table 4 – Summarising the Greeks across the 4KI, KO European Barriers below:Â
| Greeks | Delta | Gamma | Vega | ||||
|---|---|---|---|---|---|---|---|
| European Knockouts | Option type | Close to Barrier | Away from Barrier | Close to Barrier | Away from Barrier | Close to Barrier | Away from Barrier |
| Up and out Call | Reverse | large positive | positive/negative | large negative | positive/negative | large negative | positive/negative |
| Down and Out Put | Reverse | large negative | positive/negative | large negative | positive/negative | large negative | positive/negative |
| European Knock-ins | Close to Barrier | Past Barrier | Close to Barrier | Past Barrier | Close to Barrier | Past Barrier | |
| Up and In Call | Reverse | large positive | delta of vanilla call | large positive | Gamma of vanilla call | Positive | Positive |
| Down and In Put | Reverse | large negative | delta of vanilla put | large positive | Gamma of vanilla put | Positive | Positive |
Source: Pandemonium.
Since we have focused on European barriers here only the reverse category shows up in the table above. Regular options (Down and Out or In Call, Up and Out or In Put) can only be American/path dependent; reverse options can of course be American too depending on the barrier monitoring. Reverse barrier options are cheaper given the placement of the barrier at a more out of the money level of the underlying, but the preference for reverse over regular would depend on one’s view on both the direction and magnitude of the move of the underlying in near future. If there’s conviction for an orderly move towards a put or a call barrier a reverse option would be preferred over a regular one, but if the view is for the move to be disorderly/underlying to have larger swings in course of a broader move higher or lower a regular option would be the likely choice. Â
