Paradigm Insights | Exotic Options Part 1 - An Introduction

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December 14, 2022


Exotics are a staple in Tradfi, but they are still shyly shuffling into the crypto limelight. Part of this is due to the vanilla-centric nature of crypto options, but is also caused by the composability and OTC nature of exotics. However, with the diversity of outcomes these products afford users, it’s only a matter of time before they too become ubiquitous in this burgeoning industry.

This piece offers a very basic and digestible overview of Binary Options. We’ve shooed away the quants, hidden away the fancy equations/jargon and used degen-friendly charts. So with that said, let’s get stuck in.


Exotic Options are generally split into generations, indicating their length of existence in financial markets. The following list is far from exhaustive and simply serves as an introduction to the most popular styles. A side note here, as vanilla strategies are composed by combining vanilla options with different strikes/ratios, combining exotics and exotics with vanillas can create a host of different payoffs.

1st Generation Exotics

These are essentially the ‘vanillas’ of exotic options, the foundational pieces which often, but not always, have the fewest moving parts. The most prominent components are Barriers, OT (One-Touches), KIKO (Knock-In, Knock-Out), Binary and Asians amongst others. Further Generations include Faders, Target Accruals, Baskets, Variance and Vol Swaps and Windowed Barriers amongst many, many others. But let’s not get ahead of ourselves, for this piece we’re going to focus on Barriers and Binary.


Long Call payoff

Let’s start with a vanilla call option. We’re all familiar with its payoff; line goes up, PnL goes up (if you bought). As the spot rate moves through the strike rate, your PnL will reflect the degree to which you are in the money. Now let’s keep the strike but delay PnL gratification until spot crests an even higher level – 120 in this case.

Long KI Call payoff

There is a distinct kink in the payoff. The strike rate is at 100 and a barrier at 120 has been introduced. Beyond 120, the payoff is identical to an ITM vanilla, beneath 120 (inclusive of above the strike rate), the payoff is the same as if the option were OTM.

The simplest way to think of a barrier option is as an extension of a vanilla. It has a vanilla payoff but whereas vanillas only have a strike, barriers also have a barrier level. This level is a trigger that once actioned, either creates a vanilla or cancels it. This level has to be placed ‘outside’ of spot and strike e.g. for an American KI call option, the trigger level has to be either above the strike, or below the spot. If the trigger level is between the spot and the strike, it becomes the same as a vanilla call option. For a European KI call option, the trigger level has to be above the strike (or on both sides but that’s for another day). Furthermore, these levels might be observable throughout the life of the option or only at expiry.

Single Barrier Option Styles

A quick walk through on the above table:

  1. ‘European’ refers to a barrier that is only observed at the point of expiry. That is to say that whether or not it expires the vanilla or triggers it into action only depends on where the spot market is at option’s expiry.
  2. ‘American’ refers to a barrier that is observable for either the life of the product or during a predefined time window i.e. the trigger event can happen at any point during the allotted period.
  3. In’ means that if the trigger event occurs, the vanilla is created. ‘Out’ means that if the trigger event occurs, the vanilla is canceled.
  4. ‘Down’ and ‘Up’ refer to the side of the spot on which the barrier is placed.

It is worth making some distinctions here:

  • An Up and In American Call option can have the spot finish above the Strike rate but if the barrier has not yet been triggered, the option will expire worthless.
  • A Down and In European Put option can have the spot rate trade through the barrier level prior to expiry but if the spot rate fixes between the spot rate and the strike rate (i.e. not beneath the barrier level), the option will expire worthless.

The point being conveyed here is that a series of factors need to be taken into consideration:

1) What is the barrier style/observation period?

2) Has the barrier been triggered?

3) Where is the spot rate at maturity in relation to the strike.

Now that we have introduced these variants, we can now briefly discuss their nuances.

Whenever a barrier is added you essentially introduce an element of uncertainty. That is to say that your payoff is no longer solely dependent on just one element but now has another hurdle to crest. Purely as an extreme example to illustrate matters, take an American Down & Out Put option i.e. a Put option with a barrier beneath the Strike Rate. If the barrier is placed a considerable distance from spot, there is a reduced possibility of this trade knocking out and therefore pricing will better represent that of a vanilla. However, if you place the barrier sufficiently close to spot such that a small move in the spot rate will trigger the barrier and knock out the option, then this option poses little to no value as the option is as good as canceled. Granted the above is an extreme example but it serves to illustrate that this element of uncertainty works to cheapen the price of the option. In isolation this might appear misplaced but this attribute can be exploited when options are combined.

Combining Products

The beauty of options lies in their ability to let the user generate a myriad of payoffs through composability. For example let’s look at the prior discussed EKI and combine that with a Call option.

Long European KI Call and Short Vanilla Call payoff

As discussed earlier, if you are long a KI Call and the spot rate fixes ITM i.e. beyond the barrier, you will have the right to exercise the option at the strike rate. Selling a vanilla struck at the barrier level means any gains beyond the barrier will be capped because as the market moves higher, any ITM value is offset by the short vanilla. Combining both of these products produces the following net payoff.

Digital Call payoff

Beneath the 120 barrier your payoff is flat and reflects only the associated premiums and beyond 120 you have a jump in ITM’ness which is capped. This therefore becomes an all or nothing scenario:

  1. At or Below 120, you get Nothing!
  2. Above 120, you get a flat, handsome payout.

This is a binary outcome and is very simple to understand. There is no matter of degree of ITM or deliberation over which path the spot rate took, it’s a simple yes, no as to where spot fixes. Thus this combination is called a Binary. We won’t go into too much detail here but you have ‘Cash or nothing’ or ‘Asset or nothing’ styles. In this case we are looking at the former.

These products are extremely popular when packaged with other products and can act as rebates but they are also very attractive in their own right due to their simplicity and binary nature.

Trade Example

Below is a pricing grid for SOL Digitals which can be found at

Owing to the number of moving parts, we’ve taken to simplifying things in order to convey all relevant information as elegantly as possible.

SOL Digital Option Pricing — Spot Ref: 33.77

The ‘Premium as %’ column represents the USD amount you will pay/rcv relative to the Notional payout in the event of the option fixing ITM. The ‘Tenor’ represents the expiry date.

For example, a Call with 10% premium at 1mo gives a Strike (Barrier) Level of 44.46. Translating this, if you buy this Binary and SOL fixes above 44.46 on 6/11 (6/10 at time of writing), paying USD 10k, would see you walk away with USD 100k. This is just one example as these products are customisable. In addition to buying Binary Calls and Puts, you also have the ability to sell. Therefore if you reckon the market remains rangebound or perhaps will not descend further, you can sell a Binary Put.

SOL Digital Option Pricing — Spot Ref: 33.53

E.g. 20% at 3mo gives a Strike/Barrier of 20.85. Selling here for USD 20k means that if SOL fixes at or above 20.85 on 6/1/23, you walk away with the USD 20k premium. Otherwise if the market fixes beneath this level you will pay out USD 100k (Total P&L -80k: 100k payout less the 20k premium).

Shameless plug done, let’s turn our attention to risks. Buying a Binary seems simple enough but what about selling? What risks could this potentially pose? We will take a purely conceptual look at this and save the Greek deep dive for another day (when my quants awaken and can generate some nice charts).

Replication and Hedging

Due to discontinuities around the barrier level, managing digital risk can prove trying. Thankfully, you can approximate and therefore hedge Binary risk using vanillas. Recall that a Digital gives you a fixed payout once spot fixes beyond a certain Barrier level. Let’s call this payout X. Adding two vanillas with strikes K1 and K2 and notional N. You can generate an identical payout using the equation:

               X = (K2 — K1) * N

Call Spread

Looking at a Call Spread, as spot moves through the first strike, the position moves increasingly ITM. Once spot moves through the second strike, (assuming notional symmetry) PnL is capped. By compressing the distance between both strike rates, you can model a binary payoff. The tighter the spread the larger the notional needed to replicate a constant payout:

N = X / (K2 — K1)

Assuming a trader has a digital barrier at 100, more often than not you will find that K2 is placed at this barrier level and K1 is placed a certain distance ahead. Remembering the payoff for a Digital, above the barrier you receive X payout but beneath the barrier you receive nothing, a Call Spread does not truly replicate this because the payoff between the strikes is non-zero. Therefore this will always lead to over-replication. Deciding the level of over-replication is an art and depends on the c̶a̶j̶o̶n̶e̶s̶ skill of the trader.

As an example, assume the payout on the digital is 50. If K2 — K1 = 50, the notional needed to achieve this payout is 1. Although easy to manage, a 50 point wide call spread would be rather expensive so compressing this to 5 would result in a notional of 10. The rationale here is pretty straightforward, the tighter the spread the more aggressive pricing is but then again the larger the notional and associate hedging difficulty.

As promised, this piece has served as a very light touch introduction to digitals. We will be bringing you more pieces studying a variety of exotics as we celebrate and explore our partnership with OrBit Markets; the leading institutional liquidity provider of digital assets options and structured products.

As the demand for sophisticated hedging strategies increases, pioneering players offering innovative products will come to dominate the market; thereby enabling users to create tailored payouts and capitalize upon all opportunities.

Article by
Laevitas is a Bitcoin and Ethereum token data analytics platform providing quantitative insight to crypto traders. Users can sign up to get data about options, futures, and derives on custom dashboards.

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