February 9, 2023

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In the third and final part of our write-up we’ll take a look at how traders visualise the risks associated with barrier options, consider how a delta-neutral trader would hedge these risks and introduce one or two more ‘colourful’ products.

Harking back to our previous write ups, we introduced the concept of exotic options and also pt2. We talked about how a barrier option is similar to a conventional vanilla albeit with an additional ‘trigger’ level that either terminates or instigates the vanilla.

##### Up & In Call Payoff

Above is the payoff diagram for a Knock-In barrier option. As a reminder, if the spot rate fixes above the Barrier at expiry (for an Up and In European barrier), the buyer would have the right to trade at the strike rate - in principle this would be no different to a Vanilla. However, regardless of where the market trades prior to expiry, if the spot rate fixes beneath the barrier level, the trade expires worthless and the Vanilla is never triggered into action.

As with a Vanilla, there is an element of uncertainty here, ‘Will the option expire worthless or might there be an obligation?’ Unlike a Vanilla however, there is an additional element of discontinuity, evidenced by the kink around the barrier level. This poses a problem for the writer of the option as the potential cost of being wrong is figuratively and quite literally steep. Often, due to the fact that a 1bp move can mean the difference between 0 and x, we find that the sensitivity of barrier options to the market can be sharp.

**Measuring Sensitivity**

Greeks are sensitivity measures that explain changes in the price of a derivative relative to parameters such as the underlying, volatility, passage of time and so on. This is a very easy topic to get lost in as it can quickly become very complex but for my sake (moreso than yours) we’ll be taking a surface level approach and focus on the 4 most common Greeks.

##### Vanilla Call Delta

For a Vanilla, as the spot rate moves through the strike, the option will become increasingly In The Money (ITM) until its value moves in step with the underlying. That is to say, it is so far ITM that for every dollar the spot market moves, the option’s value will change by an equal amount (Delta = 1).

**Knock In Call option**

##### KI Call Delta

The Delta for a KI Call differs slightly. To begin with, the ATM delta will be influenced by the barrier level; the closer the barrier, the higher the Delta i.e. the greater the gain in value. Like many things with options this is quite intuitive, the closer the barrier, the higher the chance the option will be triggered. Whereas the ATM delta for a Vanilla will always be 0.5 (with 0 rates). Additionally, as the spot rate nears the barrier, the Delta also exceeds 1 i.e. the rate of change of the option value will exceed the rate of change in the underlying e.g. spot may increase 2% but the value of the option may change by 2.3%.

An important caveat to note is that keeping all other things constant, bringing the barrier closer to the strike, the higher the overall delta and consequently a lower overhedge (Delta>1).

##### KI Call Gamma

Gamma is best thought of as the rate of change of Delta. It measures how Delta evolves as the underlying moves and is essentially the slope of Delta. Time to expiry will greatly influence Gamma, narrowing it, increasing its peak and shifting its centre/peak towards the barrier as expiration nears. Gamma is very large at the barrier because an incremental move of a tick could mean the difference between 0 or 1 Delta.

##### KI Call Vega

Above, Vega peaks around the strike but then moves lower and towards the barrier as time lapses. The peak of Vega decreases over time because there is less time for volatility to impact the value of the option. Vega is negligible at the tails (away from the Strike/Barrier) because when the option is deep ITM/OTM, there is little impact a rise in implied vol can have but as you near the Strike/Barrier, a higher vol increases distribution and therefore the chance an option trades at or beyond the Strike/Barrier** . **It’s worth noting that Vega’s impact is largely dependent on the closeness of spot to the barrier as well as the strike relative to the barrier e.g. If spot and strike are 100 but the barrier is very far, an increase in vol will have little impact on the ITM’ness of the option because the prospect of it being knocked in is remote.

##### KI Call Theta

Theta is most negative between the Barrier and Strike and is increasingly so. This is because as time passes, the probability of the Barrier being triggered diminishes and therefore the option loses a greater amount of value. Much like Vega, the further from the strike/barrier you get, Theta’s impact diminishes.

**Risk Management**

With the Greeks visualised we will briefly touch on how a market maker might manage their Delta for an exotic.

A market maker’s job is to provide liquidity whilst mitigating risks brought about by their provisioning services. Imagine a trader sells a vanilla Call, at the point of execution they will buy the underlying such that the amount held is equal to the option’s delta e.g. Call option with a 0.5 Delta. The trader will buy the underlying at a ratio of 0.5 per option Notional sold. The hedge will need constant rebalancing in order to remain effective and as such our trader would buy the underlying as it rallies and sell as it falls. This very act means that regardless of the path of the underlying, our trader will be ultimately indifferent as to whether the option is exercised or not. Should the option end up being exercised, the trader has bought enough of the underlying to fulfill their obligations and should the option expire worthless, well they sold spot such that by expiry there will be no inventory nor obligation.

Hedging a barrier is a little trickier as the market needs to trade at/or fix beyond the barrier for the vanilla to be triggered. As per the *KI Call Delta* chart, as the spot rate nears the strike, the delta increases as the likelihood of the option being triggered further increases, therefore the trader must buy additional amounts of the underlying. This causes a feedback loop; spot higher, trader buys even more spot and so on. This can often lead to runs as the market nears barriers if the delta management requires large enough buying. Once the barrier is breached, the option is suddenly x% ITM, therefore the trader will need to overhedge to account for this immediate and accelerated ITMness. Now that the option is triggered and because the absolute delta became greater than 1, the trader will need to sell their excess deltas. This offloading will calm the rise in spot as selling pressure is now applied to the prior rallying spot.

**Knock In Put Option**

##### Down and In Put Option Greeks

The below charts illustrate the Greeks for a Down and In Put option. The displayed Greeks behave similarly to those of an Up and In Call which can be confirmed by simple rationale; both options are triggered into being live with a move in the ITM direction therefore both must exhibit similar risks to the holder.

Studying the Delta chart, close to expiry, as the underlying approaches the Barrier, a marginal move in spot (no matter how incremental) will increase the probability of the barrier being triggered, hence the drastic change. As with a KI Call, a delta neutral trader will need to overhedge in order to maintain neutrality. Assume the trader is long the Down and In Put (the client sells in a bid to collect premium, hoping the barrier is not triggered), as spot declines our trader will buy the underlying increasingly and once the barrier level is breached they will then sell their excess deltas. Consequently, it might seem that this behaviour is counterintuitive, buying spot as the market falls, thereby reducing the likelihood that spot falls further, but then selling below the level at which it was bought after the barrier triggers, thereby locking in a loss. However, it is important to note that the goal is to remain indifferent towards the barrier being triggered i.e. the spot rate’s path, and instead to generate Pnl by conscientiously providing a b/o spread and/or trading volatility.

**Exotics & Crypto**

With a little background established we can now explore how crypto and exotics are commingling in the non exhaustive quest for yield enhancement.

##### Reverse Convertible Note

This product pays a guaranteed monthly coupon on deposited currency, allowing users to earn a return on funds that might otherwise sit idly. This product is rather vanilla but readers should note that the holder gives up potential upside but earns a guaranteed coupon that is paid independently of the underlying’s performance.

##### RCN Payoff

The simplest form of a RCN is a Short Put. The holder sells a Put, and earns a return on deposited funds. The lower the strike on the put, the lower the coupon. Seeing as these are essentially short vol, they are usually traded when vol is high or expected to retrace.

##### Barrier Reverse Convertible Note

A Barrier Reverse Convertible is a variation of the above in which the holder is not exposed to downside movements unless the underlying breaches a predetermined barrier. This trade comprises a Short Down and In put option. The strike rate is usually placed OTM with the barrier 50-80% away. Intuitively, the further the barrier the lower the value of the option and therefore the lower the coupon. Trading a European barrier vs an American will also affect the option’s value. The American being the riskier of the two from the holder’s perspective and therefore affording a higher coupon.

The terms of the trade are as such:

The Autocallable (European) Barrier Reverse Convertible offers a guaranteed coupon.

Coupon Mechanism

• The Guaranteed Coupon is paid independently of the performance of the underlying (assuming no previous early redemption).

Redemption Mechanism

• On any Observation Date, if the underlying price closes above the respective Autocall Trigger Level, the Product will be early redeemed and the investor will receive 100% of the denomination plus any payable coupon.

At Maturity (if the product has not been early redeemed):

• If the bitcoin price closes above the Barrier Level, the investor will receive 100% of the denomination.

• If the Final Fixing Level of the bitcoin price is at or below the respective Barrier Level, the Investor will receive a Cash Settlement in the Settlement Currency according to the following formula: Denomination × Underlying Performance.