Range Accrual Options

The simplest accrual product is a range accrual, which pays out an accrued notional at expiry. Range accrual prices are quoted in payout currency % terms, like touch options or European digital options.

Following are some typical European double barrier range accrual contract details:

For products with fixings it is vital to specify the fixing source, confirm which holiday calendars are used to generate the fixing schedule, and agree the fixing schedule itself with the client prior to quoting a price.

Within the example range accrual, the notional is split between the fixings (usually equally), that is, USD1m/250 = USD4k per fixing. Therefore, for every fixing in the schedule where spot is between the accrual barriers, USD4k is added onto the cash payout at maturity.

These are European accrual barriers, so if spot goes outside the range (either between fixings or it fixes outside the range) and then later fixes within the range again, the notional continues to accrue. A typical European range accrual structure is shown in Exhibit 28.1.

Therefore, within a European range accrual:

where n is the number of times spot fixes within the European accrual barriers and N is the total number of fixes in the fixing schedule.

If there is just one fixing at expiry, the option becomes a European digital range that pays out the full notional if spot is within the range at expiry. Extending this idea further, European range accruals can be perfectly replicated by a strip of European digital ranges expiring at each fixing within the range accrual, each with their [Notional × ] cash payout late-delivered to the range accrual delivery date. As seen in Chapter 21, the vega profile from a long European digital range is long vega in the wings (want spot to move back into the range) and short vega between the barriers (don't want spot to move) and this same profile also applies to European range accruals:

• If spot is between the European range accrual barriers, more payout will be accrued if spot stops moving; hence short vega.
• If spot is outside the European range accrual barriers, spot needs to move into the range again in order to start accruing; hence long vega.

Exhibit 28.2 shows the long wing vega profile from a long European range accrual. Recall from Chapter 18 that contracts with long wing vega profiles usually have positive TV adjustments.

The fact that European range accruals can be replicated with a strip of European digital ranges demonstrates two important features of European range accruals. First, accrual option pricing must take the full ATM term structure into account because there is vega exposure to every date within the fixing schedule, not just to the expiry date. Second, any well-calibrated smile pricing model will correctly incorporate the impact of the volatility smile into the product. Therefore, the simplest and quickest available smile pricing model (often local volatility) can be used.

American keep range accruals stop accruing and all accrued notional is kept if spot ever goes through an American keep accrual barrier. Therefore, the European range accrual will be more expensive than the American keep range accrual with the same barriers. In turn, the American keep range accrual will be more expensive than a no-touch option in the full notional with the same barriers because the entire no-touch notional is effectively lost if a barrier level trades.

An American keep range accrual can be replicated with a strip of double no-touches with their cash payouts late-delivered to the range accrual delivery date. Therefore, the vega profile of a long American keep range accrual looks similar to a long double-no-touch option, with no vega beyond the barriers as shown in Exhibit 28.3. Again, the vega profile is long wings and will usually have a positive TV adjustment.

Importantly, the process of stripping barrier or digital risk from a single date into multiple dates reduces trading risk because it is not possible to gain or lose the entire notional on a single date. For this reason, the bid–offer spread on a range accrual is usually tighter than the bid–offer spread on the equivalent European digital range or double-no-touch option in the total notional.

The stripping process also reduces model risk on accrual options because the stripped contract has different exposures at each fixing and this is similar in some sense to taking an average. American keep accrual options with many fixings often give similar valuations under different models and therefore the simplest and quickest available smile pricing model (often local volatility) can often be used.

Accrual Forwards

The most popular accrual products are accrual forwards. These products contain a forward payout at maturity with the notional determined by how spot moves. There are many variations, but the most popular is a double accrual forward. The notional accrues:

• 1× (notional/number of fixings) for each spot fix in an area where the payoff is positive for the client at maturity
• 2× (notional/number of fixings) for each spot fix in an area where the payoff is negative for the client at maturity

Therefore, the product can accrue up to double the notional if spot fixes in the 2× accrual area at every fix. The different accrual multiples generate value for the client and hence the forward rate is moved in the client's favor, meaning that the client transacts for zero premium.

The risk from a simple accrual forward can be decomposed into two range accruals with the same accrual barriers. For example, a long EUR/USD accrual forward can be replicated using:

• Long EUR range accrual in the accrual forward EUR notional.
• Short USD range accrual in the appropriate notional (using CCY2 notional = CCY1 notional × strike).

As before, a simple smile pricing model can be used for European accrual forwards and is often sufficient for pricing American keep accrual forwards with many fixings.

Following are some typical American keep double accrual forward contract details:

By capping the upside with the American keep accrual barrier and making the downside worse with 2× accrual below the strike the client buys USD/JPY in 1yr forward at 98.80 rather than at 102.05 for zero premium. This is shown in Exhibit 28.4.

The vega risk on this American keep double accrual forward is similar to an American knock-out option, with a vega peak around the strike. This is shown in Exhibit 28.5.

These products are almost always long vega from the trading desk perspective. Traders describe the risks on accrual forwards as “vanilla,” meaning the vega and gamma are well-behaved over time. In addition, there are no large digital risks to hedge due to the payout depending on a strip of fixings. Therefore the bid–offer spread on accrual forwards with European or American keep barriers is often just derived from the vega spread.

Vega Risk

Exhibit 28.6 shows the bucketed vega exposures from the example trade at various spot levels from the trading desk perspective. The maximum total vega is usually located around the strike, especially if there is leverage in the trade:

• In the spot direction where the client has losses (and hence the target is less likely to be reached), the vega reduces like the equivalent vanilla structure.
• In the spot direction where the client has gains (and hence the target is more likely to be reached), vega converges to sharply and the vega risk moves to closer maturities.

Looking at the vega exposures at each spot level in Exhibit 28.6:

• 1.1750 spot: The bucketed vega profile looks similar to a strip of leveraged forwards, which has the same vega exposures as a strip of vanilla options in the unmatched notionals (see Chapter 8)—exactly what the structure would become if the target feature was removed.
• 1.2260 spot: Spot is closest to the strike (1.2290) and hence optionality (vega) is maximized.
• 1.2770 spot (current spot): The client is accumulating gains and therefore the structure will more likely knock out early. Overall vega and stopping time has reduced and there is increased vega at closer maturities.
• 1.3280 spot: The structure will likely knock out soon, so the vega move into closer tenors is even more exaggerated. The vega is bucketed mainly in the 2mth tenor, which suggests that the stopping time on the trade is around two months.
• 1.3790 spot: The structure will knock out at the first fixing and therefore has minimal exposure to implied volatility.

Exhibit 28.7 shows the vega chart of the TARF compared to the equivalent leveraged forward structure. The presence of the target reduces the stopping time of the structure and decreases the vega, particularly at higher spots where the structure will terminate more quickly.

Exhibit 28.8 shows a vega chart of the TARF with different targets. With a higher target, the vega on the TARF increases and the chance of knocking out decreases.

Delta and Gamma Risk

The spot ladder from the EUR/USD TARF from the trading desk perspective is shown in Exhibit 28.9.

The trade is long gamma for the trading desk because from the trading desk perspective the trade gains more value with spot lower than it loses with spot higher due to the target. Thinking about the delta exposure on the trade confirms this:

• If spot goes lower, the deal will get more and more in-the-money for the bank until the delta eventually becomes the leveraged notional and the trading risk becomes equivalent to a strip of forwards.
• If spot goes higher, eventually the delta position on the trade goes to zero as the deal is deep in-the-money for the client and the bank has no more to lose due to the target being reached.

Target Redemption Fixing Risk

Aside from the standard gamma and vega trading risks, the main risk management challenges on TARFs come from the fixings. At inception, the example EUR/USD trade will knock out at the first fixing if spot fixes at 1.3290 or above, paying the client the (1.3290 – 1.2290) × 300k USD = 30k USD target. Suppose instead spot fixes at 1.2790 at the first fix, hence generating USD15k profit. The deal will then knock out at the second fixing if spot fixes at or above 1.2790 again. Before the first fixing there was no explicit digital risk at the second fixing. However, after the first fixing, it becomes known that the future value of the remaining fixings will be lost if spot fixes above 1.2790 at the second fixing. Therefore, the trade develops digital risk at 1.2790 at the second fixing.

The more frequently fixings occur within a structure, the more digital risks must be risk managed. These digital risks will be particularly large if the termination level is close to the strike, so if spot fixes on one side of the strike, the option will knock out, while if spot fixes on the other side of the strike, it will have a far longer time to maturity. Large digital risk is also generated by strike changes within the contract details of the trade. For example, if the remaining strikes all move to a new level after the next fixing, the difference in option value between the option staying alive or expiring at the next fixing will be large.

The crystallization of digital risk at the next fixing can also cause delta changes. These delta gaps at fixings require careful monitoring and management within the TARF deal population.

Target Redemption Pricing

Target redemption options are conceptually similar to accrual options: Both have fixing schedules, and in some sense target redemptions knock out on time while accruals knock out on spot. The similarity extends to the pricing: Both products have minimal model risk. This can be confirmed by pricing the TARF using various smile pricing models; different models will often give similar valuations.

Traders describe the risks on target redemption forwards as “vanilla,” meaning that the vega and gamma are generally well-behaved through time. As noted, the major risk management challenges come from the fixings. It is therefore important that these potential digital risks at fixings are taken into account within the bid–offer spread as well as the usual vega-based spreading.