Average Rate Options

Within an average rate option, the spot at expiry within a standard vanilla payoff is replaced with an average of fixings:

where is an average of spot fixings.

The following are example average rate option contract details:

The average rate option (0.24 EUR%) is significantly cheaper than the equivalent European vanilla (0.91 EUR%). The realized spot action in Exhibit 29.3 shows why this is the case.

Spot moves sharply lower but the average moves lower far more slowly and hence the average rate put option has a lower payoff at maturity than the equivalent vanilla put option. In general, the volatility of the average of spot fixings will be lower than the volatility of spot itself.

The price profile of the average rate option compared to the equivalent European vanilla shown in Exhibit 29.4 confirms that the average rate option is cheaper than the equivalent European vanilla option.

For ATM strikes it is the case that (approximately):

as demonstrated in Exhibit 29.5.

Exhibit 29.6 shows the initial vega profiles from the average rate option and the equivalent European vanilla option. The full story of this average rate vega profile is more complicated than simply lower vega exposure. There are fixings throughout the life of the option and therefore the payoff depends not just on spot at maturity but on the path it takes. Put another way, Asian options have path dependence. This path dependence causes average rate options to have vega exposures to dates on which fixings occur. For example, a EUR/GBP 1yr ATM average rate option and a European vanilla option in EUR100m notional have the bucketed vega exposures shown in Exhibit 29.7.

This process of moving optionality toward the horizon leads to the increased initial gamma exposures within the average rate option shown in Exhibit 29.8. Increased gamma can also be seen in Exhibit 29.4 where the average rate option has more curvature in the price profile.

As time passes and the fixings set, the exposures on the average rate option decrease. Exhibit 29.9 shows vega profiles after six monthly fixings within the average rate option.

The vega has reduced on both options but it has reduced far more on the average rate option. When pricing average rate options, as a sense check, the reduced vega exposure leads to smaller TV adjustments than the zeta of the equivalent vanilla option. A local volatility model is often sufficient for pricing average rate options.

The gamma on the vanilla option increases closer to expiry as expected but the gamma on the average rate decreases over its life as shown in Exhibit 29.10.

Before any fixings have occurred, the average rate will be most sensitive to changes in spot because all future fixings are impacted by spot moves. However, as time passes and fixings occur, spot moves will have a smaller and smaller impact on the payoff and therefore exposures reduce.

Toward the end of the fixings, average rate options have virtually no gamma risk because so much of the average has already been determined. At the final fixing itself, there will be a delta jump at the spot level at which the resulting average goes through the strike. The size of this delta jump is the average rate option notional divided by the number of fixings: very small if the average rate option has daily fixings. For this reason, average rate options are often used to reduce strike pin risk at expiry by, for example, setting daily fixings for the last week of the option.

Average rate options are easier to risk manage than the equivalent European vanilla options due to reduced risk at expiry and generally reduced Greek exposures. Therefore, an average rate option is often quoted with a tighter premium bid–offer spread than the equivalent vanilla option.

Another feature of Exhibits 29.9 and 29.10 is the average rate vega and gamma peak exposures moving to lower spot levels. The reason for this is that maximum optionality occurs when the expected future value of the underlying (in this case, the average) is at the strike. After six months, the six completed fixings have an average of 0.8360 compared with a prevailing spot of 0.8250. Therefore, for the expected average at maturity to be equal to the strike (also 0.8250), spot must be around 0.8140; hence the average rate peak exposures are pulled to lower spot levels.

Understanding forward drift within average rate options is also important. Consider the drift on a 1yr AUD/USD option, along with an average derived purely from the drift, as shown in Exhibit 29.11.

The averaging curtails the forward drift. If the forward drift were linear, the averaging would cut the drift in half. This effect is particularly relevant in high-interest-rate-differential currency pairs at longer tenors.

In practice, averages within Asian options can be constructed in two different ways and it is important that traders appreciate that this difference exists. The arithmetic average is the simplest form of average: the sum of the observations divided by the number of observations:

The arithmetic average payoff is naturally quoted in CCY2 per CCY1 terms. For a CCY1 call arithmetic average rate option:

The harmonic average uses the arithmetic average of the inverse observations:

For a harmonic average payoff, the reciprocal of this inverse average is used within a CCY1 notional option payoff. For a CCY1 call harmonic average rate option:

These variations basically come down to either calculating the average of the fixings in standard terms and then inverting the average or taking the inverse of the fixings and then calculating the average. These may seem like irrelevant differences but such flexibility is required in order to provide precise payoffs for clients that exactly match their FX flows.

Average Strike Options

Within an average strike option, the strike within the standard vanilla payoff is replaced with an average of fixings:

where AV is an average of fixings and S_T is spot at maturity.

The following are example average strike option contract details:

Again, the average option (1.85 EUR%) is cheaper than the equivalent European vanilla option (2.85 EUR%) with strike set to the average of the forward path. This occurs because within an average strike put option:

• As spot moves lower, the expected strike level moves lower and therefore the average strike CCY1 put payoff decreases relative to the equivalent vanilla option.
• As spot moves higher, the expected payoff from the put option decreases.

Whereas within an average strike call option:

• As spot moves lower, the expected payoff from the call option decreases.
• As spot moves higher, the expected strike level moves higher and therefore the average strike CCY1 call payoff decreases relative to the equivalent vanilla option.

Before any fixings have occurred, the vega profile from an average strike option is flat because the (expected) strike moves with spot. As fixings occur, the strike starts to be known, although the averaging keeps the strike closer to spot as spot moves. This in turn keeps the wings of the vega profile higher than the equivalent European vanilla option as shown in Exhibit 29.12.

The gamma profile from the average strike option starts at zero before any fixings have occurred, and gradually builds up over time. After the last fixing, when the strike level is known, the gamma (and all other Greek exposures) of the average strike option will be exactly equal to the regular vanilla to the expiry date. However, the fact that the strike is not exactly known until the final fixing makes precise pre-hedging of the strike risk impossible.

In general, average strike options will be quoted with a bid–offer spread roughly equal to the equivalent European vanilla. Like average rate options, smile risks on average strike options are usually subdued compared to the equivalent vanilla option and a local volatility model is often sufficient for pricing.

Double Average Rate Options

Within a double average rate option, both the strike and the spot within the standard European vanilla payoff are replaced with averages of fixings:

where AV1 and AV2 are averages of fixings. This is shown in Exhibit 29.13. Note that these fixing periods may overlap.

The following are example double average rate option contract details:

Within this trade, the “strike” is determined first, then “spot.” Exhibit 29.14 shows the bucketed vega exposures from the trade.

While the strike is fixing during the first three months of the trade, vega is short, because if spot doesn't move, that keeps the strike average close to spot, hence maximizing the optionality. Once the strike has fixed, the trade is long vega because increasing implied volatility moves the spot average as far as possible from the strike.

So many possible products can be created within the double average framework it is hard to generalize, but TV adjustments on double average rates are often small because the double averaging subdues the Greeks even more than the single average variations.