The hedging of a barrier option is more involved than for vanilla options, as the delta near the barrier can be significantly more than ±100% near expiry. The extra hedging risk of barriers widens the bid-offer spread in comparison with vanilla options. Barrier options are always European and are traded OTC.

There are three key variables to a barrier option, each of which has two possibilities. These combinations give eight types of barrier option (8=2×2×2).

While vanilla options (and knock-in options) will increase in value as spot moves further in the money, this is not the case for knock-out options, where spot is near the strike. This effect is caused by the payout equalling zero at the barrier, which can cause delta to be of opposite sign to the vanilla option. This effect is shown below for a one-year ATM put with 80% knock-out. The peak value of the option is at c105%; hence, for values lower than that value the delta is positive not negative. This is a significant downside to using knock-out puts for protection, as their mark to market can increase (not decrease) equity sensitivity to the downside.

If a knock-out put and a knock-in put have the same strike and barrier, then together the combined position is equal to a long vanilla put (P_KO + P_KI = P). This is shown in the charts below. The same argument can apply to calls (P_KO + P_KI = P). This relationship allows us to see mathematically that if knock-out options are seen as visually cheap, then knock-in options must be visually expensive (as a knock-in option must be equal to the price of a vanilla less the value of a visually cheap knock-out option)

The payout of a knock-out put is equal to a ‘shark fin’ (see figure on the left) until the barrier is reached. A ‘shark fin’ is equal to a short digital position (at the barrier) plus a put spread (long put at strike of knock-out put, short put at barrier of knock-out put). The price of a knock-out put can therefore be considered to be the cost of a put spread, less a digital and less the value of the knock-out. As pricing digitals and barriers is not trivial, comparing the price of a knock-out put as a percentage of the appropriate put spread can be a quick way to evaluate value (the knock-out will have a lower value as it offers less payout to the downside). For reasonable barriers between 10% and 30% below the strike, the price of the knock-out option should be between c15% and c25% of the cost of the put spread.

There are two types of barriers, continuous and discrete. A continuous barrier is triggered if the price hits the barrier intraday, whereas a discrete barrier is only triggered if the closing price passes through the barrier. Discrete knock-out barriers are more expensive than continuous barriers, while the reverse holds for knock-in barriers (especially during periods of high volatility). There are also additional hedging costs to discrete barriers, as it is possible for spot to move through the barrier intraday without the discrete barrier being triggered (ie, if the close is the correct side of the discrete barrier). As these costs are passed on to the investor, discrete barriers are far less popular than continuous barriers for single stocks (approx. 10%-20% of the market), although they do make up almost half the market for indices. Jumps in stock prices between close and open is a problem for all barriers While the hedge for a continuous barrier should, in theory, be able to be executed at a level close to the barrier, this is not the case should the underlying jump between close and open. In this case, the hedging of a continuous barrier suffers a similar problem to the hedging of a discrete barrier (delta hedge executed at a significantly different level to the barrier).

Double barrier options have both an up barrier and a down barrier. As only one of the barriers is significant for pricing, they are not common (as their pricing is similar to an ordinary singlebarrier option). They make up less than 5% of the light exotic market.

The main disadvantage of knock-out barrier options is that the investor receives nothing for purchasing the option if they are correct about the direction of the underlying (option is ITM) but incorrect about the magnitude (underlying passes through barrier). In order to provide compensation, some barrier options give the long investors a rebate if the barrier is triggered: for example, an ATM call with 120% knock-out that gives a 5% rebate if the barrier is touched. Rebates comprise approximately 20% of the index barrier market but are very rare for singlestock barrier options.

Worst-of (or best-of) options give payouts based on the worst (or best) performing asset. They are the second most popular light exotic due to structured product flow. Correlation is a key factor in pricing these options, and investor flow typically buys correlation (making uncorrelated assets with low correlation the most popular underlyings). The underlyings can be chosen from different asset classes (due to low correlation), and the number of underlyings is typically between three and 20. They are always European, and normally ATM options.

Worst-of/best-of options can be any maturity. Although the most popular is one-year maturity, up to three years can trade. As an option can be a call or a put, and either ‘worst-of’ or ‘best-of’; there are four types of option to choose from. However, the most commonly traded are worst-of options (call or put). The payouts of the four types are given below:

The payout of a worst-of call option will be equal to the lowest payout of individual call options on each of the underlyings. As it is therefore very cheap, they are popular to buy. If all the assets are purely correlated, then the value of the worst-of call is equal to the value of calls on all the underlyings (hence, in the normal case of correlation less than 1, a worst-of call will be cheaper than any call on the underlying). If we lower the correlation, the price of the worst-of call also decreases (i.e. the price of a worst-of call on two assets with -1 correlation is zero, as one asset moves in the opposite direction to the other). A worst-of call option is therefore long correlation. As worst-of calls are cheap, investors like to buy them and, therefore, provide buying pressure to implied correlation.

Rumour of QE2 lifted demand for worst-of calls on cross assets
Before QE2 (second round of quantitative easing) was announced, there was significant buying flow for worst-of calls on cross assets. The assets chosen were all assets that were likely to be correlated should QE2 occur but that would normally not necessarily be correlated (giving attractive pricing). QE2 was expected to cause USD weakening (in favour of other G10 currencies like the JPY, CHF and EUR), in addition to lifting ‘risk-on’ assets, like equities and commodities. The buying of worst-of calls on these three assets would therefore be a cheap way to gain exposure to the expected movements of markets if quantitative easing was extended (which it was).

A worst-of put will have a greater value than any of the puts on the underlying assets and is therefore very expensive to own. However, as correlation increases towards 1, the value of the worst-of put will decrease towards the value of the most valuable put on either of the underlyings. A worst-of put is therefore short correlation. As selling (expensive) worst-of puts is popular, this flow puts buying pressure on implied correlation (the same effect as the flow for worst-of calls).

While worst-of options are popular, there is relatively little demand for best-of options. There are some buyers of best-of puts (which again supports correlation); however, best-of calls are very rare. A useful rule of thumb for worst-of/best-of options is that they are short correlation if the price of the option is expensive (worst-of put and best-of call) and the reverse if the price of the option is cheap. This is why the buying of cheap and selling of expensive worst-of/best-of options results in buying flow to correlation.

As the flow from worst-of/best-of products tends to support the levels of implied correlation, implied correlation typically trades above fair value. While other light exotic flow might not support correlation (eg, outperformance options, which are described below), worst-of/best-of options are the most popular light exotic, whose pricing depends on correlation and are therefore the primary driver for this market. We would point out that the most popular light exotics – barrier options – have no impact on correlation markets. In addition, worst-of/best-of flow is concentrated in uncorrelated assets, whereas outperformance options are usually on correlated assets.