Chapter 23
Touch Options
If FX derivative contracts contain a barrier that is monitored continuously against spot, the barrier is described as American or continuous. The simplest American barrier products are touch options. There are two main kinds of touch option: One-touch options pay out a fixed amount of cash on the delivery date if spot trades through a specified barrier level at any time between horizon and expiry. No-touch options pay out a fixed amount of cash on the delivery date if spot does not trade through a specified barrier level.
Touch options are the American barrier version of European digitals and there are many similarities. Prices on touch options are quoted as a percentage of the payout notional, and like European digitals, prices cannot be quoted in the non-payout currency because there is no strike to switch between notional currencies.
The Greek exposures on touch options can be thought of as being “caused by” the American barrier, just as Greek exposures on vanilla options are “caused by” the strike.
The theoretical value (TV) of a one-touch usually lies between 0% and 100%, as shown in Exhibit 23.1. Note that one-touch TV doesn't always lie between 0% and 100% because interest rates could be negative.
Exhibit 23.1 100.00 one-touch TV over time
It is important to understand that prior to expiry, in a delta hedged trading position, there is no large P&L gain or loss generated from the touch option triggering. As spot nears the barrier, the mark-to-market value of the one-touch option rises (ignoring discounting), e.g., 98% to 99% to 100% (barrier touched), and the trading risks prior to expiry can be hedged using standard Greek exposures.
On the expiry date itself, if risk is viewed in discrete daily steps (see Chapter 11) and the one-touch has not previously triggered, its value is 0%. If it then triggers, its value becomes 100%. In trading systems this is shown as an instantaneous P&L jump through the barrier level. This can be a challenging situation to risk manage, particularly if the one-touch notional is large.
Delta Risk
The delta exposure from a one-touch contract will be long into a topside barrier and short into a downside barrier since option value increases with spot closer to the barrier. Delta on a one-touch option over time is shown in Exhibit 23.2.
Exhibit 23.2 100.00 one-touch delta over time
If spot is far from the one-touch barrier, delta bleeds away from the barrier as the sensitivity to changes in spot decreases over time. If spot is near the one-touch barrier, the delta bleeds toward the barrier as the sensitivity to changes in spot increases over time. The inflection point between far and near in this context occurs around the spot level at which the one-touch option has 30% TV.
On the expiry date, the one-touch option is viewed as a cash-or-nothing payout with no delta exposure. This potentially causes a large delta bleed away from a long barrier or toward a short barrier. For example, on the day before expiry, an untriggered long topside one-touch option will have a long delta exposure, whereas on the expiry date itself it will have zero delta. Therefore, there will be significant short delta bleed into the expiry date.
Vega Risk
One-touch options are generally long vega since if spot volatility increases there is more chance of spot moving and hitting the barrier. Exhibit 23.3 shows how one-touch vega moves toward the barrier over time. The peak vega exposure stays roughly constant and there is no exposure to implied volatility through the barrier because the contract has knocked and is now a guaranteed cash payout at maturity.
Exhibit 23.3 130.00 one-touch vega over time
Far from the barrier, spot has a very small chance of trading through the barrier level and therefore the sensitivity to implied volatility is low. Similarly, very close to the barrier, spot will almost certainly touch the barrier level so sensitivity to implied volatility is low.
The maximum sensitivity to implied volatility comes at the point where the barrier knock is in the balance. This point moves closer to the barrier over time. As discussed in the delta risk section, the point is located at the spot level for which the TV of the one-touch is around 30%.
The vega exposure on a one-touch can be hedged with a vanilla option since the two vega profiles take similar shapes. However, vega on a vanilla option evolves differently over time. Vanilla vega reduces and drifts toward the strike as shown in Exhibit 23.4.
Exhibit 23.4 120.00 vanilla vega over time
Therefore, if a one-touch option is hedged with vanilla options, no static vanilla hedge will remain valid for the life of the option. The vanilla vega hedge will either be better now or better later as the exposures evolve.
To minimize rehedging costs it is usually preferable to put a vanilla vega hedge in place that becomes better over time, at least initially. Even then, there can be substantial residual risk, particularly at the barrier, as shown in Exhibit 23.5.
Exhibit 23.5 Vega of 130.00 one-touch hedged with 120.00 vanilla in proportions such that vega risk at 6mth tenor is minimized
As the vanilla vega dies away, the composite vega position of the one-touch plus the vanilla hedge starts to look more like the one-touch only. At this point, the risk management focus switches to the gamma coming from the barrier rather than the vega.
In addition, when the touch barrier is closer to spot, stopping time will reduce; the option is not expected to live to expiry and therefore vega exposures move to closer maturities than the expiry date. It is important to always view bucketed vega Greeks when risk managing contracts with American barriers.
An important subtlety is that one-touch vega, unlike vanilla option vega, is not always positive. In high-interest-rate differential pairs the vega profile shown in Exhibit 23.6 can be generated for a long downside one-touch option.
Exhibit 23.6 AUD/USD 5yr 0.8000 one-touch vega
This counterintuitive negative vega on a long one-touch option with a downside barrier occurs when CCY1 interest rates are far larger than CCY2 interest rates and therefore swap points are large negative. With spot at, for example, 0.9000, the forward is below the one-touch barrier and therefore the model expects the barrier to be touched with very high probability. If implied volatility rises, spot is likely to diverge further from the forward path and hence is less likely to touch the barrier. This recalls an important feature of the Black-Scholes framework first described in Chapter 5: Zero volatility does not mean the spot is static; it means that spot perfectly follows the forward path. This is relevant particularly in high-interest-rate differential or pegged currency pairs.
Gamma and Pin Risk
One-touch options are generally long gamma and the gamma moves toward the barrier over time and increases sharply into expiry as shown in Exhibit 23.7.
Exhibit 23.7 100.00 one-touch gamma over time
Just as a vanilla option changes from being gamma risk into being strike risk on the expiry date, a one-touch option changes from being gamma risk into being cash-or-nothing P&L risk on the expiry date.
If spot is close to the barrier into the expiry date, the theta will be equal to the remaining value of the one-touch as the value in the risk management system drops from almost 100% to 0%. This theta can be sizeable and therefore the best hedge for a long one-touch into expiry is often to sell vanillas at (or slightly in front of) the barrier level to the same expiry date. This offsets the increased gamma risk on the last few days of the option and the theta risk into the last day itself.
Traders often find themselves long one-touch barriers and short vanilla hedges at/near the barrier level to the same expiry date. If the barrier knocks prior to expiry, the exposures from the one-touch disappear and the position is left short optionality from the vanilla hedges. If the position is now too short gamma/vega, the vanilla hedges must be bought back just as spot has broken into a new range and implied volatility has often risen due to increased uncertainty. Therefore, it is often better to increase vanilla hedges over time if possible, rather than putting on the full hedge notional up front.
Touch Barrier Delta Gap
If a one-touch barrier knocks prior to expiry, the delta from the option disappears to zero. This potentially causes a delta jump (also known as a delta gap) that must be risk managed.
For example, if a long topside one-touch barrier knocks, the delta from the one-touch jumps from long delta to flat delta (no further exposure to spot) once it knocks. This causes the trading position to get shorter delta. Therefore, spot must be bought in order to replace the delta that has been lost. As described in Chapter 3, buying spot when it moves higher is called a stop-loss order.
The following rules apply to all American barrier products:
- Long the barrier (i.e., want the barrier to touch) naturally generates a stop-loss order.
- Short the barrier (i.e., don't want the barrier to touch) naturally generates a take-profit order.
Executing these spot orders can result in “slippage,” the risk that the spot market moves against the spot order as it is being executed and hence results in negative P&L. This is particularly a risk for larger-sized spot orders. See Chapter 25 for more information on how large spot orders can cause negative P&L.
In G10 currency pairs, spot orders to hedge barrier delta gaps in the position are usually calculated ahead of time and placed in an order management system so the delta exposure within the trading position remains unchanged as barriers knock. Note that when risk is viewed in discrete daily steps, on the expiry date, when risk becomes cash-or-nothing risk, the delta disappears and hence no delta gap order exists on the expiry date of options with American barriers.
Within risk management systems it is usually possible to view spot ladders assuming delta gap spot orders from American barriers are either executed or not. Traders in G10 currency pairs usually view their trading positions assuming delta gap orders from barriers are executed, but traders in emerging market currency pairs often view their positions assuming no barrier delta gap orders are executed. In all currency pairs, the levels and sizes of delta gaps within trading positions must be calculated and monitored over time.
One-Touch Pricing
For one-touch options, the vega versus TV profile is fairly consistent across tenors and currency pairs, as shown in Exhibit 23.8.
Exhibit 23.8 One-touch vega versus TV
Similarly, volga versus TV and vanna versus TV profiles are fairly consistent across tenors and currency pairs. In practice this means that simply by knowing a few details about a one-touch contract, the TV adjustment can be estimated and sense-checked versus a model price. Exhibit 23.9 shows a typical vanna versus TV profile for a topside one-touch. For a downside barrier one-touch, the vanna profile is simply the negative of Exhibit 23.9.
Exhibit 23.9 One-touch vanna versus TV (topside barrier)
The vanna in Exhibit 23.9 implies that in terms of skew exposure only, if the risk reversal is in the same direction as the barrier, lower TV one-touches will have a positive TV adjustment and higher TV one-touches will have a negative TV adjustment. Alternatively, if the risk reversal is in the opposite direction to the barrier, lower TV one-touches will have a negative TV adjustment and higher TV one-touches will have a positive TV adjustment.
Exhibit 23.10 shows a typical volga versus TV profile for a topside or downside one-touch.
Exhibit 23.10 One-touch volga versus TV
This profile implies that in terms of the wing exposure only, lower TV one-touches will have a positive TV adjustment while higher TV one-touches will have a negative TV adjustment.
How these vanna and volga exposures are reflected within a TV adjustment depends on the volatility surface in a particular currency pair.
Exhibit 23.11 shows the one-touch TV adjustments generated by various smile pricing models for a 1yr EUR/USD topside one-touch with the risk reversal for downside.
Exhibit 23.11 1yr EUR/USD topside one-touch TV adjustment under various smile pricing models
As predicted by the vanna and volga profiles, TV adjustments on these one-touches are positive for very low TV contracts due to the long volga exposure, then at low TV the TV adjustment goes negative due to the vanna exposure and at high TV the TV adjustment stays negative due to the short volga exposures.
All models except VVV have their most negative adjustment around 40% TV due to the short volga. The mixed volatility model gives prices between the local volatility and stochastic volatility models, as expected. Recall from Chapter 19 that the local volatility model undervalues volatility convexity and the stochastic volatility model overvalues volatility convexity; this can be seen in these results.
Exhibit 23.12 shows the one-touch TV adjustments generated by various pricing models for a 1yr EUR/USD downside one-touch with a risk reversal also for downside.
Exhibit 23.12 1yr EUR/USD downside one-touch TV adjustment under various smile pricing models
As predicted by the vanna and volga profiles, TV adjustments on low TV one-touches are positive due to the vanna and the long volga exposures. TV adjustments on high TV one-touches are negative due to the flipped vanna and the short volga exposures.
Again, the local volatility model undervalues volatility convexity, but values the skew component well. The stochastic volatility model overvalues volatility convexity and undervalues the skew component.
In general, the mixed volatility model gives good pricing for one-touch contracts in liquid currency pairs provided the model is correctly calibrated. There are two main situations in which the mixed volatility model may fail to give prices close to the market:
- Contracts for which there is a large exposure to interest rates
- Currency pairs in which there is a spot jump dynamic.
The pricing models examined in this chapter all assume that interest rates are constant and therefore spot volatility ≈ forward volatility. This is obviously not the case in practice (see Chapter 17 for more details). The key point is that American barriers knock out on spot, but are usually priced and valued using the ATM volatility, which is not the expected spot volatility. Therefore, when pricing a one-touch:
- If the volatility of spot is higher than the volatility of the forward, the market value of the one-touch will be higher than the smile model price since spot is more likely to knock the barrier than the model suggests.
- If the volatility of spot is lower than the volatility of the forward, the market value of the one-touch will be lower than the smile model price since spot is less likely to knock the barrier than the model suggests.
This interest rate impact can be quantified using a stochastic interest rate pricing model. There are two main elements to consider within such a model:
- Interest rate volatility. Higher interest rate volatility increases the forward volatility and hence reduces the market value of one-touch options.
- Correlation between spot and interest rates. As seen in Chapter 17, positive correlation between spot and CCY1 rates or negative correlation between spot and CCY2 rates depresses the volatility of the forward and hence increases the market value of one-touch options. Similarly, positive correlation between spot and CCY2 rates or negative correlation between spot and CCY1 rates increases the volatility of the forward and hence decreases the market value of one-touch options.
This spot volatility versus forward volatility difference is important at longer tenors (approximately beyond 2yr) or in pegged/managed currency pairs. In these cases the impact of stochastic interest rates should always be quantified within pricing.
Finally, it is worth noting that one-touch value is intuitively similar to the probability of the one-touch barrier touching. Specifically, one-touch TV is the discounted, risk-neutral probability of the barrier touching priced under single static volatility and single static interest rates assumptions. Touch options should be priced using pricing models and adjusting for the limitations of the pricing models rather than guessing barrier knock probabilities. However, if the trader's intuition about the barrier knock probability is significantly different from the one-touch valuation, it is important to investigate why this is the case.
One-Touch Bid–Offer Spread
In practice, bid-offer spreads for touch options, like bid-offer spreads for European digitals, are usually generated using grids of bid–offer spreads at different maturities maintained by traders. Since one-touch contracts can knock out prior to expiry, and hence avoid the biggest risk management challenges at expiry, they are often quoted with a tighter bid–offer spread than the equivalent European digital contract, perhaps two-thirds of the width.
One-Touch Variations
Clients often request prices on one-touch options that have subtle variations from the standard one-touch contract.
CCY1 versus CCY2 Payout
One-touch options can pay out either CCY1 or CCY2 cash at maturity. For topside one-touch options, a CCY1 payout will have relatively higher valuation than CCY2 payout because CCY1 is relatively worth more with spot higher at the barrier level. Therefore, flipping from CCY1 to CCY2 payout on a topside one-touch will cause the TV% to fall. Exhibit 23.13 shows an example of this.
Exhibit 23.13 CCY1 versus CCY2 one-touch options in pricing tool
Similarly, for downside one-touch options, a CCY2 payout will have relatively higher valuation than CCY1 payout because CCY2 is relatively worth more with spot lower. Therefore, flipping from CCY1 to CCY2 payout on a downside one-touch will cause the TV% to rise.
For two-sided touch options that could knock with spot higher or lower, there is usually minimal valuation difference between CCY1 payout and CCY2 payout.
Since the standard one-touch pays out at maturity, the effect of discounting must also be taken into account. For a standard one-touch contract, the higher the interest rates in the payout currency, the relatively lower the option value since the payout at maturity must be present valued to the barrier knock date (see Chapter 10).
Pay-at-Maturity versus Pay-at-Touch
Standard one-touch contracts pay at maturity, but clients sometimes request prices on one-touch contracts that pay out when the barrier touches (value spot). These are called pay-at-touch, or instant one-touch options.
When interest rates are positive, the instant one-touch will be more expensive than the standard version because a fixed amount of payout cash will be received sooner (when the barrier knocks) rather than later (at maturity).
For short-dated contracts, this value difference is usually negligible, but for longer expiries there can be a significant price difference, particularly if interest rates in the payout currency are high or if there is a strong correlation between spot and interest rates in the payout currency. For example, if interest rates in the payout currency tend to increase as spot heads higher toward a topside one-touch barrier, the differential between smile model TV adjustment and the market value of the instant one-touch should be higher than the standard version.
No-Touch Options
No-touch options pay out a fixed amount of cash at maturity if spot does not trade through a specified barrier level at any time between the horizon date and the expiry date.
A long one-touch option plus a long no-touch option with all other contract details the same therefore results in a guaranteed payout at maturity. The Greeks and the TV adjustment on a no-touch contract are the equal and opposite (negative) of the equivalent one-touch option.
The market convention is to trade single-barrier touch contracts as one-touch options and double-barrier touch contracts as no-touch options; therefore, double-no-touch (DNT) options are a standard exotic product in the interbank broker market and with institutional clients.
As seen in Chapter 18, the main exposure on a double-no-touch is usually volga. Intuitively this makes sense; with barriers either side of current spot, pricing depends less on the relationship between spot and implied volatility and more on the volatility of implied volatility.
A long double-no-touch has a short vega exposure since lower volatility gives a greater chance of spot not touching either barrier and hence the payout being generated. The vega profile of a double-no-touch option is shown in Exhibit 23.14.
Exhibit 23.14 Double-no-touch vega profile
As time passes (or if the barriers are placed wider apart) the separate vega exposures from each barrier become clear and this is shown in Exhibit 23.15. The vega first increases as the value of the double-no-touch rises; then it starts to split into two separate vega exposures from each barrier over time.
Exhibit 23.15 Double-no-touch vega profile over time
Lower TV double-no-touch options are long volga, but higher TV double no-touches can have negative volga. Exhibit 23.16 shows how the volga exposure on a symmetric double-no-touch changes as the barriers widen and hence TV increases.
Exhibit 23.16 Symmetric-barrier double-no-touch volga against theoretical value profile
The bid–offer spread shown on double-barrier touch options is usually similar to the bid–offer spread shown on single-barrier touch options since it is unlikely that traders will encounter risk management issues at both barriers at expiry.
Finally, note that instant versions of no-touch options are not possible: Spot must get to maturity without having triggered the barrier for a no-touch option to pay out.