Foreign exchange derivatives Commerzbank AG

2. The popularity of barrier options

Isn't there anything cheaper than a) They are less expensive than vanilla options? From an actuarial vanilla contracts: in fact, the closer point of view a put or a call is the spot is to the barrier, the an insurance against falling or rising cheaper the knock-out option. Any exchange rates, and surely a buyer price between zero and the vanilla would like to keep the premium at a premium can be obtained by minimal level. For this reason barrier taking an appropriate barrier level. options have been invented. They One must be aware however, that belong to the first generation exotics. too cheap barrier options are very The premium can be lowered by likely to knock out. shifting risk to the option holder. We give an overview of the issues Vanilla Put and Down-and-out Put Compared related to barrier options 0.040 barrier 0.035 1. What is a barrier option option 0.030 value 0.025 vanilla Varatio delectat - there are lots of 0.020 option different kinds of barrier options. A 0.015 spot = 0.90 strike = 0.92 standard barrier option can be either a option value 0.010 maturity = 3M call or a put with the additional feature 0.005 = 14% that the option becomes worthless if 0.000 domestic rate = 5% the spot hits a prespecified barrier. foreign rate = 6% Such an option is called a knock-out 0.70 0.74 0.78 0.82 0.86 0.90 Barrier call or knock-out put. Correspondingly there are options which are worthless b) They allow to design foreign unless and until the spot hits a barrier exchange risk exposure to special and at hitting time becomes a vanilla needs of customers. Instead of option (knock-in type). Clearly holding lowering the premium one can both a knock-in plus a knock-out and increase the nominal coverage of otherwise identical vanilla is the same the vanilla contract by admitting a as holding a mere vanilla option. barrier. Several customers feel Alternatively a long knock-out call can sure about exchange rate levels not be replicated by a long call and a short being hit during the next month knock-in call. which could be exploited to lower More generally, any option other than the premium. Others really only vanillas can have knock-out barriers. A want to cover their exchange rate knock-in type is often referred to as exposure if the market moves kick-in in the exotic case. drastically which would require a knock-in option. c) The savings can be used for another hedge of foreign exchange risk exposure if the first barrier option happened to knock out. d) The contract is easy to understand USD/DEM 1990-1997 if one knows about vanillas. e) Many pricing and trading systems 1.90

include barrier option calculations 1.80 in their standard. 1.70 f) Pricing and hedging barriers in the March 29 1995 Black-Scholes model is well- 1.60 10:30am 1.3800

understood and most premium 1.50 calculations use closed-form exchange rate solutions which allow fast and 1.40 low 1.3870 Sept 2 1992 1pm all time low 1.3500 April 19 1995 9:30am stable implementation. 1.30

3. Barrier option crisis in 1994-96, 02.01.90 02.07.90 02.01.91 02.07.91 02.01.92 02.07.92 02.01.93 02.07.93 02.01.94 02.07.94 02.01.95 02.07.95 02.01.96 02.07.96 02.01.97 02.07.97 date questions about exotics in source: Bundesbank general 4. Types of barriers

In the currency market barrier options a) American vs. European - became popular in 1994. The exchange Traditionally barrier options are of rate between USD and DEM was then American style, which means that between 1.50 and 1.70. Since the all the barrier level is active during time low before 1995 was 1.3870 at the entire duration of the option: September 2 1992 there were a lot of any time between today and down and out barrier contracts written maturity the spot hits the barrier, with a lower knock-out barrier of the option becomes worthless. If 1.3800. The sudden fall of the US the barrier level is only active at Dollar in the beginning of 1995 came maturity the barrier option is of unexpected and the 1.3800 barrier was European style and can in fact be hit at 10:30 am on March 29 1995 and replicated by a and fell even more to its all time low of a digital option. 1.3500 at 9:30 am on April 19 1995. b) Single, double and outside Numerous barrier option holders were barriers - Instead of taking just a shocked that loosing the entire option lower or an upper barrier one is something that can really happen. could have both if one feels sure The shock lasted for more than a year about the spot to remain in a range and barrier options had been unpopular for a while. In this case besides for a while until many market vanillas, constant payoffs at participants had forgotten the event. maturity are popular, they are Events like this often let the question called range binaries. If the about using exotics arise. Complicated barrier and strike are in different products can in fact lead to unpleasant exchange rates, the contract is surprises. However, to cover foreign called an outside barrier option or exchange risk in an individual design double asset barrier option. Such at minimal cost requires exotic options. options traded a few years ago Often they appear as an integral part of with the strike in USD/DEM and an investment portfolio. The number of the barrier in USD/FRF taking market participants understanding the advantage of the misbalance advantages and pitfalls is growing between implied and historic steadily. correlation between the two Value of a 3 month up-and-out with a barrier currency pairs. window active only for the second month c) Regular and reverse barriers - Regular barrier options are out-of- the-money at hitting time, whereas reverse barrier options are in-the- money at that time. Loosing a reverse barrier option due to the 0.20 spot hitting the barrier is more 0.18 81 0.16 painful since the owner already 68 0.14 0.12 54 0.10 has accumulated a positive Running time in

0.08 Value 41 days 0.06 intrinsic value. Hedging a reverse 27 0.04 0.02 barrier option also causes 14 0.00

0 0.81 0.83 0.85 0.88 0.90

difficulties due to large delta and 0.93 0.95 0.98 1.00 1.03 1.06 1.09 1.12 gamma values. 1.15 Spot d) Rebates - For knock-in options an amount R is paid at by the seller of the option to the holder of the option if the option failed to kick in during its lifetime. f) Step and soft barriers - Come on, For knock-out options an amount the spot only crossed the barrier R is paid by the seller of the for a very short moment, can't you option to the holder of the option, make an exception and not let my if the option knocks out. The option knock out? This is a very payment of the rebate is either at common concern: how to get maturity or at the first time the protection against price spikes. barrier is hit. Including such rebate Such a protection is certainly features makes hedging easier for possible, but surely has its price. reverse barrier options and serves One way is to measure the time as a consolation for the holder's the spot spends opposite the disappointment. The rebate part of knock-out barrier and let the a barrier option can be completely option knock out gradually. For separated from the barrier contract instance one could agree that the and can in fact be traded options nominal is decreased by separately, in which case it is 10 % for each day the exchange called a one-touch (digital) option rate fixing is opposite the barrier. or hit option (in the knock-out This can be done linearly or case) and no-touch option (in the exponentially. Such contracts are knock-in case). also referred to as occupation time e) Window or partial barriers - derivatives. Barriers need not be active for the g) Fluffy barriers - This is a way to entire lifetime of the option. One let a barrier option knock out can specify arbitrary time ranges gradually not depending on the with piecewise constant barrier time spent beyond the barrier but levels or even nonconstant the depth: For instance one can barriers. Linear and exponential specify a barrier range of 2.20 to barriers are useful if there is a 2.30 where the option looses 25% certain drift in the exchange rate of its nominal when 2.20 is caused, e.g., by a high interest rate breached, 50% when 2.25 is differential (high rates). breached, 75% when 2.275 is breached and 100% when 2.30 is 5. How the barrier is monitored breached. (continuous vs. discrete) and how h) Parisian and Parasian barriers - this influences the price. Another way to get price spike protection is to let the option How often and when exactly do you knock out only if the spot spends a check, whether an option has knocked certain prespecified length of time out or kicked in? This question is not opposite the barrier - either in trivial and should be clearly stated in total (Parasian) or in a row the deal. The intensity of monitoring (Parisian). Clearly the plain barrier can create any price between a option is the least expensive, standard barrier and a vanilla contract. followed by the Parasian then the The standard for barrier options is Parisian barrier option and finally continuous monitoring. Any time the the corresponding vanilla contract. exchange rate hits the barrier the option is knocked out. An alternative is Parisian vs. Parasian barrier to consider just daily/weekly/monthly

2.50 fixings which makes the knock-out option more expensive because 2.00 chances of knocking out are smaller. 1.50 Parasian 1.00 Parisian Discrete vs. Continuous Monitoring

0.50 0.70 0.00 0.60 value of the barrier option 1 2 3 4 5 10 15 20 0.50 0.40 continuous number of fixings leading to knock-out 0.30 finite fix 0.20 i) Resetable barriers - This is a way 0.10 barrier option value to give the holder of a barrier 0.00 option a chance to reset the barrier 1 2 3 7 14 30 during the life of the option n number of days between fixings times at a priori determined N times in the future (N³n). This kind of extra protection also 6. How breaching the barrier is makes the barrier option more determined expensive. j) Quanto barriers - In foreign The Foreign Exchange Committee exchange options markets option wishes to recommend to the foreign payoffs are often paid in a exchange community a new set of best currency different from the practices for the barrier options underlying currency pair. For market. In the next stage of this instance a USD/JPY call is project, the Committee is planning to designed to be paid in EUR, where publish a revision to the International the exchange rate for EUR/JPY is Currency Options Master Agreement (ICOM) User Guide to reflect the new determined a priori. Surely such 1 features can be applied to barrier recommendations. Some key features options as well. are · Name a barrier determination agent.

1 For details see http://www.ny.frb.org/fxc/fxann000217.html · Determination whether the spot has when the spot is near the barrier and breached the barrier must be due to the time is close to expiry. This is actual transactions in the foreign because the intrinsic value of the exchange markets. option jumps from a positive value to zero when the barrier is hit. In such a · Transactions must occur between situation a simple delta hedge is 5:00 A.M. Sydney time on Monday and impractical. However, there are ways 5:00 P.M. New York time on Friday. to tackle this undesirable state of affairs by moving the barrier or more · Transactions must be of systematically apply valuation subject commercial size. In liquid markets, to portfolio constraints such as limited dealers generally accept that leverage. commercial size transactions are a minimum of $3 million. Best hedge; after 21 days - 69 days left 0.15

· The barrier options determination 0.1 agent may use cross-currency rates to 0.05 determine whether a barrier has been breached in respect of a currency pair 0 vega that is not commonly quoted. -0.05 0.810 0.832 0.854 0.876 0.900 0.923 0.948 0.973 0.999 1.025 1.053 1.081 1.109 1.139 1.169 1.200

-0.1 7. Hedging methods, coping with -0.15 high delta and gamma spot Barrier ATM Vega Hedge Vanilla Sum Barrier options can be hedged statically with a portfolio of vanilla 8. How large barrier contracts options. These approaches are affect the market problematic if the hedging portfolio has to be liquidated at hitting time, Let's take the example of a reverse since volatilities for the vanillas may down-and-out put in EUR/USD with have changed between the time the strike 0.90 and barrier 0.85. An hedge is composed and the time the investment bank delta-hedging a short barrier is hit. Moreover, the position with nominal 10 Million has occasionally high nominals and low to buy 10 Million times delta EUR. As deltas can cause a high price for the the spot goes down to the barrier, delta hedge. For regular barriers a delta and becomes larger and larger requiring the vega hedge is more advisable. A vega hedging institution to buy more and hedge can be done almost statically more EUR. This can influence the using two vanilla options. In the market since steadily asking for EUR example we consider a 3-month up- slows down the spot movement and-out put with strike 1.0100 and towards the barrier and can in extreme barrier 0.9800. The vega minimising cases prevent the spot from crossing hedge consists of 0.9 short 3-month 50 the barrier. On the other hand, if the delta calls and 0.8 long 2-month 25 hedger runs out of breath or the delta calls. Spot reference for downward market movement can't be EUR/USD is 0.9400 with rates 3.05% stopped by the delta-hedging and 6.50% and volatility 11.9%. institution, then the options knocks out and the hedge is abandoned. Then Reverse barrier options have extremely suddenly fewer EUR are asked whence high values for delta, gamma and theta the downward movement of an exchange rate can be accelerated once The price of barrier options is a large barrier contract in the market influenced by a variety of factors: has knocked out. The reverse situation occurs when the a) Smile of vanilla options - take bank hedges a long position, in which the example of an upper knock-out case EUR has to be sold when the spot barrier: if low-delta calls are more approaches the barrier. This can cause expensive than at-the-money calls one an accelerated movement of the supposes that volatilities will grow exchange rate towards the barrier and a with rising spot. This in turn increases sudden halt once the barrier is the probability of knocking out and breached. hence makes an up-and-out barrier option less expensive than the 9. Difference between market theoretical value. A systematic prices and theoretical Black- approach to take the vanilla smile into Scholes values - explained account is building a Dupire-tree or equivalently solving a Black-Scholes Barrier options often do not trade at partial differential equation with a their theoretical Black-Scholes values. state- and time- dependent volatility Vanilla options face the same problem, coefficient, which ensures to match the but since their price grows monotone observed vanilla option volatilities. in the volatility, one can change the However, this approach has not volatility in the Black-Scholes formula explained the way the market prices and adjust the theoretical value to barrier options. match the market price. It turns out b) Price adjustments take the that in vega position of the barrier option into markets this volatility adjustment is account. typically positive for low delta puts c) The extreme values for delta and calls. This empirical phenomenon and gamma in a dynamic hedging is called the volatility smile. It still strategy described above can cause a allows to use the Black-Scholes further supplement to the price, formula for valuation, but one should particularly on the ask side. be aware that the formula serves as a mean of communication, and the 10. Summary actually traded underlying is the volatility. However, the monotone We have seen that barrier options are volatility-price relationship does not an extremely versatile tool to manage hold for barrier options and thus can foreign exchange exposure. Over the not be applied. years lots of refinements to simple barrier options have been invented, vanilla vs. barrier option values depending on such that market participants now use volatility various ways to comply with 0.025 customers' needs, expectations and risk profile. 0.020 ______0.015 vanilla value

value Jürgen Hakala and Uwe Wystup, 0.010 barrier value Quantitative Research, Commerzbank 0.005 Treasury and Financial Products

0.000

2% 3% 4% 5% 6% 8% 9% 10% 11% 12% volatility