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Foreign Exchange Derivatives Commerzbank AG 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 option 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 volatility = 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 vertical spread 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 call option 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 expiration 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 swap 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.
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