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Group V: Hervé Fadom Tchomgouo Fracisco J. Gil Bartlomiej Bartoszewicz Västerås 3 Table of cotets Table of cotets.... Itroductio Goal Defiitio of Exotic Optio Types of Exotic Optios...

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Group V: Hervé Fadom Tchomgouo Fracisco J. Gil Bartlomiej Bartoszewicz Västerås 3 Table of cotets Table of cotets.... Itroductio Goal Defiitio of Exotic Optio Types of Exotic Optios Why to write Exotic Optios? Why to buy Exotic Optios? Problems Cocerig Exotic Optios Optios Uder Thorough Ivestigatio Double Barrier Call Optio Itroductio Advatages of barrier optios Disadvatages of barrier optios Call-up-ad-out-dow-ad-out pricig formula Call-up-ad-out-dow-ad-out properties Summary Refereces Iteret Liks Asia Call Optio Itroductio Defiitio Pricig Formulae Geometric Closed Form (Kemma & Vorst) Arithmetic Rate Approximatio (Turbull & Wakema) Arithmetic Rate Approximatio (Levy) Arithmetic Rate Approximatio (Curra) Arithmetic Rate Approximatio (Mote Carlo Simulatio) Characteristics Of Asia Optios Pro & Cos Compariso of a Asia Optio ad a Vailla Call Applicatios Of Asia Optios Summary Ad Coclusio Refereces Iteret liks Iterest Rate Collar Defiitio What is a Cap? What is a Floor? Pricig Formulae Features Who uses Iterest Rate Collars? How does a Iterest Rate Collar work? Are there ay risks associated with a Iterest Rate Collar? How much does a Iterest Rate Collar cost? Example Coclusios Refereces Iteret liks Table of Figures... 4 . Itroductio Hervé Fadom Tchomgouo, Fracisco J. Gil, Bartlomiej Bartoszewicz. Goal The goal of this report is to give aswers to four followig questios: What is a Exotic Optio? What sorts of Exotics are traded? Why are these optios attractive? How to price several chose Exotic Optios?. Defiitio of Exotic Optio A Exotic Optio is a more complex cotract tha simple Europea or America call or put optio o stock, idex, foreig currecy, commodity or iterest rate. Exotic optios are the secod geeratio of optios. They have key terms differet from or additioal to those foud i Vailla (o-exotic) optios..3 Types of Exotic Optios Exotic optios iclude: Quato optios Forward start optios Compoud optios Chooser optios Barrier optios Biary optios Lookback optios Shout optios Asia optios Optios to exchage oe asset for aother Optios ivolvig several assets.4 Why to write Exotic Optios? Exotic optios offer the writer the opportuity to explore wider bid-offer spread. This is due to the fact that ot may fiacial istitutios trade exotics ad the competitio o the market is ot as strog as o Vailla optio market. This fact makes also possible for the 3 writer to maitai higher profit margi. Aother good reaso for tradig exotic optios is that they are cosidered as a sophisticated extesio to Vailla optios..5 Why to buy Exotic Optios? The mai reaso for buyig exotic optios is that they offer a tailor-made protectio for a moderate price. A trader that has a view of decliig volatility ca employ a barrier optio istead of strategy based o vailla optios sice this solutio is less expesive. Exotic optios also offer structured protectio whe Vailla optios ca t be successfully employed. Cosider a compay that has reveues i may foreig currecies. This compay is exposed to exchage rate risk i may currecies. The profits of this compay ca be protected agaist large movemets of exchage rates be a very costly strategy of buyig put optios o each of these currecies. Istead, a exotic optio o a basket of foreig currecies might be cosidered. This exotic optio offers protectio agaist large movemets of the whole basket of currecies which actually is the case here because the profits of compay uder ivestigatio deped o a joi behavior of all the currecies..6 Problems Cocerig Exotic Optios Although exotic optios have substatial advatages, they also have drawbacks. Oe of them is low liquidity o some exotic optio markets. This might make difficult or eve impossible to buy or sell sufficiet amout of exotic optios to hedge ivestor s portfolio. This might ifluece the markig-to-market process whe rehedges have to be doe. Aother disadvatage of exotic optios is that uderlyig market might become maipulative if large amouts of exotic optios are traded ad approach maturity. I some cases the writer might for istace try to kill the barrier optios o less liquid uderlyig market if this would protect him agaist large loss whe optio expires i-the-moey..7 Optios Uder Thorough Ivestigatio I this report three exotic optios will be uder thorough ivestigatio: Double Barrier Optio - Call-up-ad-out-dow-ad-out Arithmetic Average-Rate Asia Call Optio Iterest Rate Collar 4 . Double Barrier Call Optio Bartlomiej Bartoszewicz. Itroductio Barrier optios are oe of the most widely-traded exotics o some markets. Whe compared to vailla optios, they have oe additioal key term: a barrier imposed o price of uderlyig. The barrier might be below the strike or above the strike. Whe barrier is hit durig the life of the optio, a evet occurs. There are two kids of such evets: the cotract might be cacelled or the cotract might become effective. Hece there are four basic types of barrier optios: dow-ad-out the barrier is below the strike price; oce it is hit, the optio dies ad at maturity there is o payout for optio holder, although the optio might be ithe-moey at maturity; dow-ad-i the barrier is also below the strike price, but this time the cotract becomes alive whe the barrier is hit at maturity optio holder gets usual payout if ad oly if the barrier is hit durig the life of the optio; if the barrier is ot hit, the optio expires worthless, although it might be i-the-moey at maturity; up-ad-out this time the barrier is above the strike price; oce it is hit, the cotract becomes ulled there is o payout for optio holder at maturity, o matter if the optio is i-the-moey or out-of-the-moey; up-ad-i the barrier is also above the strike price; optio holder gets usual payout if ad oly if the stock price hits the barrier durig the life of the optio. These four basic features of barrier optios apply to both call ad put optios, as well as to Europea ad America optios. The uderlyig of a barrier optio might be a stock, stock idex, commodity, foreig currecy or iterest rate. Barrier optios are clearly path-depedet. The payout at maturity depeds ot oly o the price of uderlyig at maturity, but also o the way the price of uderlyig got to this particular level. There are other possible features that ca be applied to four basic types of barrier optios. The optio might be said to have two barriers istead of oe. There are two basic types of double barrier optios: up-ad-put-dow-ad-out up-ad-i-dow-ad-i 5 For the former type the cotract dies if either lower or upper barrier is hit durig the life of the optio. For the latter type the optio becomes effective if ad oly if either lower or upper barrier is hit. Aother feature that occurs with barrier optios is a rebate. Barrier optio with a rebate pays certai amout of moey to optio holder oce the barrier is reached. This is ofte the case for out optios although the cotract dies, the optio holder receives rebate to reduce his loss o losig the payout. The rebate might be paid as soo as the barrier is reached or later, eve at maturity. Sice double barrier optios metioed above are becomig more ad more popular, they are ot cosidered to be complicated ay more. For ivestors keep seekig for features that make the cotract complicated, there have bee itroduced double barrier optios with repeatig hittig feature. I this case optio is cacelled (or becomes alive) oce both barriers are hit before maturity. Raibow barrier optios are aother example of more complicated barrier optios. Additioal feature i this case is such that the barrier is imposed o oe uderlyig, but the payout at maturity is calculated o the basis of aother uderlyig. Soft barrier optios allow the cotract to be gradually kocked-out or kocked-i. The barrier i such a case is split ito two levels. Cosider as a example up-ad-out optio. Oce the lower part of the barrier is reached, the cotracts starts to become worthless, but it is doe proportioally as log as the upper part of the barrier is ot reached. If, o oe had, the maximum price of the uderlyig lies somewhere betwee lower ad upper part of the barrier, the optio is kocked-out proportioally to how deep the price wet ito the iterval. If, o the other had, the maximum price is above upper part of the barrier, the cotract is kockedout i %. The last type of barrier optios which are worth metioig are Parisia optios. The differece betwee plai barrier optios ad Parisia optios lies i for how log the uderlyig price is above (for up optios) or below (for dow optios) the barrier. I case of plai barrier optios it is sufficiet that the barrier is reached. I case of Parisia optios the price should stay beyod the barrier for a specified i advace time. O oe had this feature makes the cotract less tractable to maipulatio of uderlyig price close to the barrier ad also makes the dyamic hedgig easier. O the other had pricig of Parisia optios is far more complicated due to higher dimesioality of the problem. 6 . Advatages of barrier optios The most frequetly metioed advatage of barrier optios is that they are cheaper whe compared to vailla optios with the same strikes ad maturities. This is due to the fact that barrier optio ca ever perform better the vailla optio, uless rebate paid to optio holder whe the barrier is hit is very large. I fact barrier optios holder retais much more risk of future uderlyig price behavior tha vailla optio holder. Aother advatage of barrier optios is their flexibility i terms of settig the level of the barrier ad thus the cost of the cotract. The closer the barrier is to the strike, the higher is the probability that the barrier is reached ad the lower is the price of the cotract (for out optios). This makes it possible to adjust the terms of barrier optio to meet every particular ivestor s requiremets. The barrier optios ca be writte o every uderlyig stocks, stocks idices, commodities, iterest rates ad (especially popular) foreig currecies..3 Disadvatages of barrier optios It is ot a surprise that barrier optios have also shortcomigs. Oe of the most importace drawback is that the delta of the out optios is very ustable whe uderlyig price is close to the barrier. I fact, delta ca easily become egative whe uderlyig approaches the barrier. Gamma is also very large ad egative close to the barrier. This fact makes delta hedgig very difficult as well as very costly, especially whe the ivestor has to buy the uderlyig istead of sellig it. Aother disadvatage is that the cotract ca be killed o purpose by the writer close to maturity by makig the uderlyig price reach the barrier for a very short time..4 Call-up-ad-out-dow-ad-out pricig formula Barrier optio that will be examied i details i our report is double barrier call. The cotract has two barriers: oe upper ad oe lower to the strike price. Both barriers are out barriers oce a barrier is reached, the optio dies. Although it is easy to itroduce barriers depedet o time, to make the example easier both barriers are flat. Basic calculatios are made for a at-the-moey optio o o-divided payig stock curretly pricig dollars. Barriers are set to $8 ad $3. The time to maturity is oe quarter of a year (T=.5). The risk free iterest rate is assumed to be %. There are o costs of carry. The volatility of stock price is assumed to be 4% aually. No rebate is paid to optio holder whe a barrier is reached. 7 The value of the optio is determied by the formula [see Haug (998)]: rt T r b T d N T d N S U L T d N T d N S L L U Xe d N d N S U L d N d N S L L U Se c where: T Ue F b b T T b FSU L d T T b XSU L d T T b FL SU d T T b XL SU d l l l l ad S is curret uderlyig price, X is a strike price, U is upper barrier, L is lower barrier, r is risk-free iterest rate, b is cost of carry, T is time to maturity (i years), is aualized volatility, N(.) deotes cumulative distributio fuctio of stadardized Normal distributio ad determie the curvature of the lower barrier L ad the upper barrier U such that:, correspods to two flat barriers; correspods to a lower barrier expoetially growig with time, while the upper barrier will be expoetially decreasig; correspods to covex dowward lower barrier ad covex upward upper barrier. The value of the double barrier call optio is expressed as a ifiite series of weighted ormal distributio fuctios. However, the covergece of the formula is quite rapid. Haug (998) suggests that i most cases it is sufficiet to calculate oly two of three leadig terms. I our calculatios from to was used. 8 Importat issue cocerig pricig of barrier optios is how ofte is the uderlyig price moitored. Preseted model assumes that the price is moitored cotiuously. However, this is ot the case i practice. I practice the uderlyig price might be moitored hourly, daily, weekly or eve mothly. I this report it is assumed that the uderlyig price is moitored daily..5 Call-up-ad-out-dow-ad-out properties Usig the formula show above, the value of call-up-ad-out-dow-ad-out cotract assumig daily moitorig is $3.5. If cotiuous moitorig is employed, the value of the optio drops to $.8 due to higher probability of reachig either upper or lower barrier. Correspodig plai vailla call optio (which ca be thought of as a double barrier optio with ifiite iterval betwee poits whe the price is moitored) has a value of $7.77. Figure. shows the value of double barrier call optio as a fuctio of stock price at time T=.5. Betwee the barriers the cotract has a positive value with a peak value of $3.5 at $7. Below the lower barrier as well as above the upper barrier the optio value is zero. Optio Value 4 3,5 3,5,5, Asset Price Call up-ud-out-dow-ad-out Fig.. The value of call-up-ad-out-dow-ad-out optio as a fuctio of uderlyig price. Figure. shows the value of double barrier call as a fuctio of stock price, but as a compariso the value of plai vailla call optio is also give. It ca be see that double barrier optio is cheaper for all asset prices. The time to maturity is T=.5. 9 Figures 3. ad 4. give optio s delta ad gamma, respectively. The time to maturity is T=.5. For asset prices higher that $7 delta is egative, but it still takes values close to zero. Gamma is egative ad takes values close to zero. Util ow othig uusual has happeed, but as the maturity is approached, the value of the cotract as well as delta ad gamma start to behave i a strage maer. For asset prices sigificatly lower tha the upper barrier the call-up-ad-out-dow-ad-out optio behaves like plai vailla call, but close to upper barrier the value of double barrier optio suddely drops, due to very high probability that the barrier is reached ad optio expires worthless (see Fig. 5.). Optio Value Asset Price Call up-ud-out-dow-ad-out Vailla Call Fig. The value of call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of asset price. Delta,,5,,5 -, , -,5 -, -,5 Asset Price Call up-ud-out-dow-ad-out Fig. 3. Delta for call-up-ad-out-dow-ad-out optio at T=.5. Sice optio s value decreases, delta becomes egative ad takes very low values. The same happes to optio s gamma. As metioed above, this makes the delta hedgig very difficult. Firstly, this is due to sudde chage of positio from short to log (or from log to short; see Fig. 6.) which forces the ivestor to buy istead of sellig (or the opposite). Secodly, delta becomes very ustable close to upper barrier ad the positio has to be rehedged very ofte (gamma takes large values, see Fig. 7.). Gamma -, ,4 -,6 -,8 -, -, -,4 -,6 Asset Price Call up-ud-out-dow-ad-out Fig. 4. Gamma for call-up-ad-out-dow-ad-out optio at T=.5. Optio Value Asset Price Call up-ud-out-dow-ad-out Fig. 5. The value of call-up-ad-out-dow-ad-out optio as a fuctio of uderlyig price. (T=/365) Delta Asset Price Call up-ud-out-dow-ad-out Fig. 6. Delta for call-up-ad-out-dow-ad-out optio at T=/365. Gamma,5 8 -, ,5 - -,5 Asset Price Call up-ud-out-dow-ad-out Fig. 7. Gamma for call-up-ad-out-dow-ad-out optio at T=/365. Optio Value Strike Price Call up-ud-out-dow-ad-out Vailla Call Fig. 8. The value of call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of strike price. Figure 8. shows value of both call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of strike price betwee the barriers while uderlyig spot price remais costat (at $). The time to maturity is T=.5. Agai, it ca clearly be see that double barrier optio is cheaper whe compared to vailla optio. Optio Value Lower Barrier Call up-ud-out-dow-ad-out Vailla Call Fig. 9. The value of call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of lower barrier. Optio Value Upper Barrier Call up-ud-out-dow-ad-out Vailla Call Fig.. The value of call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of upper barrier. Figures 9. ad. show the value of call-up-ad-out-dow-ad-out cotract as a fuctio of lower ad upper barrier, respectively. Whe lower barrier is close to strike price (ad curret spot price), the value of double barrier optio is very close to zero. As the lower barrier decreases, the value of the optio icreases. For barrier at about $8 ad lower the value of 3 the optio stabilizes at about $3.. This is due to the fact that the probability that the uderlyig price reaches the barrier is very low. The value of the call optio is determied maily by the level of upper barrier which has more ifluece o the payout at optio s maturity. As upper barrier icreases (with lower barrier costat), the value of double barrier optio coicides with the value of vailla call. O oe had, icreased upper barrier meas decreased probability of reachig it. O the other had, the higher the upper barrier, the bigger the potetial payout at maturity. Optio Value 8 6 4,5,4,3,,, Time to maturity Call up-ud-out-dow-ad-out Vailla Call Fig.. The value of call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of time to maturity. Both optios are at-the-moey. 3 5 Optio Value 5 5,5,4,3,,, Time to maturity Call up-ud-out-dow-ad-out Vailla Call Fig.. The value of call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of time to maturity. Both optios are i-the-moey, uderlyig price is close to upper barrier. 4 Figure. gives the value of both call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of time to maturity. As both optios are at-the-moey ad log before maturity, double barrier optio is much cheaper tha vailla call, but its value grows as the time to maturity decreases (theta is egative), which is ever the case for vailla optios. Close to maturity, whe the probability of reachig either lower or upper barrier is very low (optio is at-the-moey), its value coicides with the value of vailla call ad both optio expire worthless. 5 Optio Value 5 5,,4,6,8 Volatility Call up-ud-out-dow-ad-out Vailla Call Fig. 3. The value of call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of volatility. Both optios are at-the-moey. Optio Value ,,4,6,8 Volatility Call up-ud-out-dow-ad-out Vailla Call Fig. 4. The value of call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of volatility. Both optios are i-the-moey, uderlyig price is close to upper barrier. 5 Figure. shows the value of both call-up-ad-out-dow-ad-out ad vailla call optios as a fuctio of time to maturity, but this time both optios are i-the moey whereas uderlyig price is close to the upper barrier. Sice the probability of reachig upper barrier is very high, the value of double barrier optio remais low (whe compared to correspodig vailla call). As time to maturity decreases, the value of double barrier optio icreases (theta is egative). At maturity both optio expire with the same payout. Figure 3. gives the value of double barrier ad vailla call optios as a fuctio of volatility. Sice both optios are at-the-moey ad uderlyig price is relatively far from barriers, for low volatilities both optios (double barrier ad vailla) have similar values. As volatility icreases, the probability of reachig either lower or upper barrier icreases ad at some poit vega (first derivative of optio value with respect to volatility) becomes egative, which is ever the case for vailla optios. Similar result ca be see o figure 4. but this time uderly

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