Analyzing Target Redemption Forward Contracts under Lévy Process

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1 Ieraoal Research Joural of Face ad Ecoomcs ISSN Issue 65 Jauary, 08 hp:// Aalyzg Targe Redempo Forward Coracs uder Lévy Process Jerry T. Yag Assocae professor he Deparme of Face Naoal Ued Uversy, Maol,Tawa E-mal: Szu-Lag Lao Professor he Deparme of Moey ad Bakg Naoal Chegch Uversy,, Tape, Tawa E-mal: Ju-Home Che Correspodg Auhor, Ph.D. of he Deparme of Moey ad Bakg Naoal Chegch Uversy, Tape, Tawa E-mal: Absrac The deprecao of he remb (RMB) he las few years had caused may defaul eves o he leveraged srucural producs called Targe Redempo Forward (TRF). Aalyzg he compoes of he TRF, we ca fd hese producs are composed of buyg ad sellg exchage opos. From he emprcal aalyses of he reurs of he exchage rae of USD/CNY, here exs o-ormal, lepokurc ad volaly cluserg pheomea. Hece, we use he me-chaged NIG-Lévy process o cosruc he dyamcs of he exchage rae.fally, we apply he Moe Carlo smulao echque o prce he TRF ad aalyze he mpacs of he clauses he erm shee of TRF. Keywords: Targe Redempo Forward, volaly cluserg, NIG-Lévy, Moe Carlo smulao. JEL Classfcao: F3, G, G3. Iroduco Targe redempo forward coracs (TRF) deomaed remb (RMB) have bee que popular due o he expeced apprecao of he RMB. However, he RMBdeprecao has led vesors o suffer sgfca losses o TRF, because of heslowg dow of GDP growh of Chaad he mpac of he Brex. A he same me, he baks Tawa wh hgher exposures o TRF face hgher defaul rsk from her cles due o he sharp deprecao of RMB durg 05 ad 06.Hece, he exposure corol seems exremely mpora for he baks. Alhough he exposure ca be mgaed hrough posg collaerals or marg requreme from he cles whle her losses over her cred le gve by he bak, he amou of collaeral or marg should be posed basg o he far value calculaed hrough model or o quoao from he marke. Neverheless, mos of baks sll rely o he quoao from he couerpary, especally for complcaed srucure dervaves such as TRF. Whle a

2 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08) 69 uexpeced eve (such as he Brex) caused he RMB agas US dollar o deprecaead resuled sgfca losses o cles TRF posos, he baks whou he ably of calculag he prce of he TRF ad relyg o he quoao from he couerpary cao mmedaely ask he cles o pos collaerals or marg requreme ad evaluae he mpac of he RMB deprecao. I radoal sudes, he dyamc of he asse prces are assumed o follow a geomerc Browa moo (GBM) proposed by Samuelso (965) ad asse reurs are assumed o be ormally dsrbued. From he emprcal leraure, he asse reurs are o ruly ormally dsrbued bu usually wh fa als ad excess kuross (Madelbro, 963; Grav, 000) caused by jumps or volaly cluserg pheomea. Hece, someresearches specfed he jump compoe he GBM for prcg opos ((Mero, 976, Am, 993, ad Kou, 00) ad ohers used he model wh sochasc volales o ehace he dmesos of he asse processes(bardorff-nelse ad Shephard, 00; Bardorff-Nelse, Ncolao ad Shephard, 00). Recely, helévy models are wdely dscussed he valuao of dervave secures (Hsu ad Che, 0; Orhaala, 04; Fajardo, 05; Jag e al., 06) o modfy he ormal dsrbuo assumpo of he GBM. Uder he Lévy model, he jump ad sochasc volales of he asse reurs ca all be ake o cosderao ad he pheomea of lepokurc, fa al ad skew of asse reursca also be depced smulaeously. Furhermore, he asse reurs Lévy model wh some specal specfcao ca be assumed o follow may ypes of o-ormal dsrbuos, such as NIG dsrbuo ad VG dsrbuo. I addo, some scholars foud ha modelg he asse reur wh Lévy model s beer ha usg he GBM (Schoues, 003). From Fgure, we ca fd he reurs of USD/CNY exchage rae s o ormally dsrbued bu wh he pheomea of lepokurc, jump ad volaly cluserg. Hece, we use me-chaged Lévy process o model he dyamc of exchage rae. The advaage of me-chaged Lévy process les depcg he jump ad volaly cluserg codos of asse reursa he same me (Carr ad Wu, 004). Fgure : USD/CNY Exchage Rae Reur The remader of hs sudy s orgazed as follows. The ex seco roduces he srucure of TRF ad he marke codo of USD/ CNY exchage rae. Seco 3 preses he dyamcs of he model. Seco 4 provdes he umercal expermes ad he comparso of TRF coracs. The fal seco shows he cocluso of hs sudy. Daa perod of USD/CNY s 05//~05//3.

3 70 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08). The Srucure of TRF The deprecao of he RMB caused may defaul eves o he leveraged producs called TRF. Aalyzg he compoes of he TRF, we ca fd hese producs are composed of buyg ad sellg exchage opos. There are several kds of TRF coracs he marke ad maly are composed of codos of coracs beg ermaed ad kockg (Europea Kock I, EKI). The ermaed codos ca also be dvded o hree ypes: () ermag whle he accumulag pos exceeds he kock ou po descrbed he corac; () ermag whle he accumulag umber exceeds he kock ou umber descrbed he corac; (3) ermag whle he exchage rae ouches he kock ou prce (Dscree Kock Ou, DKO). I he ssuers of TRF aspecs, hese ermaed clauses beg favor of hem proecg he ssuers from losg much more moey. O he corary, he seg of he kock prce s favor of he vesors. The vesors of TRF suffer he loss oly whe he fxg rae of he exchage rae exceeds he kock prce. Hece, he combao of ermaed codos ad kock codos ca geerae several kds of TRF coracs such as TRF, TRF wh EKI, TRF wh DKO ad TRF wh EKI ad DKO. Several kds of TRF coracs are summarzed Table o Table 4. Table : Type corac of TRF Type TRF There are seles hese coacs ad ooal s N. Whle he fxg exchage rae Q( ) s below he srke prce K,he vesor makes a prof ad ge (K - Q( )) N ad sar o cou he accumulagmes or po ha he vesor ears. The corac wll be ermaed whle he accumulag mes ad po ouches he po ad mes(targe) se he coracs. Whle he fxg exchage rae Q( ) s over he srke prce K, he vesor loses (K - Q( )) N Leverage. N Payoff = N [ K - Q( )] [ K - Q( )] f Q( ) K ad Leverage = max(k Q( f Q( ) > K ad = ),0) < Targe max(k Q( ),0) < Targe

4 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08) 7 Table : Type corac of TRF Type TRF wh EKI There are seles hese coacs ad ooal s N. Whle he fxg exchage rae Q( ) s below he srke prce K, he vesor makes a prof ad ge (K - Q( )) N ad sar o cou he accumulag mes or po ha he vesor ears. The corac wll be ermaed whle he accumulag mes ad po ouches he po ad mes (Targe) se he coracs. Whle he fxg exchage rae Q( ) s over he kock prce (EKI), he vesor loses (K - Q( )) N Leverage. N Payoff = N [ K - Q( )] [ K - Q( )] f Q( ) K ad = Leverage f Q( ) > EKIad max(k Q( ),0) < Targe = max(k Q( ),0) < Targe Table 3: Type 3 corac of TRF Type 3 TRF wh DKO There are seles hese coacs ad ooal s N. Whle he fxg exchage rae Q( ) s below he srke prce K, he vesor makes a prof ad ge (K - Q( )) N ad sar o cou he accumulag mes or po ha he vesor ears. The corac wll be ermaed whle he accumulag mes ad po ouches he po ad mes (Targe) se he coracs. Whle he fxg exchage rae Q( ) s over he srke prce K, he vesor loses (K - Q( )) N Leverage. I addo, whle he fxg exchage rae Q( ) s below he DKO prce, he he corac s ermaed. K - Q( ) f DKO < Q( ) K ad max(k Q( ),0) < Targe = Payoff = (K - Q( )) Leverage f Q( ) > K ad < max(k Q( ),0) Targe =

5 7 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08) Table 4: Type 4 corac of TRF Type 4 TRF wh EKI ad DKO There are seles hese coacs ad ooal s N. Whle he fxg exchage rae Q( ) s below he srke prce K, he vesor makes a prof ad ge (K - Q( )) N ad sar o cou he accumulag mes r or po ha he vesor ears. The corac wll be ermaed whle he accumulag mes ad po ouches he po ad mes (Targe) se he coracs. Whle he fxg exchage rae Q( ) s over he kock prce (EKI), he vesor loses (K - Q( )) N Leverage. I addo, whle he fxg exchage rae Q( ) s below he DKO prce, he he corac s ermaed. K - Q( ) f DKO < Q( ) K ad max(k Q( ),0) < Targe = 報報 = (K - Q( )) Leverage f Q( ) > EKI ad < max(k Q( ),0) Targe = 3. The Model ad Parameer Esmaos 3. The Specfcao of he Dyamcs of Exchage Rae I he radoal sudes, he asse prce s always assumed o follow a GBM model proposed by Samuelso (965); ha s: where ds S = µ d +σ dw () µ : he saaeously expeced asse reur, σ : he saaeously sadard devao of asse reur, ad W : he sadard Browa moo of uderlyg asse. We ca model he asse prce S uder rsk-eural measure as follows: S = S exp( r + σw ϖ ), () 0 GBM where ϖ GBM = σ, deoed as he compesaor erm of GBM model whch esures ha he r expecao of S e s a margale uder he rsk-eural measure. Uder GBM model, he asse reur s ormally dsrbued. However, as meoedprevously he dsrbuo of asse reur has fa als ad excess kuross. I he applcao of Lévy process, he kuross ad skewess of he asse reur could be depced he characer of a fe dvsble

6 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08) 73 dsrbuo. Hece, we use he Lévy model o geerae he exchage rae. Le ad s cumulabe ψ (u). Accordg o Lévy-Khche formula, we have ψ u) = a u σ u + + (exp( ux) ux { < }) Π( dx) x, ( X be a Lévy process where, a R, whch deoes a lear deermsc par of X ; σ 0, whch deoes a Browa par of X ; ad Π(dx) s he Lévy measure, whch deoes a pure jump par of X. A rple of Lévy characerscs [ a, σ, Π( dx)] s used o deoe he fely dvsble dsrbuo of Lévy process. Specfcally, gve he specfcaos rple [ a, σ, Π( dx)], oe ca deerme he dsrbuo ype of X. The, he lévy measure of he NIG s he form of β x Π NIG( x; α, β, δ ) = π δα x K( α x ) e (Bardorff-Nelse,998). The TRF ca be deermed based o he smulao of exchage raeq. I Lévy model, he exchage rae uder rsk-eural measure s descrbed as follows: Q = Q0 exp( rd rf + X ϖ ), (4) ( rd r f ) whereϖ s he compesaor erm of Lévy model whch esures he Qe s a margale uder he rskeural measure. I order o geerae he exchage rae ad o deerme he value of ϖ, we assume he Lévy process X follows he specfc dsrbuo forms; ha s he NIG dsrbuo. If X follows NIG dsrbuo (deoed as NIG X ), he we have = µ + β z( ) + z( ) W, (5) X NIG where µ s a locao parameer; β s he skewess parameer; z () follows Iverse Gaussa dsrbuo, deoed as IG( δ, α β ) 3, α, β ad δ are he parameers of NIG dsrbuo ;α (α s he shape parameer). I such case, he compesaor of NIG model s descrbed as follows: (3) ϖ = µ + δγ δ α + ( β ), (6) whereδ s he excess kuross parameer, γ = α β. Based o prevous descrpos, we ca smulae he process of exchage rae NIG dsrbuos. The smulao procedure s descrbed as follows: () we radomly selec values of W, z () ad G () from he dsrbuo of N (0,), IG ( δ, α β ) ad G ( / v, v), respecvely; () we NIG oba he value of X based o equaos (5). (3) we ca calculae he value of ϖ accordg o equao (6). (4) from Equao (6), we ca ge a smulao pah of he exchage rae uder he assumpo of NIG. Wh he smulao pahs of he exchage rae, he we ca calculae dffere ypes of TRF values. ψ ( u) = log( φ( u)), where φ(u) s he characersc fuco of X. 3 Wh γ = α β, he he desy of he IG s he form as IG f ( x) = δ 3/ γ δ x exp( ( x ) ). π x γ

7 74 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08) The Mehod of Parameer Esmaos I hs subseco, we use he maxmum lkelhood mehod o esmae he parameers ofnig dsrbuo. The probably desy fuco of NIG dsrbuo s descrbed as follows: αδ f NIG ( x; α, β, δ, µ ) = exp( δ α β + β ( x µ )), (7) π K ( α δ + ( x µ ) ) δ + ( x µ ) where x R, δ > 0, α > 0, 0 β α ; Ks he modfed Bessel fuco of he hrd kd, we have x ( ) exp( ) 4 x K x = + d, (8) 0 4 The maxmum log lkelhood fuco of NIG dsrbuo s as follows: max l L( α, β, δ, µ x, x,..., x ) = max ( log δ α β β ( x µ ) θ π = θ = αδ log( K( α δ + ( x µ )) log( δ + ( x µ ) (9). Valuao ad Comparso of he TRF Coracs There does o exs closed form soluo for evaluag he TRF, because of early ermao clauses depedg o he accumulaed pos or mes he prevous seleme dae. Therefore, we use he Moe Carlo smulao mehod ogeerae 300,000 pahs for prcg dffere ypes of TRF coracs. The daa perod for parameers esmao les bewee 05// ad 05//3 ad s colleced from he Tawa Ecoomc Joural (TEJ). From he sascal summary preseed Table 4, we ca fd he reurs of USD/CNY exchage rae exs skew ad lepokurc pheomea. The oe year rsk free rae for he RMB ad USD are.34% ad 0.45%, respecvely. The coracs clauses of he dffere ypes of TRF are preseed Table 6 ad he fxg schedule s show Table 7.Table 5 shows he esmao resuls of parameers of he NIG model. The exchage rae of USD/CNY 06// was From he Table 8, we ca fd he value of he TRF wh EKI clause s hgher ha he TRF whou EKI clause due o he EKI clause s se for proecg he vesor from losg much more moey. O he corary, he DKO clause s se favor of he ssuer. Hece, he value of TRF wh DKO clause s much less valuable ha he res of he ypes of TRF coracs. I addo, he leverage also has grea fluece o he value of TRF corac. The larger he leverage s, he much more moey he vesor wll lose whle he exchage rae exceeds he EKI. From he clauses of he TRF coracs, we also ca fd here s o proeco effec for he vesor whle he EKI codo s se very close o spo prce of he exchage rae. O he corary, f he DKO codo was se far from he spo prce, here sll s o proeco effec for he ssuer. Furhermore, from Fgure o Fgure 4, we ca fd he cash flow each fxg perod s he paer of decayg. Ths meas he vesor wll ear moey he al sage of he corac durao bu wll lose much more moey he remag sage ll he maury of he corac due o he leverage effec. Ths ype of cash flow s very smlar o hecash flow of aoher ype of facal srume amed as accumulaor. Hece, whle vesg such complcaed facal producs, he vesor should kow he characerscs of he produc ad he red of exchage rae. If he exchage rae s o he oppose sde as he vesor predc, he he vesor wll lose a grea amou of moey.

8 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08) 75 Table 4: Summary of sascs of reur of USD/CNY Currecy USD/CNY Number of observaos 360 Mea.7E-04 Sadard devao.47e-03 Skewess Kuross Table 5: Parameers esmao resuls Parameer Value of esmao(sadard devao) α (49.480) β ( ) µ.7e-06(.e-05) δ.9e-04(.59e-05) Table 6: Coracs clauses of dffere ype of TRF Type of TRF TRF TRF wh EKI TRF wh DKO TRF wh EKI ad DKO Nooal Mllo Mllo Mllo Mllo Srke Targe Po EKI DKO Leverage Table 7: Fxg schedule of TRF coracs Fxg dae Delvery Dae 06//3 06// 06//8 06/3/ 3 06/3/3 06/4/ 4 06/4/30 06/5/ 5 06/5/3 06/6/ 6 06/6/30 06/7/ 7 06/7/3 06/8/ 8 06/8/3 06/9/ 9 06/9/30 06/0/ 0 06/0/3 06// 06//30 06// 06//3 07// Table 8: Prcg resuls of dffere ypes of TRF TRF TRF Wh EKI(6.7) TRF Wh DKO(6.3) TRF Wh EKI(6.7) ad DKO(6.3) -40, , , ,44.70

9 76 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08) Fgure : Value comparso of TRF ad TRF wh EKI coracs TRF VS TRF wh EKI TRF TRF wheki Fgure 3: Value comparso of TRF wh EKI ad TRF wh DKO coracs TRF wh EKI VS TRF wh DKO TRF wheki TRF whdko

10 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08) 77 Fgure 4: Value comparso of TRF wh DKO ad TRF wh EKI ad DKO coracs TRF wh DKO VS TRF wh EKI ad DKO TRF whdko TRF wh EKI ad DKO Coclusos The TRF coracs are maly composed of wo mechasms: () proeco clause; () kock ou clause, ad combed wh dffere seleme perods ad leverage. Alhough he clauses of TRF coracs are se wh proeco prce (EKI), he vesor wll lose moey very soo whle he EKI codo se very close o he spo prce. I addo, he leverage he coracs s almos se as, such case, f he exchage rae exceeds he EKI, he vesor wll sele he deal wh mes ooal, ad hs wll make he vesor accumulae los of loss shor erm rapdly. Therefore, vesg he TRF, he vesor should kow hgher EKI ad lower leverage clauses are much more good sake for hm whle he predcs he currecy agas USD dollar wll apprecae. Furhermore, o forcas he red of he exchage rae, he vesor should also cosder he model whch ca more compleely capure he characerscs of exchage rae reur. Hece, we use he NIG me-chaged Lévy o model he dyamcs of he exchage rae o depc he o-ormal, jump ad volaly cluserg pheomea hs sudy. Refereces [] Am, K. I., 993, Jump Dffuso Opo Valuao Dscree Tme, Joural of Face, 48, [] Baes, D. S., 996, Jumps ad sochasc volaly: exchage rae processes mplc deusche mark opos, The Revew of Facal Sudes, Vol. 9, Issue, p [3] Bardorff-Nelse, O. E ad N. Shephard, 00, No-Gaussa Orse-Uhlebeck-Based Models ad Some of Ther Uses Facal Ecoomcs, Joural of Royal Sascal Socey, 63, [4] Bardorff-Nelse, O. E ad E. Ncolao ad N. Shephard, 00, Some Rece Developmes Sochasc Volaly Modelg, Quaave Face,, -3. [5] Carr, P., & Wu, L., 004. Tme-chaged Lévy processes ad opo prcg. Joural of Facal ecoomcs, 7(), 3-4.

11 78 Ieraoal Research Joural of Face ad Ecoomcs - Issue 65 (08) [6] Fajardo, J., 05, Barrer syle coracs uder Lévy processes: A alerave approach. J. Bak. Face 53, [7] Garv, J., 000, Exreme Value Theory- A Emprcal Aalyss of Equy Rsk, Quaave Rsk: Model ad Sascs, UBS Warburg. [8] Heso, S., 993, A closed-form soluos for opos wh sochasc volaly, Revew of Facal Sudes, Vol. 6, p [9] Hsu, P.P., Che, Y.H., 0 Barrer opo prcg for exchage raes uder he Levy HJM processes. Face Res. Le. 9 (3), [0] Jag, G., Xu, C., & Fu, M. C., 06,O sample average approxmao algorhms for deermg he opmal mporace samplg parameers prcg facal dervaves o Lévy processes. Operaos Research Leers, 44(), [] Kou, S. G., 00, A Jump-dffuso Model for Opo Prcg, Maageme Scece, 48, pp [] Madelbro, B., 963, The Varao of Cera Speculave prces, Joural of Busess, Vol. 45, No. 4, pp [3] Mero, R. C., 976, Opo Prcg Whe he Uderlyg Sock Reurs Are Dscouous, Joural of Facal Ecoomcs, 3, [4] Orhaala, C., 04, Levy jump rsk: evdece from opos ad reurs. J. Fac. Eco. (), [5] Samuelsom, P. A., 965, Proof Tha Properly Acpaed Prces Flucuae Radomly, Idusral Maageme Revew, Vol. 6, No., pp [6] Schoues, W., 003, Lévy Process Face: Prcg Facal Dervaves, Belgum: Joh Wley ad Sos.