General Bounds for Arithmetic Asian Option Prices

Transcription

1 The Uiversiy of Ediburgh Geeral Bouds for Arihmeic Asia Opio Prices Colombia FX Opio Marke Applicaio MSc Disseraio Sude: Saiago Sozizky s Supervisor: Dr. Soirios Sabais Augus 16 h 2013 School of Mahemaics MSc Fiacial Modellig ad Opimisaio

2 Absrac This disseraio explais i deail how Albrecher e al. 2008) developed hree diere model idepe lower bouds ad oe upper comoooic boud for Europea Asia call opio prices. The mai characerisic of hese bouds is ha Albrecher e al. 2008) oly use he observable plai vailla opio prices i he marke o calculae hem which allows for he dig of saic hedgig porfolios. The coceps behid he bouds are basically Jese's iequaliy ad some properies of comoooiciy of radom vecors. This docume will also explai how o impleme hese bouds whe he marke has a ie umber of opios available, moreover, how o do i whe hese bouds do o ivolve srikes foud i he marke. Two facs will be used: he call opio price fucio is covex wih respec o srike, ad he lower ad upper bouds for a call opio suggesed by Bersimas ad Popescu 2002). Fially, as a example, oe applicaio of hese bouds i he Colombia FX opio marke will be preseed. 2

3 Ackowledgmes I would like o hak Bacolombia for is suppor durig my maser sudies ad is help providig me some Colombia marke iformaio which gave me he opporuiy o ish his docume successfully. I also wa o hak my supervisor Dr. Sabais for his guide hroughou he disseraio period. 3

4 OWN WORK DECLARATION

5 OWN WORK DECLARATION

6 Coes 1 Iroducio 7 2 The Cocep of Comoooiciy ad Radom Vecors 9 3 Model-Idepe Bouds for Asia Opio Prices Lower Bouds Upper Boud Numerical Implemeaio - The Fiie Srike Case Lower Boud Implemeaio Black-Scholes Framework Model-Idepe Framework Upper Boud Implemeaio Numerical Resuls Colombia FX Opio Marke Case Garma ad Kohlhage Pricig Formula Dela Forward Adjusme Implemeaio Lower Boud Adjusmes Power Call Opio Local Volailiy Numerical Resuls Coclusios 39 7 Refereces 41 8 Appix - Malab Codes 42 6

7 Geeral Bouds for Arihmeic Asia Opio Prices 1 Iroducio I acial markes, derivaive producs are widely used as ivesme isrumes speculaio) or hedgig ools. Oe ype of hese derivaive 1 producs are opios, which give he buyer he righ o buy or sell) he uderlyig asse whe he marke codiios are favourable o him or her o make a pro). I could be udersadable o he reader ha hese producs do o have a iiial cash ow equal o zero here is a premium) sice he fuure ivesor's cash ows will be o-egaive possible fuure icome) because he/she has he righ o exercise i or o, oherwise i will geerae a arbirage opporuiy make moey ou of ohig or free luch are some colloquial expressios which describe he idea of arbirage). Oe impora ad o easy) ask arises from his opio characerisic o-egaive): how o d he fair opio price wihou geeraig a arbirage opporuiy. Black ad Scholes 1973) iroduced a way o price hese producs, ad i has bee developed furher by ucouable auhors sice he. I is well kow ha i he case of Europea opios he oes ha oly ca be exercised i he mauriy dae) is price is deermied by E Q e rt ) f S T ) F where Q represes he risk eural measure which esures o arbirage opporuiy implyig he exisece of a replicaig porfolio), T deoes he mauriy, he pricig dae, he fucio f is he payo fucio o-egaive), S T is he price of he uderlyig asse a mauriy ad ally F is he sigma algebra geeraed by S up o. if f is deed as follows ) 1 S i K he he opio is called arihmeic Asia call. Where i has discree moiorig imes 1,..., = T, S i is he price of he uderlyig asse a each moiorig ime, K is he srike price ad x) + = max x, 0). These sors of opios ca be coveie for ivesors or hedgers because, amog ohers, he followig reasos Wilmo, 2006: 428): ˆ The Asia opio prices are less ha he equivale plai vailla call opios sice he average price volailiy is less ha he volailiy of he price iself. ˆ The average is much harder o maipulae ha he price of he uderlyig asse i a sigle day. I ca be releva whe he marke is illiquid, ad herefore, easily maipulaed. 1 The erm derivaive is used due o is price deps o oe uderlyig asse price. The price of his produc is derived from aoher asse price. + 7

8 I ca be oiced ha he Asia opio prices do o dep o he asse price probabiliy disribuio fucio a S T, uder he risk eural measure, bu i deps o he joi probabiliy disribuio of he prices a each moiorig ime 1,...,. This pah depecy characerisic makes i harder o price his kid of produc, eve i he Black-Scholes frame, here is o closed formula for his payo. Besides, i could be much more dicul if i is ake io accou ha acial markes do o follow he BS framework 2 sice he marke recogizes he exisece of asymmery ad fa ails i he asse price probabiliy disribuio, or eve jumps, i he asse price process via he volailiy surface. Cosequely, he risk eural measure is o uique icomplee marke). So, i becomes harder o deermie which is he marke's risk eural measure, or he dyamics ha he asse price follows, which is a exra diculy o price opios i geeral. I order o clarify he volailiy surface, a brief descripio is provided. Black ad Scholes 1973) published he derivaio of he call opio price formula, where i was assumed, for isace, cosa volailiy, which is a srog assumpio sice i has bee widely sudied, ad cocluded, ha acial asse prices do o mee his codiio Heso, 1993). I his maer, Heso 1993) meioed ha here is a correlaio bewee volailiy ad spo price, ad i is he source of skewess i he desiy probabiliy fucio which aecs more he prices of he i-he-moey 3 regio ha ha he ou-of-he-moey oe. I addiio, o-cosa volailiy explais he excess of kurosis as well Heso, 1993). I fac, he marke opio quoaio ries o icorporae his feaure io he prices modifyig he volailiy srucure of he Black-Scholes model. This srucure is called volailiy surface. Assumig he exisece of he volailiy surface implies ha he marke does o follow Black-Scholes framework, ad probably, i has a o-cosa volailiy, he he marke is icomplee ad he ivesors are akig a model risk hey do o kow which he real asse price dyamic is). The if i is cosidered ha he marke is icomplee, he he Asia opio payo ivolves a challegig ask. So, i ca be impora o calculae a leas a ierval where he opio price should be. Risk maageme area could be ieresed i havig bouds for he Asia opio prices i which calculaio oly ake io accou he marke iformaio available i he mome ha he derivaive is priced. I ca help o avoid problems whe a specic model is assumed Albrecher e al., 2008). The aalysis required o d bouds for Asia opio prices ca be divided i wo: he upper ad lower boud esimaio. I case of upper bouds, Simo e al. 2000) ad Albrecher e al. 2005) worked o dig a super-replicaig sraegy porfolio) i erms of Europea opios. Albrecher ad Schoues 2005) we deep ad ivesigaed a modelidepe framework usig oly he Europea opios o he uderlyig asse which are available srikes ad mauriies which are raded i he marke). To esimae he lower bouds, he idea of usig Jese's iequaliy Curra 1994) ad Rogers ad Shi 1995)) has bee used which gives he followig relaioship Albrecher e al 2008): S i cx ES i Z where Z is a arbirary radom variable ad he iequaliy cx meas covex orderig 2 Black ad Scholes 1973) framework assumes coiuously compouded asse reurs ad i is ormally disribued wih kow mea ad variace Heso, 1993). 3 I-he-moey meas a posiive irisic value if he opio were exercised oday, he payo would be posiive). I he case of ou-of-he-moey opios he opposie happs zero payo). 8

9 relaio 4 which is described i he ex expressio for every covex fucio g: X cx Y Eg X ) Eg Y ) oe ha he Asia opio payo is a covex fucio) Albrecher e al., 2008). This disseraio is based o he research of Albrecher, Mayer ad Schoues 2008) which describes how o obai he opimal idepe lower boud for Asia opio prices usig jus he opio prices wih mauriies ad srikes available i he marke. To derivae he upper boud hey used he cocep of comoooiciy, i a similar way ha Hobso 2005) used o d he upper bouds for he Baske opio prices. This docume is divided as follows: secio 2 will describe ad prove, some properies of he comoooic ses ad radom vecors. This will help o explai why he applicaio of his cocep s up i dig he upper bouds ad helpig o esimae he lower bouds as well. Laer, secio 3 will develop i deail he lower ad upper bouds, his par will illusrae i deail he Albrecher's e al. work. Secio 4 will also explai he umerical implemeaio of hese bouds uder he mehodology proposed by Albrecher e al. 2008). Fially, secio 5 will give a real-marke applicaio usig he iformaio of he Colombia over he couer FX opio marke. 2 The Cocep of Comoooiciy ad Radom Vecors The aim of his secio is o describe he mai coceps relaed wih comoooiciy which will help o udersad why his propery, i some radom vecors, will allow o calculae he bouds for Asia opio prices. Bu rs, i is goig o be explaied why his cocep is useful for his purpose: Asia opio payo has a sum of o-idepe radom variables 1 S i K ) ), where he probabiliy disribuio of each radom + variable S i is kow Dhaee e al., 2002) i.e. usig Black-Scholes model), bu i is dicul o esimae which is he joi probabiliy disribuio i is ukow) ha allows he calculaio of he expecaio of he discoued payo uder he risk eural measure ha i is eeded o d he Asia opio price. However, i will be proved ha he probabiliy disribuio of he larges sum, i he covex sese, will have he comoooic disribuio Dhaee e al., 2002) i will be explaied i deail laer). Deiio 1 The se A R is said o be comoooic if for ay x ad y i A, eiher x y or y x holds. Where x is he oaio for x 1, x 2,..., x )ad x y for x i y i for all,2,...,. 5 Deiio 2 A radom vecor X = X 1,..., X )is said o be comoooic if i has a comoooic suppor. Theorem 1 A radom vecor X is comoooic if ad oly if oe of he followig equivale codiios holds: 1. X has a comoooic suppor; 4 This holds if ad oly if E X = E Y ad E X d) + E Y d)+ for < d < 5 All hese deiios ad heorems have bee ake from Dhaee e al. 2002) 9

10 2. For all x he followig hold: 3. For U Uiform 0, 1), his holds: F X x) = mi {F X1, F X2,..., F X } X d = F 1 X 1 U),..., F 1 X U) ) 4. There exis a radom variable Z ad o-decreasig fucios f i, i = 1,...,, such ha X d = f 1 Z),..., f Z)) where d = deoes ha boh sides have he same disribuio. Proof: 1 2) Asumme ha X has a comoooic suppor B. Le x R ad A j = { y B yj x j } for j = 1,...,. The, because of he comoooiciy of B, here exiss a i such ha A i = j=1 A j i.e. y A i : y x, are ordered compoewise). The exisece of his iersecio ca be proved as follows: Take y j = { y A j y j = max y j A j { y j } } for all j i.e. he maximum vecor y i he sece ha his vecor has he maximum compoe value amog all vecors which belog o A j for all j, oice ha his vecor exiss because i belogs o a closed se sice i coais x j, look a he deiio of A j ). { { } } Now akig yi = y yj yi = mi 1 i,j {y } j i i.e. miimum compoe j=1,..., amog all vecors yj ). The i implies ha yi yj, j = 1,...,, because he comooociy of B, ad moreover yi A i ad also o A j for all j i.e. his vecor appears i all A j sice i saises he codiio y j x j haks o he comoooic propery) he A i = j=1 A j. Hece, i ca be foud ha F X x) = Pr X ) A j = Pr X A i ) = Pr y i x i ) = F Xi x i ) j=1 ad sice A i A j he F Xi x i ) F Xj x j ) for all j, so F X x) = mi {F X1,..., F X } 2 3) Assume ha F X x) = mi {F X1,..., F X } for all x = x 1,..., x ). F X x) = Pr X 1 x 1,..., X x ) by defiiio = Pr F 1 X 1 U) x 1,..., F 1 ) X U) x = Pr U F X1 x 1 ),..., U F X x )) sice F 1 X p) x p F X x) = Pr { U mi 1 j FXj x j ) }) { is he iersecio = mi 1 j FXj x j ) } by Uiform disribuio propery 3 4) Noice ha U is by deiio o-decreasig as well as is iverse i.e. by deiio of he iverse of a disribuio is he o-decreasig ad coiuos fucio 10

11 F 1 1 X x) such ha FX p) = if {x R F X x) p} wih p 0, 1 Dhaee e al., 2002: 10)). The le U Z, so here is a radom variable Z ad o-decreasig fucio f i such ha X = d f 1 Z),..., f Z)) 4 1) Assume ha here exiss a radom variable Z wih suppor B ad o decreasig fucios f i such ha X d = f 1 Z),..., f Z)). The se of possible oucomes of X is {f 1 Z),..., f Z) Z B)} Dhaee e al., 2002: 14)). Noice ha his se is comoooic sice f i are o-decreasig, which implies ha X is comoooic because i mees he deiio of a comoooic vecor. From his heorem i is posible o coclude he followig: i) To d he probabiliy of all oucomes of a X beig less ha x, i is oly eeded o calculae he probabiliy of he leas likely X i sice X is comoooic Dhaee e al., 2002). Besides, from Hoedig 1940) ad Fréche 1951), i is kow ha mi 1 j { FXj x j ) } is a mulivariae cumulaive disribuio fucio wih he same margials of X 1,..., X ) Dhaee e al., 2002). ii) If all margial disribuio of F X are ideical, he X 1 = X 2 = = X holds almos surely whe X is comoooic Dhaee e al., 2002). iii) if X 1,..., X ) d = f 1 Z),..., f Z)) where all fucios f i are o-decreasig, X is coomooic, ad he such a model is i a sece a exreme form of a mixig model, as i his case he exeral parameer Z = z compleely dermies he oucomes of XDhaee e al., 2002:14). Bu akig io accou ha he aim of his docume is o d Asia opios prices ad is payo ivolves a sum of radom variables, he i is ecessary o iroduce ew properies i order o be able o deal wih hese kid of payos. From ow, he followig oaio X c 1,..., X c ) will be used for he comoooic radom vecor wih he margials of X 1,..., X ). Theorem 2 Cosider S c = X1 c X c which is he sum of he compoes of a comoooic versio of he radom vecor X 1,..., X ). The he iverse disribuio fucio F 1 S c of a sum Sc is give by F 1 S c p) = F 1 X i p), 0 < p < 1 Proof: Cosider he radom vecor X 1,..., X ) ad is comoooic couerpar X1, c..., X), c he S c d = = g u), u U 0, 1) g u) = X c i F 1 X i u), 0 < u < 1 oice ha each F 1 X i is a o-decreasig ad coiuos, he g u) is o-decreasig ad coiuos as well. Dhaee e al ) proved ha if g mees hese codiios he F 1 gx) p) = g F 1 X p)). So, F 1 S c 1 p) = Fgu) = g F 1 U p)) = g p) he las iequaliy due o he propery of he uiform disribuio. F 1 X i p), 0 < p < 1 Dhaee e al., 2002). The F 1 S c p) = 11

12 By his heorem he miimal value of a comoooic sum equals he sum of he miimal values of each erm, ad he maximal value is equal o he sum of he sum of he maximal values of each erm Dhaee e al., 2002: 20). So far, his secio has show some ools o allow he Asia opio payo beig more racable, however i is ecessary o iroduce oe more ool i order o be able o deal wih he posiive par fucio. Theorem 3 Cosider he sop-loss premium of a sum of he compoes of he comoooic radom vecor deoed by S c. Dee by E S c d) + he sop-loss premium. The his premium is give by E S c d) + = E X i d i ) +, F 1+ S c 1 0) < x < FS 1)) c wih ad d i = F 1 X i F S c d)), i = 1, 2,..., F 1 S F c S c d)) = d Proof: Le d F 1 1 S 0) < x < F c S 1)). Also cosider he suppor of X c beig comoooic, ha is { F 1 X 1 c p),..., F 1 X p) ) 0 < p < 1 }, he i ca have a mos oe poi i he iersecio wih he hyperplae {x x x = d} his iersecio is impora because from his poi he posiive par fucio chages is value). This saeme follows from he comoooic propery sice x y or y x mus hold. Now, i is possible o say ha F S c is sricly icreasig ad coiuos margials bewee F 1+ 1 S 0) < x < F c S 1), he F 1 c Sc x) is uiquely deermied by FS F c S c x)) = x Dhaee e al. 2002), or equivalely by F 1 X i F S c x)) = x provided by heorem 2. The d i = F S c d)) = F 1 S F c S c d)) = d F 1 X i so d i is a eleme of X c ad he hyperplae {x x x = d}. Now, i is clear ha d is he uique eleme of he iersecio bewee he suppor of X c ad he hyperplae. Sice x ad d are elemes of he laer suppor, he oe of he followig cases mus hold: x j > d j implies ha x k d k for al k = 1,..., or he opposie holds because of he comoooiciy of X c. The if x j > d j he x x d) + x 1 d 1 ) x d ) + sice x k d k for all k ad d i = d, bu if x j < d j he boh sides are equal o 0 Fially, i remais o show which is he relaioship bewee he sum of radom variables ad is comoooic couerpar. This relaio will allow he calculaio of he upper boud he lower boud will be reaed i subseque secios). Bu le rs dee precisely wha is mea by covex order. 12

13 Deiio 3 Cosider wo radom variables X ad Y. covex sese if ad oly if The X precedes Y i he E X = E Y, E X d) + E Y d) + < d < which will be deoed as X cx Y Theorem 4 For ay radom vecor X 1, X 2,..., X )he followig iequaliy holds X 1 + X X cx X c 1 + X c X c Proof: I suces o prove E X X d) + E X c X c d) + sice he posiive par is covex ad E X i = E Xi c because hey have he same margials. The cosider d F 1+ 1 S 0), F 1)) ad d c 1 + d d = d, so S c x 1 + x x d) + = x 1 d 1 ) + x 2 d 2 ) x d )) + x 1 d 1 ) + + x 2 d 2 ) x d ) + ) = x 1 d 1 ) + + x 2 d 2 ) x d ) + holds for all x 1, x 2,..., x ). So, akig expecaio i boh sides ad replaicig x wih a radom variable X E X X d) + E X1 d 1 ) X d ) + = E X i d i ) + = E S c d) + where he las equaliy holds due o heorem 3. Ad also by he same heorem E S c d) + ca be expressed as follows + E S c d) + = E Xi c d i ) + = E X c X c d) + The X 1 + X X cx X c 1 + X c X c sice E X i = E X c i ad E S d) + E S c d) +. All hese four heorems have impora resuls because hey will allow he esimaio of he lower ad upper bouds for Asia call opio prices. The way how o do so will be explaied i he ex secio, ad also, i will be described laer o i deail how o impleme such a bouds i real circumsaces. 13

14 3 Model-Idepe Bouds for Asia Opio Prices This secio will explai i deail how Abrecher e al. 2008) developed some lower ad upper bouds for Asia call opio prices usig Jese's iequaliy ad he cocep of comoooiciy explaied above. There will be hree diere expressios o calculae hree lower bouds. The dierece amog hem will be how igh he boud is. Ad he las par will show how o calculae a super-replicaig porfolio, which is he miimum upper boud ha ca be obaied uder o assumpio regardig o he dyamics of he uderlyig asse price. 3.1 Lower Bouds The aim of his secio is o d he bes lower boud for Asia opio prices by subreplicaio. Albrecher e al. 2008) used Jese's iequaliy o obai a ew payo, or eve, a sub-replicaig sraegy formed by a call opio. By codiioig he Asia call opio payo o S 1 rs moiorig ime), he followig useful expressio is obaied For oaio simpliciy le S i as S i ): AC K, ) = E e rt 1 ) + S i K = E E e rt 1 ) + S i K S 1 E e rt 1 ) + E S i S 1 K he sum of codiioal expecaios ca be expressed as follows E S i S 1 = E e r i 1 ) Si S 1 = e r i 1 ) S 1 by he marigale propery of he discoued price process S i. Cosiderig ha S 1, e r 2 1 ) S 1,..., e r 1) S 1 ) is a comoooe vecor sice e r i 1 ) S 1 is o decreasig ad he radom variable S 1 deermies he value of all compoes see heorem 1), so i is possible o apply heorem 3 AC K, ) E e rt 1 ) + E S i S 1 K = e rt 1 )) + E e r i 1 ) S 1 F 1 F e r i 1) S1 er i 1) S1 K) ad akig io accou ha F er i 1) S1 K) = Pr e r i 1 ) S 1 K 14

15 he i ca be wrie as AC K, ) e rt 1 = Pr S 1 = F S1 K er i 1 ) ) K er i 1 ) E e ri 1) S 1 F 1 F e r i 1) S1 S1 moreover F 1 e r i 1) p) ca be expressed i erms of F 1 S 1 : so K er i 1 ) F 1 e r i 1) p) x p F e r i 1) S1 x) = Pr e r i 1 ) S 1 x ) AC K, ) e rt 1 = F S1 x e r i 1 ) ) F 1 S 1 p) E e r i 1 ) S 1 e r i 1 ) F 1 S 1 F S1 x e r i 1 ) K er i 1 ) ))) + ))) + = e rt 1 )) + E e ri 1) S 1 e ri 1) K er i 1 ) = e rt 1 ) + e r i E e r K 1 S 1 er i 1 ) = e rt 1 ) e r i K C, er i 1 ) 1 = 1 ) C K, er i 1 ) 1 e rt i) =: LB 1 1) where C K, ) deoes he Europea call opio price wih srike K ad ime o mauriy. LB 1 will be used as oaio for Lower Boud which ivolves a call opio price wih mauriy 1. LB 1 is o ecessarily he bes lower boud sice i was calculaed codiioig he expeced value usig arbirarily S 1, as a codiioig radom variable, bu here ca be aoher moiorig ime which will give a much igher boud. The applyig he followig iequaliy will help o improve his boud dig such a : E 1 ) + 1 S i K E ) S i K 1 {Y c} The reaso why his iequaliy holds ca be explaied i a simple way: whe 1 S i K ad 1 {Y c} = 1 i becomes a equaliy, bu if 1 {Y c} = 0 i urs o a iequaliy. The same aalysis ca be doe whe 1 S i < K, bu i his case he righ had side's miimum value is zero ad he lef had side ca be egaive. Tha is why he iequaliy always holds. 15

16 Applyig his o geerae he lower boud, isead of usig Jese's iequaliy, will allow for o improve i maximasig over i order o choose he opimal moiorig ime whe Y = S. The assumig c 0 leads o AC K, ) e rt E 1 ) S i K 1 {S c} = 1 e rt E ) S i 1 {S c} E K1{S c} = 1 e rt E E ) S i 1 {S c} F E K1{S c} his by he ower propery of codiioal expecaios. Now leig J ) = mi i { i } ad separaig he sum bewee F -measurable par ad which is o, i is obaied AC K, ) 1 J) 1 e rt E S i 1 {S c} + E 1 {S c}e S i F E K1 {S c} = 1 e rt J) 1 i=j) E S i 1 {S c} + io he las equa- ow iroducig E 1 {S c}c i=j) er i ) io leads o AC K, ) 1 e rt = 1 e rt J) 1 = 1 e rt J) 1 E S i 1 {S c} + i=j) i=j) E 1 e rt Pr S c K c E S i 1 {S c} + J) 1 i=j) E S i 1 {S c} + E 1 {S c}e ri ) S 1 {S c}c i=j) er i ) E 1 {S c}e ri ) S i=j) E 1 {S c}e ri ) S c) i=j) e ri ) K Pr S c i=j) Pr S c K c e ri C c, ) Pr S c K c e ri ) E 1 {S c}c i=j) e ri ) i=j) e ri ) I remais o esimae E S i 1 {S c} ad Pr S c. Before coiuig, he followig erms are o be esimaed: rs, oe ha i ca be claimed ha S i ad 1 {S c} are o-egaive correlaed for > i ad c 0 6. The E S i 1 {S c} E Si E 1 {S c} = S0 e r i Pr S c ad secod, Pr S c = e r CK,) K K=c, ad deoe i as C K K, ) 7. As a resul i) ca be wrie as AC K, ) 1 J) 1 e rt e r i C c, ) Pr S c K e r i c e r i ) i=j) i=j) ) 6 Sice CovS i, S ) = e r+i) Var Si 0 he i is a reasoable assumpio because 1 {S c} ad S are posiively relaed sice whe S is high ad 1 {S c} = 1, oherwise zero. This resul holds if he risk eural measure mees he Black Scholes framework codiios eve whe is used a expoeial Lévy model). I geeral his codiio cao be checked usig oly he marke call opio prices Albrecher e al,. 2008) 7 This equaio is obaied derivaig C K, ) = e r s K)+ f S s) ds wih respec o K 16 i)

17 1 e rt 1 e rt i=j) i=j) e r i C c, ) e r i i=j) C c, ) + C K c, ) e r i ) Pr S c K c) J) 1 e r i i=j) er i ) K )) J) 1 e r i c i=j) er i ) Recallig ha he framework used i his docume is uder o-arbirage codiio, i implies ha call price fucio is covex wih respec o K. This fac will allow o maximise ii) uder c. To do so, deoe c 1) = K J) 1 e r i i=j) er i ) ad calculae he rs derivaive wih respec o c of he fucio iside he parehesis, ad he, d is roo: d ) C c, ) + c 1) C c c, ) cc c c, ) = C c c, ) + c 1) C cc c, ) cc cc c, ) + C c c, )) = 0 dc c = c 1) bu his poi ca lead o a miimum, maximum or eve a saddle poi, bu due o covexiy of C c, ), i is kow ha he followig codiio holds C c 1), ) C c, ) + C c c, ) c 1) c ) i is called supporig plae) so C c, ) has o have is maximum a c 1) because his relaioship applies for all c. From his follows ha ii) AC K, ) e rt C c 1), ) i=j) for all. So i ca be claimed ha i order o igh his boud i is possible o opimise over, i which case he boud will be AC K, ) e rt max C c 1), ) e r i =: LB 1) 2) 0 T i=j) The performace of LB 1) is a leas as good as LB 1 for all srikes Albrecher e al. 2008: 130), because leig = 1 i LB 1) LB 1 is obaied. Bu ow, les ry o esimae E S i 1 {S c} i i) usig a diere approach. Albrecher proved ha he expecaio of S i, give ad S, is bouded by some sor of weighed geomeric average of ad S Albrecher e al. 2008:130). I order o prove his claim le's assume ha S = e X where X ) 0, X 0 = 0, is a Lévy process i his docume will be assumed ha he process is a Browia moio): E S i 1 {S c} = E E e X i X = l S e r i ) 1 {S c} oice ha he idicaor fucio codiio is F -measurable which allows o ake i ou of he codiioal expecaio. By Jese's iequaliy his expressio ca be wrie as ) E S i 1 {S c} E e E S X i X =l 1 {S c} iii) ii) 17

18 S he i remais o calculae E X i X = l ). Cosider he same X ) 0 Browia moio. Dee B i ) = X i i X, which i he lieraure is called Bridge Browia moio, ad i is idepe from X E B i )X = E X i X i X2 = i i = 0 i is clear ha E B i ) = E Xi i X = 0, he he followig is rue Chag, 2007) 0 = E B i ) = E Bi ) X = E Xi X i E X X he E X i X = i X applyig his resul i iii) is obaied ha ) ) E S i 1 {S c} E e i i S l S 1 {S c} = E 1{S c} ad usig i i i) = E = E AC K, ) 1 e rt ) i S ) c i c 1{S c} 1{S c} + ) i S J) 1 1 e rt deoe for simpliciy f c) = E + 1 e rt J) 1 E c ) i S c i=j) ) i ) ) i S c ) i ) 1 {S c} 1 {S c} ) i 1{S c} ) c i + Pr S c J) 1 + e r i C c, ) Pr S c K c ) i ) 1 {S c} + ) c i Pr S c i=j) 1 J) 1 ) c i e rt Pr S c K c J) 1 E ) i S c ) i ) 1 {S c} + i=j) i=j) e r i ) e r i C c, ) i=j) e r i ) e r i C c, ). I is desirable o d he larges possible boud, so i will be applied he same approach ha was used o opimise ii) over c. For his, cosider c 2) which solves: K J) 1 ) c 2) i c 2) 18 iv) e ri ) = 0 3) J)

19 iroducig his expressio io iv): J) 1 ) c i e rt AC K, ) f c) Pr S c K c + Pr S c K = f c) Pr S c J) 1 J) 1 i=j) ) c 2) i c 2) e r i ) i=j) e r i ) ) c 2) i ) c i ) + c 2) c i=j) e r i ) replaicig f c) ad Pr S c by is correspodig expressios, i is possible o simplify he equaio as i is show i he ex lie: = J) 1 E S ) ) i c 2) i 1 {S c} + i=j) oice ha agai he maximum value of C c, ) + C K c, ) c = c 2) g c) i he same way ha ii) was solved ), he J) 1 E S ) ) i c 2) i 1 {S c} + )) e C ri c, ) + C K c, ) c 2) c := g c) i=j) so ) AC K, ) e rt f c 2) =: LB 2) which holds for all, leadig o he opimal lower boud AC K, ) max 0 T LB2) = e rt max 0 T J) 1 where c 2) is deed by 3). E S ) ) i c + 2) i + ) c 2) c e r i C i=j) c 2) e r i C is obaied whe ), = f c 2), c 2) ) ) 4) I is impora o oe ha i he rs summad a opio eeds o be evaluaed wih a coige claim of he form S x K) + which is called power opio. I he umerical secio will be described how o price his ype of payos. Noe ha E S ) ) i c 2) i + = e r E e r ) ) i S c + 2) i So far, i was esimaed hree diere expressios for he Asia call opio price lower boud, wo of hem LB 1 ad LB 1) ) use Jese's iequaliy o d he boud whe he payo is codiioed o he asse price a 1 ad where LB is maximised. The hird lower boud expressio LB 2) ) uses aoher approach which ries o igh eve more he boud. I secio 4.3 will be show how hey perform uder some marke assumpios, ad i secio 5 uder real Colombia marke codiios. 19

20 3.2 Upper Boud This subsecio will describe how o esimae he model-idepe upper boud developed by Albrecher e al. 2008) who based heir work o Simo e al. 2000) ad Hobso e al. 2005). Agai, his boud will be based o a comoooe upper boud: AC K, ) = E e rt 1 ) + S i K e rt because of heorems 3 ad 4, where S c = X i. The Si E F 1 X i F S c K)) ) + AC K, ) e rt e r i+r i E S i κ i ) + = 1 e rt i) C κ i, i ) 5) wih κ i = F 1 X i F S c K)) ad κ i = K. I pracice, implemeig 5) ca be o easy sice each κ i would o ecessarily be available i he marke, ad also i could be eeded o have some specicaio of he uderlyig asse model, or he kowledge of he complee opio price surface, o be able o price each C κ i, i ) Albrecher e al., 2008). I he umerical implemeaio par secio be preseed a opimisaio problem which will deal wih boh issues. I he same way ha he implemeaio of he lower bouds, he upper comoooic boud will be esed uder he Black Scholes model i order o compare is performace wih he oe i he Albrecher's e al. 2008) examples. The, i secio 5, his UCB will be implemeed i he Colombia FX opio marke o observe how i behaves uder real marke codiios. 4 Numerical Implemeaio - The Fiie Srike Case I he acial markes i is o possible o kow he eire Europea call price surface sice oly a few pois are kow i he srike-ime mesh, or eve, o less impora, i is o kow which he asse price model is. Tha is why model-idepe bouds are impora. I will give a guide where he Asia opio price should be, ad probably, i also oers a robus mehodology o esimae a saic hedgig sraegy ha will help o maage some of he opio porfolio's risks of a marke maker could be a bak). I a similar way of how i was divided i he las secio, his oe will cosider lower ad upper bouds separaely, sice he rs oe is referred o a sub-replicaig porfolio ad he secod oe is a super-replicaig porfolio. Before sarig, i is coveie o explai rs he oaio ha will be used i he ex wo sub secios. Assume = T ad for each moiorig ime i mauriy as well) here are available m i) Europea call prices. Noe ha opio's srikes K i,j will be ordered by size wih j = 0,..., m i), K i,0 = 0 i his case he call opio price is ) ad K i,mi) = max j K i,j. Takig io accou ha he opio prices are a covex ad o-decreasig fucio wih respec o he srike price, ad also, usig oe of he resuls published by Bersimas ad Popescu 2002), he model-idepe upper boud for Europea opio prices C K i,j, i ) = c i,j, give a mauriy i wih srike price K, is deermied by Albrecher e al., 2008) 20

21 f i K) = max {g K) g K) covex, o decreasig ad g K i,j ) = c i,j j {0,..., m i)}} 6) The maximum i 6) is obaied by coecig he give pois K i,j, c i,j ), j = 0,..., m i) liearly ad seig f i K) = c i,mi) for K > K i,mi) Albrecher e al., 2008: 135). I he ex secios i will be also required o calculae he slopes of he fucio f i K) for all mauriies i a each poi K K i,j, j = 0,..., m i) ad 0 < K < K i,mi) as follows: f i K) = c i,j c i,j 1, K i,j 1 < K < K i,j K i,j K i,j 1 bu whe K is a srike raded i he marke, he lef ad righ-had slopes ca be diere for 0 < j < m i), so i should be calculaed as: f i K i,j ) := c i,j c i,j 1 K i,j K i,j 1 c i,j+1 c i,j K i,j+1 K i,j =: f + i K i,j ) where f + i Ki,mi) ) = 0 ad fi K) = 0 for K > K i,mi). 4.1 Lower Boud Implemeaio Bersimas ad Popescu 2002) proposed a opimisaio mehodology o d he bes model-idepe boud for a plai vailla opio. Now, remember ha he Asia K call opio price lower bouds are based o oe of he followig srikes: K J) 1 e r i i=j) er i ), which will o ecessarily be available i he marke. e r i 1) or Accordig o Bersimas ad Popescu 2002) ad Albrecher e al 2008), he bes modelidepe lower boud for a Europea call opio wih srike K i K i,mi), K i,j 1 K i K i,j ad mauriy i is give by he proof, ha is based o he covexiy of f i, is preseed i boh aricles): ) { C Ki, i max c i,j 1 + f i K i,j 1) Ki K i,j 1 ), c i,j + f + i K i,j ) K i,j K )} i 7) wih f i 0) = e r i 8, ad by ) { C Ki, i max c i,mi) + f i ) ) } Ki,mi) Ki K i,mi), 0 8) However, i he case of LB 2) i is eeded o esimae a Europea power call opio price as well. For his, wo approaches will be used: Black-Scholes ad model-idepe approach. The rs oe refers o BS pricig formula for Europea power call opios, which have he advaage of kowig he exac price uder BS assumpios. The modelidepe price is based o pricig a saic sub replicaio porfolio akig io cosideraio oly he Europea plai vailla call opio prices available i he marke. 8 Sice i is eeded ha he risk eural probabiliy sums up o 1. Noe ha Pr X = x i = Pr X x i Pr X x i+1 i he discree case 21

22 4.1.1 Black-Scholes Framework This is a well-kow model, so a complee deailed explaaio will o be give bu he mai resul will be preseed: pricig formula. Assume ha he asse price follows his sochasic diereial equaio SDE) uder he risk eural measure): ad he risk-free bod dyamics is ds = rs d + σs dx db = rb d where r, σ are cosa, X ) 0 is a Browia moio ad i is also assumed ha σ > 0. By he Mai ) Theorem of he Europea Opio Pricig he price of he opio wih payo + S β T K wih β 0, 1) a = 0 ad mauriy a T is give by ) + ˆ E e S rt βt K = e rt S β 0 e βr 1 2 σ2 )T +βσ ) + T y K e 1 2 y2 dy where he expecaio is ake wih respec o he risk eural measure ad y represes he radom variable wih he same disribuio of a Browia moio wihi he ime ierval 0, 1. The he power call opio price is P C K, T ) = S β 0 e β 1)r+ β 2 β 1)σ2)T N d + βσ ) T e rt K N d) 9) N x) represes he cumulaive probabiliy of a sadard ormal disribuio ad d = l K )+r 1 2 σ2 )T σ T Model-Idepe Framework Albrecher e a. 2008) preseed a saic sub-replicaig porfolio based o he followig payo, which is domiaed by he arihmeic Asia call opio payo, S β T K ) + K β 2 K K 2 K 1 S T K 1 ) + 1 {ST K 2 } 10) + i=2 K β i K ) + Kβ i+1 Kβ i K i+1 K i S T K i ) + ) 1 {Ki <S T K i+1} where K 1 <... < K liquid srikes for Europea call opios wih mauriy T ad K β 1 K. This payo will up i he followig porfolio Albrecher e al., 2008): Srike N umber K 1 K 2 K β 3 Kβ 2 K 3 K 2 K β 2 K K 2 K 1 K i K β i+1 Kβ i K i+1 K i Kβ 2 K K 2 K 1 Kβ i Kβ i 1 K i K i 1 K Kβ K β 1 K K 1 Now, i is possible o esimae he Asia call opio price lower bouds sice i is kow how o price he power ad plai vailla call opios ivolved i he calculaio of LB 1, LB 1) ad LB 2). 22

23 4.2 Upper Boud Implemeaio I order o d he upper boud Albrecher e al. 2008) developed a approach which is based o he Hobso's 2005) soluio for Baske opios. Noe ha 6) provides he bes upper o-arbirage boud for Europea call opios whe i is oly kow c i,j, i = 1..., ad j = 1,..., m i). The his approach uses C K, i ) = f i K) which allows o calculae he upper boud for ay call opio wih srike K ad mauriy i. Remember ha by 5) i is possible o d he bes upper comoooic boud solvig mi κ i e rt i) f i κ i ) subjec o κ i K 11) which is soluio is foud by applyig he followig algorihm proposed by Albrecher e al. 2008): Alg-1 If K i,mi) K, he he soluio is give by κ i = K i,mi) else se κ i = K. Alg-2 Deermie f + i κ i ) ad f i κ i ) for al i = 1,...,. Alg-3 Se I = argmi i e rt i) f + i κ i ), = e rt I) f + I κ I) J = argmax i e rt i) f i κ i ), = e rt J ) f J κ J). Noe ha for all i, he slopes are egaive by he covexiy ad o-icreasig propery of f i, uless κ i > K i,mi) whe he slop is equal o zero. If < he se u I = K I,j+1 κ I where K I,j κ I < K I,j+1 ad u J = κ J K J, j where K < κ J, j J K J, j+1 ad se κ I = κ I + mi {u I, u J }, ad κ J = κ J mi {u I, u J }. If go o Alg-4. Else updae f + i κ i ) ad f i κ i ) for i = I, J ad reur o he sar of Alg-3. Alg-4 Se M = {i κ i K i,j j = 1,..., m i)}. If M = 1 sop. Else choose ay pair {m 1, m 2 } M M, m 1 m 2 ad calculae u 1 = K m1,j+1 κ m1 where K m1,j κ m1 < K m1,j+1 ad u m2 = κ m2 K m2, j where K m2, j < κ m 2 K m2, j+1 ad se κ m1 = κ m1 + mi {u m1, u m2 }, ad κ m2 = κ m2 mi {u m1, u m2 }. The reur o he sar of Alg-4. Afer Alg-4, here will be a mos oe i for which κ i K i,j for all j = 1,..., m i) because i is ecessary o mee he opimisaio problem cosrai). For his case, f i κ i ) ca be expressed as follows: f i κ i ) = K i,j+1 κ i c i,j + κ i K i,j c i,j+1 12) K i,j+1 K i,j K i,j+1 K i,j 23

24 for K i,j κ i < K i,j+1 sice 6) is foud coecig he c i,j pois wih lies for all i = 1,..., ad j = 1,..., m i). The covergece proof of his algorihm is i Albrecher e al. 2008) bu i may be more ieresig o udersad he behaviour of i. Le's explai he mai algorihm's feaures ad how i operaes below. From he algorihm i is kow ha κ I = K I,j+1 u I ad κ J = K + u J, j J, le deoe he porfolio cos which does o ivolve he call opio wih mauriy I ad J by Π, he i is possible o wrie ) UCB = Π + e rt I) f I K I,j+1 u I ) + e rt J ) f J K + u J, j J where UCB deoes he oal cos of he Upper Comoooic Boud. Usig 12) isead of f sice by 6) his is possible), he i ca be expressed as UCB = Π + e rt I) c I,j+1 c ) I,j+1 c I,j u I K I,j+1 K I,j + e rt J ) c c J, j+1 c ) J, j J, j K K u J J, j+1 J, j = Π + e rt I) c I,j+1 f + I K ) I,j) u I ) ) + e rt J ) c f + J, j J K J, j u J ) oe ha f + I K I,j) = f I κ I) ad f + J K J, j = f J κ J) he UCB = Π + e rt I) c I,j+1 + e rt J ) c J, j u I + u J ow assumig wihou loss of geeraliy ha u I u J, he u J = u I + u J u I ). Reorgaisig he las equaio UCB = Π + e rt I) c I,j+1 + e rt J ) c J, j + ) u I + u J u I ) sice u I is he smalles u. Now, i order o x κ I o a marke srike, i ca be shifed o he oe correspodig o c I,j+1. However here are hree scearios if < he if > he if = he ) ui 0 UCB will be reduced ) ui 0 UCB will be icreased ) ui = 0 UCB will o be chaged So his par of he proof shows he codiios which have o hold i order o apply Alg-3, ad whe i has o sop. Now, o udersad why Alg-4 exiss le's cosider ha he Alg-3 ca o loger be applied, ad also, cosider M > 1. Noice ha he miimum porfolio cos is obaied i he Alg-3 ad Alg-4 will o chage he oal cos. I order o prove las claim, assume ha here is a leas oe m i such ha e rt m i ) f mi κ mi ) > e rt m i) fmi κ mi ), m i m i where m i deoe he idex i he se M. The he followig holds = e rt m i ) f m i κ mi ) = e rt m i ) f + m i κ mi ) 24

25 > e rt m i) f mi κ mi ) = e rt m i) f + mi κ mi ) = implyig < which is a coradicio wih he ermial codiio of Alg-3. The whe M > 1, = mus hold, ad morover for all m i, also mus hold. e rt m i ) f mi κ mi ) = e rt m i) fmi κ mi ), m i m i Alhough his proved ha Alg-4 does o improve or ge wors) he oal porfolio cos, i will x a leas 1 call opios o a marke srikes which helps o porfolio maagers o hedge easily his/her risk sice for each, where κ K,j for all j, requires wo rasacios i he marke c,j ad c,j+1 ) isead of oe. 4.3 Numerical Resuls This par of secio 4 seeks o observe he performace of hese bouds uder some marke assumpios Black Scholes marke), ad also, o compare he resuls obaied by he Malab) code aexed o his docume wih he oes i he Albrecher's paper. Basically he rs hree examples explaied i Albrecher 2008) will be compared. The rs oe is based i a opio wih mauriy T = 120 days whose averagig occurs i he las 30 days, i meas a he moiorig imes 1 = 91, 2 = 92,..., 30 = 120. A daily compouded ieres rae r = l ) ad spo price S0 = 100 were used,. LB2 deoes LB 2) deoes LB 2) he boud LB 1) ad LB 2). usig Black Scholes model o price he power opios ad LB2 Id usig he model idepe approach. Each refers o he opimal for Table 1: Reproducio of he Table 2 i Albrecher 2008) The dierece bewee he prices from he above able wih he oes i Table 2 i Alcrecher e al. 2008) is foud i he hird decimal, implyig ha he Malab) compuer code is workig well. Bu oe ha LB2 Id is very close o he oe which uses he Black Scholes model, meaig ha i ca provide a good esimaio if eough call opios are available a each mauriy. Also observe ha he model idepe approach gives a worse boud comparig wih LB2 ad i also ges worse whe fewer srikes are available i he marke. From a praciioer poi of view, i ca be argued ha he LB2 Id 25

26 will help o hedge some risk sice i ca be reproduced oly usig he available opios, bu i ca also be said ha he marke kows he implied volailiy for a give se of call opios, so whe he srike used o price LB 2 is available, his volailiy ca be used o price he power opio uder Black Scholes framework, so i ca be up o he rader o decide which of he approaches apply i such a case. The followig ables also reproduce able 3 ad 4 from Albrecher's paper. Here i is cosidered a Asia call opio priced usig = 100, r = 0.04, volailiy σ = 0.25 ad moly averagig. Table 3 i Albrecher e al. 2008) cosider a opio wih ime o mauriy e years ad able 4 hree years. The accuracy of he code i hese cases is similar, or eve beer: i mos of he cases he dierece is foud i he fourh decimal, ad also, mos of he coicide wih Albrecher e al. 2008) calculaios. Table 2: Reproducio o he Table 3 i Albrecher 2008) Table 3: Reproducio of he Table 4 i Albrecher 2008) The i is possible o rus i his Malab implemeaio i order o apply he mehodology o he Colombia FX opio marke. The imporace of he applicaio of his work sem from he ecessiy of model validaio, because i ca provide a academical approach o validae he cosisecy of some specic models whe some marke maker was o iroduce Asia opios io is acial derivaive produc porfolio, which is he case of Colombia sice oly oe bak oers his kid of exoic opios. 26

27 5 Colombia FX Opio Marke Case Up o ow, his docume has explaied i deail he developme of he lower ad upper bouds for Asia opio prices uder he model-idepe framework proposed by Albrecher e al. 2008). Bu, he umerical examples have bee based o simple assumpios, such as cosa volailiy. Fiacial markes do o operae uder Black- Scholes model assumpios, however hey are sill usig is formula o price diere kid of opios. To correc he volailiy assumpio, hey developed a cocep called Volailiy Surface, which is oe way o prese wha is he volailiy ipu ha has o be used i he BS formula for a give srike ad mauriy. Reboao 1999) calls his as usig he wrog umber for he wrog model i order o ge he correc call opio price. Markes are aware of he real reur disribuios are o ormal disribued, ha is why hey ry o chage he Black Scholes model disribuio usig he well-kow Smile i he volailiy quoaio. Icreasig he Ou of The Moey OTM) volailiies call ad pu) markes are able o creae fa ails i he risk eural probabiliy disribuio, ad also, usig a asymmeric smile hey ca iroduce he correlaio eec which ca exis bewee asse price ad is volailiy Heso, 1993). I is impora o oe ha Over The Couer OTC) markes do o quoe he volailiy surface direcly, his has o be deduced from he producs ha he raders rade: sraddle, risk reversal verical spread) ad buery. They use hem o be able o hedge or speculae) possible movemes i he marke risk eural measure Malz, 1997) because he followig: ˆ A The Moey ATM) Sraddle log call ad shor pu a he same srike) are used o rade he level of he volailiy of he risk eural measure. Noe ha his sraddle allows o rade he risk eural volailiy sice he moeyess of he opios used is he ATM oe. I is called ATM opio he oe which has he forward rae as srike price his ATM deiio ca vary from marke o marke). This volailiy level he same volailiy for boh call ad pu opios) is deoed by σ AT M. ˆ Risk reversals log call ad shor pu) are quoed usig he followig coveio: Buyig a call opio which has dela equals o 25% ad sellig a pu wih he same dela. This is also doe usig 10% dela. This produc allows raders o hedge chages i he symmery of risk eural probabiliy skew ad ails how fa hey are) akig a posiio i he call-pu volailiy dierece. This ca be see as he way he raders hedge he symmery of he smile wih respec o he srike. This volailiy spread is quoed as σ RR = σ 25Call σ 25P u where he umber i he sub idex deoe he value of he dela of he opio. ˆ Fially, he Buery log call a 25 dela, log call a 75 dela ad wice shor ATM call) helps o marke makers o hedge chages i he curvaure of he smile. This spread is quoed as σ BF = 0.5σ 25Call + 0.5σ 25pu σ AT M sice he classical Black Scholes formula sock markes) implies ha σ 25call = σ 75P u. This is o rue i he FX markes bu i will be explaied how i is adjused by he marke. 27

28 Alhough he volailiy surface cao be observed from hese quoaios, i ca be calculaed from he las formulas: σ 25Call = σ AT M σ RR + σ BF, σ 25P u = σ AT M 1 2 σ RR + σ BF 13) which allows o calculae 5 pois i he volailiy smile for each mauriy. This is easily implemeed i he sock markes where he symmery i he smile holds for he dela formula, bu for FX marke i does o loger hold. I order o explai how i is adjused i is ecessary o explai he Black-Scholes framework i FX marke developed by Garma ad Kohlhage 1983)). 5.1 Garma ad Kohlhage Pricig Formula The currecy price, uder he risk eural measure, has he followig dyamics ds = r d r f ) S d + σs dw where r d is he domesic ieres rae, r f is he foreig ieres rae, W deoes he Browea moio uder he risk eural measure. Moreover, he risk free bod follows db = r d B d ad by holdig oe ui of S a ieres paymes P, which follows dp = r f S d, will be received. Cosider he hedgig sraegy ψ, ϕ) for he payo h := S T K) +, where ψ A 0, T ) ad ϕ S 0, T ) 9, such ha he porfolio value V ψ, ϕ) saises he followig V ψ, ϕ) = ψ B + ϕ S, V ψ, ϕ) 0, 0, T ad V T ψ, ϕ) = h where las equaliy holds by he Mai Theorem o Pricig Europea Opios 10 ad dv ψ, ϕ) = ψ db + ϕ ds + ϕdp which is called self-acig codiio. The, by Io's lemma, he discoued porfolio value Ṽ ψ, ϕ) = e r d V ψ, ϕ) has a diereial equals o dṽ = σϕ S dw ha is a o-egaive local marigale, moreover, i is a supermarigale uder he risk eural measure. The, by he Mai Theorem o Pricig Europea Opios 11, he Call opio price is C K, T ) = E e rdt S T K) + ˆ = e r dt e r d r f 1 2 σ2 )T ++σ T y) + e 1 2 y2 dy where he expecaio is ake uder he risk eural measure ad y has he same probabiliy disribuio of a Browia moio i he ierval 0, 1. The pricig formula is C K, T ) = e r f T N d) Ke rdt N d σ ) T 14) 9 A 0, T ) is he se of F -adaped processes g = g ) T 0,T such ha 0 g d <. S 0, T ) deoes he se of F -adaped processes f = f ) T 0,T such ha 0 f 2 d < a.s.) 10 Le h be a o-egaive F T -measurable radom variable such ha E h 2 <, he h is replicable ad he value of he replicaig porfolio V ψ, ϕ) a = 0 is E e r dt h Gyogy, 2013) 11 Sice Ṽ is a local marigale ad i esures he exisece of ϕ by he marigale represeaio heorem Gyogy, 2013) 28

29 where Nx) is he cumulaive sadard ormal disribuio ad d = l K )+r d r f σ2 )T σ T I is kow ha i he Black Scholes seig he replicaig porfolio ψ, ϕ) a = 0 is foud as follows Gyogy, 2013) ad I addiio, he ϕ 0 dela) formula is ψ 0 = C K, T ) ϕ 0 ϕ 0 = S C K, T ) =: Call Call = e r f T N d) 15). ad for a pu is P u = e r f T N d) see Garma ad Kohlhage 1983)). 5.2 Dela Forward Adjusme Noe ha Call = e r f T 1 N d)) = e r f T + P u 1 + P u which was he relaioship eeded o have symmery i he volailiies explaied before i.e. σ 25call = σ 75P u ). However, whe a dela hedgig is performed by a marke maker, i ca be argued ha i is o ecie o impleme such a hedgig sice is implemeaio implies rasacios coss more ha usual), credi exposure limis, or eve, some regulaio rules which impose exposure limis i he foreig posiio amou Colombia case), so usually raders use forward coracs o hedge opios porfolios. Bu i implies o adjus he formula for he dela i order o have he correc hedgig replicaig) porfolio. Remember ha he value of a forward corac is V f = e r dt F T ) K) = e r f T Ke r dt 16) where F T ) is he forward rae wih delivery a T. The he dela of a opio ca be calculaed wih respec o he forward corac value Bosses e al., 2010) Clark, 2011) f = C = C S = N d) 17) V f S V f sice S V f ca be calculaed because V f is iverible. This ew dela is called Dela Forward. Now, he marke has a symmeric dela which allows o apply he volailiy surface decomposiio explaied above i equaio 13). I order o d he correspodig srike o each ode i he volailiy surface, i is jus eeded o calculae he iverse of dela fucio as follows: K f,,σ,r d,r f,t) = e N 1 f)σ T +r d r f σ2 )T 18) 29

30 Table 4: Marke Quoaios implyig he followig volailiy surface Table 5: Volailiy Surface 30

31 Figure 1: Volailiy Surface - Delas where boh ieres raes domesic ad foreig oes) were assumed cosa. Alhough his is o rue i he real markes, he impac i opio prices due o chages i he ieres raes is low for shor mauriies if i is compared wih he chages i he volailiy. Tha is why his work will be focused oly i how o impleme a volailiy surface isead of a ieres rae curve. If he las volailiy surface is expressed i erms of srikes, i is obaied Figure 2: Volailiy Surface - Srike where hese srikes were used 31

32 Table 6: Volailiy Surface's Srikes he as a resul, he Europea call opio price mesh is Table 7: Call Opio Surface 5.3 Implemeaio Lower Boud Adjusmes Noe ha i his framework E S = e r d r f) mus hold Schoues, 2003), as a cosequece, he calculaio of LB 1, LB 1) ad LB 2) should be adjused. So, he sum of codiioal expecaios used i LB 1 should be expressed as follows les deoe r d r f ) as µ) E S i S 1 = E e µ i 1 ) Si S 1 = e µ i 1 ) S 1 he AC K, ) e r dt 1 )) E e µ i 1 ) S 1 e µ i 1 ) K + eµ i 1 ) 32

33 cosequely = e r dt 1 e r d i r f i 1 ) E LB 1 = 1 ) C K, eµ i 1 ) 1 e rd1 S 1 ) + K eµ i 1 ) e r dt i ) r f i 1 ) 19) Similarly, LB 1) ca be calculaed as follows: AC K, ) 1 e r dt J) 1 = 1 e r dt J) 1 E S i 1 {S c} + i=j) E S i 1 {S c} + io he las equa- ad iroducig E 1 {S c}c i=j) eµ i ) io leads o AC K, ) 1 e r dt = 1 e r dt J) 1 = 1 e r dt J) 1 J) 1 E S i 1 {S c} + i=j) E i=j) 1 e rt Pr S c K c E S i 1 {S c} + i=j) E S i 1 {S c} + i=j) E 1 {S c}e S i F E K1 {S c} E 1 {S c}e µi ) S 1 {S c}c i=j) eµ i ) E 1 {S c}e µi ) S i=j) E 1 {S c}e µi ) S c) e µi ) K Pr S c i=j) Pr S c K c e r d i r f i ) C c, ) Pr S c K c e µi ) E 1 {S c}c i=j) Esimaig E S i 1 {S c} ad Pr S c i he same way ha i secio 3: E S i 1 {S c} E Si E 1 {S c} = S0 e µ i Pr S c ad Pr S c = e r d CK,) K AC K, ) 1 e r dt e r d i r f i ) C c, ) Pr S c 1 e r dt i=j) i=j) K=c i will resul ha i ) ca be wrie as e r i r f i ) C c, ) + C K c, ) K J) 1 i=j) e µi ) e µ i c e µi ) i=j) K )) J) 1 e µ i c i=j) eµ i ) i ) e µ i ) Deoig c 1) = K J) 1 e µ i i=j) eµ i ) ad he opimisig over c ad i is obaied LB 1) = e r dt max C c 1), ) 0 T i=j) e r d i r f i ) 20) 33

34 Fially, akig io accou ha he derivaio sarig poi of LB 2) is LB 1), so 3) should be as follows K J) 1 ad he he lower boud LB 2) LB 2) = e r dt max 0 T J) 1 E is S ) c 2) i c 2) e µi ) = 0 21) J) ) ) i c + 2) i + i=j) e r d i r f i ) C c 2), ) 22) Power Call Opio Now, havig a volailiy surface, i is possible o calculae he call opio surface ad apply he lower ad upper boud mehodology explaied i secio 3. Bu, o be able o d he LB 2) he power call opio price is eeded. I will be cosidered he same wo ways o calculae hem used i secio 4: he Black Scholes formula ad model idepe approach. I order o d he formula uder Garma ad Kohlhage framework BS oe) he same approach applied o he plai vailla opio will be used, obaiig P C K, T ) = S β 0 e β 1)r d βr f + β 2 β 1)σ2)T N d + βσ ) T e rdt K N d ) 23) wih d = l K )+r d r f 1 2 σ2 )T σ. For he model idepe price he same call opio T porfolio described i secio 4 is used. I addiio, o compare he price ierval obaied by he bouds, he Asia call opio price will be calculaed usig Moe Carlo simulaio, bu ow i mus be ake io accou ha here is a volailiy surface, so oe way o do i is o use a local volailiy model calibraed o he surface Local Volailiy Dupire 1994) proposed oe way o d he local volailiy surface. He foud ha he price's dyamic uder he marke's risk eural probabiliy has a uique diusio process which is cosise wih he Europea opio prices; he called as local volailiy fucio: ds = µs d + σ S, ) S dw 24) where µ is he risk eural drif r d r f ), σ S, ) is he local volailiy fucio ad W ) 0 is a Bowia process uder he risk eural measure. I is kow ha he risk eural probabiliy π ST,T ) evolves accordig o he Kolmogorov forward equaio Gaheral, 2006): S 2 T σ 2 S,) ST 2 ) π π ST,T )) S µst π ST,T ) = S T T ad sice C = e r dt ˆ K S T K) π ST,T )ds T 34

35 he C T = r dc + e r dt ˆ K S T K) ) 1 µst π ST,T ) + S T 2 le's solve hese wo iegrals separaely, dee 2 S 2 T σ 2 S,) S2 T π ST,T )) ds T I 1 = µ I 2 = ˆ K ˆ K S T K) S T K) 2 S 2 T ST π ST,T ) dst S T σ 2 S,) ST 2 π ST,T ) dst so, iegraig by pars ad leig u = S T K ad v = I 1 = ˆ µ S T K) S T π ST,T ) ) K µ S T π ST,T )ds T = µ K ST S T π ST,T ) ˆ sice whe S T = K, S T K) = 0 ad S T, π ST,T ) 0. Noe ha he Ce r dt = ˆ K S T π ST,T )ds T + Ke r dt C K I 1 = µe r dt C + µke r dt C K ad doig he same procedure for I 2, leig u = S T K ad v = 2 S 2 T I 2 = S T K) K S T π ST,T )ds T σs,) 2 S2 T π S T,T ) ˆ σ 2 S S,) ST 2 π ST,T ) K σ 2 T K S S,) ST 2 π ST,T ) dst T I 2 = σs,)k 2 2 e r dt 2 C K 2 as a resul C T + r dc = µc µk C K σ2 S,)K 2 2 C K 2 he σ 2 S,) = C + r T fc + r d r f ) K C K 1 2 K2 2 C K 2 25) which is he expressio ha will be implemeed usig he ie diereces scheme ceral derivaives for he oes wih respec o K, ad forward derivaive for T ) ad cubic splies as a ierpolaio mehod o ll missig iformaio wih respec o srike volailiy smile). 35

36 Figure 3: Local Volailiy Surface Havig he local volailiy surface, ow i is possible o impleme a cosise Moe Carlo simulaio wih he marke volailiy surface. This docume will use he Derma e al. 1995) suggesio o perform he simulaio, sice accordig o hem, he local volailiy surface idicaes he fair value of he local volailiy a fuure imes ad marke levels Derma e al., 1995: 13), he Moe Carlo simulaio ca be implemeed as follows: usig Euler's approximaio for a SDE 20) ca be wrie as S + = S + r d r f ) S + σ S, ) S Z where Z N 0, 1) ad σ S, ) is approximaed usig he local volailiy mesh esimaed wih 21) Numerical Resuls This subsecio is based o he marke iformaio preseed before ad he Asia call opio, which is priced i here, has he followig feaures: ime o mauriy is 12 mohs wih mohly averagig.le aalyse he resuls, as i ca be observed i able 8, LB 2) has a beer performace ha he oher kid of lower bouds, showig he same behaviour ha he example explaied i secio 4. Model idepe LB 2) is o as good as he oe i which Black Scholes formula was used for pricig he power opio, as expeced, sice oly 5 srikes are available for each moh. This eec ca be see i he Upper Comoooic Boud as well, which shows a big gap wih respec o lower bouds. 12 Remember ha Euler's aproximaio has a error ε α ) 1 2 where α is a cosa, so he price preseed i ex secios usig his approach implies his error. 36

37 Table 8: Asia Call Opio Price Bouds - Colombia FX Marke All hese prices excep Moe Carlo price) have a poor performace because here are oly 5 call opios o hedge each mauriy. I ca oly be improved if i is possible o price more srikes, bu OTC opio markes aroud he world jus use 10 ad 25 dela pois for calls ad pus, plus he ATM opio, so he oly opporuiy o improve hese bouds is assumig oe model. Table 9: Compariso - Marke Prices Thaks o Bacolombia S.A. he bigges bak i Colombia), he work show i his docume has he opporuiy o validae he performace of hese bouds comparig hem agais he real marke prices idicaive oes) quoed usig mid prices accordig o he volailiy surface implemeed before, bu hey ake io accou he ieres rae erm srucure. Noe ha all real prices are foud iside he price ierval, eve whe he bouds are calculaed assumig a cosa ieres rae, which ca be oe of he reasos why real prices are quie diere o he oe which uses local volailiy Moe Carlo simulaio. Aoher explaaio for he las dierece is ha he real marke prices are esimaed wih Black Scholes framework usig mome approximaio o d he joi risk eural probabiliy measure. I seems ha he resuls of he boud calculaio mehodologies are reasoable for he Colombia FX marke alhough he price ierval ca be wide his ca be expeced sice he amou of call opio available i he marke is low). I order o coiue aalysig 37

38 he behaviour of hese bouds, le see wha ca happe if he mauriy is reduced. Some srikes will oly be cosidered because for hose deep i ad ou of he moey oes could o make sese for small mauriy cases sice hey could have a very low probabiliy ad maiaiig he mohly averagig): Table 10: Mauriy Compariso I ca be observed ha he prices behave as expeced: hey decrease as he mauriy decreases, i is a rivial resul bu i is also observed ha he spread bewee he UCB ad LB2 decreases as well. This resul also makes sese sice he risk eural probabiliy disribuio decreases i dispersio sese) makig possible opio's payo oupus o be more compac as he mauriy becomes small i follows from he Browia moio volailiy propery). Now le s check if he resuls also behave as expeced whe he mauriy remais he same bu he averagig period is chaged agai, mohly averagig): 38

39 Table 11: Averagig Period Compariso I geeral, he prices icrease whe he averagig period ges small. I is logical because i has o o he correspodig plai vailla opio as he averagig period s o 1. Wih his las checkig resul, he algorihms ad he Malab codes seem o be workig well. The, deiely here is evidece ha his mehodology of calculaig lower ad upper bouds for Asia opio prices ca be used o validae model implemeaio, sice hese prices are model-idepe ad all possible prices have o lie iside hese iervals. 6 Coclusios As i could be see i he umerical resuls of secios 4 ad 5, ad also i Albrecher e al. 2008), i is possible o sae he followig relaioships proved by Albrecher e al. 2008)): LB 1 LB 1) sice LB 1 belogs o he se where LB 1) is opimised. Ad also LB 1) LB 2) if K er i holds Albrecher e al, 2008), he he followig holds LB 1 LB 1) LB 2) AC UCB 39

40 alhough Albrecher e al. 2008) foud ha if K er i does o hold, he las relaioship ca be saised as well. This opes he door o some marke applicaios ha will be described below. The model idepe framework preseed here has show ha i is a useful ool for esimaig sub or super-replicaig porfolios, which performace will dep o how may call or pu) opios are available i he marke. As observed i he umerical resuls of secios 4 ad 5, whe he opio price surface has more odes i is possible o improve ge closer) he lower ad upper bouds. I he case of secio 4, where here are may prices available, eve his mehodology ca be suiable for pricig Asia opios whe hey are deep OTM or ITM, because he lower ad upper bouds are close eough. This feaure is o rue whe he assumpio regardig o available srikes does o hold, as i could be see i secio 5. Oe implicaio of he explaaio above is ha his mehodology ca oly be useful i liquid markes sice i is ecessary o have eough call prices o apply his way of calculaig price bouds. Despie he fac ha i OTC markes i is dicul o icrease he umber of call opio prices available for each mauriy, his bouds sill a useful guide for opio porfolio maagers. However, i markes such as EUR/USD or GBP/USD could be possible o quoe more pois sice here are may baks, or marke makers, paricipaig i he marke, so i could be foud or approximaig) o arbirable marke prices amog he radiioal dela pois. Noe ha i he model idepe bouds behave as expeced whe he mauriy, or he averagig period, is chaged because he ipus for is calculaio are Europea call opios. So, whe he mauriy is chaged, he algorihms cosider more call opios i he sample, so i s up i a diere boud price sice he call opios icrease is prices as he mauriy is icreased or he opposie if he mauriy is decreased). The clearly boh lower ad upper bouds for Asia opio prices will have a similar behaviour ha Europea call opio prices. The same explaaio applies whe he averagig period is chaged. Alhough his could soud rivial, i is a impora feaure whe a marke maker was o validae is valuaio models. Fially, as meioed before, he lower ad upper bouds proposed by Albrecher e al. 2008) ca be used o validae he model implemeaio by a marke maker sice he price has o lie iside his ierval. I is impora because mos of he models eed o be calibraed. This procedure is o easy o perform ad i is possible o lose some of he marke opio prices durig he calibraio. This due o he square error miimisaio usual objecive fucio) ha does o guaraee ha all marke prices available ca be reproduced by he calibraed model for his i is ecessary o have a objecive fucio value equal o zero). The, as a coclusio, he lower ad upper bouds, developed by Albrecher e al. 2008), seem o be a ecie ad accurae way o calculae a model idepe price ierval, which provide saic sub ad super-replicaio sraegies ha will help raders ad risk maagers o validae specic-model prices ad corol ad hedge heir risks. 40

41 7 Refereces Albrecher, H., Mayer, P.A. ad Schoues, W. 2008) Geeral bouds for arihmeic asia opio prices. Applied Mahemaical Fiace. 15 2). p Bersimas, D. ad Popescu, I. 2002) O he relaio bewee opio ad sock prices: a covex opimizaio approach. Operaios Research. 50 2). p Black, F. ad Scholes, M. 1973) The pricig of opios ad corporae liabiliies. Joural of Poliical Ecoomy. 81. p Bosses, F., Rayée, G., Skazos, N., Deelsra, G. 2010) Vaa-volga mehods applied o FX derivaives: form heory o marke pracice. Ieraioal Joural of Theoreical ad Applied Fiace. 13 8). p Chag, J. 2007) Sochasic Processes Olie. Available from: hp:// Accessed: 11 h Jul Clark, I. 2011) Foreig exchage opio pricig: a praciioer's guide. Chicheser: Joh Wiley ad Sos. Derma, E. Kai, I. ad Zou, J. 1995) The local volailiy surface: ulockig he iformaio i idex opio prices. Quaiaive Sraegies Research Noes. Goldma Sachs. Dhaee, J. Deui, M, Goovaers, M.J., Kaas, R. ad Vycke, D. 2002) The cocep of comoooiciy i acuarial sciece ad ace: heory. Isurace: Mahemaics ad Ecoomics. 31 1). p Dupire, B. 1994) Pricig wih smile. Risk. 7. p Garma, M. ad Kohlhage, S.W. 1983) Foreig currecy opio values. Ieraioal Moey ad Fiace. 2 3). p Joural of Gaheral, J. 2006) The volailiy surface: a praciioer's guide. New Yersey: Joh Wiley ad Sos. Gyogy, I. 2013) Sochasic aalysis i ace: Black Scholes marke. Lecure oes. School of Mahemaics. Uiversiy of Ediburgh. Heso, S.L. 1993) A closed-form soluio for opios wih sochasic volailiy wih applicaio o bod ad currecy opios. The Review of Fiacial Sudies. 6 2). p Malz, A. 1997) Opio-implied probabiliy, disribuios ad currecy, excess reurs. Sa Repors. 32. Reserve Bak of New York. Reboao, R. 1999) Volailiy ad correlaio: i he pricig of equiy, FX ad ieresrae opios. Eglad:Joh Wiley ad Sos. Schoues, W. 2003) Lévy processes i ace: pricig acial derivaives. Chicheser: Joh Wiley ad Sos. Wilmo, P. 2006) Paul Wilmo o quaiaive ace. 2 d ediio. Eglad:Joh Wiley ad Sos. 41

42 8 Appix - Malab Codes This Aex provides all ecessary codes o impleme he Lower ad Upper bouds mehodology. Noe ha i may be ecessary o modify some lies i he mai scrip i order o adjus i accordig o he opio characerisics ad marke. 42

43 Malab Code 1 Mai Scrip %% GENERAL BOINDS FOR ARITHMETIC ASIAN OPTION PRICES %% Plai Vaila Call Opio Prices Geeraio % This code will cosider OTC opio prices. I is well kow ha he % marke uses Black-Scholes formula o price opios where he volailiy % is defied i a volailiy surface. % Black Scholes Formula Ipus defied mohly) %% Reproducio Tables 3 ad 4 Albrecher 2008) ic % Srikes which will be valuaed Srike= ; able=; Ave_period=12*3; % Average Period Legh spo=100; sk=0:1:250; srikes=oesave_period,1)*sk; % This deps o volailiy surface irae=0.04; % coiuos compouded vol=0.25*oessizesrikes)); %Time x srikes ime=1:1:ave_period'/12; % Time srucure of he Volailiy Surf. %% Call Price Surfice C=zerossizesrikes)); for :sizesrikes,1) for j=1:sizesrikes,2) d1=logspo/srikesi,j))+irae+voli,j)^2)/2)*imei)))/voli,j)*sqrimei))); d2=d1-voli,j)*sqrimei)); N1=ormcdfd1,0,1); N2=ormcdfd2,0,1); Ci,j)=spo*N1-srikesi,j)*exp-iRae*imei))*N2; for q=1:leghsrike) K=Srikeq); %% UPPER BOUND - Albrecher e al 2008) % Sep 1 if sumsrikes:,))<= Ave_period*K k=squeezesrikes:,)); Id=1; else k=k*oesave_period,1); Id=0; %Sep 2 if Id==0 Dif=-1; %I esures he while loop will ru alea oes Dsup=0; while Dif<Dsup posiios=zerosave_period,3); DelaP=zerosAve_period,1); DelaN=zerosAve_period,1); for :Ave_period % This par fid where he ki) is ubicaed i he srikes "srucure" l1=fidsrikesi,:)<ki)); if umell1)==0 posiiosi,1)=1; else posiiosi,1)=l1); l1=fidsrikesi,:)>ki)); if umell1)==0 posiiosi,3)=sizesrikes,2); else posiiosi,3)=l11); l1=fidsrikesi,:)==ki)); if umell1)==0 posiiosi,2)=posiiosi,3); else posiiosi,2)=l11); % Slope calculaios if posiiosi,2)<posiiosi,3) if posiiosi,3)==sizesrikes,2) DelaPi)=0; else DelaPi)=Ci,posiiosi,3))-Ci,posiiosi,2)))/... srikesi,posiiosi,3))-srikesi,posiiosi,2))); DelaNi)=Ci,posiiosi,2))-Ci,posiiosi,1)))/... 43

44 srikesi,posiiosi,2))-srikesi,posiiosi,1))); else DelaNi)=Ci,posiiosi,3))-Ci,posiiosi,1)))/... srikesi,posiiosi,3))-srikesi,posiiosi,1))); DelaPi)=DelaNi); %Sep 3 DelaP=exp-iRae*ime)-ime)).*DelaP; DelaN=exp-iRae*ime)-ime)).*DelaN; I1=fidDelaP==miDelaP)); I=I11); Dif=DelaPI); J1=fidDelaN==maxDelaN)); J=J11); Dsup=DelaNJ); if Dif<Dsup if srikesi,posiiosi,1))<=ki) && srikesi,posiiosi,3))>ki) Ui= srikesi,posiiosi,3))-ki); if srikesj,posiiosj,1))<kj) && srikesj,posiiosj,3))>=kj) Uj= kj)-srikesj,posiiosj,1)); ki)=ki)+miui,uj); kj)=kj)-miui,uj); %Sep 4 M=; for :Ave_period I=fidsrikesi,:)==ki)); if umeli)==0 M=M;i; if leghm)>1 while leghm)>1 U1=srikesM1),posiiosM1),3))-kM1)); U2=kM))-srikesM),posiiosM),1)); km1))=km1))+miu1,u2); km))=km))-miu1,u2); if miu1,u2)==u1 M=M2:); else M=M1:-1); f=zerosave_period,1); for :Ave_period fi)=ci,posiiosi,2)); if leghm)==1 fm)=srikesm,posiiosm,3))-km))/srikesm,posiiosm,3))-... srikesm,posiiosm,1)))*cm,posiiosm,1))+km)-... srikesm,posiiosm,1)))/srikesm,posiiosm,3))-... srikesm,posiiosm,1)))*cm,posiiosm,3)); CosUpper=sumexp-iRae*ime)-ime)).*f)/Ave_period; l2=fidsrikes1,:)<=k); if umell1)==0 p=2; else p=l2); %% LOWER BOUNDS - Albrecher e al 2008) %% LB1 % Fidig he closes K o K1 K1=Ave_period*K)/sumexpiRae*ime-ime1)))); CosLB1=CallPriceEs1,iRae,ime,C,srikes,K1); CosLB1=CosLB1*1/Ave_period)*sumexp-iRae*ime)-ime))); %% LB1 CLB1=zerosAve_period,1); K1=zerosAve_period,1); for i=2:ave_period K1i)=Ave_period*K-sumexpiRae*ime1:i-1))*spo))/... sumexpirae*imei:)-imei)))); K1i)=maxK1i),0); CLB1i)=CallPriceEsi,iRae,ime,C,srikes,K1i)); CLB1i)=CLB1i)*sumexpiRae*imei:))); CLB1=exp-iRae*ime))*CLB1/Ave_period; 44

45 CLB11)=CosLB1; CosLB1=maxCLB1); aux=fidclb1==coslb1); poslb1=aux1); %% LB2 LB2=zerosAve_period,1); LB2lw=zerosAve_period,1); % c3=; for :Ave_period x0=300; % Nex fucio solves he eq. 13 i p 131 c2=fzero@c2) fidc2c2,i,ave_period,k,spo,ime,irae), x0); % c3=c3 c2; % This is a corol variable %PowerCalli,x,K,srikes,C) powcall=zerosi-1,1); % I uses B-S framework powcall2=zerosi-1,1); % I uses Idepe model approah % PC=zerosi-1,1); PCbs=zerosi-1,1); for z=1:i-1 aux=fidsrikesi,:)<=c2); PCbsz)=PowerCallBSimez)/imei),spo,iRae, c2, voli,aux)), imei)); powcallz)=spo^1-imez)/imei)))*expirae*imei))*... PCbsz); PCz)=PowerCalli,imez)/imei),c2,srikes,C); powcall2z)=spo^1-imez)/imei)))*expirae*imei))*... PCz); callc2=callpriceesi,irae,ime,c,srikes,c2); LB2i)=sumpowCall)+sumexpiRae*imei:)))*... callc2; LB2lwi)=sumpowCall2)+sumexpiRae*imei:)))*... callc2; aux1=maxlb2); CosLB2=exp-iRae*ime))/Ave_period*aux1; aux=fidlb2==aux1); poslb2=aux1); aux2=maxlb2lw); CosLB2lw=exp-iRae*ime))/Ave_period*aux2; aux3=fidlb2lw==aux2); poslb2lw=aux31); %% Asia Call Opio Price - Black-Sholes Framework - MC Simulaio aux=fidsrikesi,:)<=k); MC=ACbsMCspo,iRae, K, vol,aux)), ime), Ave_period, 'moh'); %% Resuls Table 3 ad 4 Albrecher e al 2008) able=able; K CosLB1 CosLB1 poslb1 CosLB2 poslb2 MC CosUpper CosLB2lw poslb2lw; xlswrie'able.xlsx',able); oc Malab Code 2 - Fid Srike c2 fucio f=fidc2c2,i,,k,spo2,ime2,irae2) if i==1 f=*k-c2*sumexpirae2*ime2-ime21)))); else f=*k-sumspo2*c2*oesi-1,1))/spo2).^ime21:i-1)/ime2i))))-c2*... sumexpirae2*ime2i:)-ime2i)))); Malab Code 3 Model Idepe Power Opio Pricig %% This fuciio calculae he subreplicaig call opio porfolio eeded %% o hedge a power call opio. % I is required o provide a call opio price surfice C ) % x: deoes he expoe isidehe payoff fucio % i: deoes he"ime" posiio of he mauriy dae i he Call Prices % Surfice fucio value=powercalli,x,k,srikes,c) pos=fidsrikesi,:)>=k); amou=zerosleghpos),1); if K<srikesi,-1) for j=1:leghpos) if j==1 amouj)=srikesi,posj)+1)^x-k^x)/srikesi,posj)+1)-srikesi,posj))); %if j==2 && pos)<2 45

46 if j==2 && leghpos)>2 amouj)=srikesi,posj)+1)^x-srikesi,posj))^x)/srikesi,posj)+1)-srikesi,posj)))-... srikesi,posj))^x-k^x)/srikesi,posj))-srikesi,posj)-1)); %if j==2 && pos)==2 if j==2 && leghpos)==2 amouj)=-srikesi,posj))^x-srikesi,posj)-1)^x)/srikesi,posj))-srikesi,posj)-1)); if j>2 && j<leghpos) amouj)=srikesi,posj)+1)^x-srikesi,posj))^x)/srikesi,posj)+1)-srikesi,posj)))-... srikesi,posj))^x-srikesi,posj)-1)^x)/srikesi,posj))-srikesi,posj)-1)); % if j==leghpos) if j==leghpos) && j > 2 amouj)=-srikesi,posj))^x-srikesi,posj)-1)^x)/srikesi,posj))-srikesi,posj)-1)); value=ci,pos1):)*amou; else value=0; Malab Code 4 Black Scholes Power Opio Formula %% This fucio calculae he price of a Europea Power Call Opio which payoff % x: deoes he expoe iside of he payoff fucio fucio price=powercallbsx,spo,irae, K, vol, ime) d=logspo/k)+irae-0.5*vol^2)*ime)/vol*sqrime)); N1=ormcdfd+x*vol*sqrime),0,1); N2=ormcdfd,0,1); price=spo^x)*expx-1)*irae+0.5*x*vol^2)*x-1))*ime)*n1-exp-irae*ime)*k^x)*n2; Malab Code 5 Moe Carlo Simulaio %% This fucio calculaes he Asia Call Opio Prices usig Moe Carlo %% Simulaio % Period ca be 'moh' or 'day' ommied) % I uses Euler Aprox. o geerae price pahs accordig o he Black % Sholes framework fucio price=acbsmcspo,irae, K, vol, ime,, period) if srcmpperiod,'moh') period=12; else period=365; ier=700000; =period*ime; Z=radier,); S=spo*oesier,+1); for i=2:+1 S:,i)=S:,i-1)+iRae*1/period)*S:,i-1)+S:,i-1).*vol*sqr1/period)*Z:,i-1)); payoff=meamaxmeas:,--1):)')-k,0)); price=exp-irae*ime)*payoff; Malab Code 6 Black Scholes Formula for FX Power Opio Formula %% This fucio calculae he price of a Europea Power Call Opio which payoff % x: deoes he expoe iside of he payoff fucio fucio price=powercallbscurrx,spo2,irae2, iraef2, K, vol, ime2) d=logspo2/k)+irae2-iraef2-0.5*vol^2)*ime2)/vol*sqrime2)); N1=ormcdfd+x*vol*sqrime2),0,1); N2=ormcdfd,0,1); price=spo2^x)*expx-1)*irae2-iraef2*x+0.5*x*vol^2)*x-1))*ime2)*n1-exp- irae2*ime2)*k^x)*n2; Malab Code 7 Local Volailiy Moe Carlo Simulaio %% This fucio calculaes he Asia Call Opio Prices usig Moe Carlo %% Simulaio % Period ca be 'moh' or 'day' ommied) % I uses Euler Aprox. o geerae price pahs accordig o he Local Volailiy framework fucio price=acbsmccurrspo2,irae2,iraef2, K, vol, srikes2, ime2,, period) Call_lc srikes_lc Vol_lc=CallPricesMesh_locVoliRae2, iraef2, ime2, spo2, vol, srikes2); vol_local sr_local=vol_localcall_lc, srikes_lc, ime2, irae2, iraef2, Vol_lc); if srcmpperiod,'moh') period=12; 46

47 else period=365; ier=150000; =period*ime2); Z=radier,); S=spo2*oesier,+1); for i=2:+1 for j=1:ier pos=fidsr_local<=sj,i-1)); if isempypos)==1 && sr_local)<sj,i-1) v=vol_locali-1,); if isempypos)==1 && sr_local1)>sj,i-1) v=vol_locali-1,1); if isempypos)==0 v=vol_locali-1,pos)); Sj,i)=Sj,i-1)+iRae2-iRaef2)*1/period)*Sj,i-1)+Sj,i-1).*v*sqr1/period)*Zj,i-1)); payoff=meamaxmeas:,--1):)')-k,0)); price=exp-irae2*ime2))*payoff; Malab Code 8 Call Opio Mesh for Calculaig Local Volailiy Surface fucio Call_lc srikes_lc Vol_lc=CallPricesMesh_locVoliRae2, iraef2, ime2, spo2, vol, srikes2) K1=1750:1:2300; V=zerosleghime2),leghK1)); Call_lcAux=zerosleghime2),leghK1)); for :leghime2) for j=1:leghk1) %if K1j)<srikesi,2) % Vi,j)=voli,2); % %if K1j)>srikesi,) % Vi,j)=voli,); % %if K1j)>=srikesi,2) && K1j)<=srikesi,) Vi,j)=spliesrikes2i,:),voli,:),K1j)); % d1=logspo2/k1j))+irae2-iraef2+vi,j)^2)/2)*ime2i)))/vi,j)*sqrime2i))); d2=d1-vi,j)*sqrime2i)); N1=ormcdfd1,0,1); N2=ormcdfd2,0,1); Call_lcAuxi,j)=spo2*exp-iRaef2*ime2i))*N1-K1j)*exp-iRae2*ime2i))*N2; srikes_lc=k1; Call_lc=Call_lcAux; Vol_lc=V; Malab Code 8 Call Opio Mesh for Calculaig Local Volailiy Surface fucio vol_local sr_local=vol_localcall_lc, srikes_lc, ime2, irae2, iraef2, Vol_lc) %% This fucio esimaes he local volailiy surface usig Dupire %% Formula wih impus Call Prices ad Srikes K = srikes_lc; C = Call_lc; T = ime2; r = irae2; q = iraef2; K = sizek,2); T = sizec,1); % Defie dela for T dela_t=zeros1,t); for :T-1 dela_ti)=ti+1)-ti); % Defie dela for K dela_k=zeros1,k); for j=1:k-1 dela_kj)=kj+1)-kj); % Parcial derivaive of Call respec o T % Forward Fiie Diferece d_c_t=zerost,k); for :T-1 for j=1:k 47

48 d_c_ti,j)=ci+1,j)-ci,j))/dela_ti); % parcial derivaive of Call respec o K % Ceral Fiie Diferece d_c_k=zerost,k); for :T-1 for j=2:k-1 d_c_ki,j)=ci,j+1)-ci,j-1))/2*dela_ki)); % Secod parcial derivaive of Call respec o K % Ceral Fiie Diferece dd_c_k=zerost,k); for :T-1 for j=2:k-1 dd_c_ki,j)=ci,j-1)-2*ci,j)+ci,j+1))/dela_ki)*dela_ki)); Loc_vol=zerosT,K); for :T-1 for j=2:k-1 if dd_c_ki,j)==0 d_c_ti,j)==0 d_c_ki,j)==0 Loc_voli,j)=0; else Loc_voli,j)=sqr2*d_C_Ti,j)+r-q)*K1,j)*d_C_Ki,j)))- q*ci,j))/k1,j)^2*dd_c_ki,j))); vol_local=vol_lc1,2:-1);loc_vol1:-1,2:-1); % Sice he firs Iplici Volailiy is Local as well sr_local=k2:-1); Malab Code 9 Mai Scrip Colombia Applicaio %% GENERAL BOINDS FOR ARITHMETIC ASIAN OPTION PRICES %% Colombia Applicaio ic Srike=1700:50:2250; x0=3000; able=; srikes=xlsread'srikes.xlsx'); % This deps o volailiy surface irae=0.0339; iraef= ; % coiuos compouded vol=xlsread'volailiy.xlsx'); %Time x srikes ime=30:30:330,365'/365; spo=1922; Ave_period=12; %% for q=1:leghsrike) K=Srikeq); %% Call Prices C=zerossizesrikes)); for :sizesrikes,1) for j=1:sizesrikes,2) d1=logspo/srikesi,j))+irae-iraef+voli,j)^2)/2)*imei)))/voli,j)*sqrimei))); d2=d1-voli,j)*sqrimei)); N1=ormcdfd1,0,1); N2=ormcdfd2,0,1); Ci,j)=spo*exp-iRaef*imei))*N1-srikesi,j)*exp-iRae*imei))*N2; %% UPPER BOUND % Sep 1 if sumsrikes:,))<= Ave_period*K k=squeezesrikes:,)); Id=1; else k=k*oesave_period,1); Id=0; %Sep 2 if Id==0 Dif=-1; %I esures he while loop will ru alea oes Dsup=0; while Dif<Dsup posiios=zerosave_period,3); DelaP=zerosAve_period,1); DelaN=zerosAve_period,1); for :Ave_period l1=fidsrikesi,:)<ki)); 48

49 if umell1)==0 posiiosi,1)=1; else posiiosi,1)=l1); l1=fidsrikesi,:)>ki)); if umell1)==0 posiiosi,3)=sizesrikes,2); else posiiosi,3)=l11); l1=fidsrikesi,:)==ki)); if umell1)==0 posiiosi,2)=posiiosi,3); else posiiosi,2)=l11); if posiiosi,2)<posiiosi,3) if posiiosi,3)==sizesrikes,2) DelaPi)=0; else DelaPi)=Ci,posiiosi,3))-Ci,posiiosi,2)))/srikesi,posiiosi,3))- srikesi,posiiosi,2))); DelaNi)=Ci,posiiosi,2))-Ci,posiiosi,1)))/srikesi,posiiosi,2))- srikesi,posiiosi,1))); else DelaNi)=Ci,posiiosi,3))-Ci,posiiosi,1)))/srikesi,posiiosi,3))- srikesi,posiiosi,1))); DelaPi)=DelaNi); %Sep 3 DelaP=exp-iRae*ime)-ime)).*DelaP; DelaN=exp-iRae*ime)-ime)).*DelaN; I1=fidDelaP==miDelaP)); I=I11); Dif=DelaPI); J1=fidDelaN==maxDelaN)); J=J11); Dsup=DelaNJ); if Dif<Dsup if srikesi,posiiosi,1))<=ki) && srikesi,posiiosi,3))>ki) Ui= srikesi,posiiosi,3))-ki); if srikesj,posiiosj,1))<kj) && srikesj,posiiosj,3))>=kj) Uj= kj)-srikesj,posiiosj,1)); ki)=ki)+miui,uj); kj)=kj)-miui,uj); %Sep 4 M=; for :Ave_period I=fidsrikesi,:)==ki)); if umeli)==0 M=M;i; if leghm)>1 while leghm)>1 U1=srikesM1),posiiosM1),3))-kM1)); U2=kM))-srikesM),posiiosM),1)); km1))=km1))+miu1,u2); km))=km))-miu1,u2); if miu1,u2)==u1 M=M2:); else M=M1:-1); f=zerosave_period,1); for :Ave_period fi)=ci,posiiosi,2)); if leghm)==1 fm)=srikesm,posiiosm,3))-km))/srikesm,posiiosm,3))-srikesm,posiiosm,1)))*... CM,posiiosM,1))+kM)-srikesM,posiiosM,1)))/srikesM,posiiosM,3))- srikesm,posiiosm,1)))*... CM,posiiosM,3)); CosUpper=sumexp-iRae*ime)-ime)).*f)/Ave_period; 49

50 l2=fidsrikes1,:)<=k); if umell1)==0 p=2; else p=l2); CosUpperTrivial=sumexp-iRae*ime)-ime)).*C:,p))/Ave_period; %% LOWER BOUNDS %% LB1 % Fidig he closes K o K1 K1=Ave_period*K)/sumexpiRae-iRaef)*ime-ime1)))); CosLB1=CallPriceEs1,iRae,ime,C,srikes,K1); CosLB1=CosLB1*1/Ave_period)*sumexp-iRae*ime)-ime)-iRaef)*ime-ime1)))); %% LB1 CLB1=zerosAve_period,1); K1=zerosAve_period,1); for i=2:ave_period K1i)=Ave_period*K-sumexpiRae-iRaef)*ime1:i-1))*spo))/sumexpiRaeiRaef)*imei:)-imei)))); K1i)=maxK1i),0); CLB1i)=CallPriceEsi,iRae,ime,C,srikes,K1i)); CLB1i)=CLB1i)*sumexpiRae*imei:)-iRaef*imei:)-imei)))); CLB1=exp-iRae*ime))*CLB1/Ave_period; CLB11)=CosLB1; CosLB1=maxCLB1); aux=fidclb1==coslb1); poslb1=aux1); %% LB2 LB2=zerosAve_period,1); % c3=; for :Ave_period c2=fzero@c2) fidc2c2,i,ave_period,k,spo,ime,irae-iraef), x0); % c3=c3 c2; %PowerCalli,x,K,srikes,C) powcall=zerosi-1,1); powcall2=zerosi-1,1); % PC=zerosi-1,1); PCbs=zerosi-1,1); for z=1:i-1 aux=fidsrikesi,:)<=c2); PCbsz)=PowerCallBScurrimez)/imei),spo,iRae, iraef, c2, voli,aux)), imei)); powcallz)=spo^1-imez)/imei)))*expirae*imei))*... PCbsz); PCz)=PowerCalli,imez)/imei),c2,srikes,C); powcall2z)=spo^1-imez)/imei)))*expirae*imei))*... PCz); %CallPriceEsi,iRae,ime,C,srikes,K) callc2=callpriceesi,irae,ime,c,srikes,c2); LB2i)=sumpowCall)+sumexpiRae*imei:)-iRaef*imei:)-imei))))*... callc2; LB2lwi)=sumpowCall2)+sumexpiRae*imei:)-iRaef*imei:)-imei))))*... callc2; aux1=maxlb2); CosLB2=exp-iRae*ime))/Ave_period*aux1; aux=fidlb2==aux1); poslb2=aux1); aux2=maxlb2lw); CosLB2lw=exp-iRae*ime))/Ave_period*aux2; aux3=fidlb2lw==aux2); poslb2lw=aux31); %% Asia Call Opio Price - Local Volailiy Dupire) Framework - MC aux=fidsrikes1,:)<=k); MCcurr=ACbsMCcurrspo,iRae,iRaef, K, vol, srikes, ime, Ave_period, 'moh'); able=able; K CosLB1 CosLB1 poslb1 CosLB2 poslb2 MCcurr CosUpper CosLB2lw poslb2lw; xlswrie'able.xlsx',able); oc 50