Pricing Asian Options with Fourier Convolution

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1 Prcg Asa Opos wh Fourer Covoluo Cheg-Hsug Shu Deparme of Compuer Scece ad Iformao Egeerg Naoal Tawa Uversy

2 Coes. Iroduco. Backgroud 3. The Fourer Covoluo Mehod 3. Seward ad Hodges facorzao 3. Re-ceerg he deses 3.3 The erpolao formula 3.4 The prcg algorhm 3.5 The choce of parameers 4. Numercal Resuls 4. Dscree case 4. Couous case 5. Coclusos Bblography

3 Absrac Ths hess vesgaes he fas Fourer rasform-based prcg algorhm for dscree Asa opos by Behamou []. We compare wh oher mehods ad combe wh exrapolao o crease umercal accuracy. We also apply o he couous case by usg exrapolao. Rug he algorhm wh dffere umbers of grd pos, we observe he covergece of opo values boh he couous case ad he dscree case. The dsadvaages of he algorhm are also dscussed. 3

4 Chaper Iroduco Asa opos are pah-depede coge clams whose payoff s based o a average of uderlyg prces, eres raes, dces, or couless ohers. They have become popular hedgg perodc cash flows ha hey cos far less ha sadard opos o he same uderlyg asses. As poed ou by Levy [7], he reaso Asa opos are cheaper ha he oherwse decal sadard opos s explaed by he fac ha he varace of he Asa opo s smaller ha ha of he uderlyg asse s prce process uder he Black-Scholes model. Asa opos ca mgae he possbly of spo mapulaos or exreme movemes of uderlyg prces a seleme. Ths feaure s especally useful hly raded asses markes whch case he prce mapulao o or ear he exprao dae has a sgfca mpac o he payoff of sadard opos. I summary, he averagg feaure of Asa opos become aracve for hedgg because ca avod he large volaly of he prce chage ad ca also remove exreme sesvy of sadard opos payoff o he uderlyg prce. There are wo ma classes of Asa opos: floag-srke ad fxed-srke. The floag-srke Asa opo pays he dfferece bewee he average ad he spo prce of he uderlyg. The fxed-srke Asa opo pays he dfferece bewee he average prce of he uderlyg ad he pre-specfed srke prce. Asa opos ca also be classfed as dscree or couous accordg o he way he average s calculaed. Whe he average s calculaed from uderlyg asse s prces a dscree mes, s called a dscree Asa opo. If all of he uderlyg s prces o he me le ake par he calculao of average, s called a couous Asa opo. Whe he al uderlyg asse s prce does o ake par he calculao of he average, he opo s 4

5 called a forward-sarg Asa opo. We wll deal exclusvely wh forward-sarg dscree Asa opos hs hess uless saed oherwse. Uder he Black-Scholes opo model, he asse prce follows a geomerc Browa moo. Thus he asse prce a ay fuure me s descrbed by he logormal desy fuco. If he Asa opo s based o geomerc average, he average s sll logormally dsrbued because he produc of logormal radom varables remas logormal. I hs case, s possble o derve a explc formula for geomerc averagg Asa opos (see, for example, Kema ad Vors [6] ad Zhag [5]). However, f he Asa opo s based o arhmec average, here s o explc represeao for he dsrbuo of he average of he uderlyg asse s prces because he sum of logormal radom varables s o logormally dsrbued ay more. Thus, here s o explc prcg formula for arhmec averagg Asa opos as of ow, ad hs s he source of dffculy prcg hem. Ths hess focuses o arhmec averagg Asa opos. The ma goal s o approxmae s probably desy fuco by usg dscree pos o he desy fuco s doma o represe he dsrbuo fuco of average. Several approaches have bee proposed he leraure o ackle he dffculy of prcg Asa opos. We classfy hem as follows.. Moe-Carlo smulaos wh varace reduco echques. Kema ad Vors [6] derve a prcg formula for geomerc-based dscree Asa opos ad used as a corol varae o reduce he varace of he dscree Asa opo prces.. Bomal ree. Hull ad Whe [5] augme a addoal sae varable o each ode he ree o record he possble averages of he uderlyg asse s prce realzed 5

6 bewee me zero ad he me of ha ode. Cho ad Lee [3] mprove by dervg he maxmum ad mmum averages for each ode. Hsu ad Lyuu [4] furher mprove by usg o-uform allocao scheme of saes each ode accordg o s probably. All above are umercal mehods for couous Asa opos. 3. Approxmao of he desy fuco of he average. Turbull ad Wakema [3] apply he Edgeworh seres expaso up o he fourh erm aroud he logormal dsrbuo fuco o approxmae he desy fuco of he average for dscree Asa opos. Levy [7] derves a approxmae prcg formula for dscree Asa opos by machg he frs wo momes of he desy of he average wh ha of he logormal desy. Carverhll ad Clewlow [] use dscree pos o he desy fuco s doma o represe he desy fuco ad evaluae he covoluo of desy fucos o approxmae he desy fuco of he average for dscree Asa opos. Behamou [] mproves by corporag a re-ceerg sep o he algorhm. 4. Paral dffereal equaos Zhag [4] derves a aalycal approxmae formula ad a correco erm govered by a paral dffereal equao, whch requres umercal evaluao, for couous Asa opos. Rogers ad Sh [] compue he prce of couous Asa opos by reducg he prcg problem o ha of solvg a PDE wh he fe-dfferece mehod. The mehodology adoped hs hess o prce Asa opos belogs o caegory 3. The res of he hess s orgazed as follows. I chaper, we se up he framework ad he roduce basc facs for laer use. Chaper 3 develops a procedure o calculae 6

7 he desy fuco of he average represeed by dscree pos he desy fuco s doma ad descrbes he prcg algorhm. The cosderaos for he choce of parameers he algorhm are also dealed. Chaper 4 preses he umercal resuls. From hem mpora characerscs of he Fourer covoluo mehod ca be draw. Coclusos are gve Chaper 5. 7

8 Chaper Backgroud Cosder a Asa call opo wh maury T, srke K, ad fxg daes durg s lfe. The uderlyg s prce a me s deoed by S. Oly he prces S o fxg daes ake par he calculao of he arhmec average. We dvde he oal legh of he dervave s lfe o me ervals of equal legh. There s a fxg dae bewee wo cosecuve ervals. Assume hese fxg daes are deoed by,,, wh = T ad he al me of he opo s deoed by 0. The average prce s he defed by A = = S () We also defe he rae of reur for each of hese ervals as R S = log S () We assume he Black-Scholes model, where he uderlyg s prce follows a geomerc Browa moo, ds = μ Sd + σsdz where dz s a Browa moo whose cremes are ucorrelaed, σ s he volaly of he uderlyg s prce, ad μ s s expeced rae of reur. The assumpo mples ha he uderlyg s prce a ay me S ca be expressed erms of he precedg prce S as S μ σ ( ) + σz = S e (3) where z s a Browa moo. I s mos uvely o hk of z as beg ormally 8

9 dsrbued wh mea 0 ad varace. We ca vew hs expresso as he produc of he precedg prce ad he rae of reur. I parcular, R S = S e (4) Comparg equaos (3) ad (4), we see ha he rae of reur σ () follows a ormal dsrbuo wh mea μ ( ) R defed by equao ad varace σ ). ( Tha s, E[ R Var[ R σ ] = μ ] = σ ( ) ( ) Noe ha f each erval s o of equal legh, he each rae of reur R correspodg o each erval sll follows a ormal dsrbuo bu has dffere mea ad varace. Aleravely, equao (4) ca also be expressed erms of he al prce S as 0 0 R R L R S = Se (5) Subsue he above equao (5) o expresso () of he average o ge R + R + L+ R Se 0 = A = (6) I complee markes wh o arbrage opporues, here exss a uque rsk-eural probably measure uder whch he prce process of he dervave s a margale. I hs case, he mea of he rae of reur R wll be r ( ) σ ad he prce of he Asa call C s he expeced payoff uder he measure dscoued by he rsk-free eres rae r : Q rt C = E [ e ( A K) + ] (7) 9

10 where max(x + X sads for,0). As saed above, he dsrbuo of he average s ukow; here s o smple closed-form formula o calculae equao (7). Isead, we wll umercally compue he desy fuco backwards he me le by meas of represeg he desy fuco a dscree pos. The mehod coverges o he real desy fucos as he umber of such pos eds o fy. The covoluo of wo fucos f (x) ad g (y) s defed as x= C ( z) = f ( x) g( z x). The par of Fourer rasform ad s verse rasform for fuco f (x) s defed as F( k) = f ( j) = N N N j= N k = f ( j) e F ( k) e π ( k ) ( j ) N π ( k ) ( j ) N We sae he followg wo facs for laer use. Fac Suppose ha X ad Y are depede radom varables wh he jo dsrbuo fuco f ( x, y), ad le Z = X + Y. The dsrbuo fuco of Z s he covoluo of he dsrbuo fuco f (x) for X ad he dsrbuo fuco g (y) for Y. Fac The Fourer rasform of he covoluo of wo fucos f (x) ad g (y) equals he produc of F (x) ad G (y), whch are he Fourer rasforms of f (x) ad g (y), respecvely. The covoluo of he wo fucos ca be obaed by akg he verse Fourer rasform of he produc of F (x) ad G (y). 0

11 Chaper 3 The Fourer Covoluo Mehod The Fourer covoluo mehod represes he desy fuco by dscree grd pos wh a fxed-wdh wdow o he desy fuco s doma. Fgure Fgure Fgure shows he graph of he desy fuco for a sadard ormal radom varable X. The horzoal axs represes he possble values of X, ad he vercal axs represes he correspodg desy value. Noe ha he desy values for large absolue values of X ed o zero. Fgure shows he represeao of he desy fuco correspodg o Fgure he Fourer covoluo mehod. The mehod frs eeds o deerme he wo parameers: he umber of grd pos ad he wdow wdh. I he case of Fgure, here are 3 grd pos he wdow, ad he wdow wdh s 6 ragg from 3 o +3. Noe ha he Fourer rasform requres ha hese grd pos be equally spaced. Oly hose desy values o grd pos are recorded he represeao of he desy fuco. Obvously, he umber of grd pos has sgfca mpac o he accuracy of represeg he desy fuco. Errors wll evably be produced f we eed he desy values for o-grd pos. The fxed-wdh wdow defes he doma of he desy fuco. The desy values a

12 grd pos ou of he wdow wll be assumed o be zero. Thus he wdow should be wde eough o coa he bulk of he desy fuco, bu should o be oo large as o cosume oo much resource. To sum up, he desy fuco wll be lmed o hese desy values o dscree grd pos, ad compug he desy fuco s equvale o compug hese desy values o dscree grd pos. A he begg, he mea of he al desy fuco wll be locaed a he ceer of he wdow. As he process of compug he desy fuco for he average progresses, he wdow wll be shfed o make sure he locao of he mea of he desy fuco remas roughly a he ceer. The followg develops he procedures o compue he desy fuco for he average deal. 3. Seward ad Hodges facorzao The average s expressed equao (6) as a fuco of he raes of reurs. Each rae of reur R follows a ormal dsrbuo wh mea ( r σ )( ) ad varace σ ). From he formao abou he dsrbuo for he rae of reur ( each erval, we ca compue he desy fuco for he uderlyg asse s prce a ay parcular me, bu he desy fuco for he average s sll hard o compue. Because he summads he average are o depede, Fac cao be appled o compue he desy fuco for he average. I order o apply Fac, he followg proposo s eeded. Proposo (Seward ad Hodges facorzao) The average equao (6) ca be expressed as: Proof: From equao (6), we have S0 A = exp( R + l( + exp( R l( l( exp R )))))

13 R + R + L+ R S e 0 = A = S 0 R R R R R R R 3 R = ( e + e e + e e e + L+ e L e ) S 0 R R R R 3 = e ( + e ( + e + ( + L + ( + e )))) S = e e R R R 3 0 R l( + e ( + e + ( + L+ ( + e )))) S = e R R R 3 0 R l( ( ( ( + + e + e + + L+ + e )))) S = e R R R 3 0 R l( exp(l( ( ( + + e + e + + L+ + e ))))) S = e 3 0 l( exp( l( R R R ( ( + + R ))))) + + e + + L+ + e = S R l( exp( l( exp( l( l( 0 R + + R )))))) + + R + + L+ + e 3 e (8) Afer applyg Proposo, he summads he expoeal erm of he equao (8) for he average are depede ad ow we ca apply Fac recursvely ad backwards me o compue he desy fuco for he average. Before proceedg, we defe he sequece B as B R + l( + exp B ) +, =,3, K, (9) = wh B =. The equao (8) ca be compacly wre as R A S 0 B = e (0) Our objecve s fac o compue he desy fuco for B sep by sep from B, whch s kow o be ormally dsrbued as dscussed before. Equao (9) gves a formula ha ca be used o compue he desy fuco of B from ha of B 3

14 recursvely. 3. Re-ceerg he deses + Noce ha he erm l( exp B ) he rgh had sde of equao (9) would cause he desy fuco compued a he precedg sep o shf. More specfcally, he probably a ay po b would be mapped o he probably a he po l( + expb), whch resuls he shf locao of he desy fuco. I order o preve he desy fucos from shfg ou of he wdow, a each sep we wll move he wdow so as o f he desy fucos as much as possble ad make he mea of he desy roughly a he ceer. We ex deerme how dsa he wdow would move. If we kew he mea of l( exp B ), we could drecly move he ceer of he wdow o ha locao. + Suppose ha we kow he mea m of B whereas he mea of l( + exp B ) s o avalable. We approxmae he mea of l( exp B ) by l( exp m ). The + + we ca defe he followg sequece o approxmae he mea of B defed by equao (9): m u + l( exp m ) () = + + wh m = u, where u = E[ R ], whch s ( r σ )( ) uder he Black-Scholes model. Remember ha he oal lfe of opo s dvded o ervals of equal legh ad all R are ormally dsrbued wh he same mea ad varace. Therefore, all u + wll be he same ad equal ( r σ )( ). If hese ervals are o of equal legh, he hese R are sll ormally dsrbued bu have dffere meas ad varaces. Equao () gves a approxmaed E[ B ] from he prevously approxmaed E[ B ]. The ceer of he wdow s he se o he approxmaed E[ B ]. 4

15 Noe ha he fuco l( + exp x) s covex ad hus he approxmaed mea uderesmaes he rue mea, whch ca be proved by Jese s equaly: f (E[ X ]) E[ f ( X )]. We defe he ceered sequece for he equao (9) as A = B m. I effec, hs moves he ceer of he wdow o he approxmaed m. The expresso for he average defed by equao (0) ca be expressed erms of he ceered sequece as A = S 0 e A + m () where A s derved as follows A = B m = R + l( + exp B ) m + = R + l( + exp A exp m ) m + (3) wh al codo A = R m. Wh he re-ceerg sep corporaed o he algorhm, we compue he approxmae desy fuco for A sep by sep from A. Noe ha he al radom varable A s kow o be ormally dsrbued ad s mea s ceered he wdow. Equao (3) gves a formula o compue he desy fuco of A from ha of A whch has bee compued a he precedg sep. 3.3 The erpolao formula Suppose we are compug he desy fuco of A a sep. From equao (3), we eed he covoluo of he desy fucos of R + ad l( A m m + exp exp ). The dsrbuo fuco for R + s kow ad he dsrbuo fuco for A has bee compued he precedg sep. The desy 5

16 fuco for l( A m m + exp exp ) s also kow ad represeed by grd pos equally spaced he doma of A, bu hese grd pos are o equally spaced he doma of l( A m m + exp exp ) because he fuco s o-lear. Bu he Fourer rasform requres ha he grd pos represeg he desy fuco be equally spaced. Thus wha we eed o compue s he desy values for l( A m m + exp exp ) o o-grd pos before applyg covoluo so ha hese ew grd pos are also equally spaced. I geeral, he desy value for a fuco y of a radom varable x s f x ( y ) d dy y, where f x represes he desy fuco for x ad y represes he verse fuco for y []. Ths leads o he followg erpolao formula: f a+ m e a+ m exp ) ( ) (l( ) ) = m m a f a+ m A e m (4) e l( + exp A whch gves he desy value of l( A m m + exp exp ) a grd po a he doma of l( A m m + exp exp ) from ha of a+ m A a po l( e ) m he doma of A. Noe ha he erpolao formula (4) wll roduce errors because of he dscrezao of he desy fucos. Specfcally, f a s o a grd a+ m po he doma of A, he l( e ) m wll o be o a grd po he doma of A. Thus applyg formula (4), erpolao wll be used o ge he a+ m desy value a po l( e ) m bewee he wo eares grd pos he doma of A. The errors caused by erpolao wll accumulae as he umber of applyg formula (4) creases. Alhough he formula (4) s exac, he erm a+ m f A (l( e ) m ) s o exac ad ca oly be obaed by erpolao []. 6

17 3.4 The prcg algorhm All he eeded procedures o compue he desy fuco for he average have bee developed prevous secos. Now we merge hem o oba he prcg algorhm. The algorhm ally calculaes he approxmaed mea m = u for B = R ad he ceered desy fuco for A = R m. Noe ha he ceered desy fuco for A s dscrezed ad represeed by dscree grd pos a fxed-wdh wdow. The objecve s o ge he desy fuco of A + m, ha s, he desy values a grd pos. All he operaos performed wll be o hese grd pos. Iducvely, suppose we kow he values of he m ad he desy fuco of A compued a sep. We he recursvely execue he followg procedures o compue he ex approxmaed mea ad ceered desy fuco ul we ge he value of m ad he desy fuco of A :. Ierpolae he desy fuco for A usg formula (4) o ge he desy fuco for l( + exp A expm ) m.. Compue he desy fuco for A by Fac. Noe ha A s he sum of he wo depede radom varables R + ad l( + exp A expm ) m. Oce we have goe he approxmaed desy fuco for he average, we compue he expeced payoff by umercal egrao ad he dscou by he rsk-free eres rae o oba he opo value. The above complees Behamou s algorhm []. 3.5 The choce of parameers 7

18 There are wo parameers, whch are he umber of grd pos ad he wdow wdh, he algorhm. Eher of he wo parameers ca affec he accuracy of he algorhm. How o choose a approprae umber of grd pos depeds o he requred accuracy. If oe desres a more accurae value, he ca crease he umber of grd pos a he expese of compuao me. The wdow wdh should be chose a he begg of he algorhm ad he fxed so as o be large eough o coa he bulk of he desy fuco. However, he larger he wdow wdh s, he less accurae he opo value s whe he umber of grd pos s fxed. I Behamou s orgal paper, he chooses he wdow wdh as T 9 σ. Ths choce produces less accurae resuls our expermes because s wdh s oo large whe gve he same umber of grd pos. So we ry o reduce he wdow wdh so ha he accuracy ca be mproved. From emprcal rule, we kow ha abou 99.7% of probably s wh he erval bewee +3 ad 3 sadard devaos. Because he desy fuco of he al radom varable A = R m s ormally dsrbued, he wdow wdh should be a leas larger ha T 6 σ order o coa he bulk of he al desy fuco. Whe he umber of covoluo operaos creases, more ad more probably wll be a he al of he wo eds of he dsrbuo ad hus he wdow wdh should also crease. Through exesve expermes, we fd ha he crease he wdow wdh should be roughly proporoal o o acheve good accuracy. So we choose he al wdow wdh as 6 σ T so as o coa he bulk of he desy fuco. Ths choce s dffere from Behamou s choce, ad produces accurae resuls. 8

19 Chaper 4 Numercal Resuls I hs chaper, we compare he Fourer covoluo mehod wh oher mehods prcg dscree Asa opos. I order o furher speed up he covergece of opo values, we corporae exrapolao o he Fourer covoluo. We fally apply exrapolao o he dscree verso of Asa opos o oba prces of he couous verso. 4. Dscree case Table shows he resuls wh dffere mehods for prcg dscree Asa opos wh dffere srkes. The wdow wdh of Fourer covoluo (FC) Table s he same as Behamou s choce. Is resuls are far less accurae compared o he oher mehods because he wdow wdh s oo wde. Table shows he resuls correspodg o Table excep ha he wdow wdhs of FC follow our choce. As he resuls show, s accuracy s mproved, alhough s sll o as accurae as he oher mehods. Bu whe he umber of grd pos doubles, he covergece rae also doubles wh a cocurre mproveme accuracy. Ths lear relaoshp bewee he umber of grd pos ad covergece rae suggess ha exrapolao ca be used order o crease accuracy. Exrapolao s a echque o speed up he covergece rae by usg wo approxmaed opo values. The formula s f = ( ) f f ( ) where ad are wo dffere choces of umbers of grd pos. Whe =, he exrapolao s called he Rchardso exrapolao. Table 3 shows he resuls whe he Rchardso exrapolao s corporaed o he FC mehod. The umbers of grd 9

20 pos uder FC Table 3 are he values of. The FC mehod whou exrapolao s o accurae eough; however, s accuracy ouperforms he oher mehods f s combed wh exrapolao. 4. Couous case All he uderlyg asse s prces o he me le ake par o he prcg of couous Asa opos. Thus, heory he lfe of opos should be dvded o a fe umber of me ervals. Exrapolao echque s used o approxmae he resul. As he dscree case, he same formula s used wh ad beg wo dffere choces of umber of ervals. Table 4 shows he resuls whe = 00 ad uses Rchardso exrapolao. Table 5 shows he resuls whe = 80 ad = 40. There are wo exrapolao sages he couous case. We kow from he resuls he dscree case ha he FC mehod whou exrapolao s o accurae eough. So we use Rchardso exrapolao he frs sage o oba approxmaed values as he umber of grd pos eds o fy. The umbers of grd pos uder he FC Table 4 ad Table 5 are he values of for exrapolao he frs sage. The we apply exrapolao he secod sage by usg he resuls he prevous sage o oba he approxmaed values as he umber of ervals eds o fy he couous case. The resuls boh Table 4 ad Table 5 show ha he opo value coverges o some value, alhough s dffere from he exac value. Bu f hgher ad he secod exrapolao sage are used, he resuls ca be more accurae. Because he resuls Table 5 use more me ervals o exrapolae ha Table 4, s more accurae f he umber of grd pos creases. However, as he umber of me 0

21 ervals creases, he umber of mes he erpolao formula (4) s appled also creases, whch resuls more accumulaed errors. Tha he approxmaos Table 5 usg 3 are slghly less accurae ha hose Table 4 wh he same umber of grd pos verfes hs fac. The accumulaed errors ca be lowered by usg more grd pos, ad hs also ca be see from approxmaos Table 5 as he umber of grd pos creases.

22 Chaper 5 Coclusos The FC s a effce prcg algorhm for dscree Asa opos. Le m be he umber of me ervals ad be he umber of grd pos used he algorhm. The complexy of FC s O ( m l ). I pracce, m s coracual ad raher small compared o he dscree case. The combed verso of FC wh exrapolao becomes a fas ad accurae prcg algorhm for dscree Asa opos. However, m wll be large for he couous case, whch resuls more applcaos of he erpolao formula (4). I order o compesae he accumulaed errors caused by erpolao, mus also be creased. Because boh m ad crease a he same me, he compuao me also creases rapdly. Ths s he ma dsadvaage of he FC f s o be appled o he couous case.

23 Bblography [] E. Behamou, Fas Fourer Trasform for Dscree Asa Opos. Joural of Compuaoal Face 6, 00, [] A. Carverhll ad L. Clewlow, Flexble Covoluo. Rsk 3(4), 990, 5 9. [3] H.Y. Cho ad H.Y. Lee, A Lace Model for Prcg Geomerc ad Arhmec Average Opos. Joural of Facal Egeerg 6(3), [4] W.W.-Y. Hsu ad Y.-D. Lyuu, A Coverge Quadrac-Tme Lace Algorhm for Prcg Europea-Syle Asa Opos. I Proceedgs of LASTED Ieraoal Coferece o Facal Egeerg ad Applcaos, 004. [5] J.C. Hull ad A. Whe, Effce Procedures for Valug Europea ad Amerca Pah-Depede Opos. Joural of Dervaves, 993, 3. [6] A.G.Z. Kema ad A.C.F. Vors, A Prcg Mehod for Opos Based o Average Asse Values. Joural of Bakg ad Face 4, 990, 3 9. [7] E. Levy, Prcg Europea Average Rae Currecy Opos. Joural of Ieraoal Moey ad Face, 99, [8] E. Levy ad S. Turbull, Average Iellgece. Rsk 5(), 99, 5 9. [9] S.-L. Lao ad C.-W. Wag, Prcg Arhmec Average Rese Opos wh Corol Varaes. Joural of Dervaves, Wer 00, [0] F.A. Logsaff, Hedgg Ieres Rsk wh Opos o Average Ieres Raes. Joural of Fxed Icome, March, 995, [] J. A. Rce, Mahemacal Sascs ad Daa Aalyss, d Edo. Belmo, CA: Duxbury Press, 994. [] L.C.G. Rogers ad Z. Sh, The Value of a Asa Opo. Joural of Appled Probably 3, 995, [3] S. Turbull ad L. Wakema, A Quck Algorhm for Prcg Europea Average Opos. Joural of Facal ad Quaave Aalyss 6, 99, [4] J.E. Zhag, A Sem-aalycal Mehod for Prcg ad Hedgg Couously Sampled Arhmec Average Rae Opos. Joural of Compuaoal Face 5(), 00, [5] J.E. Zhag, Prcg Couously Sampled Asa Opos wh Perurbao Mehod. Joural of Fuures Markes 3(6), 003,

24 Approxmaos FC wh dffere umbers of grd pos Srkes MC SD Levy TW Vors Hsu-Lyuu Table : Comparso of dffere prcg mehods: he case of 3-year dscree Asa opos. The opos are forward-sarg Asa calls wh S = 00, T = 3, σ = 0.5, r = 0.04 ad = 36 (mohly averagg). The wdh of he FC mehod s 9 σ T, whch s 3.3 hs case. 4

25 Approxmaos FC wh dffere umbers of grd pos Srkes MC SD Levy TW Vors Hsu-Lyuu Table : Comparso of dffere prcg mehods: he case of 3-year dscree Asa opos. The opos are forward-sarg Asa calls wh S = 00, T = 3, σ = 0.5, r = 0.04 ad = 36 (mohly averagg). The wdh of he FC mehod s.6. 5

26 Approxmaos FC wh dffere umbers of grd pos Srkes MC SD Levy TW Vors Hsu-Lyuu Table 3: Comparso of dffere prcg mehods: he case of 3-year dscree Asa opos. The opos are forward-sarg Asa calls wh S = 00, T = 3, σ = 0.5, r = 0.04 ad = 36 (mohly averagg). The wdh of he FC mehod s.6. The Rchardso exrapolao wh dffere = values s corporaed o he FC. 6

27 Approxmaos FC wh dffere = Srkes σ Exac AA AA3 Hsu-Lyuu , Table 4: Comparso wh Zhag (00, 003) wh a wde rage of volales: he case of -year couous Asa opos. The parameers are from Table of Zhag (003). The opos are calls wh S=00, r=0.09, ad T =. The wdh of he FC mehod s 6 σ T. The Rchardso exrapolao wh dffere = values s corporaed o he FC. The couous opo values for FC are approxmaed by dvdg T o 00 perods ad he usg Rchardso exrapolao. 7

28 Approxmaos FC wh dffere = Srkes σ Exac AA AA3 Hsu-Lyuu , Table 5: Comparso wh Zhag (00, 003) wh a wde rage of volales: he case of -year couous Asa opo. The parameers are from Table of Zhag (003). The opos are calls wh S=00, r=0.09, ad T =. The wdh of he FC mehod s 6 σ T. The Rchardso exrapolao wh dffere = values s corporaed o he FC. The couous opo values for FC are approxmaed by dvdg T o 80 perods ad usg exrapolao wh =40 perods. 8

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