Pricing maximum-minimum bidirectional options in trinomial CEV model

Peng, Bin; Peng, Fei A icle P icing maximum-minimum bidi ec ional op ions in inomial CEV model Jou nal o Economics, Finance and Adminis a i e Science P o ided in Coope a ion wi h: Uni e sidad ESAN, Lima Sugges ed Ci a ion: Peng, Bin; Peng, Fei (2016) : P icing maximum-minimum bidi ec ional op ions in inomial CEV model, Jou nal o Economics, Finance and Adminis a i e Science, ISSN 2218-0648, Else ie España, Ba celona, Vol. 21, Iss. 41, pp. 50-55, h ps://doi.o g/10.1016/j.je as.2016.06.001 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/179776 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ Jou nal o Economics, Finance and Adminis a i e Science 21 (2016) 50–55 Jou nal o Economics, Finance and Adminis a i e Science www.else ie .es/je as A icle P icing maximum-minimum bidi ec ional op ions in inomial CEV model夽 Bin Penga,∗, Fei Pengb aSEME, Beijing Uni e si y o Ci il Enginee ing and A chi ec u e, Beijing, P. R. China bElec ical & Compu e Enginee ing, UBC, Vancou e , Canada a i c l e i n o A icle his o y: Recei ed 10 June 2016 Accep ed 15 June 2016 JEL classifica ion: C14 F17 Keywo ds: T inomial CEV model Recu si e algo i hm Maximum-minimum bidi ec ional op ions a b s a c Maximum-minimum bidi ec ional op ions a e a kind o exo ic pa h dependen op ions. In he cons an elas ici y o a iance (CEV) model, a combining inomial ee was s uc u ed o app oxima e he non- cons an ola ili y ha is a unc ion o he unde lying asse . On his basis, a simple and e ficien ecu si e algo i hm was de eloped o compu e he isk-neu al p obabili y o each di e en node o he unde lying asse eaching a maximum o minimum p ice and he o al numbe o maxima (minima) in he inomial ee. Wi h help o i , he compu a ional p oblems can be e ec i ely sol ed a ising om he inhe en complexi ies o di e en ypes o maximum-minimum bidi ec ional op ions when he unde lying asse e ol es as he inomial CEV model. Nume ical esul s demons a e he alidi y and he con e gence o he app oach men ioned abo e o he di e en pa ame e alues se in he inomial CEV model. © 2016 Published by Else ie Espa˜ na, S.L.U. on behal o Uni e sidad ESAN. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/). P ecios de opciones bidi eccionales máximas-mínimas en el modelo ECV inomial Códigos JEL: C14 F17 Palab as cla e: Modelo ECV inomial Algo i mo ecu si o Opciones bidi eccionales máximas-mínimas e s u m e n Las opciones bidi eccionales máximas-mínimas son un ipo de opciones exó icas dependien es de la ayec o ia. En el modelo de elas icidad cons an e de la a ianza (ECV), se es uc u ó un á bol inomial combinado pa a ap oxima la ola ilidad no cons an e, que es una unción del ac i o subyacen e. En base a es o se desa olló un algo i mo sencillo y eficaz pa a calcula la p obabilidad de neu alidad al iesgo de cada nodo del ac i o subyacen e llegando a un p ecio máximo o mínimo y el núme o o al de máximos (mínimos) del á bol inomial. De es a mane a, los p oblemas compu acionales pueden esol- e se eficazmen e a aíz de las complejidades inhe en es a los dis in os ipos de opciones bidi eccionales máximas-mínimas cuando el ac i o subyacen e e oluciona como el modelo ECV inomial. Los esul a- dos numé icos demues an la alidez y con e gencia del en oque an e io men e mencionado pa a los pa áme os de alo es es ablecidos en el modelo ECV inomial. © 2016 Publicado po Else ie Espa˜ na, S.L.U. en nomb e de Uni e sidad ESAN. Es e es un a ´ ıculo Open Access bajo la licencia CC BY-NC-ND (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/). 夽This pape is suppo ed by Na ional Na ual Fouda ion o China (NO: 71002098, NO: 61102009), Excellen You h ounda ion o Beijing, China (YETP1652) and he Na u al Sciences and Enginee ing Resea ch Council o Canada (NSERC). ∗Co esponding au ho . E-mail add ess: [email p o ec ed] (B. Peng). 1. In oduc ion Black and Scholes (1973) de i ed he well-known op ion p icing o mula by assuming he unde lying asse p ice ollows a geome ic B ownian mo ion. Unde he cons uc ion, he p ice dis ibu ion is logno mal and ola ili y is cons an . Howe e , in he ma ke eal- i y, whe e o en he asse p ice beha io is a ec ed by ola ili y smile e ec , his implies ha he asse p ice is gene ally unlikely o be logno mally dis ibu ed he unde lying asse ola ili y ends h p://dx.doi.o g/10.1016/j.je as.2016.06.001 2077-1886/© 2016 Published by Else ie Espa˜ na, S.L.U. on behal o Uni e sidad ESAN. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons. o g/licenses/by-nc-nd/4.0/). B. Peng, F. Peng / Jou nal o Economics, Finance and Adminis a i e Science 21 (2016) 50–55 51 o change as asse p ice mo es up and down. Cox (1975) ini ially obse ed ha he cha ac e o he ola ili y o he unde lying asse was linked o i s p ice le el. He no ed ha he o igin o he ola il- i y smile was he nega i e co ela ion be ween asse p ice changes and ola ili y changes, on his basis Cox de eloped cons an elas- ici y o a iance (CEV) model. As documen ed by Becke s (1980) and Da il (1982), he e a e heo e ical a gumen s o and empi i- cal e idences o he alidi y o CEV model. Many empi ical s udies in Macbe h and Me ille, 1980 and Emanuel Mache h (1982) ha e shown ha CEV model be e desc ibe he e olu ion o asse p ice. Hence i is ins uc i e o apply CEV model o op ion p icing. The p oblem o p icing a s anda d Eu opean op ion, when he unde lying asse alue is d i en by a CEV model was sol ed by Cox and Ross (1976) who de i ed an analy ical solu ion o he op ion alue. Things a e mo e complica ed in he case o pa h- dependen op ions. The analy ical solu ion o he p icing p oblem is no a ailable and nume ical app oxima ions mus be used. Boyle and Tian (1999) used Mon e Ca lo simula ions. Subsequen ly, he ex ended Babbs me hod on he basis o Black-Scholes model o cons uc a inomial ee me hod o e alua ing he lookback op ions unde he CEV model, as was ansla ed in o Chinese by He (2001). Du ydo and Line sky (2001) de i ed he close- o m o mulae o ba ie op ions unde he CEV model wi h help o he nume ical in e sion o he Laplace ans o m o he op ion p ice ol- lowing o dina y di e en ial equa ion. Bin and Fei (2006) applied he in ui ion binomials ee me hod o p ice Asian op ion unde he cons an elas ici y o a iance. Among hese nume ical ech- niques, inomial ee e alua ion app oach s ill plays a ema kable ole bo h o i s highly flexible and implemen a ion in sol ing he p icing p oblem o pa h-dependen op ions unde he CEV model. Howe e , in he amewo k o he inomial ee o app oxima e he cons an elas ici y o a iance model, he ansi ion p obabili- ies a e no longe cons an o e he ee. This p ope y in oduces a u he complica ion in he e alua ion p ocess. In addi ion, he e is li le wo k on he exo ic pa h-dependen such as maximum- minimum bidi ec ional op ions. The objec i e o his pape is o s udy he applica ion o he inomial ee app oach o maximum-minimum bidi ec ional op ions when he asse p ice e ol es as a CEV model. Especially, ecu si e algo i hm based on a o wa d induc ion p ocedu e is de eloped o compu e he isk-neu al p obabili y o each di e en node o he unde lying asse eaching a maximum o minimum p ice in he binomial ee. The emainde o he pape is o ganized as ollows: Sec ion 2 illus a es he inomial me hod used o app oxima e he CEV model unde he equi alen ma ingale measu e. In Sec ion 3 he inomial app oxima ion is applied o p ice maximum and minimum bidi ec ional op ions and nume ical esul s a e gi en. Conclusions a e p esen ed in he final sec ion. 2. A inomial CEV model I is known ha Black-Scholes model wi h cons an ola ili y does no hold empi ically o asse p ices. Al e na i e s ochas ic models ha e been s udied and applied o op ion p icing. Fo exam- ple, Cox (1975) and Da il (1982) s udy a gene al class o s ochas ic model known as he Cons an Elas ici y o Va iance (CEV) di usion model in a isk-neu al wo ld. dS = Sd + SdB  = ıS0.5a−1(1) whe e B is a s anda d Wiene p ocess unde he Q-measu e, is isk- less a e o in e es , he s anda d de ia ion o e u n  is a unc ion o he unde lying p ice ins ead o a cons an ı and a a e cons an s, 0 ≤ a < 2. We conside a disc e e app oxima ion o he asse p ice e o- lu ion desc ibed in (1) using he inomial app oach. Le n be he numbe o ime in e als be ween he ime and he ma u i y T, [ ,T] is di ided in o n equal pieces, each o wid h  . The unde lying asse may mo e up a le el, down a le el o s ay he le el wi h geome ic a e age be ween up le el and down le el. In he case o he asse p ice dynamics d i en by CEV model, he ola ili y is no a cons an bu a ies wi h he le el o he unde lying p ice. This implies ha he app oxima ion inomial ee is non- ecombining and he numbe o nodes p oduced a each e ical laye is 3i i = 0. . .. . ..n, hus compu a ional complexi y becomes unmanage- able e en wi h a small numbe o ime s eps. In o de o e ec i ely sol e he compu a ional p oblems a ising om he inhe en complexi ies o he cons an elas ici y o a iance model, we need o ans o m he a iable S go e ned by (1) wi h non-cons an ola ili y so ha he ans o med p ocess has cons an ola ili y. Conside ing he ans o med p ocess X = S1−0.5a/(1 − 0.5a)ı and using he I o’s Lemma, we ha e: dX =∂X ∂S dS +∂X ∂ d +1 2 ∂2X ∂S2(ıS0.5a)2d =S−0.5a ıdS −1 2 0.5aS−0.5a−1 ı(ıS0.5a)2d =S−0.5a ıdS −0.5aıS0.5a−1 2d (2) Subs i u ing equa ion (1) in o he abo e equa ion and using he ac ha S =ı(1 − 0.5a)X1/(1−0.5a). Equa ion (2) becomes: dX =S−0.5a ı Sd + ıS0.5adB−0.5aıS0.5a−1 2d = (1 − 0.5a)X −a2 4(1 − 0.5a)Xd + dB (3) Now, i is easy o build up a compu a ionally simple inomial ee o app oxima e he X-p ocess. The alue X o he p ocess a ime , a e one pe iod a ime + 1, can ise o X +√ , o dec ease o X −√ , o s ay he same wi h X . Con inuing in his way, we see ha he alue o he inomial X-p ocess a e equal o Xj +i = X + (j − i) , i = 0, 1, . . ., n, j = 0, 1, . . ., 2i(4) Whe e Xj +i ep esen s he alue o he inomial X-p ocess a ime + i a e j/2 up s eps and i-j/2 down s eps. o i = 0,1,. . ..,n and j = 0,1,. . ..,2i. Gi en he alues on he X la ice, we can eco e he dynamics o he unde lying p ice on i s la ice. As be o e, conside he alue S jo he p ocess a ime , a e one pe iod a ime + 1, can ise o Sj+ +i1 = (X +√ ), o dec ease o Sj− +i1 = (X −√ ), o s ay he same wi h S . Con inuing in his way, we no e ha he alue o he binomial app oxima ing S-p ocess a ime + i a e j/2 up s eps and i-j/2 down s eps. is exp essed as ollows: Sj +i = (Xj +i ) = (X + (j − i) ) (5) Once we ha e de eloped he inomial ee ha app oxima es he S-p ocess, i emains o compu e he p obabili ies o an upwa d and downwa d mo e o he e olu ion o he asse p ice in he app oxima ing inomial ee. To ensu e non-nega i e and less han one p obabili ies, i is op imal o ha e he unde lying asse p ice o cen al mo e equi alen o he expec ed asse p ice o he unde lying s ochas ic p ocess we define S¯ j +i , j = 0, 1, . . .2i o be g ea es wi h up jumps such ha e  Sj +(i−1) = S¯ j +i < 0 wi h up p obabili y pj +(i−1) and Sˆ j +i o be he smalles wi h down jumps such ha e  Sj +(i−1) − Sˆ j +i > 0. wi h down p obabili y 52 B. Peng, F. Peng / Jou nal o Economics, Finance and Adminis a i e Science 21 (2016) 50–55 qj +(i−1) and S˜ j +i o be middle mo e be ween he g ea es and he smalles , such ha e  Sj +(i−1) − S˜ j +i = 0. The p obabili ies ha he unde lying asse wi h p ice Sj +(i−1) makes an up jump and down jump a e specified by he ollowing se o equa ion, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ pj +(i−1) S ៝ j +i − S˜ j +i − qj +(i−1) S˜ j +i − Sˆ j +i = 0 pj +(i−1) S¯ j +i 2 −S˜ j +i 2 + qj +(i−1) Sˆ j +i 2 −S˜ j +i = 2 Sj +(i−1) ˛ (6) Sol ing o he p obabili ies om he abo e equa ion, one ob ain: pj +(i−1) = 2 Sj +(i−1) ˛ S¯ j +i − S¯ j ¯ +i S¯ j +i − Sj ¯ +i , qj +(i−1) = 2 Sj +(i−1) ˛ S¯ j ¯ +i − Sj ¯ +i Sj +i − Sj ¯ +i (7) Ob iously, he p obabili y ha he unde lying asse wi h p ice Sj +(i−1) makes pa allel mo e is 1 − pj +(i−1) − qj +(i−1) . The abo e calcula ion o he ansi ion p obabili ies pj +(i−1) and qj +(i−1) ep esen s legi ima e p obabili ies which allows o mul- iple jumps in he app oxima ing inomial X-p ocess. This is impo an because, in he egion nea o S = 0, he magni ude o each jump could be e y small and, as a consequence, he ansi ion p obabili ies could exceed one o smalle han ze o. 3. P icing maximum-minimum bidi ec ional op ions We now apply he inomial CEV model depic ed in abo e sec ion o p icing he maximum-minimum bidi ec ional op ions In oking he no-a bi age p inciple, he alue a ime o op ions is gi en by discoun ing a he isk- ee in e es a e he sum o all he op ion payo s mul iplied by he co esponding p obabili y o occu ing. Thus we need o see he .payo co esponding o each node o he ee a ma u i y. Fo floa ing s iking maximum- minimum bidi ec ional op ions, i s payo a ma u i y is equal o he di e ence be ween he cu en unde lying asse and he mini- mum p ice egis e ed by he unde lying asse du ing he op ion li e ime plus he di e ence be ween he maximum unde lying ass p ice egis e ed du ing he op ion li e ime and cu en asse p ice. Con e sely, he fixed s ike maximum-minimum bidi ec ional op ions pay o he maximum be ween ze o and he di e ence be ween he maximum unde lying asse p ice and a fixed s ike p ice plus he maximum be ween ze o and he di e ence be ween a fixed s ike p ice and he minimum asse p ice. Howe e , hings a e complica ed by he ac ha , o each e minal node o he ee co espond, in gene al, di e en alues o he op ion payo . This is because, e en when some ajec o ies o he unde lying asse p ice end wi h he same e minal alue, hey could ha e egis e ed a di e en minimum and maximum p ice du ing he op ion li e ime. In he amewo k o he inomial CEV model de eloped by Sec- ion 2 we see ha he ansi ion p obabili ies a e no cons an o e he ee, and as a consequence, all he pa hs o he unde lying asse wi h he same e minal alue may ha e he di e en p obabili ies o occu ing. In his amewo k, he e alua ion p oblem is no sol ed e en i we can compu e he numbe o pa hs wi h he same e minal alue bu wi h a di e en minimum o maximum p ice egis e ed du ing he op ion li e. This p ope y in oduces a u he complica ion in he e alua ion p ocess. To copy wi h his p oblem, we de i e a ecu si e algo i hm based on a o wa d induc ion p o- cedu e ha allows o simply compu e he isk-neu al p obabili y o each di e en payo o he op ions a ma u i y. The algo i hm wo ks as ollows: a fi s , we w i e (i, i-j) o he inomial la - ice whe e he unde lying asse has a p ice Sj +i (i = 0,1,. . ..,n; j = 0,1,. . ..,2i). No e ha i-j is he di e ence be ween he down s eps and he up s eps aken by he unde lying asse p ice and, as we see below, i is use ul ep esen a ion o posi ion o he asse p ice a ime i. Fu he , we use index k and h (k, h = 0,1,. . ..i) espec i ely o speci y he lowes and highes laye o ho izon al nodes eached by he unde lying asse p ice a e i ime s eps. In o he wo ds, k = 0,1,. . ..i means ha he asse wi h cu en p ice Sj +i has egis e ed minimum p ice equal o S0 +k , h = 0,1,. . ..i means ha he asse wi h cu en p ice Sj +i has egis e ed maximum p ice equal o S2h +h we use he iple (i,j, k) and (i,j, h) o espec i ely speci y he s a e o he wo ld in which he unde lying asse eached a minimum p ice equal o S0 +k and a maximum p ice equal o S2h +h ,. o e a pa h wi h j/2 up s eps and i-j/2 down s eps. Clea ly, gi en i and j, he alue o he index k anges o e he in e nal max (i − j, 0),i − j/2; and he alue o he index h anges o e he in e nal max (j − i, 0),j/2, whe e ◦ e u ns he lowe in ege is closes o a eal numbe . Wi h help o hese impo an ela ions, we can iden i y all he di e en minimum p ices, S0 +k and all he di e en maximum p ices, S2h +h egis e ed by he unde lying asse p ice ha has a cu en alue Sj +i . We s a by compu ing he p obabili y o each s a e o na u e ep esen ed by he iple (i,j, k) in he inomial ee. To do his, we need o dis inguish wo cases, he fi s conside s he s a es o na u e cha ac e ized by he iple (i, j, k) such ha k = i-j. In his case he unde lying asse p ice was a he node (i, i − j), Sj +i is loca ed on he same ho izon al se o nodes o he minimum p ice eached by he unde lying asse du ing he fi s ime s eps. This means ha a ime + (i − 1) he unde lying asse p ice was a he node (i − 1, i − 1 − j) and he las s ep om (i − 1, i − 1 − j) o (i, i − j) is a down s ep. I a ime + (i − 1) he unde lying asse p ice was a node (i − 1, i − 1 − (j − 1)) he lowes laye o nodes eached by he unde lying asse p ice would be loca ed a he le el k = i − j and his is equi alen o he hypo hesis k = i − j. his means he unde lying asse could each he minimum p ice S0 +k a ime + i wi h he las pa allel mo e. Hence he s a e o na u e (, i, j, k) is: (i, j, k)=(i − 1, j, k − 1)∩ Ld∪(i − 1, j, k)∩ Ld ∪(i − 1, j − 1, k)∩ Lm(8) Whe e Ldand Lm espec i ely ep esen s he e en ha he las ime s ep aken by he unde lying asse p ice is a down s ep and s ay he same. Hence, he p obabili y o he s a e o na u e (, i, j, k) is p ob (i, j, k)= p ob[(i − 1, j, k − 1)∩ Ld∪(i − 1, j, k)∩ Ld ∪(i − 1, j − 1, k)∩ Lm] = p ob (i − 1, j, k − 1)× qj +(−1)i + p ob (i − 1, j, k) ×qj +(i−1) +p ob (i − 1, j − 1, k)×1 − pj−1 +(i−1) − qj−1 +(i−1)  (9) The second case conside s he s a es o na u e such ha k > i − j. in his si ua ion he unde lying asse p ice a he node B. Peng, F. Peng / Jou nal o Economics, Finance and Adminis a i e Science 21 (2016) 50–55 53 (i, i − j), Sj +i is g ea e han he minimum p ice eached du ing he fi s i − 1 ime s eps. Indeed, he lowes laye o nodes ouched by he unde lying asse is loca ed a le el k, ha is, below he cu en node (i, i − j) loca ed a le el i − j. in his si ua ion, he le el k was al eady eached by he unde lying asse p ice a ime + (i − 1) . I he asse p ice is a he node (i − 1, i − 1 − j) and he i h s ep is a down s ep and i he asse p ice is a he node (i − 1, i − 1 − (j − 1)) and he i h s ep is pa allel s ep and i he asse p ice is a he node (i − 1, i − 1 − (j − 2)) and he i h s ep is an up s ep. Thus he e en (i, j, k) occu s when he e en (i − 1, j, k) occu s and he i h s ep is a down s ep o when he e en (i − 1, j − 1, k) akes place and he i h s ep is pa allel s ep o when he e en (i − 1, j − 2, k) akes place and he i h s ep is an up s ep, i.e. (i, j, k)=(i − 1, j, k)∩ Ld∪(i − 1, j − 1, k)∩ Lm ∪(i − 1, j − 2, k)∩ Lu(10) whe e Lu ep esen s he e en ha he las s ep aken by he unde - lying asse p ice is an up s ep. Hence, he p obabili y o he s a e o na u e (i, j, k) is p ob (i, j, k)= p ob (i − 1, j, k)∩ Ld∪(i − 1, j − 1, k)∩ Lm ∪(i − 1, j − 2, k)∩ Lu = p ob (i − 1, j, k)× qj +(−1)i + p ob (i − 1, j − 1, k) ×1 − pj−1 +(i−1) − qj−1 +(i−1)  +p ob (i − 1, j − 2, k)× pj−2 +(i−1) (11) The case o he p obabili y o each s a e o na u e ep esen ed by he iple (i, j, h) in he inomial ee can be compu ed as be o e. We mus dis inguish be ween wo cases, he fi s is when he cu - en p ice o he unde lying asse , Sj +i is equal o he maximum alue o he asse p ice, i.e. h= j-.i. In his case he las s ep o he asse p ice mus be an up s ep o pa allel s ep and maximum may be eached wi h he las ime s ep o du ing he fi s i-1 ime s eps. Hence, (i, j, h)=(i − 1, j − 2, h − 1)∩ Lu∪(i − 1, j − 2, h)∩ Lu ∪(i − 1, j − 1, h)∩ Lm(12) and p ob (i, j, h)= p ob (i − 1, j − 2, h − 1)× pj−2 +(−1)i + p ob (i − 1, j − 2, h)× pj−2 +(i−1) +p ob (i − 1, j − 1, h)×1 − pj−1 +(i−1) − qj−1 +(i−1)  (13) On he con a y, when he maximum unde lying asse p ice is g ea e han he cu en asse p ice, i.e. h > j − .i, we ha e (i, j, h)=(i − 1, j, h)∩ Ld∪(i − 1, j − 2, h)∩ Lu ∪(i − 1, j − 1, h)∩ Lm(14) and p ob (i, j, h)= p ob (i − 1, j, h)× qj +(−1)i + p ob (i − 1, j − 2, h)× pj−2 +(i−1) +p ob (i − 1, j − 1, h)×1 − pj−1 +(i−1) − qj−1 +(i−1)  (15) I is wo h no ing ha he ecu si e algo i hm o compu ing he isk neu al p obabili y o each s a e o na u e a he beginning o he ee, assigns he isk-neu al p obabili y 1 o he s a e o na u e (0,0,0) and a isk-neu al p obabili y 0 o he s a es o na u es ha can ne e occu , i.e, when k < 0 o k > i − j/2i − j/2and h < 0 o h >j/2. To cla i y how he p ocedu e wo ks we conside asse wi h p ice S = 100 ha mo es in he inomial la ice illus a ed in Sec ion 2. We conside he numbe o ime s eps n = 3, he isk = ee in e es a e = 10%, he ime o ma u i y T = 6 mon hs and elas ici y ac o a = 05. he ola ili y pa ame e is adjus ed in such a way ha he ins an aneous ola ili y o he asse p ice a incep ion is he same o all he alues o he pa ame e a conside ed. We se he ini ial ins aneous ola ili y o he unde lying asse a e o e u n equal o 0.25 and selec he alue o ␴ such ha ıSa/2−1 = 0.25. Figu e 1 illus a es how he o wa d induc ion based p ocedu e wo ks in compu ing he isk neu al p obabili y o each minimum and maximum in he inomial ee. In he Figu e 1, we depic he inomial ee ha desc ibes he e olu ion o he unde lying asse p ice and we epo ed he isk-neu al p obabili y o each up s ep and o each down s ep and middle s ep. Fo example, 0.4572 is he p obabili y, p0 +1/3 ha he unde lying asse p ice jumps om he alue S0 T+1/3 o he p ice S2 +1/2 ep esen ed by he node (3,2). The le hand side o Figu e 1 illus a es he isk-neu al p obabili y o each possible minimum eached by he unde lying asse ha cu en ly is loca ed a he node (i, i-j). Fo example, a he node (3,2), he unde lying asse p ice has a cu en p ice S2 +1/2. The minimum p ice eached by he unde lying asse p ice i he fi s o second s ep was an down s ep is S0 +1/6, hence k = 1 and he s a e o na u e occu ed is (3,2,1) because i − j = k, p ob(3, 2, 1) = [p ob (2, 2, 0)+ p ob (2, 2, 1)] × q2 +1/3 +p ob (2, 1, 1)×1 − p1 +1/3− q1 +1/3 =(0.1889 + 0.1953)× 0.3186 + 0.1150 × 0.2196 = 0.1479 Con e sely, in he case o wo consecu i e down s eps o he unde lying asse p ice a incep ion, he minimum p ice eached by he unde lying asse p ice is S0 +1/3, k = 2, and he s a e o na u e occu ed is (3,2,2). Because k > i − j p ob (3, 2, 2)= p ob (2, 2, 2)× q2 +1/3+ p ob (2, 1, 2) ×1 − p1 +1/3− q1 +(1/3 +p ob (2, 0, 2)× p0 +1/3= 0.1067 × 0.4512 = 0.0734 No e ha he s a e o na u e (2,2,2) and (2,1,2) can ne e occu . The igh hand side o Figu e 1 illus a es he isk-neu al p obabili y o each possible maximum eached by he unde - lying asse ha cu en ly is loca ed a he node (i, i − j). Fo ins ance, a he node (3,4), he unde lying asse p ice has a cu - en p ice S4 +1/2. The maximum p ice eached by he unde lying asse p ice i he fi s o second s ep was an up s ep is S2 +1/6, hence h = 1 and he s a e o na u e occu ed is (3,2,1) because j − i = h, p ob (3, 4, 1)=p ob (2, 2, 0)+ p ob (2, 2, 1)× p2 +1/3 +p ob (2, 3, 1)×1 − p3 +1/3− q3 +1/3 = (0.2151 + 0.1425) × 0.4631 + 0.2005 × 0.2313 = 0.2114 On he con a y, in he case o wo consecu i e up s eps o he unde lying asse p ice a incep ion, he maximum p ice eached by he unde lying asse p ice is S4 +1/3, h = 2, and he s a e o na u e occu ed is (3,4,2). Because h > j − i p ob(3, 4, 2) = p ob 2, 4, 2× q4 +1/3+ p ob (2, 2, 2)× p2 +1/3 +p ob (2, 3, 2)×1 − p3 +1/3− q3 +1/3= 0.2205 × 0.2875 = 0.0634 No e ha he s a e o na u e (2,2,2) and (2,3,2) can ne e occu . 54 B. Peng, F. Peng / Jou nal o Economics, Finance and Adminis a i e Science 21 (2016) 50–55 (0,0,0) (1,2,1) 0.4665 (1,1,0) 0.2098 (1,0,0) 0.3237 (2,4,0) 0.2205 0.4728 0.4793 0.2216. 0.3278 0.3056. 0.3318 0.4632 0.4655 0.4602 0.4572 0.3186. 0.3206. 0.3156. 0.2110 0.4598 0.2875. 02332. 02972. 02373 0.2212 0.2120 0.2196. 0.4631. 0.2183. (3,6,0) 0.1057 (2,2,0) 0.1889 (2,2,1) 0.1953 (3,0,3) 0.0354 (3,1,2) 0.0225 (2,0,2) 0.1067 (3,4,0) 0.1097 (3,4,1) 0.1368 (3,5,0) 0.1447 (3,3,0) 0.0997 (3,3,1) 0.0956 (2,1,1) 0.1150 (3,2,1) 0.1479 (3,2,2) 0.0734 (2,3,0) 0.2005 (0,0,0) (1,2,1) 0.4665 (1,1,0) 0.2098 (1,0,0) 0.3237 (2,4,2) 0.2205 0.4728 0.4793 0.2216. 0.3278 0.3056. 0.3318 0.4632 0.4655 0.4602 0.4572 0.3206. 0.3156. 0.2110 0.4598 0.2875. 02332. 02972. 02373 0.2212 0.2120 0.2196. 0.4631 0.2183. (3,6,3) 0.1057 (2,2,0) 0.2151 (3,0,0) 0.0354 (3,1,0) 0.0225 (2,0,0) 0.1067 (3,5,2) 0.1447 (2,1,0) 0.1150 (2,3,1) 0.2005 (3,4,1) 0.2114 (3,4,2) 0.0634 (3,3,0) 0.0998 (3,3,1) 0.0907 0.3186 (2,2,1) 0.1425 (3,2,0) 0.0740 (3,2,1) 0.0454 Figu e 1. The ansi ion p obabili y o each s a e o na u e (i, j, h) and (i, j, h). Once we compu e he isk-neu al p obabili y o eaxh s a e o na u e a ma u i y, (n, j, k) and (n, j, h), j = 0,1,. . ..2n, k = max (i − j, 0), ....... i − j/2h = max (j − i, 0), ........ j/2; we a e eady o e alua e he bidi ec ional hindsigh op ions since o each possible s a e o na u e a ma u i y (n,j,k) and (n, j, h) a e espec- i ely associa ed he op ion payo Sj +n −S0 +k and S2h +h − Sj +n ; hence, he p ice a ime o a floa ing s ike op ion is FOP = 2n  j=9 n−j/2  k=max(n−j,0) p ob (n, j, k)Sj +n −S0 +k  + 2n  j=9 j/2  h=max(j−n,0) p ob (n, j, h)S2h +h − Sj +n (16) Subs i u ing he s iking p ice K in o Sj +n we can ob ain a fixed s ike op ion as ollow OP = 2n  j=9 n−j/2  k=max(n−j,0) p ob (n, j, k)max 0,K − S0 +k  + 2n  j=9 j/2  h=max(j−n,0) p ob (n, j, h)max 0,S2h +h − K (17) Finally we u n o he e alua ion o he compu a ional cos o he ecu si e algo i hm o p icing maximum-minimum bidi ec- ional op ions in inomial CEV model. we de e mine he numbe o possible minima o maxima eached by he unde lying asse a each node (i, j) (i = 0,1,. . ..,n; j = 0,. . .. . ..,2i) o he ee. Wi h- ou loss o gene ali y, we conside a inomial ee wi h an e en numbe , n, o ime s eps. We need dis inguish wo cases. The fi s conside s he s a es o na u e cha ac e ized by he iple (i,j,k) and (i,j,h) such ha i ≥ j, ha I o say j = 0,1,. . ..,i. In Figu e 1, we can easily obse e ha each node eached by a ajec o y wi h j/2 up s eps o i-j/2 up s eps is cha ac e ized by j/2 + 1 minima o maxima o j = 2 m. and (j + 1)/2 minima o maxima o j = 2m + 1 (m = 0,1,. . ..,i/2). Since in he ee he e exis n − j + 1 nodes cha ac- e ized by j/2 + 1 and (j + 1)/2 di e en minima o maxima. These nodes co espond o (n − j + 1)*(j/2 + 1) and (n − j + 1)*(j + 1)/2 min- ima o maxima. The second case conside s he s a es o na u e cha ac e ized by he iple (i,j,k) and (i,j,h) such ha i < j i.e. j = i + 1,. . ..,2i, in he ee he e exis j − n nodes cha ac e ized byn- j/2 + 1 and n-(j + 1)/2 + 1di e en minima o maxima espec i ely o j = 2w and j = 2w-1(w = i/2 + 1,. . ..,i). These nodes co espond o (j − n)*(n − j/2 + 1) and (j − n)*(n − (j + 1)/2 + 1) minima o maxima. Thus he o al numbe o minima o maxima in a inomial ee wi h n ime s eps is equal o: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  j=2m(n − j + 1)j + 1 2+ j=2m(n − j + 1)j 2+ 1= i/2  m=0 (2n − 4m + 1)(m + 1) =n3 12 +5n2 8+17n 12 + 1; i ≥ j  j=2w(j − n)n −j 2+ 1+ j=2w−1(j − n)i −j + 1 2+ 1= i  w=i/2+1 (4w − 2n − 1)(n − m + 1) =n3 12 +2n2 8+5n 12; i < j To compu e he isk-neu al p obabili y o each minimum o maximum in he ee, acco ding o he o wa d induc ion p oce- du e de eloped abo e, we ha e o implemen wo mul iplica ions and one addi ion, hence, n3/6 + 7n2/4 + 22n/6 mul iplica ions and n3/12 + 7n2/8 + 22n/12 addi ions ( he node (0.0.0) has a isk-neu al p obabili y equal o 1). Mo eo e , o he isk-neu al p obabili y o each possible minimum p ice a he las ime s ep, n, we ha e o associa e he co esponding op ion payo a ma u i y and hen sum hem up. Table 1 illus a e he nume ical esul s o he app oach desc ibed abo e o e alua ing floa ing s ike and fixed s ike maximum-minimum bidi ec ional op ions in he inomial CEV model. The ocus o he able is o check he app oach pe o mance and con e gence. .in o de o allow compa isons wi h o he al- ua ion me hods, we choose he same op ion pa ame e s as hese used in Boyle and Tian (1999) illus a ed in Du ydo and Line sky (2001). The cu en asse p ice S may be 90 100 110, he ime o ma u i y is T = 6 mon hs, he annualized isk- ee in e es a e equals 10% and he ola ili y o he unde lying asse a e o e u n B. Peng, F. Peng / Jou nal o Economics, Finance and Adminis a i e Science 21 (2016) 50–55 55 Table 1 Valua ion o he maximum-minimum bidi ec ional op ions in he inomial CEV model. nFixed s ike op ion floa ing s ike op ion S a = 0.5 a = 1 a = 1.5 a = 0.5 a = 1 a = 1.5 25 90 20.3697 20.3937 20.4647 30.1459 30.2299 30.3291 100 19.0725 19.2022 19.3697 28.6135 28.7147 28.8511 110 18.1462 18.1375 18.4731 29.1786 29.2291 29.3975 100 90 20.3734 20.3954 20.4681 30.1452 29.7188 30.3438 100 19.0827 19.2009 19.3722 28.6128 28.7143 28.8548 110 18.1439 18.1299 18.4724 29.1399 29.2511 29.4146 250 90 20.3723 20.3945 20.4644 30.1437 30.2185 30.3107 100 18.9898 19.1925 19.3707 28.6078 28.7071 28.8355 110 18.1405 18.1285 18.4699 29.1232 29.2486 29.4063 Close- o m solu ion 90 20.3727 20.3942 20.4644 30.1435 30.2185 30.3255 100 19.0799 19.1923 19.3707 28.6078 28.7075 28.8353 110 18.1405 18.1285 18.4699 29.1233 29.2485 29.4065 Own elabo a ion. is 25% pe annum. The s ike p ice is 100. The numbe o ime s eps n used anges among 25, 100, 250. The elas ici y ac o a = 0.5, 1 o 1.5. Fu he mo e, a close- o m solu ion is compu ed wi h Da ydo and Line sky me hod expanded o compa ison. As wi h o he nume ical me hod applica ions, p ices om ou inomial CEV model is con e ge apidly wi h all a alue o he closed- o m solu ion as he numbe o s eps n inc eases. When n = 25, on a e age, he alue om ou inomial op ion p icing app oach is close o close- o m solu ion han app oxima ion solu- ions using o he nume ical me hod de eloped by Boyle, Tian and Bin. Fo n = 100, he esul s show ha he di e ence be ween he inomial me hod and he closed- o m solu ion is less han o equal o 0.01. The esul wi h ime s ep 250 is almos app oach o close- o m solu ion. The accu acy o esul s is simila o o he nume ical me hods and he e ficiency o ecu si e algo i hm in he inomial CEV model is highe han o he nume ical algo i hm. 4. Conclusions A inomial ee app oxima ion o he CEV model was con- s uc ed o desc ibe he e olu ion o he unde lying asse wi h non-cons an ola ili y. Once he inomial ee has been buil up, we p oposed a ecu si e algo i hm based on o wa d induc ion scheme o calcula e ansi ion p obabili y o possible minimum o maximum eached by he unde lying asse a each node o e he i- nomial ee when ola ili y a ied wi h asse p ice le el, u he we use i o alue di e en ypes o maximum-minimum bidi ec ional op ions. Nume ical esul s show ha he inomial algo i hm has sa is ac o y con e gence and p oduces accu a e p ices o a wide ange o pa ame e alues o he CEV model compa ed wi h o he nume ical me hods such as close- o m solu ion me hod. In addi- ion, o exo ic pa h-dependen op ions ha depend on he ex eme o he p ocess, he p ices a e qui e sensi i e o he specifica ion o he p ocess. Re e ences Becke s, S. (1980). The consan elas ici y o a iance model and i s implica ions o op ion p icing. Jou nal o Finance, (35), 661–673. 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