Markov Chain applications to non parametric option pricing theory

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1 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June Marov Chan alcaons o non aramerc oon rcng heory Summary In hs aer we roose o use a Marov chan n order o rce conngen clams. In arcular, we descrbe a non aramerc marovan aroach o rce Amercan and Euroean oons. Frs, we dscuss he rs neural valuaon of he non aramerc aroach. Secondly, we examne he roblems of he comuaonal comlexy and of he sably wh resec o he number of he saes of he Marov chan. Fnally, we roose an ex os comarson beween he Marovan model and he Blac and Scholes one. Key words: Marov Chan, Rs eural Valuaon, sae deenden valuaon, sae ndeenden rce.. Inroducon Afer he Blac-Scholes oon rcng model many sudes have aemed o coe wh he dfferen conradcons emerged n he emrcal ess of hs model. Whle many researchers ndcae he lognormal dsrbuon hyohess of he fnancal reurn as no oo sasfyng hyohess, many ohers fnd he consan volaly of he fnancal rce as he grea wea on of he model. There exs a wde leraure on he mrovemens erformed on hs oneer model. Many effors have been desned o mae sochasc he volaly and ohers o mae he dsrbuonal hyohess more realsc on he rce rocess. Ths aer shows a smle non-aramerc way o model he conngen clams whou assessng a dsrbuonal form a ror for he asse rce and whou he necessy of a valuaon of arameers such as he volaly. The mehodology resened eners n he class of he Marovan oon rcng models. Among marovan models we essenally dsngush wo caegores: aramerc models (see, among ohers, Duan and Smonao (200; Duan, e al.(2003 and among sem-marovan models see Lmnos, Orsan (200, Blas e al.(2003, D Amco e al. (2005, and nonaramerc models. In he frs caegory he Marovan hyohess s used for dffusve models of he underlyng reurns. In he second caegory of models only he hsorcal seres are used o esmae he oon rces. Thus nonaramerc models have he man advanage n Sergo Orobell and Gaeano Iaquna, Unversy of Bergamo, Dearmen MSIA her caacy of adang o he underlyng reurn dsrbuons. Ths aer deals wh a nonaramerc marovan model, ha dfferenly by he nonaramerc dervaves models deal n leraure (see, among ohers, Huchnson, e al.(994, A-Sahala (996, 998, Suzer (996,. exlcaes drecly he marovan hyohess assumng ha he me evoluon of he reurns s descrbed by a Marov chan. Thus our nonaramerc aroach s dfferen resec o hose based on he arameer esmaons, hose ha use he Marov Chans o aroxmae such arameers and hose ha use neural newor or only he hsorcal seres o aroxmae he oon rces. Wh hs model we are able o rce Amercan/ Euroean and ah deenden oons and usng he Marov chan roeres we are able o smlfy he comuaon of he dervave rces n reasonable mes. Generally he resulng rces are dfferen from hose obaned wh he Blac and Scholes model even f hs dfference s srongly reduced when we use smulaed Gaussan log-reurns. Therefore he ducly of he model suggess ha one of he man alcaons should be for energy dervaves whch are srongly nfluenced by he seasonaly of he rce. The aer frs resens he model dscussng he rs neural valuaon when we consder eher sae deenden rces or sae ndeenden rces. Secondly we dscuss he comuaonal comlexy of he algorhm and he sably of he rces wh resec o he number of he saes of he Marov chan. Fnally we examne he emrcal dfferences beween he Blac and Scholes model and he nonaramerc marovan one. 2. onaramerc Marovan Trees Le us assume he me evoluon of underlyng asse reurn follows a Marov chan wh saes. Dong so, we wan o consruc a mulnomal recombnng ree of he asse rce wh more degrees of freedom han he classcal bnomal ree. In arcular, we assume ha he gross reurn has suor on he nerval mn z ;max z, ( where z S+ / S s he -h reurn observaon and S s he value of he secury a me. By convenon, hrough all he aer, we coun he saes begnnng from ha wh Manuscr receved June 5, Manuscr revsed June 20, 2008.

2 200 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June 2008 he greaes value. Then we buld he ranson marx as follows:. we share n nervals I ( a; a (small enough he reurn suor mn z ;max z where ( a0 max z, a u max z, u mn z / max z and,..., ; I 2. we assume ha nsde he nerval he reurn s gven by he geomerc mean of he exremes, ( 0.5.e. z a a u max z ; 3. we buld he ranson marx Ps [, ;( s ], where he robably, ;( s ons ou he robably valued a me s o rans from he ( ( sae z o he sae z condonal of beng n he -h sae. Snce he ree recombnes a each se, he number of nodes ncreases lnearly wh he number of he me ses. For hs reason we can conrol and lm he comuaonal comlexy. Thus, afer Δ nervals of me we have ( + nodes (.e., he mulnomal ree growhs lnearly wh he me. Sarng o coun from he hghes node, afer ses he -h node has: - value n he nerval: ( I u 2 ( max z ; u 2 (max z ; - gross reurn: ( + 2 z : u (max z ; ( 0z - soc rce: S,...,( + ex we consder an homogeneous Marov chan wh ranson marx P [. In hs case we assume he maxmum lelhood esmae of robably whch s smly gven by he rao of he coun of he, arorae cells,.e.,, ], n, where n s he number n of mes he reurn rans from he sae -h o he sae - h and n n s he number of mes he reurn s n he -h sae. However, we could consder a non homogeneous Marov chan ang no accoun he behavor of he rces n dfferen erods. Therefore, we can model and value dfferenly he ranson marxes when he underlned rces change s behavor durng he maury erod. For examle, f we have a seasonal rce, le hose observed n he energy mares, we can comue dfferen ranson marxes n order o consder he wee-end effec and/or he season effec. Once we ge he ranson marx, we have o fnd adequae answers o he followng hree ssues ha should be obec of he nex secons: a how one obans he rs neural valuaon sarng from he mare-based ranson robably; b wha we can say abou he valuaon rocedure for Euroean and Amercan conngen clams and he man gree leers; c dscuss he sably of he soluons wh resec o he number of he saes. 3. Rs neural valuaon Le us assume he dvdend of one uny of wealh nvesed n a gven asse durng he erod [ 0, ] s gven by ex( q( 0 where q descrbes he nensy of he dvdend and suose ex( r( 0 s one uny of wealh dscouned a me 0 where we assume ha r defnes a fxed shor erm neres rae. Wh marovan rees we can generally dsngush wo ossble rs neural valuaons:. a rs neural rce ha s sae ndeenden; 2. a rs neural rce ha s sae deenden. The wo cases requre a dfferen valuaon of he rs neural ranson marx, ha we should denoe resecvely wh Pˆ and P 3. Sae ndeenden Rs eural Valuaon Le us assume no arbrages are allowed. Then here exss a rs neural measure such ha he value oday s equal o he execed value of he fuure wealh dscouned wh he rs-free gross reurn. Wh a Marov chan hs s equvalen o wre: ˆ Eˆ( z / z I ex( r q ( where 0 0, Eˆ( z / z I s he rs neural execed value of he fuure reurn condonal on beng n he -h sae, and ˆ s rs neural robably of beng n he -h sae. Clearly n ncomlee mares could exs more han one rs neural measure sasfyng he no arbrage creron. One creron roosed n leraure consders he mnmal enroy marngale measure (see Suzer (996, Frell (2000 and he reference heren. On he oher hand, he use of he mnmal enroy marngale measure can be movaed by maxmum execed uly argumens (see Frell (2000. In our conex, we fnd he mnmal enroy marngale measure wh resec o he uncondonal robably measure P {, }, where, s he uncondonal robably o rans from he sae o he

3 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June sae. As observed by Frell (2000, n order o ge he mnmal enroy marngale measure n he dscree case, we have o comue he value θ, unque for all he saes, ha s obaned as a soluon of he equaon: ex( r q ( z ex( θz ex( θz ( ( (2 Then he rs neural uncondonal robably o rans from he sae o he sae s gven by: ( ex( θ z ˆ ˆ ( ex( θ z where θ s he soluon of equaon (2. Therefore, we wre he rs neural ranson marx consderng he followng ranson robables ( ex( θ z ˆ (3 ( ex( θ z and he robably of beng n he -h sae s gven by ˆ ( ex( θ z ( ex( θ z. Therefore, once we esmae he ranson marx P [, ], we can fnd he corresondng rs neural ranson marx Pˆ [ ˆ, ], ha could be used n he rs neural valuaon of conngen clams. Le [ ˆ,..., ˆ ] be he row vecor of rs neural uncondonal robables of he dfferen saes. Then f we on ou wh ( ( z [ z,..., z ]' he vecor of he ossble saes he fundamenal heorem of asse rcng afer one erod s smle gven by Pz ˆ ex( r q. oe ha n he dscree case he mnmal enroy marngale measure concdes wh he mnmal varance marngale measure and we can easly ln also o he Esscher rasform rs neural measure (see Gerber, Shu (994, 996 ofen used o rce conngen clams wh Levy rocesses (see Schouens (2003. Moreover, snce we aly a rs neural valuaon ha s ndeenden on he sae, we have no necessarly o correc he ranson marx as we do n he nex sae deenden valuaon. 3.2 Sae deenden Rs eural Valuaon Le us assume ha he gross reurn z a me 0 0 s n he -h sae. When no arbrage oorunes are allowed, hen Eˆ ( z / z I ex( r q (4 where Eˆ( z / z I s he rs neural execed value of he fuure reurn condonal on beng n he -h sae. However, we can fnd a rs neural measure ha sasfes condon (4 only f ( ( ex( [ r q z, z ] (5 where max { 0} and mn { > 0}. > In arcular, could haen ha for some exreme saes we canno guaranee condon (4 holds snce we have no enough observaons of hese exreme saes and he robably aroxmaons n he ranson marx are no suffcenly accurae. In order o overcome hs roblem, we can oorunely correc he orgnal ranson marx P [ such ha condon (5 s sasfed. For, ], ( ( examle, suose for he sae "" ex( r q [ z hen we correc he -h row of marx P as follows:, z ] ( a Suose ex( r q < z. We assume ε f ε, f :, > 0 m 0 oherwse where ε s an oorunely lle value belongng o he mn {,, > 0} nerval 0,, m s he number of m ndexes n he -h row of P such ha, > 0 and ( max { z < ex( r q} ( b Suose ex( r q > z. We assume ε f ε,, f :, > 0 m 0 oherwse where ε s an oorunely lle value belongng o he mn {,, > 0} nerval 0,, m s he number of m ndexes n he -h row of P such ha, > 0 and mn { z ( > ex( r q}.

4 202 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June 2008 The correced marx (ha wh abuse of noaon we call agan P erms o overcome some msalcaons roblems dervng by a non suffcenly accurae aroxmaon of he ranson marx. Once we correc ranson marx P condon (5 s sasfed for every sae. Thus, for every sae we can deermne he mnmal enroy marngale measure ha sasfy condon (4. Hence, for every sae,..., we comue he value θ (, obaned as a soluon of he equaon: ex( r q, z, ( ex( θ( z ex( θ( z ( ( Then he rs neural ranson marx P [, ], should conan he rs neural condonal robables gven by: (, ex( θ( z, ( m, ex( θ( z m where θ ( s he soluon of equaon (6. Usng hs rs neural ranson marx we ge ha P z ex( r q, where z [ z (,..., z ( ]' s he vecor of he ossble saes afer one erod and s he uny vecor column. 4. Valuaon rocedure for Euroean and Amercan conngen clams and comuaonal comlexy Gven an asse wh gross reurn z, hen we can buld he ree of he underlyng rce. Thus sarng from a rce n a generc node, we could generae ossble fuure rces (deendng on he ossble fuure saes. On he oher hand, he orgnal rce should be condoned from he sae of rovenance ( ossble bacward saes. Ths asec s fundamenal n he sae deenden valuaon because he rocedure mus ae no accoun of he revous ses. The sae deenden and he sae ndeenden rs neural valuaons allow us o deermne a bacward comuaon of conngen clams arcularly useful for Amercan dervaves. However, for Euroean conngen clams, we can also roose an alernave sae deenden forward valuaon ha s generally dfferen from he revous ones. Thus, we can generally consder wo dfferen yes of valuaon rocedures: forward and bacward. The frs one s used for Euroean conngen clams, whls he second one s a much more versale (6 aroach ha can be used for Amercan, Euroean and ah deenden dervaves. 4. Sae deenden forward valuaon of Euroean conngen clams Recen sudes have roosed a smle algorhm o deermne he reurn dsrbuon funcon on a recombnng marovan ree afer erods of me (see Iaquna and Orobell (2006. Therefore an easy way o comue he value of Euroean conngen clams consss n usng he Iaquna and Orobell s recursve algorhm ha resens comuaonal comlexy of O( 3 2 order. In hs framewor we aly he same algorhm o he ranson marx P of an homogeneous Marov chan n order o oban he dsrbuon afer erods of me. The forward rocedure of he algorhm bulds a sequence of marxes Q of dmenson (( + such ha, afer erods of me, he reurn robables n he ( + nodes of he ree are gven by he vecor Q where s he uny vecor column. oe ha each node of he ree s smulaneously achevable from dfferen saes. Thus each node could be n dfferen saes and hs deends on he rovenance sae. In arcular Q [ q, ( ] ( + where q ( s he robably of beng n -h sae and n he -h node (counng from he hghes node afer erods of me. Therefore, f we suose he nal sae s he -h one, hen he frs ranson marx s he dagonal marx wh he dscouned robables corresondng o he -h row of P on he dagonal,.e., Q dag(,..., Insead, he oher marxes are gven by Q dagm( Q P, where he dagm oeraor erforms a dagonalzaon rocess conssng n he followng wo oeraons aled o Q P :. shf below he s-h column of s- saces for s2,,, creang a new marx (( + ; 2. fll all he new saces wh zeros. In order o ge he mnmal enroy marngale measure ha s rs neural wh resec he dsrbuon afer erods of me, we have o comue he unque value θ soluon of he equaon: ex(( r q ( + q ( ( + q ( ( z ex( θ z ex( θ z ( ( Then he rs neural uncondonal robably of beng n he -h node afer ses s gven by: (7

5 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June ( q( ex( θ z q (,, ( + ( + ( q( ex( θ z where θ s he soluon of equaon (7. Le f f,..., f ]' he vecor of conngen clam ( T [ ( T, ( T,( T + value a maury T. Then he rce of he Euroean conngen clam s smly gven by: ' ex( ( r q T q T f. ( ( T Ths s a forward rs neural valuaon of he rce wh comuaonal comlexy of O( 3 T 2 order. In hs valuaon we do no correc he ranson marx as we sugges n he revous sae deenden valuaon, snce we mlcly assume ha ex(( r q T belongs o he suor of he gross reurn afer T erods of me.e.: ( ( zt ex(( r q T zt where max a q( T > 0 ( T + and mn a q( T > 0 ( T + Ths nequaly s generally verfed when T s bg enough. Even for hs reason we could exec some dfferences n he rce valuaons when we do no correc he ranson marx before alyng he recursve algorhm o comue he reurn dsrbuon afer T erods of me. 4.2 The bacward valuaon rocedure for Amercan and Euroean conngen clams The bacwards rcng of dervaves roceeds as any oher sandard bacward rocess, dsngushng he sae deenden and sae ndeenden valuaons. Sae deenden bacward valuaon: In he sae deenden valuaon each node reresens dfferen values n deendence on he revous rovenance sae. I s he case o observe ha hs seemngly comlcaon of a node wh mully values, due o he recombnng urose of he ree, allows a grea advanage n order o save comuaonal me and he memory usage. Snce he ree s mulnomal, he sngle node consdered has ossble fnal nodes reresenng he fnal ayoff of he dervave. A sngle bacward se n he execed dscouned rocess consss of he marx mullcaon beween he dscouned ranson marx ransformed (as revously exlaned and he vecor of he fnal ayoff. The resul s a vecor of elemens whch reresen he dfferen values of he node n deendence of he rovenance sae. The descron of he enre Euroean oon rcng rocess s offered hrough s algorhm form. Le consder a recombnng mulnomal rce ree comosed by M me ses and branches for each node. Then we can buld he ree of he conngen clam.. Suose we have he fnal ayoff a M-h se (he - h ayoff from above s gven by f M :. Sarng o coun from he hghes node hen a he -h node (for,...,( ( M + we consder he vecor of ayoffs f M : [ fm :..., fm : + ]'. Thus, a he (M-- h se we consder he ( ( M + vecors of dscouned ossble rces: f M : ex( q r PfM: However, n hs se we ge more rces han hose we have n he ree. In order o elmnae he rces whch are no on he ree, we have o reorder he vecors ha should be dscouned n he bacward rocess. 2. We buld he new vecors : ( ( fm : [ f,..., f ]' for,,(-(m- 2+where M : M : + ( M : s f M : s ex( q r PfM : s f s he -h comonen of vecor: 3. Afer s ses we have a he -h node (sarng from above he vecor: f s: ex( q r Pfs+ : and he new recombnng ( ( s + vecors ( ( fs: [ fs :,..., fs : + ]',,(-(s A he frs se we have only one vecor ( ( f : [ f:,..., f: ]' The value of he conngen clam deends on he sae I m we begn from and s gven by he m-h comonen of ex( q r P f:. The comlexy of hs algorhm s he same of he sae deenden forward valuaon (.e., of O( 3 2 order. As a maer of fac, n he bacward rocedure he algorhm above can be summarzed as follows. We buld a sequence of marxes of ayoffs F [ f:,..., f:( ( + ] of dmenson (( ( +. Thus, gven he fnal ayoff marx F M he oher marxes are gven by F reducm (ex( q r PF +, where he reducm

6 204 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June 2008 oeraor erforms a reducon rocess conssng n he followng wo oeraons aled o ex( q r PF + :. a he s-h row, cancel he frs s- values for s2,..., and he las -s for s,...,-; 2. shf on he lef he s-h row of s- saces for s2,...,, creang a new marx ((-(-+ (whou consderng he cancelled saced of he frs oeraon. Fnally he conngen clam rce s gven by he m- h comonen of ex( q r PF when we suose ha he nal sae s he m-h one. Observe ha hs reducon rocess s n some sense he nverse oeraon of he dagonalzaon rocess and has he same comuaonal comlexy. Ths algorhm can be easly adaed o comue Amercan oons. For examle, f we value an Amercan u wh exercse rce X for every s less han he me o maury (.e., s M-, n he bacward rocedure, we have o consder he vecor ( ( ( ( + fs: [max( fs :, X S0zs,...,max( fs : +, X S0zs ]',,(-(s-+. Moreover, even f hs aroach s no aramerc we can also aroxmae he Gree leers whch are ofen used o hedge he nvesors osons.. However, n hs case we ae no accoun he ncremenal raos wh her rs neural robably. Suose a me zero we are on he m-h sae, hen afer one erod we have he vecor of conngen clams f : whose he -h comonen s realzed wh he rs neural robably m,. In order o esmae he dela ( Δ Δf ΔS of he oon, we have ( ( f: f: ncremenal raos Δ wh 2 ( ( S0 z z m, + m, robably esmaes q for,,-, +,,. Thus, an esmae of dela afer one erod condoned by he sarng sae m s gven by he average have ( m Δ q Δ + To deermne gamma (Ã f S 2 esmaes of dela Therefore afer one erod we have esmaes of gamma 2 Γ s 0.5S0 2 noe ha we Δ afer one erod. Δ 2 2 Δ s ( ( ( s ( ( z + z z z q + qs wh robably esmaes qs 2 for 2,s,,-, +,,, s+,, and, s,. Thus, an esmae of gamma afer one erod condoned by he sarng sae m s gven by he average: Γ ( m s s s+ + q Γs, where we have no consdered, s,. snce when s we ge Γ 0. Sae ndeenden bacward valuaon: Wh he sae ndeenden valuaon we ge a rce a each se nsead of a vecor of rces snce we have no o ae no accoun he sae of rovenance. Le consder a recombnng mulnomal rce ree comosed by M me ses and branches for each node. Then we can buld he ree of he conngen clam.. Suose we have he fnal ayoff a M-h se (he - h ayoff from above s gven by. Sarng o f M : coun from he hghes node hen a he -h node (for,...,( ( M + we consder he vecor of ayoffs f [ f,..., f ]'.Thus, a he (M - M : M : M : + h se we consder he ( ( M + rces: ex( q r Pf f M : M : and we buld he new vecors f [ f,..., f ]' for,,(-(m- M : M : M : Thus, afer s ses we have a he -h node (sarng from above he rce: f s: ex( q r P ˆ fs+ : and we buld he new (-(s-+ vecors fs: [ fs:,..., fs: + ]', ( (s A he frs se we have only one vecor f : [ f:,..., f: ]' and he value of he conngen clam s gven by ex( q r P f:. Observe ha he comlexy of hs algorhm s he same of he sae deenden one (.e., O( 3 2 order and, even n hs case, we can easly ada he algorhm o value an Amercan conngen clam. So n order o value an Amercan u wh exercse rce X for every s less han he me o maury (.e., s M, n he bacward rocedure, we have o consder he vecor ( ( + f [max( f, X S z,...,max( f, X S z ]' s: s: 0 s s: + 0 s,,(-(s-+. Smlarly o he sae deenden valuaon of Gree leers we ge he ncremenal 2

7 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June raos qˆ f ˆ : f: Δ wh robably esmaes S ( ( z z ˆ m m ˆ 0 m, + ˆ m, Thus, an esmae sae ndeenden of dela afer one erod s gven by he average Δ q ˆ Δˆ + In order o comue a sae ndeenden valuaon of Gamma we use he esmaes of gamma 2 2 Δˆ ˆ ˆ Δs Γ s wh robably ( ( ( s ( 0.5S0( z + z z z qˆ + qˆ s esmaes qˆ s for,s,,-, +,,, 2 2 s+,, and, s,. Thus, an esmae of gamma afer one erod s gven by he average: Γ q ˆ Γˆ s s s s+ + subsanally change wh greaer han 50. In arcular, we consder hsorcal daa from January 995 o Augus 2005 of Dow Jones Indusrals, S&P500 and asdaq and we comue he rce of several Euroean us and calls changng he number of he saes (from 0 o 200 he sres (fve n he money and fve ou he money and he me o maury (7 for a oal of 20 ossble oons. We comue he rces wh a sae deenden valuaon and wh a sae ndeenden valuaon. Whle here exs dfferences n he rces, we generally do no observe dfferences n sably beween he wo rocedures. Moreover, for all he exermens we oban he sably of he rce wh around 40, whle, for lower han 40, we no always have a sable rce. Fgure and 2 summarze wo of hese exermens for a call and a u on he S&P500. The grahs show clearly how ncreasng he number of he saes he rces end o be sable and maes sense o consder a leas 50 saes. On he oher hand, he valuaon of he rce of conngen clams wh he above algorhms requres few seconds usng a noeboo dual cenrno wh one Gb of Ram. As a maer of fac, Fgure 3 reors he grahs wh he seconds necessary o comue he rce of an Euroean call wh a bacward sae deenden valuaon and consderng he mean of 0 rces wh maury 20, 40, 60, 90, 20 radng days and saes varyng beween (4-50, (5-60 (6-70 (7-80. In vew of hs smle emrcal analyss, nex we consder as conngen clam rce, he average of he rces obaned wh ranson marxes from 40 o 60 saes. Fg. : Ths fgure reors he values of an Euroean call on he S&P500 comued wh he nonaramerc marovan rees varyng he number of saes of he Marov chan. 5 Sably of he rce valuaon Fgure 2: Ths fgure reors he values of an Euroean u on he S&P500 comued wh he nonaramerc marovan rees varyng he number of saes of he Marov chan. From he analyss of he bacward valuaon rocedure we undersand ha he sably of he rce deend on he oorune number of saes used n he rcng valuaon. From a smle emrcal analyss we could observe ha he rce of conngen clams do no

8 206 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June 2008 nown ha log reurns are no Gaussan dsrbued (see Rachev and Mn (2000. In an analyss of long me dsrbuons Iaquna and Orobell (2006 have recenly shown ha he aroxmaon of he long me reurn dsrbuons wh a nonaramerc marovan ree resens much beer f han ha obaned assumng log-normal dsrbued reurns. Therefore we exec ha he rces comued wh marovan rees are more recse han Table : Ths able shows some dfferences n he oon rces wh T60, 90 days o maury and Sre Prce equal o E(S T we observe on Euroean oons valued for he S&P500 when we consder or he nonaramerc Marovan model (boh bacward and forward manner or he Blac and Scholes one. Fg. 3: Ths fgure reors he numbers of seconds necessary o comue he mean of 0 rces o f Euroean calls wh dfferen maures comued varyng he number of he saes of he Marov chan. 6 An emrcal comarson In hs secon we roose a comarson beween nonaramerc marovan oon rces and he rces obaned under he Blac and Scholes assumon. Frs of all we comue he dfferences of valuaon when we assume he hyoheses of he Blac and Scholes model holds. Then we descrbe he dfferences of rces comued usng real daa. In order o value he erformance of our model when we assume he same hyoheses of he Blac and Scholes model, we roose a MoneCarlo smulaon comarson. In arcular, we generae 0000 Gaussan scenaros (0.002,0.03 of log reurns. We assume ha he rs free rae s 4% and he rce of he soc oday s 50 USD. Then we comue he rces of call oons wh 20, 40, 60 days o maury consderng dfferen exercse rces X (X42, 44, 46, 48, 50, 52, 54, 56, 58. For all he oons we comue he real Blac and Scholes rce he Blac and Scholes esmaed rce, he rce esmaed wh he bacward sae deenden and sae ndeenden valuaon. Then we comue he average of he dfferences observed by esmaed models and he real Blac and Scholes rces. We observe ha he esmaed Blac and Scholes rce dffers n average of abou USD from he real one, whle boh he bacward sae deenden and sae ndeenden valuaon dffer n average of abou USD. Therefore, hs analyss confrms ha he nonaramerc marovan models well f he underlne dsrbuon and we do no observe sgnfcave dfferences beween he sae deenden and sae ndeenden valuaons. On he oher hand, s well hose obaned wh he Blac and Scholes model. Usng hsorcal daa from January 995 o Augus 2005 of S&P500, Dow Jones Indusrals and asdaq we comue some of hese dfferences n Tables,2 and 3. In arcular Table refers o S&P500, Table 2 o Dow Jones Indusrals, Table 3 o asdaq. Each able reors he values of Euroean calls and us valued n dfferen wees beween March and Arl We use wo dfferen maures T 60 and T90; exercse rce X E ( S T and rs-free rae equal o he Treasury Bll 3 monhs. Snce we have no observed sgnfcave dfferences beween he sae deenden and sae ndeenden valuaon (boh bacward and forward mehod, n hs able we consder only he sae ndeenden one. As we can observe from he ables here exs sgnfcave dfferences beween he rcng models much hgher ha hose observed under he Blac and Scholes assumons. Therefore maes sense o consder hs modelzaon as alernave o he classc Blac and Scholes one.

9 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June Concludng remars Table 2: Ths able shows some dfferences n he oon rces wh T60, 90 days o maury and Sre Prce equal o E(S T we observe on Euroean oons valued for he Dow Jones Indusral when we consder or he non-aramerc Marovan model (boh bacward and forward manner or he Blac and Scholes one. Table 3: Ths able shows some dfferences n he oon rces wh T60, 90 days o maury and Sre Prce equal o E(S T we observe on Euroean oons valued for he asdaq when we consder or he nonaramerc Marovan model (boh bacward and forward manner or he Blac and Scholes one. We have roosed a Marovan model o rce conngen clams. The model s nonaramerc, ducle and resens a reasonable comuaonal comlexy. Usng he mnmal enroy marngale measure as rs neural valuaon, we have suded he sably of he rce wh resec o he number of he saes. Moreover we have roosed an ex-os emrcal comarson wh he Blac and Scholes model showng he ducly of he model wh resec o he underlne dsrbuon. The model here roosed consder only a homogeneous Marov chan o value Euroean and Amercan dervaves. However, can be easly exended assumng non homogeneous Marov chans o value lan vanlla and ah deenden oons. We also observe ha he ranson robably marx assocaed wh he Marov chan s usually sarse. I means ha many elemens of hs marx are numercally neglgble. Ths roery s moran because deely reduces he comuaonal cos of he algorhm (see Zlaev (99 and Broyden, Vesucc (2004. Therefore we beleve ha he comuaonal me of O ( 3 2 order could be furher reduced ang no accoun hs fac. References [] A-Sahala Y., Lo A.W onaramerc esmaon of sae rce denses mlc n fnancal asse rces, Journal of Fnance 53, [2] A-Sahala Y.996. onaramerc rcng of neres rae dervave secures, Economerca 64, [3] Blas, A., Janssen, J., Manca, R., umercal reamen of homogeneous and non-homogeneous relably sem- Marov models. Communcaons n Sascs, Theory and Models. [4] Broyden, C. G., Vesucc, M. T., Krylov Solvers for Lnear Algebrac Sysems. Elsever, ew Yor. [5] D Amco, G., Janssen, J., Manca, R., June onhomogeneous bacward sem-marov relably aroach o downward mgraon cred rs roblem. Proceedngs of he 8h Ialan Sansh Meeng on Fnancal Mahemacs, Verbana, June [6] Duan, J., Dudley, E., Gauher, G., Smonao, J., Prcng dscreely monored barrer oons by a Marov chan. Journal of Dervaves 0, [7] Duan, J., Smonao, J., 200. Amercan oon rcng under GARCH by a Marov chan aroxmaon. Journal of Economc Dynamcs and Conrol 25, [8] Frell M., The mnmal enroy marngale measure and he valuaon roblem n ncomlee mares, Mahemacal Fnance 0, [9] Gerber H.U., Shu, E.S.W., 994. Oon rcng by Esscher ransform,transacons of he Socey of Acuares 46, 99-40; Dscusson 4-9.

10 208 IJCSS Inernaonal Journal of Comuer Scence and ewor Secury, VOL.8 o.6, June 2008 [0] Gerber H.U., Shu, E.S.W., 996. Acuaral brdges o dynamc hedgng and oon rcng. Insurance: Mahemacs and Economcs 8, [] Huchnson, J.M., Lo A. W., Poggo T., 994. A non aramerc aroach o he rcng and hedgng of dervave secures va learnng newors, Journal of Fnance 49, [2] Iaquna G. and Orobell S., Dsrbuonal Aroxmaon of Asse Reurns wh onaramerc Marovan Trees, Inernaonal Journal of Comuer Scence and ewor Secury, VOL.6 o.0. [3] Lmnos,., Orsan, G., 200. Sem-Marov rocesses and relably modelng. World Scenfc, Sngaore. [4] Rachev, S., Mn, S., Sable Parean Models n Fnance. John Wley & Sons, ew Yor. 7 [5] Schouens W Levy Processes n Fnance: Prcng Fnancal Dervaves, Chcheser, John Wley & Sons. [6] Suzer, M. J A Smle onaramerc Aroach o Dervave Secury Valuaon, Journal of Fnance 5, [7] Zlaev, Z., 99. Comuaonal Mehods for General Sarse Marces. Kluwer, Boson. 8 Sergo Orobell Lozza Assocae Professor a Unversy of Bergamo (Ialy. Currenly he wors n he Lorenzo Mascheron Dearmen of Mahemacs, Sascs, Comung and Alcaons. Hs research neress nclude mahemacal fnance, sochasc rocesses and comuaonal fnance. Gaeano Iaquna Currenly s Posdocoral researcher a Unversy of Bergamo (Ialy n Sochasc Omza-on aled o Economc and Fnancal roblems. Hs research neress nclude comuaonal fnance, oon rcng heory and rs managemen.