Comparative Analysis of Stock Price Simulation and its European-Styled Call Options

Transcription

1 Appl Math Inf 6-3, () 887 Comparatve Analyss of tok Pre mulaton an ts European-tyle Call Optons Yuje Cu¹ an Bngxng Lu² College of sene,north Chna Unversty of Tehnology, Bejng 44 P R Chna Emal: yj@nuteu n The enter of Computer an Net Management, North Chna Unversty of Tehnology, Bejng, Chna Emal: lu@nuteu n Reeve: Mar ; Revse Jul ; Aepte Aug Reeve: Jul 8, ; Revse Ot 4, ; Aepte Jan 6, Abstrat: Wth the popularzaton of omputer applaton tehnology, smulaton n fnane has beome an mportant metho for fnanal rsk management We ll start wth the applaton of the ranom walk moel n smulaton of stok pre Then, we ll prove an analyss of the rate of return of Chna Petroleum, the bggest lstng orporaton n Chna, an a smulaton of t Furthermore, we obtan expressons of European-style all optons base on Ito s lemma An then a smulaton s mae We analyze the effet of the varane of stok smulate pre an the tme for European-style all optons whh have a rtal an pratal sgnfane n bulng refne prng moel for the fnanal ervaton Keywors: European tyle all optons, smulaton, Ito s lemma Applaton of the Moel of Ranom walk n tok Pre mulaton Wth the popularzaton of omputer applaton tehnology, smulaton n fnane beomes an mportant metho for fnanal rsk management[,4] Here we hoose a typal stok for researh, for the reason that there s a lose relatonshp between the pre of European style all optons an the varane of stok pre[,3] An meanwhle we prove a smulaton[7] for the stok s pre Thereafter, base on the European style all optons of the smulate pre of the stok, we smulate the pre varane of the optons Ths smulaton has a pratal sgnfane n evelopment an applaton of the fnanal ervaton[], an also a pratal sgnfane for the fnanal ervaton n exertng a reatve effet n moern fnanal market[4] In the ase of orporaton s goo operaton an management an goo operaton of seurty market, base on experene, we sover a lose relatonshp between annual average rate of return of a stok on nvestment an nvestment tme[5] Frstly, we analyze the relatonshp between the mean value of rate of return an the tme pero, so that we fn a onrete relatonshp between annual M M M t M t T t e ln e e (4) Apple Mathemats & Informaton enes An Internatonal Journal average rate of return of a stok an tme nterval step Let t be the smaller tme nterval step How oes tme nterval affet return strbuton when the step turns to be short nterval samplng? Wthout any nvolve of ranomness, we have known that there s a proporton relatonshp between bons return an tme step Let be ts annual average rate of return of a stok on nvestment, then unt rate of return wth a step of t an be: t Or: t () Let be the ntal stok pre, then stok pre after one tme step turns to be: t () tok pre after two tme steps NP Natural enes Publshng Cor t (3) We an easly erve that stok pre after M tme steps turns to be: Ths s the approxmate exponental relatonshp between stok pre an tme, where T Mt

2 888 Cu, Lu: Comparatve Analyss of tok Pre mulaton Wth further onseraton for volatlty of rate of return, no matter how small we subve the tme, the varane of volatlty for rate of return n a fxe observaton pero wll not nrease nfntely By average, when tme step t turns to be very small, the expresson for stanar evaton of rate of return s: Where s N N R R (5) R s the rate of return n a sngle step In orer to ensure the fnte of varane, every tem n the braket n (5) must have the same orer wth t, n other wor, that s O t By average, the stanar evaton shoul have the same orer wth t, n other wor, that s O t In orer to show the volatlty of the rate of return, we let the volatlty satsfes the onton of normalty an refne the moel as follows: t t (6) where s the stanar evaton of annual return on assets, we use to show the unertanty of return; s the ranom number of stanar normal strbuton Therefore we an get the expresson of the moel of ranom walk: t t (7) or: / t t (8) Where, represents renewals of Let t be X, then X has a mean value of zero, an a varane of t Aorng to Ito s lemma, expresson (8) an be transforme nto: log t X (9) Let be the ntal value, then the stok pre at tme t s: t exp t t X () When the tme step nrease by nrement of t wll be: t t t t t, the exp t t () eleton of tok an mulaton of tok Pre We selet Chna Petroleum (6857)[8], whh exert a sgnfant effet on eonomy an whh has a bggest market value, as researh objet The eonom sgnfane of ths kn of stuy s very mportant We hoose the losng pre from Janst 9 to Apr 7th of Chna Petroleum as unerlyng stok pre,,(on the ate we hoose, there must be a trae) The ata erve from Netease atabase There are 8 val ata pont exept the ate wthout traes Frstly, we take on the test for normal strbuton, we hoose k-s One-ample Kolmogorov-mrnor test for these 8 ata pont, an get results as follows

3 Cu, Lu: Comparatve Analyss of tok Pre mulaton 889 Wth a sgnfane level of 5, we onser that the aly return of ths stok vares sgnfantly Table :One-ample Kolmogorov-mrnov Test Normal Parametersa Most Extreme Dfferenes N 8 Mean 567 t Devaton 4998 Absolute 7 Postve 7 Negatve -58 Kolmogorov-mrnov Z 998 Asymp g (-tale) a Test strbuton s Normal Let be the annual average rate of return, our naton s one year fxe epost rate s 35%, wth onseraton of nflaton, we assume rsk-free nterest rate nto three levels: hgh (7%), meum (5%), low (3%) For the reason that our naton s now n the pero of nflaton, we set rsk-free nterest rate over one year The step s, flutuaton rate s ae up to 5% We smulate n three stuatons, an hoose one path n every stuaton, as shown n follow fgure 3 Therefore, by means of a large number of smulatons, we an alulate the average path of stok pre varaton, whh exert an mportant effet n fnanal rsk management[9] x The average stok pre aorng to hgh nflaton rate, meum nflaton rate, an low nflaton rate after one year (n other wors, 5 trae ay) oul be Yuan, 5776 Yuan, an 5398 Yuan 3 Funamental Analyss of European tyle all Optons Base on Ito s lemma, we erve the formula of Blak- holes: V t V V r rv () V, s the present value of the ervaton whh has a value of an vares by tme t We foun the European tyle all Optons expresson as follows uner the onton of the formula of Blak-holes, the maturty ate T of European tyle all Optons satsfes: Where t max X, (), we foun: T t X e r N, aorng to expresson N (3) Where s the present value of the all optons, t s the present tme log T t / X r T t log T t / X r T t

4 89 Cu, Lu: Comparatve Analyss of tok Pre mulaton where X s the strke pre of maturty ate T of the unerlyng stok strbuton funton N s stanar normal 4 Calulatons for Relate Parameter of Equty Optons The onton for bulng equty optons s rgor, therefore we an only bul approxmate the moel of prng by means of Blak-holes formula Here we only analyze the prng moel of European style all optons Aorng to our statstal researh, we onser that stok whh has a relatvely small market value annot satsfy the onton of Blak-holes formula beause of ts volatlty s relatvely bg Therefore we stll use Chna Petroleum whose market value s bg as the unerlyng of the equty optons to bul analyss moel We gve a onverson for the rate of return n Part as follows: r 68, 5 47 Let T t be a typal analyss pero of one year, X be the strke pre of all optons We assume that s % hgher than reent average pre as the funamental analyss onton Base on t, we prove smulatons for all optons 5 mulaton Analyss of Call Optons After alulatons of relate parameter, we bul the moel of the stok of Chna Petroleum by means of Blak-holes Prng Moel an A software, an aheve relevant fgures as follows 5 the nfluene of varane of stok pre on the value of all optons We selet the losng pre, 993 Yuan, of Chna Petroleum on Apr 8th as stanar pre, an strke pre as X 93 wth an nrement of % for the losng pre Therefore we prove the value varane of European style all optons nfluene by unerlyng stok pre We fnshe expete smulaton fgure as follows fg 4 5 affetons of tme an pre of the stok hanges n all Optons Let the tme pero be months, n other wors, 5 trae ays, where rt t X e N N We shoul note that an are stll funtons of tme T t We prove the smulaton fgure of relatonshp between the value of European style all optons an the pre of unerlyng stok, as well as relatonshp between the value of European style all optons an tme (as follows fg 5) C Fgure 4 Relatonshp between European style all optons an the value of stok s 8 Fgure5 Relatonshp between European all optons an the pre of unerlyng stok(s), tme (t) 6 Comparatve Analyss of Results of the two Methos an the Conluson of our Researh () Aorng to the efnton of European style all optons, the maturty ate T satsfes max X,, an annot be operate n avane Wthout the onseraton of trang fees, base on the efnton of European style all optons we an erve that the value of European style all optons on maturty ay are () (hgh nflaton, meum nflaton, low nflaton) t

5 Cu, Lu: Comparatve Analyss of tok Pre mulaton 89 whh means that the varaton of nflaton an nfluene the value of equty optons (3) Value of optons alulate by the expressons of European style optons We prove fgures uner hgh, meum, an low nflaton, an smulaton results are as follows: (4) Comparatve analyss between absolute error an relatve error The smulaton results base on European-style optons are slghtly larger than those retly erve from the efnton of equty optons The absolute error s below % whle the relatve error s below % The omparson tells us that these two results are losely relate, but whether the smulaton results are relate to atual values epens on the evaluaton of nflaton Of ourse, the smple assumpton of hgh, meum, an low nflaton s far from enough an from ths perspetve, the alulaton expresson of Europeanstyle optons s better than that by efnton Of ourse, all optons an be nfluene by the rate of nterest, the volatlty of unerlyng stok pre, the senstvty of all optons to unerlyng stok pre an so on These sussons are pratally benefal n bulng omplete prng moel of fnanal ervaton an fnanal market whh wll exert effet on mprovng eonomy Aknowlegements Ths work s supporte by the entf Researh Common Program of Bejng Munpal Commsson of Euaton #KM9, Thanks for the help Referenes [] Zhu hwu Fnanal Calulaton an Moelng: Theory, The Algorthm an A Program 53 (Tsnghua Unversty Press, Bejng, 6) [] Paul Wlmott translaton by Lu lxn et Fnanal Engneerng an Rsk Management Tehnques, 369(Mehanal Inustry Press, Bejng, 9) [3] Blak,F an M holes The Prng of Optons for Alternatves tohast Proesses Journal of Poltal Eonomy, 5~ 6 (8) (973) [4] Blak F,Jensen M,holes M The Captal Asset Prng Moel: ome Empral Tests tues n the Theory of Captal Markets, (Praeger, New York, 97) [5] Wllam F harpe, Goron JAlexaner an Jeffery VBaley: Investments 5th e translaton by Zhao xjun, 34 (Prente Hall In,996) [6] D Yeung an L Petrosyan, Cooperatve tohast Dfferental Games (prnger, New York, 6) [7] helson M Ross, mulaton, 98 (Turng press, 6) [8] Yuje Cu s an expert n Apple mathemats statsts an s presently employe as NCUT assoate Professor at Bejng, Chnahe has been aware Chna sentf researh n statsts Awar (seon prze) by Natonal Bureau of tatsts of Chna n 4, he has been an nvte speaker of number of onferenes an has publshe more than 4 researh artles n repute journals of mathematal an management senes Bngxng Lu s an nstrutor of the Computers an Network Management Center, North Chna Unversty of Tehnology He earne hs Master Degree of ene an engneerng from North Chna Unversty of Tehnology Hs researh nterests nlue network amnstraton, network seurty an network arhteture analyss