The Application of Fractional Brownian Motion in Option Pricing


 Amos Sutton
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1 Vol. 0, No. (05), pp The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn Abstract In ths text, Fractonal Brown Moton theory durng random process s appled to research the opton prcng problem. Frstly, Fractonal Brown Moton theory and actuaral prcng method of opton are utlzed to derve BlackScholes formula under Fractonal Brown Moton and form correspondng mathematcal model to descrbe opton prcng. Secondly, based on BYD stock, estmaton model on volatlty of ths stock s gven to analyze and calculate the stock prce volatlty. Fnally, make nstance analyss for BYD s opton. Based on market data of BYD s stock and opton, calculate the actual opton prce and theoretcal prce of BYD by Black Scholes formula under Fractonal Brown Moton. Compare the forecast prce of ths stock opton gven by model wth actual prce, relatvely good effect s obtaned, and then conclude that the model has relatvely strong applcablty. Keywords: Stock Prces; BlackScholes Model; Brown Movement; Fractonal Brownan Moton; Optons; Volatlty. Introducton Brown Moton theory s usually used for researchng the change of asset prce n prevous opton prcng theory, whle comparatvely speakng, Fractonal Brown Moton theory has related nature between ncremental quanttes, and usng t to research asset prce can more reflect some characterstcs of stock yeld wth more extensve sgnfcance. In ths text, Fractonal Brown Moton theory durng random process s appled to research opton prcng problem. [4]. Modern opton prcng theory began n 990. The French mathematcan Lous Bachele frst put forward opton prcng model, and ths theory thnks that stock prce should be subject to Brown Moton, whch s consdered as the nassance mark of asset prcng theory. In 973, on the premse of noarbtrage prncple, Fsher Black and Myron Scholes derved partal dfferental equatons of European style opton prcng under a seres of deal assumptons that stock prce should be subject to logarthmc normal dstrbuton and so on, that s famous BlackScholes formula. Almost at the same tme, Robert Merton mproved BlackScholes model to make the model more practcal. In 976, J.C.Cox and S.A.Ross put forward rsk neutral prcng theory. In 979, J.M.arrSon and D.Krep put forward to descrbe noarbtrage market and ncomplete market by martngale method, whch had a profound effect on future development of fnancal mathematcs. In 988, Mogens Blat and Tna vd Ryberg derved BlackScholes prcng formula of European style opton by only usng lttle actuaral knowledge, whch s the begnnng of actuaral prcng method appled n opton prcng[57]. ISSN: IJMUE Copyrght c 05 SERSC
2 Vol. 0, No. (05) Nowadays, many domestc scholars utlze BlackScholes formula under Fractonal Brown Moton to research opton problem.. Basc Theory Defnton[8] If random process X t, t 0 satsfes: () X 0 0; () X t, t 0 s statonary ndependent ncrement; (3) For every t 0, t has X t N (0, c t ), then random process X t, t 0 s Wener Process or Brown Moton. Defnton [8] Suppose B t, t 0 s standard Brown Moton X t e t B t, t 0 s called Geometry Brown Moton wth obedence drft parameter,fluctuaton parameter. Defnton 3 [9,0] Suppose (, F, P) denotes a probablty space, s a constant between ( 0,) ( also be called urst Parameter ), f random process B ( t ) satsfes () B 0 0 ; ()For random t R, B ( t) s a random varable, and makes E B ( t ) 0 ; (3)For random t, s R,t has E B t B s t s t s then call random process B ( t) as Fractonal Brown Moton. Lemma The most common form of Fractonal Brown Moton s B ( t ) t ( ) t 0 ( ) w ( ) d where ( x ) s functon, w ( ) sgauss Whte Nose wththe mean of 0 property : The ncrementof Fractonal Brown Moton s: B ( t, s ) B ( t ) B ( s ), Property: E ( B ( t, s )) = E ( B ( t ) B ( s )) = E ( B ( t )) E ( B ( s )) 0 t, s R It can be seen that E ( B ( t, s )) as nothng to do wth the current moment. property3: E (( B ( t, s )) ) = E (( B ( t ) B ( s ))( B ( s ) B (0 ))) = E ( B ( t ) ) E ( B ( s ) ) E ( B ( t ) B ( s )) = t s t s t s 74 Copyrght c 05 SERSC
3 Vol. 0, No. (05) = t s Property4: E ( B ( t, s ) B ( s, 0 )) E (( B ( t ) B ( s ))( B ( s ) B (0 ))) E ( B ( t ) B ( s )) E ( B ( t ) B (0 )) E ( B ( s ) ) E ( B ( s ) B (0 )) ( t s t s ) When, Fractonal Brown Moton s Brown Moton. Obvously, Brown Motons are mutual ndependence at dfferent tmes, but Fractonal Brown Moton has more contnuty, so t has more unversal applcablty to use Fractonal Brown Moton to explan real natural phenomenon. Lemma [] Suppose rskfree nterest rate s r, expected yeld of underlyng assets prce s, ts prce s S ( t) at tme t,then expraton tme s T, the prce V ( t ) of European style call opton of exercse prce X at tme t t T ( )s ( T t ) r ( T t ) ( T t ) r ( T t ) V ( t ) E (( e S ( T ) e X ) I ( e S ( T ) e X )) Lemma [] Suppose expraton tme s T,Exercse Prce s X, The ntal value for the stock prce s S ( 0 ),and the prce satsfes the equaton d S ( t ) S ( t ) d t d B ( t ), where s expected yeld of underlyng assets prce, s Stock volatlty, and random process B ( t ) s Fractonal Brown Moton, then the prce of European style call opton at tme ( t t T )s where and d d * * * * r ( T t ) * V S ( t ) N ( d ) e X N ( d ) ln S ( t) r ( T t ) ( T t ) X T t ln S ( t) r ( T t ) ( T t ) X T t Formula () s BlackScholes formula under Fractonal Brown Moton and, n ths text, the opton prcng formula defned by formula () s called Model I. Defnton 4 European style prcng formula of call opton about contnuous tme BlackScholes formula The BlackScholes formula s as follows: Where T C ( S, T, K ) S (0 ) ( ) K R ( T ) T ln ( S R / K ) T T Usually, tme s the sngle moment R r, but for practcal problems, moments r are near contnuous, then R e, so the formula becomes: r T C ( S, T, K ) S (0 ) ( ) K e ( T ) () Copyrght c 05 SERSC 75
4 Vol. 0, No. (05) where r T ln ( S (0 )e / K ) T. T 3. Applcaton of Model I n BYD Opton Prcng Volatlty s also called mutablty, whch s the estmaton for rsk degree of stock market. The stock prce volatlty s a knd of measurement for uncertanty of stock prce trend, and the magntude of stock prce volatlty can affect opton value. In the opton prcng formula of ex dvdend payment, the man fve nput varables are basc stock prce, exercse prce of opton, expraton tme of opton, rskfree nterest rate and volatlty of stock prce. The frst three nput varables can be obtaned by opton contract clause, rskfree nterest rate can also be represented by general depost and loan nterest rate or treasury bond rate, and crtcal nput varable related to successful use of Black Scholes opton prcng formula s volatlty [, 3]. 3.. Method of Usng story Data to Estmate Stock Prce Volatlty Stock prce s subject to logarthmc normal dstrbuton [4], that s: ln S ( T ) ln S ( t ) T t, T t Among whch S ( T ) s the stock prce at future tme T ; S ( t ) s the stock prce at present tme t; s expected ncrease rate of stock prce; s stock prce volatlty; a, b means normal dstrbuton of mean value a, standard devaton b. Usng hstory data to estmate stock ndex volatlty, and tme nterval of observaton ndex s fxed. The defnton s as follows: n s n tme nterval, S s stock ndex at end of No. tme nterval,,.. takes year as unt, and expresses the length of tme nterval. Make u ln (S / S ), due to S S ex p u , so u s contnuous compoundng yeld wthn No. tme nterval. The usual estmated value of standard devaton of u s S ( u u ) n n where u n above formula s average value of u. Known from above dscusson, the standard devaton of u s S s estmated value of, can be estmated as S S ' S, then, so the varable ' The standard error of ths estmaton approxmates S / n. When usng hstory data to estmate volatlty, the followng stuatons should be explaned: () The choce of value n s very mportant. As a general rule, f t has more data, the result s more accurate. But n realty, the value s tmevaryng, and very old hstory data has no effect on predctng future, so latest 8 days closng ndexes are chosen as data materal. The volatlty at certan tme that wll be estmated and materals that wll be adopted should stay the same n the aspect of tme. 76 Copyrght c 05 SERSC
5 Vol. 0, No. (05) () When estmatng and usng volatlty ndex, experence has shown that transacton days should be adopted, and market closng days of exchange should be deducted when calculatng volatlty [4]. 3.. Estmaton of BYD Stock Market Volatlty In ths text, the closng prce ndexes of BYD transacton days from January 6,04 to May 9, 04 are taken as research object. The data on frst day s closng prce of January 6, 04, number of tme nterval s 8, observaton tmes are n 8, and length of tme nterval expressed by unt of day s the recprocal of tme nterval number contaned n ths day. Fgure. BYD Stock Informaton from January 6,04 to May 9, 04 Calculate by hstory data and 8 8 u u The estmated value of standard devaton of daly return rate s n n S u ( ) = u n n ( n ) The estmated value of volatlty s Its standard devaton s s ' S 0 0 % = % ' s 0 0 % =. 0 % n Then ts stock volatlty of BYD estmated from above data s 8.06% and ts standard devaton s.0% Applcaton of Model I n BYD Opton The below s emprcal analyss of BYD opton prce of Model I. Copyrght c 05 SERSC 77
6 Vol. 0, No. (05) Frstly, estmatng volatlty of above secton can obtan basc nformaton of BYD opton s shown n Table specfcally. Table. Opton Contract of BYD Opton. The Item Abbrevaton More Content BYD Stock Code Exercse Proporton Exercse Prce.6 Settlement Method Delvery Settlement Transacton Perod From January 6, 04 to May 9, 04 Expraton Date May 7, 04 Table.Comparson of Actual Prce and Theoretcal Prce of BYD Opton Tme Opton Actual Prce Opton Theoretcal Prce Copyrght c 05 SERSC
7 Vol. 0, No. (05) From above data: BYD opton transacted from January 6, 04 to May 9, 04, prce change s relatvely smooth durng ths perod, whle the fluctuaton of transacton theoretcal value calculated accordng to fractonal Black Scholes model s relatvely large. Draw Fgure wth Matlab and make comparson, seeng detals n Fgure. Fgure. Trend Chart of Opton Actual Prces and Opton Theoretcal Prces of BYD Opton Among whch the volatlty ndex s 0.806; the ntraday closng prce of BYD s taken as stock prce; the exercse prce ndex s.6 Yuan; rskfree return rate s 5.36% (above data s from Great Wsdom); and Suppose = Result Analyss of Model I From lne graph, t can be obvously seen that the prce change of BYD opton s relatvely smooth durng ths perod, whle the transacton theoretcal value calculated accordng to fractonal Black Scholes model s relatvely fluctuatng, but surround the actual value to change. The overall prce dfference s not bg. The dfference between actual value and theoretcal value s very small n front secton, but as tme grows, the predcted effect s worse and worse, so predcted opton prce can not explan actual change trend of opton, that s to say, the change of opton market s unpredctable. Analyze dfference exstence reason of BYD actual opton prce and theoretcal opton prce, I thnk that t can be dvded nto the followng ponts: () The select of value s not accurate enough, and value should be a changeable value all the tme, for convenence, n ths text, only one numercal value s selected as representaton, so compared to actual value, a certan devaton s caused. () Stock volatlty s not real numercal value, t s predcted by model, so dfference s caused between theoretcal value and actual value; (3) The select of start data can also cause correspondng devaton on general predcton; From above reasons, t can be fully explaned the dfference exstence reasons of BYD actual opton prce and theoretcal opton prce. Accordng to some reasons descrbed above, I thnk the followng adjustments can make the theoretcal value and actual value beng more dentcal: () Selectng value by subsecton for many tmes, and dfferent applcable value s chosen for every secton of data to carry out predcton and calculaton of model; () The predcton model wth best stock volatlty s selected to carry out predcton and select more sample data as far as possble to make the predcton more accurate; (3) Selectng a new start data agan to carry out model predcton for every some data to make the theoretcal value and actual value beng more dentcal; Copyrght c 05 SERSC 79
8 Vol. 0, No. (05) (4) Try to select one stock whch more conforms to model condton to research or choose some tme of one stock whch conform to model condton to research, analyze and predct. Through some solvng strateges whch put forward above, the problems researched n ths text can be solved better and try to avod devaton. 4. Concluson In ths paper, a stochastc process Brown motons and sports scores construct a mathematcal model descrbes the dfferent prcng optons. Prcng for solvng the problem of rsky assets and ts dervatves s one of the elements of the study of mathematcal fnance, and opton prcng problem s one of the most mportant content. Wth the contnuous development and mprovement of the stock market, more and more people use some new mathematcal tools to analyze and study of opton prcng. Prevous theores commonly used to study the Brownan Moton changes n asset prces, and sports scores relatvely Brown wth ncremental nter related nature, Usng t to study some of the characterstcs of asset prces whch better reflect the stock returns, and there s a broader sense. Ths paper s to study the nature of the opton prcng theory of stochastc processes. The man fndngs are as follows: The frst part s ntroducton, ntroducng the research background and ts sgnfcance. The second part descrbes the Brown campagn, geometrc Brown, Brown defntons and nature of sports score. The thrd part s the applcaton of ths artcle, based on the model of BYD optons example analyss, we used the model to predct the prce of the stock optons and compared wth actual prces. Seen from Fgure 3, the effect s relatvely consderable. The general value dfference s not bg, but volatlty dfference s relatvely large, although dfference of ths method exsts through several experments, t just reflects that stock market s changeable and unpredctable. Acknowledgement Ths research was supported by the Department of Educaton of elongjang Provnce under Grant No , Grant No. 548 and the arbn Unversty of Commerce Natural Scence Foundaton of Young Teachers under Grant No CUL030. References [] K. Wang, Foregn Exchange Opton Prcng Model and Emprcal Research Based on Fractonal Brown Moton, Chongqng Unversty, (008). [] C. J. e, Research on Opton Prcng Model n Fractonal Brown Moton Envronment, efe Unversty of Technology, (009). [3] J. C. Feng, Research on Some Problems of Opton Prcng under Fractonal Brown Moton Envronment, Lanzhou Unversty, (009). [4] X. T. Lu, Opton Prcng Problem under Fractonal Brown Moton Envronment, uazhong Unversty of Scence and Technology, (008). [5] J. Wang and J. Wang, Applcaton on Random Process Theory n Prcng Opton, Bejng Jaotong Unversty, (007). [6] S. L. L, Research on Opton Prcng Problem under Fractonal Brown Moton, arbn Engneerng Unversty, (0). [7] L. Zhang, Applcaton and Research on Actuaral Studes n Opton Prcng, Shaanx Normal Unversty, (008). [8] N. Wener, The Average of an Analytc Functonal and the Brownan Movement, Norbert Wener: Collected Works wth Commentares, MIT Press, vol., (976). [9] N. Aronszajn, Theory of Reproducng Kernels, Trans. Amer. Math. Soc., (960), pp [0] G. S. Ladde and L. Wu, Development of modfed Geometrc Brownan Moton models by usng stock prce data and basc statstcs, Nonlnear Analyss, (009), vol. 7, no., pp Copyrght c 05 SERSC
9 Vol. 0, No. (05) [] L. Bacheler, Theory of Speculaton, Cooter P. The random Character of Stock Market Prces, Cambrdge: MIT Press, (900), pp [] F. Black, M. Scholes, The Prcng of Opton and Corporate Labltes, Journal of Poltcal Economy, (973), pp. 8, [3] P. A. Samuelson, Ratonal Theory of Warrant Prce, Industral Management Revew, (965), pp. 6, 33. [4] X. Gabax,X. Gopk, P. Rshnan, V. Plerou, A Theory of Power Laws n Fnancal Fluctuatons, (003), pp. 43, Copyrght c 05 SERSC 8
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