A STOCHASTIC MODELING FOR THE UNSTABLE FINANCIAL MARKETS

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1 A STOCHASTIC MODELING FOR THE UNSTABLE FINANCIAL MARKETS Assoc. Pof. Romeo Negea Ph. D Poliehnica Univesiy of Timisoaa Depamen of Mahemaics Timisoaa, Romania Assoc. Pof. Cipian Peda Ph. D Wes Univesiy of Timisoaa Faculy of Economics and Business Adminisaion Timisoaa, Romania Pof. Ioan Lala Popa Wes Univesiy of Timisoaa Faculy of Economics and Business Adminisaion Timisoaa, Romania Absac: : An alenaive appoach o sochasic calculus fo a financial model on some impefec and unsable financial makes is poposed. Following he mos ecen insumen fo he financial modeling, we sudy he solvabiliy of a class of fowad-backwad sochasic diffeenial equaions (FBSDE) in he famewok of McShane sochasic calculus, in some geneal hypohesis on he iniial value and he coefficien funcions. JEL classificaion: C (Pimay), G3 (seconday) Key wods: belaed inegal, fowad-backwad sochasic equaions, pahwise uniqueness, financial modeling.. INTRODUCTION Coninuous developmen of he finance asses yields moe adequae mahemaical models which ae supposed be good enough o descibe he complex behavio of he financial make. Moe sophisicaed models ae available on some esicive financial hypohesis, bu his hypohesis ae no saisfied on some ansiion financial makes as in Eas Euope, whee ae moe unpublished infomaion, ove quoed iniial values and govenmen financial inevenions. Moeove, he evoluion on hese makes is chaaceized by some "smoohed" life-ime and some vey "noises" life-ime and his ime peiods ae had unexpeced. Fo hese easons, we popose an appoach somehow, moe geneal as hee fo a fee financial make. The classic sochasic appoach fo he financial models has used he famewok developed by Io o deal wih he esuling sochasic diffeenial equaions (SDE), based on he idea ha a Wiene sochasic pocess is used fo he exenal disubances. Then, moe auhos supposed an semimaingale pocess fo he exenal noises which make vey complicaed sochasic calculus. On he ohe side, E.J.McShane developed a moe simple inegaion calculus using he Io-belaed inegals. In 979, Ph.Poe showed ha he McShane calculus is equivalen wih he

2 inegaion wih espec o a semimaingale pocess. Somehow, his siuaion is simila wih he fac ha a Riemann-Sieles inegal can be consideed as a Lebesque inegal in some adequae famewok, bu pacically we pefe o use he Riemann inegaion calculus as o be moe simple. I's known ha he sochasic appoach fo he financial modeling is saed wih he famous papes of Meon and Black and Scholes in he 7's. The pincipal insumen fo he sochasic modeling is he backwad sochasic diffeenial equaions (BSDE) o fowad-backwad sochasic diffeenial equaions (FBSDE). This leads us o conside a fowad-backwad McShane sochasic diffeenial equaions.. PRELIMINARY RESULTS In fis yea of 7's, E.J.McShane inoduced so called belaed inegals and sochasic diffeenials and diffeenial sysems which enoying he following hee popeies: inclusiveness, consisency and sabiliy. McShane's calculus had poved o vey valuable in modeling and is finding applicaions in physics, engineeing and economics. A sochasic inegal equaions by McShane ype is one of he following fom: X ()= x() f(, s X()) s ds g (, s X()) s dz () s k, = = h (, s X()) s dz () s dz () s k k whee he inegals ae belaed o McShane inegals. On he above equaion, we ecall some specifically esuls of he McShane sochasic calculus. Le ( Ω, FP, ) be a complee pobabiliy space and le { F, a} be a family of complee σ subalgebas of F such ha s a hen Fs F. Evey pocess denoed by z wih difeen affixes will be a eal valued second ode sochasic pocess adaped o { F, a} (i.e. z () is F measuable fo evey [, a] ) and E[( z( ) z( s)) m / Fs ] K( s) a.s., wheneve s a, m=,,4, fo a posiive consan K having a.s. coninuous sample funcions (and we say ha he pocess saisfies a K -condiion). I is known (see [7]) ha if f :[, a] L is a measuable pocess adaped o he F and if f( ) is Lebesgue inegable on [, a ], hen if z and z saisfy a a a K -condiion, he McShane inegals f () dz () and f () dz () dz () exis and he following esimaes ae ue a f ( dz ) ( ) C{ f( ) d} a () () a (3) f () dz () dz () C { f () d} a

3 whee C = Ka K. An impoan class of McShane sochasic diffeenial equaions is he class of equaion which have a canonical exension o a canonical fom (as in McShane a), i.e. he equaion () wih he special case when g (, X()) h X g X X X X (4) n k (, ())= n k (, ()), =(,, ). i i= X Among o his fowad equaions, in he opimal sochasic conol appea some backwad diffeenial equaions as he following k, = Y( ) f( s, Y( s), Z( s)) ds [ g ( s, Y( s), Z( s))] dz ( s, ω) = [ h ( s, Y( s), Z( s))] dz ( s) dz ( s) = Y k k whee { z ( ), }, =,,, is a sochasic pocess defined on he pobabiliy space ( Ω, FP, ) wih he naual filaion { F, } and Y is a given F -measuable andom vaiable such ha E Y <. Moeove, f is a mapping fom Ω [,] R R o R which is assumed o be P B B\ B-measuable, whee P is he σ -algeba of F -pogessively measuable subses of Ω [,]. Also g is a mapping fom Ω [,] R o R which is assumed o be P B\ B- measuable. We emak ha in he case of backwad sochasic diffeenial equaions by he McShane ype we have a canonical exension when eplace he funcions h k as above. In his conex we conside he following fowad-backwad sochasic diffeenial equaion by he McShane ype (5) X = X a ( s, X( s), Y( s), Z( s)) dz ( s) Y f s X s Y s Z s dz s = b (, (), (), ()) () () k s X s Y s Z s dz s dzk s k, = = (, ( ), ( ), ( )) ( ) = g k, s X s Y s dz s dzk s h Y k, = (, (), ()) () () ( ) wih a, b, f, g: Ω (,) R R R R, h: R R and he following hypoheses (which exend he esul of Ahanassov 99 [] fo odinay diffeenial equaions and includes ohe esuls on FBSDE): (6)

4 i) ab,, f and g is P B B B measuable funcions; ii) ϕ(,,,) M ((,), R), whee ϕ is any funcions ab,, f o g ( M (,) is he se of all sochasic pocess which ae squae McShane inegable on [,] and F -measuable fo ); iii) hee exiss u () a coninuous, posiive and deivable funcion on < wih u () =, having nonnegaive deivaive u ( ) L([,]), wih u (), such ha u () ϕ( x,, y, z) ϕ( x,, y, z) min( x x, y y, z z ), Ku() u () hy ( ) hy ( ) y y, (7) Ku() fo all x, x, y, y, z, z R,, posiive consan K and ϕ is any funcion ab,, f o g ; iv) wih he same funcions u () as above, ϕ (, x, y) u ( )min( x, y, z ), and h( y) ( y ),(8) and X is a finie andom vaiable and Y is given F -measuable andom vaiable such ha E Y <. A simila fowad-backwad equaion can be obain using he canonical fom.. 3. MAIN RESULTS In his secion, we pove ha he sochasic (fowad-backwad) diffeenial sysem has a soluion on some ineval [ δ,] fo any posiive consan δ in some geneal hypoheses of he coefficien funcions ab,, f and g as above. This case, wih some disconinuiy in he iniial ime momen = is is in accoding siuaion of he ansiion financial makes, whee he undelying asses (which is modeling wih a fowad sochasic diffeenial equaion) is ove-quoed. and 3.. Exisence and uniqueness We have he following heoem: Theoem. Le be ab,, f, g and h saisfying he hypoheses i) iv) Y L ( Ω, F, P, R), hen hee exiss a iple ( X, Y, Z) ( M ((,) M ((,)) M ((,)), R) which saisfy he sysem (6) in he canonical fom, fo Poof: In a simila way as [4].

5 3.. Opion picing The valuaion of coningen claims is poeminen in he heoy of moden finances. Typical claims such as call and pu opions ae significan no only in heoy bu in eal secuiy makes. The opion picing model developed by Black and Scholes [], fomalized and exended in he same yea by Meon [3], enoys gea populaiy. We conside a Black-Scholes make MBS =( S, B, φ ) (see [],[4]) whee: i) S ={ S }, [, T], is he pice pocess of a sock and we suppose ha i saisfies he following diffeenial sochasic equaion by McShane ype: ds = μ(, S) d σ(, S) dz ρ(, S)( dz),(9) whee b S b β β b β μ(, S)=, σ(, S)= cs, ρ(, S)= c S,() β wih < b <, < β, c R; ii) B ={ B }, [, T] is he pice pocess of a bond and we conside ha i saisfies he diffeenial sochasic equaion by McShane ype: db = d l( dz ) ;() iii) φ is a ading saegy (see [4]) i.e. a pai φ =( φ, φ ) of pogessively measuable sochasic pocesses on he undelying pobabiliy space ( Ω, FP, ). I is known (see [4]) ha a ading saegy φ ove he ime ineval [, T ] is self-financing if is wealh poces V ( φ ), which is se equal V( φ)= φs φ B, [, T] saisfies he following condiion ( )= ( ) V φ V φ φ ds φ db, [, T ] u u u u whee he inegals ae undesood in he McShane sense. Remak. We obseve ha he coefficien funcions of he sochasic diffeenial equaion (5) saisfy he condiions of ou Theoem, fo u ()=, (, T]. b We conside a Euopean call opion wien on a sock S, wih expiy dae T and sike pice K. Le he funcion c: R [, T] R ( ) given by he fomula Bs () α C () cs (,)= DAs ( () Ke ),() whee A, BC, :[, T] R ae some coninuous funcions, D is a posiive consan and α = β. In [Negea 3b, []] is poved he following esuls (using classical mehod of PDEs)

6 Theoem. The abiage pice a ime [, T] of he Euopean call opion wih expiy dae T and sike pice K in he Black-Scholes make is given by he fomula C = c( S, T ), [, T], (3) whee he funcion c: R [, T] R is given above and. Remak. I is easy o check ha he fomula (4) is ue using he FBSDEs mehod (given in Theoem.). 4. SOME APPLICATIONS AND EXAMPLES We conside ha we have an Euopean call opion on a conveible cuency (such ype of deivaive asses ae ansacioned on he Sibiu Moneay-Financial and Commodiies Exchange). Moe specifically we have an opion on he epo EUR/RON (Euo/Romanian Leu) fom..9 o The behavou of his pocess is given in he gaph fom bellow and we can sea moe vey smohed pa of his simple pah and his explain a non-andom noise on he make (in fac hese ae he esuls of some financial policies of Romanian govenmen) Figue no. We compue he pice wih ou fomula and we obain C (, T )=.768 which is less han he pice of he opion a he Sibiu Moneay-Financial and Commodiies Exchange (compue wih classical fomula of Black-Scholes), bu ou pice is moe closed o ealiy.

7 5. CONCLUSIONS.We poposed a model fo he behavio of he financial asses on some unsable finacila maekes. The evoluion on hese makes is chaaceized by some "smoohed" life-ime and some vey "noises" life-ime and his ime peiods ae had unexpeced. Fo hese easons, we popose an appoach somehow, moe geneal as hee fo a fee financial make. Ou sudy is us a he begining, bu, as inhe example fom above, he obained esuls susain ou modeling fo applicaions on he Romanian finacial make whee he noise make is no a classical Gaussian noise, hee exiss moe ohes andom o non-andom peubaions. REFERENCES. Ahanassov, Z.S... Black,F., Scholes, M. 3. Blenman, L. P., Uniqueness and convegence of successive appoximaions fo odinay diffeenial equaions, Mahemaica Japonica, 99. The picing of opions and copoae liabiliies, Jounal of Poliical Econom., 973 An alenaive appoach o sochasic calculus fo economic and financial models, Jounal of Economic Dynamic & Conol, 995. e al. 4. Consanin, Gh. An Applicaion of he Sochasic McShane's Equaions in Financial Modeling, Poc. Conf. "Inenaional Symposium on Applied Sochasic Models and Daa Analysis, Ma, J., Poe, P., Yong, J. Solving Fowad-Backwad Sochasic Diffeenial Equaions Explicily A Fou Sep Scheme, Pobabiliy Theoy and Relaed Fields, Ma, J., Yong, J. Fowad-Backwad Sochasic Diffeenial Equaions and Thei Applicaions, Spinge-Velag, Belin Heidelbeg, Gemany, McShane, E.J.. Sochasic Calculus and Sochasic Models, Academic Pess, New Yok, McShane, E.J.. Sochasic Diffeenial Equaions, Jounal of Mulivaiae Analysis, Meon, R.C.. Theoy of aional opion picing, Bell Jounal.Econom. Manag. Sci., Musiela,M., Maingale Mehods in Financial Modelling, Spinge, Belin Rukowsky, M. Heidelbeg New Yok, Negea, R. On he convegence of successive appoximaions fo McShane sochasic inegal equaions and financial applicaions, Bull. Sci. U.P.T., 3.. Negea, R., On he exisence and uniqueness of soluions fo a class of backwad Peda, C. sochasic diffeenial equaions, (o appea), Negea, R., On ceain class of Backwad Sochasic Diffeenial Equaions by McShane Caunu, B.. ype, Anal. Univ. Timisoaa, Negea R., An alenaive appoach o sochasic calculus fo impefec financial Peda C., makes, (o appea),. Popa L. I. 5. Padoux, E., Peng, S. G. Adaped soluion of a backwad sochasic diffeenial equaion, Sysems & Conol Lees, 99.

8 6. Poe, P. A compaison of sochasic inegals, The Annals of Pobabiliy, Toh, T. L., The Riemann appoach o sochasic inegaion using non-unifom Chew, T. S meshes, Jounal fo Mah. Anal. Appl., 3.