A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural ingredien in sudying he evoluion of he marke sock prices. Inroducion. From physical poin of view, in many cases, we are ineresed o sudy he behavior of a cerain sysem S. Tha sysem may be coupled o anoher sysem, le i be a reservoir R, or may be no. If we have o do wih coupling, hen our sysem feels he influence of he reservoir so, we have o consider he behavior of he enire sysem S+R. Finally, being ineresed in properies of he sysem S only, i is necessary o find a way o eliminae he variables of R. A horough analysis has been done by Haken [] showing ha an equaion of moion of he sysem S coupled ly o he reservoir R is of he Langevin ype: x() = D(x (), ) f() () + where x is a variable (or a se of ) of our sysem S, D some ime dependen force and f() a flucuaing force coming from reservoir R. One can show [] ha a random force f() can be aken as: f() = s j ( ) n j δ( ) j () wih s being he size of he impulses, n j being zero or depending on he fac ha he random impulses are in forward or backward direcion and δ( j), he Dirac s funcion. Wha we need is he correlaion funcion f() f( ) which can be aken [] as : f() f( ) = Cδ( ) (3) wih C a consan and he saisical average. A sraighforward coninuaion is he Fokker Planck equaion: Ψ ( x, x, ) = x ( D (x)ψ ( x, x, )) + x,x (D (x)ψ( x, x, )) (4)
where Ψ (x, x, ) is he probabiliy densiy funcion (), ha is he probabiliy o have he variable x a ime if i had he value a a preceding ime. In fac, (x, x, ) can be aken as Ψ (x, ), namely he Ψ probabiliy o have x in he range x and x + dx a a ime. D (x) coefficien defined as: x is he drif D (x) = lim x () x (5) and D (x) he diffusion coefficien: D (x) = lim x () x (6) The socks marke and he Fokker Planck equaion I seems o be very naural o consider ha if a a cerain ime he price of a sock being x, o ask wha i will be a he subsequen ime, and he answer o his quesion o be done by he a probabiliy densiy funcion saisfying a Fokker Planck equaion. If we look a Google s sock prices [], he sock price, x(), can be considered as a coninuous variable, and as a random one, also. Neverheless, he reservoir may have an influence upon prices: a favorable (or no) aricle in a very known newspaper, an appreciaion of a consuling company, a poliical ineres a a cerain momen of ime, ec. So, he flucuaing force done by () can have n j =, or n j =, and as a consequence, he correlaion funcion is of he form done by (3). Firs version, dependence Following Haken [], we are in a hermodynamics approach of a Brownian moion where he coheren force D (x) can have a cerain expression and D (x) = C. Le us ake C = as a hypohesis. The ime evoluion of Google s socks price has a, raher, complicaed dependence on ime, and i is no specific for Google, i s generally valid. In he limis of experimenal daa, as physiciss say, we can simplify ha dependence assuming a one. The sock price, x(), can be aken as: x () = a + b (7) Allowing x() being of he form (7), D (x) = b, and reminding D (x) =, he Fokker Planck equaion akes he form: Ψ ( x, ) = b xψ ( x, ) + x, xψ( x, ) (8)
The soluion of he above equaion (8) is: Ψ(x, ) = π e (x a b) (9) Using daa [] regarding close and high sock s prices, an average beween hem, x() can be aken as: x () = 9 +.63 () Fig. shows such a dependence and he confidence inerval, as well. is he s number, namely for he firs (Aug.9, 4), for he second working and so on. The daa cover he inerval beween = (Aug.9, 4), = 573 (Nov.4. 6). goog 5 goog 4 prediced 3 uperror lowerror 3 4 5 Fig. Second version, quadraic dependence Wha will happen if he sock s price will reach a maximum value and hen sars o decrease? Obviously, he ime evoluion shall no have he form done by (7). Processing he daa concerning he Google s prices a close session, we can ge an expression like his one: x() = a + b + c () Such a form will change he drif coefficien D,and D as well x(x+ a b c) = D (x,) = + (C + x C ) e () D (x, ) a + b 3
Wih (), he soluion of he Fokker-Planck (4) is: Ψ ( x, ) = π e (x c b - a ) (3) Using daa [] regarding close sock s prices, x() can be aken as: x() =.3 +.65 + 35 (4) Fig. shows he quadraic dependence and he confidence inerval, as well. is he s number, namely for he firs (Aug.9, 4), for he second working and so on. The daa cover he inerval beween = (Aug.9, 4), = 573 (Nov.4. 6). goog 5 4 3 googclose prediced uperror lowerror 3 4 5 Fig. The cumulaive disribuion funcion, namely he probabiliy o have a price x equal or smaller han a cerain value, aking ino accoun he saring value of, for approximaion is:.77(x.63 9).77(.63 9) cdf = ( Erf [ ] Erf [ ]) (5) and for quadraic approximaion: cdf quadraic =.77(x ( Erf [ Erf.77( [ +.3 +.3.65 35) ].65 35) ]) (6) 4
Resuls The mos probably o have a price, per sock, of around 5 $, is shown in he Fig.3 and i really happened around he no 4..8.6.4. quadraic 4 6 8 Fig.3 The corresponding cumulaive disribuion funcion is: probabiliy.8.6.4. x 5 quadraic 4 6 8 Fig.4 The mos probably o have a price, per sock, of around 5 $, is, according wih his approach, shown in Fig.5, and i really happened around he no 9..5..5..5 quadraic 5 5 3 35 Fig.5 5
The corresponding cumulaive disribuion funcion is: probabiliy.8.6.4. x 5 quadraic 5 5 5 3 Fig.6 The mos probably o have a price, per sock, of around 5 $, is, according o his approach shown in Fig.7, and i really happened (perhaps,oo early), around he no 57..5.5..75.5.5 55 6 65 7 75 8 85 quadraic Fig.7 The cumulaive disribuion funcion corresponding o his siuaion is: probabiliy.8.6.4. x 5 4 6 8 Fig.8 quadraic 6
An esimaion for a price of around 55 $ per sock gives:.5.5..75.5.5 quadraic 6 7 8 9 Fig.9 The corresponding cumulaive disribuion funcion being: probabiliy.8.6.4. Conclusions x 55 4 6 8 Fig. quadraic To apply he Fokker Planck equaion in order o sudy he marke price evoluion seems o be an appropriae one. As one can see from wha was menioned above, we canno predic he jumps in prices, bu who can? Wha can do such an approach is o give a reasonable idea of wha probable will happen, and i for no a very long period. Daa accumulaion will improve he esimaion. References []. H. Haken, Rev. Mod. Phys., Vol. 47, No, January 977 []. hp://finance.yahoo.com/q/hp?s=goog 7