Many-objects interaction Hamiltonian and Schrödinger equation in application to financial modeling.

Stochastic processes and time series 1 Many-objects interaction Hamiltonian and Schrödinger equation in application to financial modeling. Rostislav A. Taratuta corresponding author, email: [email protected]; Abstract This mathematical modeling can be considered primarily as a contribution to applications of quantum mechanics. Consider many-objects interaction systems on sampling of a financial market. In this article presented is a general expression of the interaction Hamiltonian for pair interaction, which can be used in Schrödinger’s equation and can be applied to any manyobjects mathematical modeling. Consideration is given to a mathematical description of a model, which uses quantum mechanics formalism to determine trajectories depicting an average of observable prices for i-th securities and the index. Noted results can be used as a ‘signal’ for high frequency and quantitative trading. Also calculated is the influence of economic conditions on stocks behavior, and specified condition for arbitrage. Keywords: quantum mechanics POLYTECHNIC INSTITUTE of NEW YORK UNIVERSITY Stochastic processes and time series 2 Introduction A central problem of Quantitative Finance is formulating a probabilistic model of the time evolution of asset prices allowing reliable predictions on their future volatility. Theoretical and statistical analysis of data questions martingale model, i.e. the use of a random walk and more general martingale techniques to model behavior of asset prices. Prices do not completely follow the random walk (a Winner process is the scaling limit of random walk in dimension 1) Mantegna et al [1], Rickles [2]. Noted model can be considered as approximation limit of the approach 0note , which is based on the Schrödinger equation Khrennikov [4], Dürr et al [5]. Such relation between the Brownian motion and the Schrödinger equation is specified by the Feynman-Kac formula Kloeden et al [6]. In this article the presented quantum mechanics approach is also in publications Seagal et al [7], Nelson [8], Petroni et al [9] which is based on a hidden analogy between modeling for the securities and financial market on one side and the many-objects problem in quantum mechanics on the other. The behavior of the i-th security on the market is affected by market economic conditions and other securities. One can consider the market as a physical system Ilinski [10], Bak et al [11], which undergoes the same manyobjects interactions, but with securities in place of the bodies. Using this analogy the way to account for security-to-security interactions and influence of different market economic conditions on security, which has affected security and market over time, is to introduce mathematical wavefunction formalism of quantum mechanics. The main goal is to introduce mathematical formalism for a wavefunction of a whole market P and a specific i-th security on the market i P . This approach allows to define time dependence for an average of observable prices for i-th securities i q and a financial market q . Individual behavior of i q and q related to the presented model used for forecasting various univariate GARCH-type time series properties in the conditional variance and an ARMA specification in the conditional mean Wurtz et al [12]. :note   0,1 developed approach does not address to the Bohmian financial wave modeling.   0,2 also Choustova [3] distinguish between different studies from the perspective of relevancy to the aforementioned problem and to decouple from the consideration of other applications of quantum mechanics formalism for financial research, e.g. Haven [13]; or implementation of the path integral methods Baaquie [14] within the framework of the developed formalism. Stochastic processes and time series 3 Part 1. Perturbation theory Financial capital market can be viewed as a sum of two parts: a replication part and a replicable part. The replication part should consist of a minnumber of securities, which is sufficient to represent the capital market. If a wavefunction of a market is P and wavefunctions of the replication part is Pr and replicable part is Pr' (r -replication, r' -replicable), then, based on a superposition principle Davidov [15]   1   2 , : (1.1) P = r P + Pr' will present as a linear approximation (such that accuracy of the approximation tends to improve as a mass of shares for a specific security gets smaller) (1.2) P = 1 ni r iP   + ' 1 mi r in P   wherein n represents a replication security and m represents all securities in the given market, i -state (security on the market). According to Davidov [15] in the first term (1.2), the wavefunction Pi r is multiplied by coefficients of expansion (any complex numbers, not depending on time), corresponding to the same state, and, analogically, in the second term (1.2) for Pi r' , therefore, expansion coefficients in (1.2) may be omitted: ' 11 nm ii i r i r i i n P a P a P      Consider the first term, replication part, as the index and the second term as the replicable part, as perturbation to the index, which is how all securities on the market, not included into the first replication part, are affected by it. This is a standard consideration of perturbation theory Keldysh [16]. - Suitable criteria for the security to be entered in the replication or the replicable part correspondently. Consider an orthonormal basis Lang [20]. When adapted to finance it requires determination of securities, which forms a securities system coordinate, in which all other securities can be presented. Analogous to a mathematical system coordinate this financial system coordinate must consist from orthogonal securities (not correlated)   1,1 . It is a matrix of orthogonal securities, which will create this system and size of matrix - n from the equation (1.2) and will determine an index size. Stochastic processes and time series 4 Such system as a mathematical normal system will be independent. It will not necessarily be stable or self consistent all the time, because of the market dynamics a market can frequently go out of equilibrium Allen et al [17]. Part 2. A model The general method of calculating a wavefunction of a normal (orthogonal) part –the index was described in Part 1. This method is based on eliminating a not-normal (non-orthogonal) part from the index. Based on the foregoing, one can formulate an approach to the financial market as follows: Lemma (2.1): a) Consider the financial market as a set of securities, characterized by wavefunction Pi for i-th market security. b) Each wavefunction Pi represents a security from an orthogonal system (index) or non-orthogonal system, which can be represented through expansion by use of index securities. c) Wavefunction P of the financial market at t comprise superposition of wavefunctions Pi for each market securities in t . d) A dynamic of market wavefunction P is arrived at by superposition of dynamic (sum of dynamics) Pi wavefunctions for each of the market securities. e) A dynamic of wavefunction Pi for i-th market security described by HamiltonSchrödinger equation. - Representation of non-orthogonal input through an orthogonal system of securities. According to Part 1 one can find a wavefunction of orthogonal part Pr , but a real market distorts at any given moment t out of equilibrium because of the contribution of a wavefunction of the non-orthogonal part Pr' of the market. To estimate a total contribution of Pr' into P in equation (1.1) uses form (1.2) (2.2) Pr' =   m ni i r P 1 ' where Pi r' is a wavefunction of an i-th security from a non-orthogonal part. Stochastic processes and time series 5 Can Pi r' express through wavefunctions of the i-th security from the index: (2.3) Pi r' = P kb r n b i b  1 where ki b -expansion coefficient for wavefunction of the i-th security from the non-normal set Pr' on orthogonal basis of wavefunctions for securities from normal set b r P . - Representation of a market dynamic. Presented is the market as a combination of an equilibrium part, characterized by a normal system-index and represented by the H0 - orthogonal Hamiltonian: (2.4) 1 nii rr rr i PP HF    where  F operator of equilibrium system meaning orthogonal; and a non-equilibrium part characterized by a non-normal system -other securities not included into the index (dependable on the index) and represented by the Hint - nonorthogonal Hamiltonian. Dependable securities playing a role of getting index out of equilibrium, so they play a role of an outside perturbation related to the index, can be presented as: (2.5) '' '' 1 mii rr rr in PP HF    where Fr' - operator of a non-equilibrium system meaning a non-orthogonal. Then total Hamiltonian is represented based on (1.2), (2.4) and (2.5) as (2.6) P H  = H0 + Hint where H HP      ; the operators Fi r and Fi r' will depend on time t and Pi r , Pi r' will not depend on time t (Heisenberg picture). Future consideration will be based on the Schrödinger picture, where Fi r and Fi r' do not depend on time t , but Pi r , Pi r' are time dependent. Stochastic processes and time series 6 Part 3. Consider Hamiltonian Applying the following properties for the orthogonal part Pi r and the non-orthogonal part Pi r' correspondently: a) configuration-space of prices for the orthogonal part Q = Rn  , R -real set domain q ( q1 , …, qn ), where qi is the price of i-th security on interval i    n,...1 ; configuration-space of prices for non-orthogonal part Q'  = Rnm   , q ( qn1 , …, qm ), where qi is the price of i-th security on interval i   mn,..., where the whole set Qm = Q  Q' (  Rn  Rnm ), R -real set; b) time t - Calculation of the respective contribution of the orthogonal and non-orthogonal parts. Will represent operators of shares  F i r of the i-th orthogonal security in the form (operator  F i r is the operation of multiplication by mi ): (3.1)  F i r Pi r mi  Pi r , i    n,...1 mi - number of shares i-th security (orthogonal) on i    n,...1 ; and correspondently:  F i r' - operators of shares of the i-th non-orthogonal security, mi ' - number of shares of the i-th security (non-orthogonal) on i   mn,..., . Stochastic processes and time series 7 - Calculating a contribution of economic conditions. Into (2.6) Hint also include market economic conditions (such as price of oil, gas, meteorological, etc), which contribute as outside perturbation related to the index and can be presented in the form: (3.2)   qn q Hi,..., 1 int where i    n,...1 , because do not have to take into account ' q properties of a nonorthogonal system, which can be represented through an orthogonal security system characterized by properties q . Consider configuration-space for sources of perturbation Ns   , N -normal set, then (3.3) ) (,...,1  s  where  l - l-th perturbation defined on interval   s l,...,1  and corresponded to economic conditions (also ref. Appendix E). Allow that economic condition and changing of price i-th security independently (locally) affects each security on the market, represented in the form: (3.4)     l liqn q H,,..., 1 int =   n ji 1   qq ji i j H, int A li i    qq ii, where 1-st term - allow, that changing of price of an i-th security affects the behavior of a j-th security i  j (non-local interaction); 2-nd terminclude contribution of economic conditions  l and needs not contain a nonlocal interaction of securities (i.e. local ij ). - Representation of a non-factorized securities interaction term. Lemma (3.5): a). For a non-local interaction, if there is any change in price, one security will affect prices of all other agents on the market and represent a symmetric form: Stochastic processes and time series 8 (3.5.1)   qq ji i j H, int =   qq ij j i H, int which encompasses a symmetry property of Hi j following from the invisibility of the i, j-th securities on the complete space Qm . b). Economic conditions, which do not contain interactions, can be represented in a bilinear form: (3.5.2) A li i    qq ii, = A li i    2 qi where Hamiltonian has symmetric and bilinear properties to satisfy both of these property forms, must be a symmetric bilinear form, which is equivalent to a quadratic form, then obtains (3.5.2). c). Form (3.5.2) must be real by definition and this property corresponds to a Hermitian form and a Hermitian form (symmetric sesquilinear form) is a sesquilinear form Lang [20] on a complex vector space V is a map V  V  C, a susquelinear form generalizes to the Euclidian form: (3.5.3)   qq ji i j H, int =   qq ji i j H int d). Because correspondence between quadratic forms on V and symmetric forms on V given by (over a ring where 2 invertible): (3.5.4)          )()()( 2 1 ,q q q q qq j i j i ji QQQ B and on V  VC can be presented by a wavefunction of an orthogonal part for the i-th security as (in a form according to Lemma (2.1, a)): (3.6)   eiq qi i i P defined on the configuration-space of prices Q and satisfying conditions (a)-(d). Then satisfying the Lemma (3.5,) obtain an interaction term between securities i  j in the form: (3.7)   qq ji i j H, int = i m j m eiqiq ji         qq ji i j H, int = i j m eqq ji Stochastic processes and time series 9 influence of economic conditions (oil price change, etc) can present in the form: (3.8) A li i    qq ii, = ll ii mm  2 () i l iq e  A li i    qq ii, = i li m  2 i lq e   where there are different perturbations to the interaction Hamiltonian   set (3.3) (independent or local). Substituting in (2.6) with (3.4), (3.7), (3.8) is computed as: (3.9) H = em qi i i n i  1 +   m ni 1 mi ' eqi i' +   s l1   n i1    nj ji 1 em qq ji i j m li i  2 i lq e   where Hi          P Hi defined on set Qm . Substituting in (2.6) with (3.9) can present a market in terms of an orthogonal security (index): (3.10) H = em qi i i n i 1   +   m ni 1   n b1 mi '     ' b i rr R yy yd PP    b iq e +   s l1   n i1    nj ji 1 em qq ji i j m li i  2 i lq e   the coefficients of the expansion in (2.3) of the wavefunction Pi r' by the normal orthogonal system of functions Pi r determined by the formula Davidov [15]: (3.11) ki b =     ' b i rr R yy yd PP    Stochastic processes and time series 16 Then can present (5.2 ' ) in the form: (5.3) m   2 tm   + 2 tm   + 1 p i m i i     + 2 1 l t m i i tm      For an AR model, modified Yule-Walker equations Prado et al [18] provide a fit. From (5.3) obtains: (5.4) m   1 p i m i i     + 2 tm   + tm   for 0m . For 0m : (5.4 ' ) 2 1 p m i m t m mi it             For 0:m 0 1 2 11 2 1 0 1 2                                          solving all  . For 0m : (5.4.1) 2 01 p k k t kt            solving  . Stochastic processes and time series 17 2. Follow the 2-stage LSM Sandgren et al [19] calculate t F in the form: (5.5)   tt F B q where (5.5.1)   1 1pi i i BB      B -backshift operator; (5.5.2) tt F   solving t  ; than obtains: (5.6)   1 1... t t t l tl F qF F          solving all  . The model order p selected via the Akaike’s information criterion; 2pl . Stochastic processes and time series 18 Conclusion In this article presented is the model for predicting future values in time series. Using parallelism (Nelson, 1966) between stochastic approach and quantum mechanics a Schrödinger equation can be applied to incorporate complicated behavior of financial market, which is considered as manyobjects interaction system. Within the model framework obtained is the theoretical expression in lover limit approximation ( 11, 1l   ) and int 1i H  in the form of degree n=2 for an average of observable q and a standard deviation  of q , which specifies how the actual value (closing price) is dispersed from the average value (mean closing price) attn. Appendix B, B-I and incorporates heteroskedasticity features. 1. Introduced is the corresponding stochastic process to model stochastic behavior of the system. As a result, the formulation of the stochastic process   2 ,t F  underlying the model incorporates the behavior of financial markets through a Hamiltonian of the Schrödinger equation. This process represents an input of the model and is not restricted by the i.i.d. assumption and is related to the correlated, non-normally distributed data. 2. Obtained an expression for conditional mean of price of i -th security, which depends on time, meaning it allows for non-stationarity. 3. Calculated an expression for conditional variance of price of i -th security, which depends on time, meaning it incorporates heteroskedasticity. 4. Shocks (economic conditions) are assumed to be uncorrelated, but not necessarily i.i.d. conditioned. Calculated influence of shocks on stocks behavior. 5. Specified condition for arbitrage. 6. Noted applicable time interval for the model. 7. Obtained expression for standard deviation of price can be considered as a ‘signal’ for high frequency and quantitative trading noting fleeting moves in stocks behavior. Stochastic processes and time series 19 Appendix A. Expression for a normalized wavefunction. Normalization condition for the wavefunction of the 1-st security considering that the integral diverges Prudnikov et al [22] (A.1) 11 1 0 2sintm q hdqe    i.e.   121 1, ; ,tq q P  is not square-integrable can present in the form: (A.2)   011 1 0 0 1 1 1 1 0 1 2 sin 2 :2 tm q qn h q q q n q tdqe       = 1 2 2i tm nh     0 J , nN only on the interval 00 1 1 1 ,2q q q n     , the  - probability is defined on the interval 1 q . The integral is defined on the space   2 LQ where Q is the configuration price space n QR   , let denote map 00 2 1 1 ,L q q k q     00 2 1 1 ,2L q q n     , ,n n N k R   then (A.2 ' )   011 1 0 0 1 1 1 1 0 1 2 sin : tm q q k q h q q q k q q tdqe       = kq 1 2i tm h    0 J , 1n Transition from the equation (A.1) to (A.2) means that on the interval 1 q a continuous spectrum can be approximated as discrete, so the wavefunction can be normalized, then can represent a normalized wavefunction of the 1-st security in the form: (A.3)   121 1, ; ,tq q   = 1 N   121 1, ; ,tq q P  where 2 N - the norm of the wavefunction   121 1, ; ,tq q P  . Stochastic processes and time series 20 Equation (A.2) satisfies a condition on the 0 tt -interval; from a trajectory of the price for the 1-st security   1tq at 0 t defined as 11 :tqq can obtain: 0 1 q =   10 tq  0 t =   10 11 qq  where 1 1 q - inverse function of   1tq on the interval 1 q ( 0 1 q  1 q ), 0 t  0 1 q and bijective on the domain t and the codomain 1 q if considered for one exchange; then can represent (A.2) in the form: (A.2.1)   00 0 1 1 1 :2q q q t     01 2 2i t m h     0 J =   10 1 1 1 2 2i q q m h       0 J = 2 N Consider (A.2) in the small vicinity of   0 11qn , then 0 11 2q q k q      and from (A.2.1) obtains: (A.2.1 ' )   00 0 1 1 1 :q q q k q t     kq 01 2i t m h    0 J = kq   10 1 1 1 2q q m h      0 I Considering (A.2.1 ' ), a normalized wavefunction (has a unit norm) of the 1-st security is computed as: (A.4)   121 1, ; ,tq q   =   10 1 1 1 2 1 q q m kq h      0 I   121 1, ; ,tq q P  and is defined on the interval 1 q . Stochastic processes and time series 21 A-I. Constant h in eq. (4.1). Ref. Davidov [15] if associate fields   it  in classical field theory with price of the security   i qt , then transition from the classical to the quantum field theory is provided by interpreting conjugate variables, i.e.   i qt and its conjugate   i qt • as subjected to the canonical commutation relations (A.5)       , ij q t q t ih i j  •     where ,ij index of security; so h is const in commutator for price and price time derivative of the security . B. Expression for an average of observable price. In explicit form, the average of observable price of the 1-st security 1 q , in fixed state   121 1, ; ,tq q P  is computed as: (B.1) 1 q =   10 1 1 1 2 1 q q m kq h      0 I 11 11 0 2sintm q hdqeq   where 1 q   0, , 1 tm - tuning parameter for the system (refer to the linear approximation in (1.1)). Considering that the integral diverges, i.e. can only be computed on the interval,   10,qn   obtains: (B.2) 1 10:qn q   =   10 1 1 1 2 1 q q m kq h      0 I 1 11 1 2 sin 0 0 tm q nhqdqe    Stochastic processes and time series 22 let denote map   20,L l q    20,Ln  , ,n n N l R   Consider (B.2) in the small vicinity of   11qn , then 1 q l q     and from (B.2) obtains (B.2 ' ) 1 10:q l q q =   10 1 1 1 2 1 q q m kq h      0 I 1 11 1 2 sin 0 0 sin tm q lq hqdqe   , 1n Can represent (B.2 ' ) in the form: (B.2.1) 1 10:q l q q =   1 11 1 10 1 1 1 2 cos 2 2 2 1cos lq tm q h q q m kq h qdqe           0 I and from (B.2.1) obtains: (B.2.1 ' ) 1 10:q l q q =   1 10 1 1 1 2 2 1 i tm il q h q q m kq h          1 0 J I Can present in the form: (B.3) 1 10:q l q q =   10 1 1 1 2 1 q q m kq h      0 I 1 2tm lq h    1 I Stochastic processes and time series 23 B-I. Expression for standard deviation of price. Variance of price i q in the fixed state   1,..., 1, 1,..., , ,; i i l i i n P q t q q q q   is expressed as: (B.4)           2 22 2 1,..., 1, 1,..., , 1,..., 1, 1,..., , 2 1,..., 1, 1,..., , 1,..., 1, 1,..., , , ; , ; , , ; , ; , ii i i iii ll i i i n i i i n iii ll i i i n i i i n Pq q q qtt q q q q q q q q q q qtt q q q q q q q q q q PP PP                     For the 1-st security 1 q in the fixed state   121 1, ; ,tq q P  obtains (in Schrödinger picture): (B.4 ' )           1 2 22 1 1 1 2 111 2 1 2 1 11 2 111 2 1 2 1 11 , ; , , ; , , , ; , , ; , , Pq q q qt q t q qq qt q t q qq PP PP             Let’s compute a standard deviation for the 1-st security 1 q in the fixed state   121 1, ; ,tq q P  ,calculate each term in (B.4 ' ) separately: (B.5)   11 2 11 10 11 1 1 2 2 2 sin 1 0 tm q h q q m kq h e q dq q       0 I where 1 q   0, ; Stochastic processes and time series 24 on the interval,   10,qn   obtains: (B.6)   1 11 2 11 10 11 1 1 2 0: 02 sin 20 1 n ntm q h qq q m kq h q dq qe           0 I let denote map   20,L l q    20,Ln  , ,n n N l R   Consider (B.6) in the small vicinity of   11qn , then 1 q l q     and from (B.6) obtains: (B.6 ' ) 2 110:q l q q =   11 211 10 1 1 1 02 sin 20 1 sin lq tm q h q q m kq h qdqe        0 I , 1n Can represent (B.6 ' ) in the form: (B.6.1) 2 110:q l q q =   1 11 21 10 1 1 1 2 2 2 cos 2 1 cos lq tm q h q q m kq h qdqe           0 I and from (B.6.1) obtains: (B.6.1 ' ) 2 110:q l q q =   10 1 1 1 2 1 q q m kq h      0 I 1 1 2 3 2 i tm h lq itm h     1 J Stochastic processes and time series 25 Can present in the form: (B.7) 2 110:q l q q =   10 1 1 1 2 1 q q m kq h      0 I 1 1 2 3 2 tm h lq tm h     1 I Substituting (B.3), (B.7) in (B.4) obtains: (B.8)   1 1 Pq  =   10 1 1 1 2 1 q q m kq h      0 I 2 11 1 22 3 2 tm tm h l q l q tm h h                    11 II where there is an existed dependence on 0 i q     00ii q q t ; randomness is included through parameter q (for the i-th security 0 i i i q q q   ). C. Expression for E ,  . As F is a stochastic process with a state space q (ref. Part 5-1), the random vector F takes values in the column vector q q = 1 k q q      then   cov F represents a covariant matrix of the vector q in the form: (C.1) qq K =   cov q =         T E E E    q q q q =   TT E  qq is Hermitian symmetric and positive semidefinite, can diagonalize the matrix as: (C.1 ' ) qq K =   cov q = T EE Stochastic processes and time series 32 Footnotes:   1 superposition principle, i.e. system 'rr can exist in the states described by Pr and Pr' , then it can exist in the state described by superstate P ; and this is equally related to parts r  'rr described by i r P and r'  'rr described by ' i r P for Pr , Pr' correspondingly; so superstate P , Pr , Pr' are wavefunctions of states which can attain to the system 'rr and r, r' parts;   1,1 Pr is (not correlated) orthogonal concerning non-orthogonal Pr ' , thus representing basis for Pr ' according to the superposition principle, sum of ‘pure’ states is again ‘pure’ state, so a certain physical system is expanded to orthogonal and non-orthogonal terms ( ‘pure’ group states), eq. (1.1); Pr , Pr ' is not factorized as an interaction of securities i  j is considered each group state is expanded to a set of i-th system states, where i    n,...1 on orthogonal set; i   mn,..., on non-orthogonal set correspondently, eq. (1.2).   2 wavefunction Pi for i-th system state expressed in a form of solution of the Schrödinger equation (4.4). Viz. as an approximation limit of the Schrödinger equation; ansatz a general expression of the interaction Hamiltonian for pair interaction, which can be used in the Schrödinger’s equation. i P is factorized for non-interaction approximation, i.e.   l niii liqqqqq H   ,,..,,,..,;111 int  =0; probability distribution is characterized by probability density function   fy defined by     * ii P q P q . Pi is not factorized (the behavior of the i-th state of the system is affected by system perturbations l  and other states j ), so any changes of measurement i q will change the measurement of other states i  j; i.e.   l niii liqqqqq H   ,,..,,,..,;111 int  0 conditional probability distribution is characterized by conditional probability density function   |l Y f y X x   defined by     *|. |. i i i i P q P q where i def Y related to   0i Hq ; i yq ; |.) def   1 1 1 ,.., , ,.., , ll i i n Xq q q q    related to   l niii liqqqqq H   ,,..,,,..,;111 int  ; l xX   , j xq . View publication stats