Optimization of Financial Instrument Parcels in Stochastic Wavelet Model
OOptimization of Financial Instrument Parcels in Stochastic Wavelet Model
A.M.Avdeenko [email protected] Introduction
Exchange rate fluctuation, values of shares, futures and options are random or quasirandom processes, and many “objective” (or perceiving as such) and random or seeming (perceiving) as random factors influence on them. Here a number of questions arise. Whether it is possible on the basis of analysis of limited numerical sequence of financial instrument cost to draw a conclusion about the nature of statistical processes underlying the observed event, specifically, to decide a question about stationarity (quasi-stationarity) of the process, to restore probability density upon observed data, to calculate probability of certain events etc.? Preceding the result of suggested work, it is necessary to point out that the answer is positive, specifically, within the framework of not so critical axiomatics, creation of desired statistics on the basis of analysis of random or quasirandom sequences of financial instrument costs is possible. Furthermore, according to the author, solution of the problem is essentially eased because we do not speak about simulation of complicated natural processes, but only about simulation of decision making method used by market participants, who have the same initial information and comparable computational and analytical capabilities, which pursue well-defined and, generally speaking, obvious target.
2. Model of nonlinear stochastic wavelets (NSW model)
Let random process ),( tX be the cost of shares, options and futures in the point of time Rt , where is a point in probability space with measure function P . Let’s introduce a population of analyzing functions-wavelets )2(2)( tt iii , meeting the condition dttLLt and forming the orthonormal basis for )( RL with compact carrier. Small value of index j conforms to low-frequency component of the random function ),( tX , and the large one to high-frequency component [1]. Let’s express the random function in the form of linear combination of functions dtttXtY ii )(),()( . We designate ))(),..(()( tYtYt J Y , where - J is maximum order of wavelet ransformation. We presume that new J - dimensional random vector )( t Y is in accord with stochastic equation of Ito: ωYGYFY ddd ))(())(()( , (1) where )( td i is random Wiener process ,0)( td i Jjitttdtd ijji ...1,),()()( . Symbol ... means averaging as per random process ),( tX with measure function P , and ))(()),(( YGYF are drift and diffused components accordingly, we assume that ))(())(( YY GG ijij . Conventional transition probability ))(/)(( ttf yy can be represented as a path integral [2]. Nonlinear effects can be taken into account using diagrammatic method. Specifically, we can say that for nonlinear model the spectral potency (Fourier transform of two-point twin correlation function), some times has power asymptotics, i.e. observed "fractalness" of signal is only the result of nonlinearity of the model. In single-mode case J , time-independent density of distribution defined as )()( y s dyyfyYP can be represented in the form of quadratures: )(1 )( yWs eyf , where
10 121 111 )( )(2)( dyyG yFyW y . In general J - dimensional case fulfillment of additional requirement, i.e. condition of detailed balance [3], is required and it is natural. However, if stationary solution exists, then it always can be represented in exponential form ))(( )( y y k Ws ef , where )()...( yy k are certain functions of random variables y . This result can be received based on the principle of maximum informational entropy [3] or using inequation of Rao-Cramer. In this case the moments )()...( yy k are limitations (links), under which we find extremum )(ln y s f . Let )( nk xH be adequate set of orthogonal functions, for example, Ermiht polynoms of k order. Orthogonal decomposition )( YF , )( YG per polynoms )( nk YH is given by [2]: )(...)()( )(...)()( ... ... .. ,...... ... ...... plpn lk nkpn jlkjj plpn lk nkpn lk yHyHQG YHYH YYF (2) im of further stochastic simulation is reconstruction of motion equation (1) and conditional probabilities within the framework of made suppositions based upon available observations (experimental information) or, that is equivalent, determination of decomposition values in proportion (2). To solve the problem we shall use extremal principle, i.e. we shall define the value pnn lk ...... , pnn jlk Q ... ,... in such a manner that minimize functionals that are norms of differences of observed and simulated values averaged per random process with measure function P . After differentiating these expressions with respect to pnn lk ...... , pnn jlk Q ... ,... , multiplying by )( nk yH and integrating on pn yy ... we have two systems of linear equations for unknown variables pnn lk ...... , pnn jlk Q ... ,... . Solution of these systems gives us expression for drift and diffused factors of non-Markov stochastic equation, i.e. the most probable path function of the process and statistic of fluctuations round it. We shall name corresponding sequence of random variables as setting up sequence of T length. On the basis of these results we can construct optimal policy of portfolio management. It follows from compactedness and condition dtt that factors of wavelet decomposition characterize mean change of value ),( X on the length of carrier j . Therefore, in elementary case, moment of optimal purchase of securities can be represented as )1,0inf( s Pdyt and moment of optimal sale as ),0inf( s Pdyt , where )( dyyfP ss . Sign of dy has been chosen for Haar wavelet, and value is risk level. The model under consideration has non-Markov nature. Conditions ,1 ss PP are required for exception (on the average) of the purchase in case of nnn dYYY and sale in case of nnn dYYY . Alternative variant is use of convolution )()()( dyzyfyfzf Tsss . In this case criteria ,1 ss PP mean resale- or repurchase moments. In both cases it is necessary to exclude statistically nonsignificant differences between stationary distributions in time displacement Ttt . The last is possible when using non-parametric Kolmogorov criterion with risk level , i.e. synthesized stationary distributions )(),( yfyf ss in displacement Ttt with probability p are considered o be indistinguishable, if NktyKtyK nn /)())(()((max , where KK , are cumulative probabilities for this condition and condition received as a result of displacement T . The constant k depends on risk level , N is the number of experimental points (samples) per which approximation of stationary distribution has been constructed. Conditions of purchase/ sale of financial instruments in NSW model are shown schematically in figure 1. Figure 1. Conditions of purchase/ sale of financial instruments in NSW model; here is compactedness interval of analyzing wavelet, T is length of setting up sequence, T is interval of quasi-stationarity of distribution with risk level , T is interval of parcel redistribution decision making, and BA , are points of purchase and sale of financial instruments. Experimental proof of NSW model
Data of three types that are stock prices of Russian emitters (Sberbank, SurgutNefteGas and Gasprom) and stocks of U.S. companies (AT&T Inc, Dell Inc. and the Bank of America) over a period since January 1, 2009 till December 31, 2009 were used as initial database for model run.
X(t) t T T A B T T t was analyzed the non-Markov dual-mode model )2( J , scanning was implemented in increments of 1 min., calibration sequence amounts to 32-64 indications, and the order of Hermite polynom not to exceed .3 K Analyzing function - Haar wavelets H , Daubechies wavelets , and Bettle-Lamerie spline wavelets ,, BBB [1]. Although the last ones do not have compact carrier, they come down quickly enough under large time. Cost of securities at the start time was assumed as tC . Objective function of the model is profit at the point of time t : )(/)()( tCtCtZ , decision making for purchase/ sale in accordance with proposed criterion and calculation algorithm for values )( YF , )( YG at the point of time tt was implemented in automatic mode based upon prior information. Risk levels were taken as . Results of calculation for three groups of active securities are given in figure 1. The results are designated at intervals of five acts of purchase and sale. Figure 2 Profitability when using NSW algorithm within the period since January 1, 2009 till December 31, 2009 for stocks of Sberbank (SBER), Gazprom (GP) and SurgutNefteGaz (SNG) (а) and for AT&T, Dell Inc. and the Bank of America (b); and risk levels are . Average fraction of decisions made (number of purchase and sale acts towards length of purchase and sale field) were in the interval 0.01...0.025. Daubechies wavelets were optimal for the stocks of Russian emitters and stocks of USA companies. Z n GP SNG SBER Z n AT TDell Inc BAC . Optimization of security portfolio on the basis of NSW algorithm Let )( tz i be the cost of instrument Mi ...1 at the point of time t in accordance with abovementioned decision algorithm. Let )(/)(ln)( tztztx iii be logarithmic earning power and let’s define average value i X and two-point correlation function at the point of time Tt : tTt jjiiij tTt ii dtXtxXtxT dttxTX ))()()((1 )(1 Lower estimate of interval T is maximum value per i of distribution quasi-stationarity with predetermined risk level (fig. 1). Nondiagonal elements ji take account of intercorrelation of profitability of instruments ji , in accordance with NSW algorithm. If )( tn i is fraction of instrument i in parcel at point of time t , then in Gaussian approximation average profitability of the parcel is Mi ii tXtntZ )()()( and its expected mean square is )()()()()( tXtXtntnty jijMj Mi i . Moments of higher order ji ... can be set aside, as far as by estimate for all instruments under consideration the following is right:
22 ... jjii ji . Let’s define the most effective parcel at point of time t , as a set of instruments i with fraction )( tn i , which maximizes value of probability extrdssyzP еZ ),,()( )( under limitations Mi i n . The parameter implements the compromise between maximum earning power and minimum risk. When maximum of earning power is preferable, and when risk minimum is preferable. Thus, the task about the most effective portfolio is the typical task of nonlinear optimization. Necessary and sufficient ondition of extremum existence is provided with Kun-Takker theorem. To solve the problem we shall use gradient procedure. It is possible to take Mn i /1 as initial points. In Figure 3 there is relation between the average profit and the share fraction in parcel against the time for optimal portfolio of AT&T Inc, Dell Inc. and the Bank of America over a period January 1, 2009 to June 30, 2009. We use Daubechies wavelets. As opposed to Figure 2, cost of the parcel is represented at point of time, when decision about redistribution of instruments in the parcel is made. Trading was carried out in computer-aided mode without readjustment of the algorithm over a period of a half year. Only risk levels , and compromise parameter between profitability and minimum risk were pre-set as the input. Z n n i n BAC AT& T Dell Inc Figure 3. a) relation between average profitability of parcel of AT&T Inc, Dell Inc. and the Bank of America and time. Risk levels , and compromise parameter ; b) share fraction in optimal parcel depending on time. In ordinary moment the parcel is shared equally between AT&T Inc, Dell Inc and the Bank of America. However, in the process of trading according to NSW algorithm, redistribution of the parcel in favor of the Bank of America for the account of Dell Inc. takes place. Accumulated profit of the parcel is less than in case of dealing in the Bank of America shares (Figure 2), but it is achieved with the less risk.
4. Discussion about the results
In table 1 the NSW algorithm is compared with the following well-known algorithms: Price Channel, Bollinger Bands (BB), Moving Average Convergence/Divergence (MACD) and Relative Strength Index over a period of January 1, 2009 to December 31, 2009. arameters of standard algorithm were chosen to the effect that maximum earning power is provided. In this context, when comparing the algorithms, the well-known models had the advantage a priori; i.e. the parameters were chosen according to the process, which had been already realized (in reality it is natural that "future" is unknown). Proposed NSW algorithm was implemented as close as possible to reality; predictions and control were fulfilled on the basic of knowing the previous system status only. Advantages of proposed algorithm are essentially for all financial instruments. Table 1 Comparison of profitability of various algorithms, January 1, 2009 to December 31, 2009 PC BB MACD RSI NSW The Bank of America 1.596 1.631 1.273 1.683 1.852 Dell Inc 1.131 1.377 1.311 1.185 1.524 AT&T Inc 1.433 1.431 1.254 1.439 1.721 In table 2, comparison of algorithms was carried out in poor for the Bank of America market conditions. There was significant negative trend with strong volatility. Standard algorithms turn out to be unprofitable. NSW algorithm allows receiving of small, but stable profitability. Table 2. Comparison of profitability of various algorithms for significant negative trends with great volatility. The Bank of America, May 1, 2010 to June 30, 2010. PC BB СС ОС NSW
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