MATHEMATICAL MODELLING METHODS FOR TIME SERIES

Transcription

1 MATHEMATICAL MODELLING METHODS FOR TIME SERIES Eurorsk Systems Ltd. 31, General Kselov str. BG-9002 Varna, Bulgara Phone Fax

2 Contents 1. Preface Clusterng of tme seres Purpose Input data Output data Propertes Non-normal dstrbutons Purpose Input data Output data Propertes Multfactor modellng Purpose Input data Output data Propertes Predcton Purpose Input data Output data Propertes Monte Carlo Smulaton Purpose Input data Output data Propertes Volatlty Brdge Purpose Input data Output data Propertes Lterature Mathematcal Modellng Methods for Tme Seres

3 1. Preface Ths report outlnes mportant aspects of models for tme seres processng methods. It ams at proposng a set of advanced approaches for the represenaton and modellng of tme seres, that can be used for prcng, senstvty calculaton and smulaton of Value at Rsk, on short and long term horzons. These approaches should ensure easy ntegraton nto the evaluaton and management systems, to cover functonaltes from front to back offce. Moreover, the ntenton behnd them s to develop approaches for mult-factor and mult-step smulaton on several future tme ponts, to produce dstrbutons based on smulaton paths, and to allow prcng and rsk estmaton for nstruments that work wth underlyng tme seres. In addton to the model defnton, development tasks nclude the creaton of prototype evaluaton, as well as analyss of studes, thereby dentfyng the functonal content and results of the evaluaton and smulaton. The development s based on hstorcal prce tme seres (prce ndex per unt) and current prces of futures (forwards) of tme seres. The evaluaton approaches nclude the followng man steps: Representaton of expected future developments of tme seres, usng predcton based learnng or neural networks, as well as auto-regressve logc. Representaton of mpled or hstorcal volatlty of the seres, accordng to maturty. 2D performance (Mult-Factor and Mult-Step) of the Monte Carlo smulaton on selected market factors and future tme ponts, n order to obtan development paths and dstrbutons. Runnng of path-dependent prcng expressons that produce manly non-normal dstrbutons and obtan means as expectatons and VaR statstcs. A set of addtonal approaches should enhance the qualty of the prcng and VaR estmaton, ncludng: Stochastc nterpolaton (Volatlty Brdge), used to produce daly prce movements for a future tme perod. Non-normal dstrbutons (ncludng automatc dentfcaton of the dstrbuton type and ts parameters) of market tme seres and copula approach for market factors n the Monte Carlo smulaton. 3 Mathematcal Modellng Methods for Tme Seres

4 Multfactor models that represent prce development of unknown factors by known market factors. Ths dependence s automatcally calbrated on a prcng expresson that can then be used for the calculaton of prces, senstvtes and VaR smulatons. Clusterng of tme seres, used for groupng tme seres n such a way that seres wth smlar hstorcal behavour are categorzed nto the same group. Ths approach reduces the analyss and smulaton space and s able to make orthogonal removng dependences. For further nformaton, please contact our Hotlne at or send an e-mal to nfo@eurorsksystems.com. 4 Mathematcal Modellng Methods for Tme Seres

5 2. Clusterng of tme seres 2.1 Purpose The purpose of clusterng s creatng groups of tme seres n a way that seres wth smlar hstorcal behavour are organzed nto the same group. 2.2 Input data A set of tme seres. Number of clusters. 2.3 Output data Clusters of tme seres. Every cluster conssts of: The seres belongng to the cluster. A generated synthetc seres, called prototype seres, consdered a representatve of the cluster. Ths synthetc seres has the same dmensonalty as all other seres and s usually near the cluster s centre. In addton to clusters together wth ther seres and prototypes clusterng qualty statstcs are generated [21] [22] [23] n the followng manner: Inter and ntra cluster statstcs, adjuster R squared, average lnkage, etc. These elements represent the clusterng qualty. Some of these statstcs can be used to determne the optmal number of clusters. The clusterng module s able to reduce large sets of thousands of ndvdual tme seres to small sets of several dozens synthetc seres [21] [22]. The man reason for ths s the usng of prototypes nstead of real seres for tme and memory consumng operatons, such as correlatons estmaton, wth as mnmal error as possble. In ths way, the number of tme seres s reduced, as any calculaton should be done wth a gven tme seres, t s dentfed n whch seres the cluster has been classfed and the cluster prototype s used nstead of the real seres. Thus, the number of all avalable seres s reduced to the number of clusters. That way, calculatons that needed to be performed wth the seres decrease and can be performed practcally. Otherwse, the huge data causes too many calculatons, whch often cannot be fnshed wthn an acceptable tme perod. Some examples of clusters, along wth ther seres and prototypes, are shown n Fg. 1. The red bold lnes represent prototypes that are near clusters centres and substtute the real seres n tme consumng calculatons. 5 Mathematcal Modellng Methods for Tme Seres

6 Fg. 1 Examples of clusters, ther prototypes llustrated as a bold red lne 2.4 Propertes The hstorcal behavour of the seres s consdered Whle determnng clusters, the smlarty of the seres behavour s nvestgated, not only for actual rates, but for all other hstorcal rates, as well. In ths way, t s possble to put a gven seres, wth hgh or specfc current rates, nto a cluster wth lower rates seres. Ths may occur when a seres has dfferent actual rates for the current date, but has had smlar hstorcal developments and behavours n the past. 6 Mathematcal Modellng Methods for Tme Seres

7 Seres weghts Indvdual seres may also have weghts. If the weghts are all equal to 1, then all seres are equally treated n the clusterng process. Otherwse, some of the seres may be consdered more or less mportant than others. If the weght of a seres s two tmes bgger than the weght of another seres, the former s regarded as two seres wth the same values. Weghts can not only be nteger, but any arbtrary real values as well. The effect of the usage of weghts s shown n Fg. 2. The centre between Seres 1 and Seres 2 s llustrated wth a dashed red lne. If those two seres have the same weght, the centre wll be very close to the prototype. Seres 2, however, has the greater weght and thus attracts the prototype to tself. 1.0 Seres 1 Seres 2 Prototype Center 0.5 y(t) t Fg. 2 The effect of a seres weght to the cluster prototype Hstorcal values of the seres can be taken nto account The comparson of seres n the clusterng process can be acheved by usng a decay factor. Most commonly, the seres comparson s done va the Eucldean dstance (1), where the correspondng values are subtracted and powered. The decay factor, however, reduces the weght of the older hstorcal values and ncreases the weght of more recent values (2). Decay λ s defned to be between 0 and 1. The greater the decay, the more weght s gven to the last values n the seres. Eucldean dstance wthout decay: d N 1 2 ( x y ) (1) N 1 Eucldean dstance wth decay λ: d N 1 1 N 1 1 N 2 ( x y ) (2) where x and y are the seres between whch the dstance s determned. 7 Mathematcal Modellng Methods for Tme Seres

8 Determnaton of the optmal number of clusterng The optmal number of clusters could be found by clusterng n 2, 3,, n clusters and by estmatng an ndcator for the clusterng qualty (e.g. adjusted R 2 ) A graphc demonstrates the plot where axs X s the number of clusters and axs Y s the clusterng qualty. The sharp that s represented n the graph, determnes the optmal number of clusters (s. Fg. 3) The optmal number of clusters can be acheved n the followng manner: Performng the clusterng many tmes over, ncreasng the number of clusters wth a fxed value k>1. For example, clusterng n 5, 10,, n clusters When the sharp drop (startng at pont ) s found, addtonal clusterng n +1, +2,,+k-1 clusters s done, n order to fnd the optmal number of clusters more precsely, Ths s shown n Fg. 4. Num Clusters Adjusted R squared Error Tme (sec.) Error Number of clusters Fg. 3 Clusterng of 1200 shares n a dfferent number of clusters and error calculaton based on an adjusted R 2 to fnd the optmal number of clusters 8 Mathematcal Modellng Methods for Tme Seres

9 If there are too many data elements, the computaton tme could last too long. For ths reason, searchng for the optmal number of clusters could be performed on two levels, as shown n Fg. 5. Num Clusters Adjusted R Squared Error Error Number of clusters Num Clusters Adjusted R Squared Error Error Number of clusters Fg. 4 Overall number of clusterng s = 12, compared to 26 n the successve search 9 Mathematcal Modellng Methods for Tme Seres

10 Statstcs about clusterng qualty are computed showng how good the clusterng has been performed Adjusted R squared All data Cluste Clust Clust M SS ( ) H tot x m m1 1 2 SSsum R 1 SS tot 2 2 M 1 SS AdjustedR 1 ( 1 R ) 1 M p 1 SS 2 σ 1 σ 2 σ p M Number of all elements (shares or curves) μ center of all data elements M j Number of elements n cluster j H Number of hstorcal values μ j Synthetc share/curve of cluster j sum tot SS sum M j M 1 M p 1 H m1 1 p j1 j ( x jm j ) j 2 Eucldean dstances Every to every Gven synthetc curves wth H hstorcal values, the Eucldean dstance between synthetc curves and j s calculated as: d j H c k c jk k1 2 (3) 10 Mathematcal Modellng Methods for Tme Seres

11 Thus, a symmetrc square matrx s obtaned, showng the dstances between synthetc curves/ndexes. Synthetc curves Symmetrc square matrx of dstances (every to every) between the synthetc curves/ndexes d 11 d 12 d 13 d 14 d 21 d 22 d 23 d 24 d 31 d 32 d 33 d 34 d 41 d 42 d 43 d 44 Every to centrod A sequence of values can be calculated, showng the dstances between each curve/ndex and ts calculated centre (centrod). The centrod has the same dmensonalty as the curves/ndexes and every value of the centrod can be defned as the average number of the correspondng values of synthetc curves/ndexes. Synthetc curves Centro d d1 d2 d3 d4 Sequence of dstances between the synthetc curves/ndexes and the centrod 11 Mathematcal Modellng Methods for Tme Seres

12 Average lnkage Average lnkage s a method of calculatng the dstance between clusters. The dstance between two clusters s the average dstance between objects (shares/ndexes) from the frst cluster and objects from the second cluster. The dstance between cluster X and cluster Y s: Nx Ny 1 d xy d( x, y j), x X, y j Y N N x y 1 j1 (4) d(x, yj) - Eucldean dstance between curve from cluster X and curve j from cluster Y. Cluster s Square matrx of average lnkage dstances d 11 d 12 d 13 d 14 d 21 d 22 d 23 d 24 d 31 d 32 d 33 d 34 d 41 d 42 d 43 d 44 The usage of clusters prototypes After performng the clusterng, cluster prototypes are used for the correlaton matrx and volatlty vector n the Monte Carlo smulatons. However, the volatltes are calculated for real market factors, too, so that Beta factors to prototype seres can be obtaned. In fact, Monte Carlo smulates the correlated prototypes, but each real factor s related to the prototype factor va Beta factors,.e. the correlaton s used from prototypes and the volatlty s mapped by Beta factors. Thus, the data shrnkng s performed for the correlaton matrx, but real volatltes are also used n the smulaton. 12 Mathematcal Modellng Methods for Tme Seres

13 3. Non-normal dstrbutons 3.1 Purpose The purpose of the non-normal dstrbutons dentfcaton module s an automatcal determnaton of theoretcal dstrbuton types and ther parameters, for a gven tme seres. It utlzes a Copula approach, to smulate market factors n Monte Carlo VaR smulatons, usng mapped dstrbutons. Thus, the best modellng of emprcal dstrbuton shape s acheved by reproducble theoretcal dstrbuton shape. 3.2 Input data Tme seres. Chosen standard dstrbuton types (e.g. Beta, Cauchy, Student, Webull, etc.). 3.3 Output data An dentfed dstrbuton type that best fts the gven tme seres. Parameters specfc to the dentfed dstrbuton type. Numercal estmaton of the dstance between the emprcal dstrbuton and all other dstrbuton types. Thus, the dstrbuton types can be ordered and other, better fttng dstrbuton types can be manually selected. 3.4 Propertes The best dstrbuton type determnaton The assumpton that hstorcal seres of market rsk factors are normally dstrbuted s not necessarly always true. Real dstrbutons are mostly skewed, havng flat tals, whch s related to the underestmaton of market rsk for mprobable large losses. That s why the dstrbuton dentfcaton s used to mprove the Monte Carlo VaR smulaton by non-normal dstrbutons. Automatcal dentfcaton and mappng of 15 theoretcal dstrbuton types based on market factor tme seres. Estmaton of parameters of the selected, best dstrbuton. The dstrbuton type and ts parameters, as well as the comparson to emprcal market dstrbuton, can be manually set. Two dstance types are used to select the dstrbutons: hstogram and cumulatve. Usng mapped dstrbutons, the Copula approach enables the smulaton of market factors n Monte Carlo VaR. The non-normal dstrbuton dentfcaton mproves the Monte Carlo VaR smulaton (Fg. 5) by usng correlated non-normal dstrbuton samples, nstead of correlated normal ones. Steps on fndng the best dstrbuton type for a gven data are: 13 Mathematcal Modellng Methods for Tme Seres

14 1) Emprcal dstrbuton of the tme seres of a market factor s bult n the market rsk matrx. 2) For all theoretcal dstrbuton types, the followng actons are performed: a) Parameters of the dstrbuton type are estmated usng avalable market data. b) The theoretcal dstrbuton s bult usng estmated parameters. c) Theoretcal and emprcal dstrbutons are compared, and model dstances are calculated. These calculaton can be performed n two alternatve ways, usng ether hstogram or cumulatve dstances. 3) Order the dstrbutons accordng to mnmal dstances and select the most fttng dstrbuton, store t and for later usage n the Copula smulaton. Confdence level a quartle Expected value Q Market VaR(a) Fg. 5 VaR calculaton In Fg. 6, an example s gven llustratng a beta dstrbuton that better fts the tme seres dstrbuton, than a normal dstrbuton. The Beta Dstrbuton wll produce larger confdence rsk because of the flat tal Normal Dstrbuton Fg. 6 Flat tal dstrbuton 14 Mathematcal Modellng Methods for Tme Seres

15 The hypothess goodness of ft test generally ndcates whether or not the data represented as tme seres belongs to a prelmnary specfed dstrbuton type. Examples of such methods are ch squared, Kolmogorov-Smrnov, Anderson-Darlng and others. In our case, the best dstrbuton type s determned by choosng dstrbutons from a lst of specfed pre-defned dstrbutons. Ths can be acheved usng two alternatve measures: hstogram measure and cumulatve measure. Hstogram measure Here, the average squared dstance between the hstogram bn frequences of emprcal and theoretcal hstograms s calculated (3). k 2 1 d ( O E ) k 1 2 (5) The dstances are shown n Fg. 7. Ths type of measure s smlar to the ch squared goodness of ft test statstc, expressed by formula (4) [17] [20]. It requres the buldng of emprcal and theoretcal hstograms, as well as work on bnned data. 2 k 1 (O E ) E 2 (6) Dstances between theoretcal and emprcal hstograms Theoretcal hstogram Emprcal hstogram mn max Fg. 7 Hstogram measure The emprcal hstogram can be easly bult from sample data, but the theoretcal hstogram uses the Cumulatve Dstrbuton Functon (CDF) for every dstrbuton type. Ths means that, for each dstrbuton type, the dstrbuton parameters should be 15 Mathematcal Modellng Methods for Tme Seres

16 estmated beforehand, n order to be appled to the CDF formula. Table 1 shows the most common dstrbutons types wth ther parameters [12] [13] [14] [16] [19]. Dstrbuton Dstrbuton parameters Addtonal parameters Parameter 1 Parameter 2 Parameter 1 Parameter 2 Beta Shape Shape Shft Scale Cauchy Locaton Scale Exponental Rate --- Shft --- Inverse Normal Mu Lambda Shft --- Log Normal Log Scale Shape Shft --- Normal Mean Varance Shft --- Pareto Scale Shape Raylegh Sgma --- Shft --- Student Nu --- Shft Scale Webull Scale Shape Shft --- Logstc Locaton Scale Gumbel Locaton Scale Maxwell-Boltzmann A --- Shft --- Gamma Shape Scale Shft --- Pearson type 7 Shape Scale Table 1 Dstrbuton Parameters There are one or two dstrbuton specfc parameters, along wth one or two addtonal parameters. Dstrbuton specfc parameters. For each dstrbuton type, specfc parameters are estmated from the emprcal sample data, usng the method of moments, least squares regresson, or the maxmum lkelhood estmaton. In most cases, there s more than one possble way for the estmaton. Thus, the choce s based on the establshed researches about ther effectveness. 16 Mathematcal Modellng Methods for Tme Seres

17 Addtonal parameters. From a practcal pont of vew, t s possble for the sample data, or a part of t, to fall nto a regon n whch the dstrbuton s not defned. When takng nto account that the man goal s to dentfy the dstrbuton based on the probablty dstrbuton curve shape, the data s shfted and scaled, n order to compare the emprcal to the theoretcal shape (s, Fg. 8). mn max mn (new) Shft max (new) mn max Scale mn (new) max (new) Fg. 8 Addtonal parameters After obtanng both dstrbuton specfc and addtonal parameters. the theoretcal hstogram s bult. The startng pont s the known CDF: F x ( x) P( X x) (7) The theoretcal hstogram s bult as the probablty dstrbuton functon (PDF) from a dstrbuton wth known CDF (s. Fg. 9). P( a X b) Fx ( b) Fx ( a) (8) where Fx(x) s the CDF. 17 Mathematcal Modellng Methods for Tme Seres

18 Theoretcal hstogram P(a < X = b) a b Fg. 9 Theoretcal hstogram bns The emprcal hstogram s bult by countng the values belongng to the bns. If a value s on the border between two bns, then half of the value s added to both of them. The number of bns n a theoretcal hstogram concdes wth that of an emprcal hstogram. There are dfferent ways to determne the number of bns. In our mplementaton, the followng formula s used: B 5log( N) (9) and B 5 B 5 (10) B 25 B 25 (11) where: N - seres sze. Cumulatve measure Here, cumulatve dstrbuton values are used. Ths approach ams at ncreasng the accuracy of the best dstrbuton dentfcaton, takng nto account all values n the dstrbuton dentfcaton process. The cumulatve values approach works on non-bnned data and therefore more computatonal operatons are needed, leadng to more tme consumaton, compared to the hstogram measure. The method s as follows. Havng the sample values n axs Y (shown on the left n Fg. 10), the dfferences between them are accumulated and the graphc of ths functon s consdered. A shft of the functon s performed, n order for ts mean to concde to the mean of the dentcal emprcal data functon. 18 Mathematcal Modellng Methods for Tme Seres

19 Data values y Cumulatve dstances between values Cumulatve dstances graph p p 1 = d 1 d 2 p 2 = d 1 + d 2 p 3 = d 1 + d 2 + d 3 d 1 Fg. 10 Cumulatve dstances The most dffcult task n ths approach s obtanng the sample wth the theoretcal dstrbuton that corresponds to characterstcs of the emprcal sample data. Here agan, as n the hstogram measure, parameter estmatons are used. The full theoretcal sample s generated usng the nverse CDF (12). x F 1 () (12) The same sample values generator s also used to test the dstrbutons dentfcaton system. The generaton of the sample, dstrbuted accordng to a specfed dstrbuton type, s performed usng unformly dstrbuted values n an nterval from 0 to 1, generated ether randomly or n equal dstances. Graphcs of the hstogram and cumulatve measures are shown n Fg. 11 and Fg. 12. Fg. 11 Graphcs of the emprcal and all theoretcal hstograms 19 Mathematcal Modellng Methods for Tme Seres

20 Fg. 12 Graphcs of emprcal and cumulatve values for all theoretcal dstrbutons The dstrbuton of random numbers n a Monte Carlo smulaton s mportant for the accuracy of the computed rsks, as well as for the calculaton performance. The market envronment conssts of many dependent rsk factors and they are modelled as a multvarate random varable. Here, non-normal dstrbutons can be used for the descrpton of varous rsk factors. The Monte Carlo smulaton then apples the mapped dstrbuton wth ts parameters, to calculate the Value at Rsk. Fg. 13 shows the usage of non-normally dstrbuted factors n the Monte Carlo smulaton. 20 Mathematcal Modellng Methods for Tme Seres Fg. 13 Usage of non-normal dstrbutons When the smulaton s performed, the dstrbuton types of dfferent market factors are already determned. Random, unformly dstrbuted values are generated for each seres. Applyng the nverse cumulatve functon of the prelmnary dentfed dstrbuton type for the seres, the correlated non-normally dstrbuted samples are used for the Monte Carlo smulaton, nstead of the normally dstrbuted samples, generated usng the rsk correlated

21 matrx. In ths way,var dstrbuton s obtaned more accurately. Ths process should be performed for every seres. Immedately after the determnaton of the dstrbuton type, the CDF s numercally bult and preserved, n order to mprove the performance for a smulaton of a large number of samples (Fg. 14). Tme seres Beta Cauchy Exponental Inverse Normal Log Normal Normal Pareto Raylegh Student Webull Logstc Gumbel Maxwell- Gamma Pearson 7 Best dstrbuton type Probablt y Bnary search Interpolato n Buld and preserve numercally CDF Value Monte Carlo smulaton Fg. 14 Determnaton and usage of the best dstrbuton type Thus, the CDF resdes n the memory and can later be used n the Monte Carlo smulaton stage. Gven that CDF values are sorted, ascendng the next equally dstrbuted value s transformed nto a specfcally dstrbuted value (x=f -1 (y)) by a bnary search n the numercally bult CDF. 21 Mathematcal Modellng Methods for Tme Seres

22 4. Multfactor modellng 4.1 Purpose The purpose of multfactor modellng s to buld formulas that descrbe tme seres by other multple tme seres. The descrbed seres s called target seres (or factor) and the seres that descrbe the target are called explanatory seres (factors). 4.2 Input data Hstorcal tme seres of the target factor. Other avalable hstorcal tme seres to be used as explanatory factors. Settngs regardng the formula: functons that can partcpate, coeffcents restrctons, etc. 4.3 Output data Polynomal formula, descrbng the dependency of target factors from explanatory factors. Target factor generated by the formula. 4.4 Propertes Multfactor modelng requres both target, as well as all explanatory factors, n order to be on the same tme horzon. However, because of ther dfferent natures, these factors often have mssng and ncomplete data values. In cases where the tme seres are receved from dfferent sources n dfferent countres, that have dfferent holday or non-workng days, or n cases where the database contans some corrupted values, nterpolaton and volatlty brdge approach can be used. When tme seres of the modeled target factor and modelng explanatory factors are avalable, the goal s to use a multfactor approach for the target. Ths approach s buld up on an estmaton of a polynomal-based expresson, found by regresson and other ntellgent technques over known market factors and functons over them. The prcng expresson should reproduce the orgnal target tme seres as best as t possbly fts. After the expresson estmaton step, the prcng expresson can be used n the near future as a prcng model for the target factor, whch allows prcng, VaR, volatlty estmaton and prce projecton. If the future prce estmaton devates enough from the market prce of the target seres, a new expresson calbraton s performed /Fg. 15). 22 Mathematcal Modellng Methods for Tme Seres

23 Avalable market Target seres Formula buldng Calbrato Target seres values Target seres by formula Explanatory factor values Other factors The target seres s calculated by formula Fg.15 Multfactor modelng The formula (expresson estmaton) looks as follows: y = β1f1(x1) + β2f2(x2) βnfm(xn) + βn+1+ ε (13) where: y - target factor x1, x2,, xn - explanatory factors β1, β2,, βn, βn+1 - regresson coeffcents f1, f2,,fm - bass functons ε - error Stages of the modelng process are shown n Fg. 16. Start Target factor selecton Explanatory factors suggeston/selecton Bass functons combnaton determnaton Regresson coeffcents determnaton Fnal formula determnaton and error calculaton End - chosen by human - determned by system and/or human - determned by system Fg. 16 Stages of the multfactor modellng 23 Mathematcal Modellng Methods for Tme Seres

24 The frst step n the modellng s target seres selecton. After that, explanatory factors should be selected va an automatc suggeston and/or hand choosng. The whole set of explanatory canddates would be too bg and some of the seres are not as descrptve as others. An automatc suggeston can be performed n one of the followng ways. Clusterng. Explanatory factors are obtaned from the cluster n whch the target factor s classfed. If the number of explanatory factors, determned n ths way, s nsuffcent, the number of clusters can be decreased n order to ncrease the number of elements n the cluster. Ths approach of automatc suggeston s based on the clusterng that s descrbed earler n ths document. Mnmal covarances between canddate factors. The covarances between all factors are calculated, where frst mnmal values are used to determne explanatory factors. Maxmal covarances between canddate factors and the target factor. The frst maxmal values are used to determne the explanatory factors n ths case. The suggested factors can be manually changed,or other factors can be added or removed. When both target and explanatory factors are selected, the formula buldng process can be started n whch the system should fnd a combnaton of basc functons to the explanatory factors and regresson coeffcents β calculaton. The basc functons are used to mprove the accuracy and possbly to avod lnear dependences between factors that cause matrces equatons problems. Gven target and explanatory factors and basc functons combnaton, regresson coeffcents β are estmated by solvng the matrces equatons (14-16) [4] [10] [11]. date 1 date 2 date k Target factor Explanatory factors Regresson coeffcents factor 1 factor 2 factor n y 1 y 2 y k f 1 (x 11 ) f 1 (x 12 ) f 1 (x 1k ) f 2 (x 21 ) f 2 (x 22 ) f 2 (x 2k ) Bˆ ( A A) T 1 A T Y f m (x n1 ) f m (x n2 ) = x f m (x nk ) β 1 β 2 β m Y A B (14) (15) Yˆ A Bˆ (16) 24 Mathematcal Modellng Methods for Tme Seres

25 Explanatory factors Bass functons Functon name Date 1 Date 2 Date t Date K f1 f2 f3 f4 fm exponent logarthm pow(-1) pow(1) pow(3) Combnaton of bass functons appled to the explanatory factors y ŷ ε=(y- ŷ) 2 x1 x2 x3 xn SI2935 Bond DE87 Bond DE05 Share DE18. Index SX12 Target factor SI2935 estmaton Dstance ŷ =β 1f3(x 1) + β 2f1(x 2) βnf2(xn) + βn+1 + ε Fg. 17 Multfactor modellng The formula s bult n such a way that the dstance ε must be mnmzed. To attan a noton of the modelng qualty, addtonal statstcs for the fttng qualty should be shown correlaton, R squared, adjusted R squared, target and synthetc volatlty. In Fg. 18, settngs of the multfactor are presented. 25 Mathematcal Modellng Methods for Tme Seres

26 Formula Target factor Explanatory factors Statstcs Fg. 18 Multfactor modelng Formula terms Remove small coeffcents. Small formula term coeffcents can be removed, as they do not sgnfcantly nfluence the formula results. Include free term wthout explanatory factor. The free term does not nclude an avalable factor. It s a constant and allows more degrees of freedom n mprovng the formula accuracy. Include the target factor nto explanatory factors. In most cases, ths s not acheved, because the formula s not applcable to real-lfe stuatons. Auto seed. If the seed stays fxed, dfferent attempts produce the same results wth the same data. By choosng the auto seed opton, each attempt wll generate dfferent formulas. The reason s that, n the formula generaton approach, a random generator used that can reproduce the same or dfferent results n every generaton. Factors Performance. Factors can be pre-processed pror to the modelng and post-processed after the modelng stage. Performance for clusterng. Factors can be pre-processed only for the factors selecton stage, va the clusterng approach. Bass functons selecton The basc functons that partcpate n the formula must be selected beforehand. For some functons t s possble to not partcpate n the fnal formula, but they are all used n the formula fndng step. If no bass functons are selected, factors partcpate wth ther raw values. Dependng on the factors nature, t s possble for some bass functons to be 26 Mathematcal Modellng Methods for Tme Seres

27 unsutable for factor values (for example, a large factor value appled to an exponental bass functon wll produce a value near to nfnty and would cause an error). Formula terms could be reduced by removng the ones whose regresson coeffcents are smaller than the gven threshold. In ths case, the correspondng factor should be excluded from the formula. Fg.19 Multfactor settngs A constant (free regresson coeffcent wthout a factor) could also be ncluded n the formula. It s represented as a factor wth all hstorcal values equal to 1.0 and wthout a bass functon appled. The followng standard bass functons can be ncluded: Sne Cosne Tangent Hyperbolc tangent Hyperbolc sne Hyperbolc Cosne Square root Exponent Logarthm Arc sne Arc cosne 27 Mathematcal Modellng Methods for Tme Seres

28 Power functon, etc. Fg. 20 llustrates an example of a target seres wth synthetc seres generated by applyng the formula n whch they are juxtaposed. A good modelng s seen, where a decay factor s appled addtonally, enablng the actual values (near the end of the seres) to be more mportant n the modelng. Thus, the gven and generated seres are more alke at the end of the seres than n the begnnng. Multfactor calbraton Ths formula calbraton s shown n Fg. 21. Frst, target and explanatory factors are selected and loaded. The frst verson of the formula s created and calculatons are performed wth ths formula verson. Afterwards, the generated target begns to devate from real market values and the formula must be corrected. The selected explanatory factors, that are used for buldng the frst formula verson, are used agan, but the formula s calbrated and new functons and coeffcents for the formula terms are produced. The last modeled values are closer to target values because of the decay. Fg. 20 A gven target seres and a synthetc seres generated by the multfactor module 28 Mathematcal Modellng Methods for Tme Seres

29 If the formula s not sutable anymore and new factors Date 1 Date 2 should be selected Date 3 Frst verson of the formula Loadng and suggeston of the factors Calculatons wth the frst verson of the formula from Date 1 to Date 2 Second verson of the formula Reload the same factors and fndng a new formula wth the same factors Calculatons wth the second verson of the formula from Date 1 to Date 2 Tm e Thrd verson of the formula Loadng and suggeston of the factors. Fndng a new formula wth new factors Fg. 21 Formula calbraton For ths reason, a second formula verson s created and t s further used to generate the target, untl ts values start to devate from the real target values. If ths devaton s too sgnfcant, new explanatory factors should be selected. The factors are loaded agan, a selecton of new factors s performed by one of the above mentoned automated approaches (clusterng, mn or max correlated) and they can be manually changed. The selected factors are used to generate the thrd formula verson, that, as opposed to prevous formula versons, ncludes dfferent explanatory factors, bass functons and coeffcents. Ths formula can later be used n further calculatons, untl new calbraton s needed. There are two knds of calbratons: 1) Preservng currently selected explanatory factors and changng only the functons appled to them, as well as and term coeffcents, ncludng the free term. 2) Selectng new explanatory factors. In ths case, new factors can be added, exstng factors removed, or both. The formula has completely changed regardng bass functons and terms coeffcents. In each formula calbraton, settngs can be changed,.e. the set of bass functons can be used n the formula, removng terms wth small coeffcents, etc. Senstve analyss Multfactor analyss can be used for the senstvty analyss. Coeffcents β1, β2,... βn n formula (13), also shown n Fg. 18 and Fg. 22, can be used to analyze the factor partcpaton n the modelng of the target factor. 29 Mathematcal Modellng Methods for Tme Seres

30 Coeffcent s Generated formula Fg. 22 The generated formula Ths allows the relatonshp between prce changes (P) the rsk factors to be expressed. Accordng to the delta-gamma approach, n whch the change n the prce s: P δ T Z ZT Γ Z (17) where δ s the frst dervatve of the rsk factor: δ = P z (18) estmated by P(,z +h, ) P(,z h, ) 2h = s + s (19) 2h The dagonal elements of Γ are: Γ = 2 P z 2 (20) estmated by: P(,z +h, ) P(,z, ) h2 P(,z h, ) P(,z, ) h2 = s + s h2 (21) 30 Mathematcal Modellng Methods for Tme Seres

31 The cross elements of the Γ are: Γ k = 2 P z z k, ( k) (22) estmated by: P(, z + h,, z k + h k, ) P(, z + h,, z k h k, ) 4h h k + P(,z h,,z k h k, ) P(,z h,,z k +h k, ) 4h h k (23) The changes and ther relatonshps can be modeled and analyzed by a multfactor modelng approach. 5. Predcton 5.1 Purpose The purpose of the predcton module s forecastng a gven tme seres for a gven tme horzon, by analysng the seres' hstorcal development. 5.2 Input data Hstorcal tme seres. Settngs accordng to the used approach (for example, learnng teratons, tme wndow sze, etc.). Predctons of the tme horzon (the length of the predcted sub-seres). 5.3 Output data Tme seres, wth addtonally predcted values added at the end. Predcton qualty statstcs. 5.4 Propertes The tme seres predcton s a feld wth too many predcton methods. Some of the most commonly used are: Averages (Movng Average (MA), Weghted Movng Average (WMA), Exponental Weghted Movng Average (EWMA), etc. Autoregressve methods (Autoregressve (AR), Autoregressve Movng Average (ARMA), Autoregressve Integrated Movng Average (ARIMA), Seasonal 31 Mathematcal Modellng Methods for Tme Seres

32 Autoregressve Integrated Movng Average (SARIMA), Autoregressve Movng Averages wth Exogenous Input Model (ARMAX), Self-Exctng Threshold Autoregressve (SETAR), etc.) wth Box-Jenkns methodology. Trend-extrapolaton (based on Least Squares Error (LSE), trend polynomal fndng, etc.). Neural Networks (Multlayer perceptron, Radal bass functons, Self-organzng map, Adaptve resonant theory, recurrent Elman/Jordan networks, etc.). Other regresson based (e.g. Observers) and econometrc models. Kalman, Wener and other flters. Wavelet based methods. Holt-Wnter decomposton. Hybrd approaches. Predctons can be used for techncal analyss, algorthmc tradng and other smlar purposes. Excludng the exact predctons, the confdence bands could also be used to determne ntervals n whch the predctons can fall, as shown n Fg. 23, marked n grey colour around the predcted values. Ths fgure llustrates an example of hstorcal values of a tme seres and ts predcton for the horzon of some future values. Predctablty ndcators can be suggested (Hurst exponent, etc.). When a predcton s performed, some ndcators for the predcton qualty should be calculated for prelmnary separated test values. Fg. 24 shows one of the most commonly used approaches for tme seres predcton usng the sldng wndow [7] [8] [9]. There are dfferent mathematcal models usng ths approach. Box-Jenkns approach, autoregressve methods, as well as neural networks are amongst them. The work ncluded n these methods can be separated nto two stages: model dentfcaton and predcton. The former s based on hstorcal values analyss, whle the latter uses a bult mathematcal model from the frst stage n order to generate predctons. The model parameter dentfcaton can be consdered an optmzaton problem and the predcton s propagaton of the nput through the model. 32 Mathematcal Modellng Methods for Tme Seres

33 Predcton Hstorcal values Horzon Fg. 23 Tme seres predcton for a gven tme horzon wth confdence bands Fg. 24 Tme seres predcton va the sldng wndow approach Ths approach s used n both neural network and autoregressve technques. One of the most dffcult tasks here s the wndow sze determnaton that concdes to the number of model parameters. For ths purpose, one of the followng methods can be used: Analysng the autocorrelaton functon (ACF) and Partal Autocorrelaton functon (PACF). These functons are bult for the tme seres. The pont n whch they become smaller than a gven sgnfcance level s chosen to determne the tme wndow. Ths approach emerges from the Box-Jenkns methodology. It s commonly used n the autoregressve predcton, but also n other smlar methods, such as neural network predcton. Brute force searchng. Ths method performs a predcton for all possble wndow szes and errors are calculated accordng to dfferent crtera. 33 Mathematcal Modellng Methods for Tme Seres

34 Trend brute force searchng If there s t hstorcal values x1 xt the lnear trend La s bult for them and t s extrapolated n the future tme horzon for whch predctons wll be performed. Here, the future tme horzon s equal to the hstorcal seres sze t. After that, for every possble wndow sze p from mnwndowsze to maxwndowszet predctons xt+1 x2t are generated n the future and the lnear trend Lf s bult for the whole seres x1 x2t, consstng of both hstorcal and predcted values. Value p. for whch the Eucldean dstance dt between the extrapolated La and Lf s mnmal, s chosen to be the wndow sze. After our nvestgaton, mnwndowsze and maxwndowsze are chosen to be t 5 and t 2 respectvely. The closer the trend lnes are to each other, the smaller the error s. Fg. 25 Trend brute force searchng Varaton brute force searchng In ths approach, varaton Sa s calculated for the hstorcal values x1 xt. For every possble wndow sze p, from mnwndowsze to maxwndowszet, predctons xt+1 x2t are generated and varaton Sf s calculated for xt+1 x2t. Value p s chosen to be the wndow sze, for whch the dstance dv= Sa - Sf between hstorcal and predcted values varaton, s mnmal. 34 Mathematcal Modellng Methods for Tme Seres

35 203,00 198,00 193,00 Hstorcal Predcted W=3 188,00 183,00 Varaton ,00 173,00 Varaton , Varaton 6.41 Fg. 26 Predctons n red colour are better because ther varaton s closer to the varaton of the hstorcal tme seres Trend and varaton brute force searchng Ths approach represents a combnaton between the two prevous approaches. Both dt and dv are calculated and ther sum ds=dt+dv s calculated for every p from mnwndowsze to maxwndowsze. Value p, for whch ds s mnmal, s chosen to be the wndow sze. Here, dfferent weghts can be appled to both dt and dv n the calculaton of ds. An nconvenence n the tme seres predcton methods s the great number of settngs requred for the mathematcal model. Most of them are precsed usng a tral and error approach. Settngs also depend on tme seres propertes. as shown n Table Mathematcal Modellng Methods for Tme Seres

36 No Addtve seasonalty Multplcatve seasonalty No Addtve Multplcatve Table 2 Tme Seres Probabltes In order to enable predctons of tme seres, not dependng from ts characterstcs, some pre-processng should be performed, pror to the buldng of the model, as well as some post-processng after the predcton Tme seres Trend Tme seres Trend Fg. 27 An orgnal seres wth ts trend on the left and the seres after pre-processng on the rght The seres s rarely used n ts pure values. Most often t s pre-processed by: x t = y t y t 1 y t 1 (24) 36 Mathematcal Modellng Methods for Tme Seres

37 37 Mathematcal Modellng Methods for Tme Seres Thus, the seres x s one value shorter than the orgnal seres. After the predcton, the reverse operaton s performed by: y t = x t, t=0 (25) y t = (x t + 1)y t 1, t > 0 (26) The am of the tme seres s to predct each value by some prevous values of the seres n the followng way: 1 t 1 p t p 1 t 2 t 1 1 t e x x x x... (27) Ths s, n fact, an autoregressve (AR) process [4] [5] [6] [10] [11]. In order to use t, order p and coeffcents should be determned. One of the methods to acheve do that s to solve the matrx equaton (28) and fndng of Φ. p p A p s t 1 p s t 2 s t 1 s t 2 p t 3 p t t 1 t 1 p t 2 p t 1 t t b s t 2 t 1 t x x x x x x x x x x x x x x x (28) Another way s to use the Yule-Walker equatons [3] [5] [6]. The order p s equal to the wndow sze shown n Fg. 24. Thus, after determnng coeffcents they can be appled to the last values x of the seres, obtanng the frst predcton. Takng t as a real value, the next predcton can be obtaned usng the same formula, and so on. Applyng ths method, the generated predctons are shown n Fg. 28, where the black lne shows the hstory and the red lne expresses the predcted future.

38 Fg. 28 A tme seres predcton Tme seres s shown wth seasonalty. The autoregressve predcted values are shown n red colour. It can be seen that the seasonal effect s successfully predcted, whch s mportant for data processng. Addtonally, wth pre-processng and post-processng, the trend s removed and restored before and after the model buldng respectvely, thus obtanng qualtatve predcton and takng nto account both seasonalty and the trend, ndependently of ther addtve or multplcatve behavour. 38 Mathematcal Modellng Methods for Tme Seres

39 6. Monte Carlo Smulaton 6.1 Purpose The purpose of the Monte Carlo smulaton s to fnd the Value-at-Rsk (VaR) for a gven data and portfolo. 6.2 Input data Data represented as tme seres and portfolo data. Settngs, such as the number of runs, confdence nterval, and default categorzaton n case of mssng data and rsk type. 6.3 Output data Market rsk dstrbutons graphcally dsplayed va hstograms or reports. 6.4 Propertes A market rsk nvolves the uncertanty of future earnngs, resultng from changes n varous ndependent seres n the market envronment for a certan future tme pont (rsk horzon) [2]. The market rsk of a seres s measured by means of a sngle value, called market Valueat-Rsk (VaR). Market VaR represents unexpected losses at adverse market movements and measures the rsk based on a probablty of loss and a specfc tme horzon n whch ths loss can be expected to occur. The regulators use VaR to defne the requrements for the tradng, snce VaR models can be used to estmate the loss of captal due to the market rsk. There are dfferent methods to calculate Market Rsk, amongst whch the most powerful are parametrc VaR/CoVaR, hstorcal smulaton and structured Monte Carlo smulaton [1] [15] [18]. The module that carres out the structured Monte Carlo Smulaton, can create scenaros for the market development n the future and can compute the seres to be analyzed successvely, by usng scenaros (Fg. 29). Fg. 29 Generated scenaros 39 Mathematcal Modellng Methods for Tme Seres

40 The resultng outcome seres represents a prce dstrbuton, ndcatng the rsks wth confdence. The systematc evaluaton of the Monte Carlo Smulaton s shown n Fg. 30 and s based on the steps descrbed below. Correlaton Matrx Market Tme Seres Cholesky Transformaton Monte Carlo Smulaton Structure Volatlty Vector S Cholesky Matrx Random Generator Vector Stand. Normal Dstrbuton X Scenaro Vector + Correlated Vector Value at Rsk Prce Dstrbuton Prce Dstrbuton (Sub)Portfolo Calculaton Smulaton Tree Market Factor Vector Fg.30 Structured Monte Carlo smulaton 1) Volatlty vector (V) and correlaton matrx (R) for a set of market rsk drvers, such as nterest rates, FX rates, prces and spreads, are ether produced from statstcs of ts own hstorcal seres n the PMS data base, or are mported from JPMorgan standard rsk data sets. 2) Covarance matrx (C) s obtaned by a matrx multplcaton of volatlty vector (V) and correlaton matrx (R). 3) Cholesky matrx S s constructed from the covarance matrx (C), so that S x S' = C,c where S' s the transposed matrx. 4) Matrx Perturbaton or Spectral Decomposton adjusts the covarance matrx to be postve defnte, n case the orgnal matrx s not postve defnte, as a result from dependences of rsk drvers. 5) A random generator produces ndependent samples for each market drver, usng standard normal dstrbuton SND (0,1) stored nto devaton vectors (D). A SVD correcton and normal form correcton appled to random vectors ensures normal dstrbuton, where mean = 0, standard devaton = 1, skewness = 0, kurtoss = 0 and correlaton between 40 Mathematcal Modellng Methods for Tme Seres

41 factors = 0. The hgh qualty of the random seres reduces the number of Monte Carlo runs to runs are usually enough to obtan good results. 6) Devaton vector (D) s multpled by the Cholesky matrx (S), takng nto account the covarance between every rsk drver par. Ths produces delta vectors (B) of normal dstrbuted as well as correlated devatons. 7) Vectors (B) are appled to actual rsk drver values (nterest rates, FX rates, spreads, etc.) n the asset vectors (A). Ths produces scenaro vectors (E) for Monte Carlo runs that compute the prcng expresson value, usng prcng trees to represent nstruments and portfolos suppled by a Prcng Tree Generator. Prcng trees are specalzed data structures ensurng a smulaton s hgh performance. 8) Steps 6 s repeated a number of tmes ( ). The large scenaro set affects the calculaton of prce dstrbutons of prcng expresson. The resulted prce dstrbuton s constructed by countng the appearance of the values wthn many adjacent small ranges. Thereafter, Value at Rsk s calculated va a numercal ntegraton of the dstrbuton densty, usng a confdence percentage, for example 1%, for the upper lmt of the prce dstrbuton. The Monte Carlo smulaton steps descrbed above use a set of well-known mathematcal operatons: Calculaton of the covarance matrx where: C, R, * V * V, = 1...n (matrx wdth), j<= (trangle matrx) (29) j j j Cj - Element from covarance matrx Rj - Element from correlaton matrx (R =1) V, Vj - Element from volatlty vector Constructon of the Cholesky matrx S where: S x S' = C (30) S' - Transposed matrx, aj - Elements of the matrx S 41 Mathematcal Modellng Methods for Tme Seres

42 For a matrx where the wdth = 3, the followng matrx operaton apples: c c c C c c c c c c a a a a a a a a a a a 0 0 a (31) The matrx multplcaton nvolves formulas for the calculaton elements of the Cholesky matrx: 1 a c a a 2 k k 1 j 1 1 cj ak a jk j 1, 2,..., N a jj k 1 1/ 2 (32) Postve defnte matrx A matrx s sad to be symmetrc f A=A, where A s the transpose of A. Therefore, covarance and correlaton matrces are real and symmetrc. A real symmetrc matrx A s sad to be postve defnte f x Ax s postve for every vector x dfferent from 0. Gven ths defnton, one can prove that a real symmetrc matrx s postve defnte f, and only f, all ts egenvalues are greater than zero. The queston s: How do the egenvalues of a matrx A change f t s subject to a perturbaton A A+E? Because the egenvalues are contnuous functons of the entres of A, t s natural to thnk that, f the perturbaton matrx E s small enough, the egenvalues should not change too much. For our purposes, t would be undesrable to drastcally change a matrx n order to turn t nto a proper covarance matrx. There are two methods that can transform a non-postve defnte matrx nto postve one. Matrx Perturbaton The Monte Carlo engne uses the followng perturbaton: AA+e(I-A) = B, where e s a scalar and I s the dentty matrx. If e = 0 then AA and f e = 1, then AI, whch s clearly a postve defnte matrx. Therefore, snce the egenvalues of a matrx are contnuous functons of ts entres, there s an nfnte number of scalars e n the nterval [0, 1] that, gven the equaton above, are a postve defnte matrx. The task s to obtan the mn of e, whch s calculated from the mn of egenvalues. Spectral Decomposton 42 Mathematcal Modellng Methods for Tme Seres

43 Gven the rght-hand-sde egensystem S of the real and symmetrc matrx R and ts assocated set of egenvalues {} such that R. S =. S wth = dag(), (34) defne the non-zero elements of the dagonal matrx as : 0 ': ' 0 : 0 (35) If the target matrx s not postve-sem-defnte, t has at least one negatve egenvalue and at least one of the wll be zero. In addton, defne the non-zero elements of the dagonal scalng matrx T wth respect to the egensystem S by T : 2 t s m m ' m 1 (36) B' : S ' (37) By the constructon, B : T B' T S ' (38) R T : BB (39) s now both postve sem-defnte and has unt dagonal elements. A procedural descrpton of the method above may clarfy what, n fact, has to be done: Calculate the egenvalues and the rght-hand-sde egenvectors s of R Set all negatve to zero. Multply the vectors s wth ther assocated corrected egenvalues and arrange as columns of R. Fnally, R results from R are obtaned by normalsng the row vectors of R to unt length. Correcton of random seres 43 Mathematcal Modellng Methods for Tme Seres

44 1) SVD Correcton of expected value, standard devaton and correlaton For the frst correcton of expected value, standard devaton and correlaton s created by a common post-processng. The approach s based on calculatng the statstcal fgures of random numbers and applyng the dfferences to the deal case for the correcton. Formal Task Gven the matrx X(n,k), where n lnes contan the generated k-dmensonal normally dstrbuted random samples. The statstcal parameters of these samples m(k) vector of mean values and K(k,k) covarance matrx can be calculated. The dstrbuton, caused by Matrx X, s represented and follows the normal dstrbuton N (k) (m, K). It also requres that the sample dstrbuton of the target N (k) (v, C) should have specfc parameters. In ths partcular case, v = (0, 0,..., 0) and matrx C should be the unt matrx (overall 0 and everywhere else on the man dagonal = 1). There are two formal challenges: The task of shftng the statstcal means of pont m to pont v; The task of transformng the statstcal structure, represented by the covarance matrx K, to the target matrx, represented by C. Soluton to the tasks The frst challenge s easly resolved by a shft of the statstcal means by the dfference vector r. The second challenge requres the converson of a statstcal sample X nto a new Y, that owns the target parameter. The soluton s n the form of a lnear operator, wth the converson matrx S sought. Every vector Y s determned n the followng manner: y T S x r, 1 n (40), Statstcal parameters of sample Y are calculated as follows: The vector of mean values: E{ y } v E{ S T r S T x r} r E{ S E{ x } r S T T m x } (41) The dfference vector r can be determned usng the above expresson: r v S T m (41) TCovarance matrx elements are: 44 Mathematcal Modellng Methods for Tme Seres

45 T E{( y v) ( y v) } C E{( S E{[ S E{[ S S S T T T E{( x m) ( x m) } S T T T K S x r v) ( S x r ( r S ( x m)] [ S T T T T x r v) } m)] [ S T ( x m)] T x r ( r S } T m)] T (42) For the exstence of the matrx S, t s necessary that matrces K and C are commutatve. Ths condton s satsfed, because the covarance matrces are symmetrcal. The correcton mplemented for random seres uses a SVD algorthm (Sngular Value Decomposton) n determnng the converson matrx S. 2) Correcton for normal dstrbuton form The purpose of the form correcton s to correct fgures of the second order (Kurtoss = 0, Skewness = 0),.e. the dstrbuton form of the deal normal dstrbuton should be reached, but the results of the frst correcton shall be retaned. The followng aspects are mportant n the mplementaton of ths correcton: The nverse functon of a standard normal dstrbuton SND (0.1) s known and represents a strct lnear functon,.e. a synthetc lnear numercal seres could produce an deal standard normal dstrbuton after the converton. The necessary correlaton between random seres, whch s equal to zero, s gven by the sequence of random numbers and through ther mutual order n the random seres. The sequence of each random number n a random order, wth regards to the sequence of all other random numbers n other random seres, should be preserved. From the above stated, an approach follows to the mplementaton the correcton, usng the followng steps: Each tme seres s converted by the nverse functon. Converted random numbers should be equally dstrbuted, because of the orgn standard normal dstrbuton. Of course, ths s not the case, because of the random generator, and the task of correctng the form to deal equal dstrbuton. The random numbers are then sorted, wth ther orgnal order saved. The new order should follow a lnear functon. Agan, ths s not the case; the numbers vary somewhat wthn the lnear functon. The fgures are then overwrtten by a synthetc lnear functon (from 0 to 1) and converted back to normal dstrbuton agan, thus enablng the rse of random seres wth a very good qualty. 45 Mathematcal Modellng Methods for Tme Seres

46 As a fnal step, the orgnal order s restored, n order to reconstruct the ntal seres correlaton. Obtanng the delta normal dstrbuton vector B S D, where S - Cholesky matrx D - standard normal dstrbuton vector B - delta normal dstrbuton vector Calculaton of the scenaro vector The scenaro vector s obtaned by applcaton the delta normal dstrbuton vector on the asset vector, that contans nterest rates, foregn exchange rates, etc. for nterest rates: A Scenaro A 1 B (43) for prces and foregn exchange rates A Scenaro B t At e (44) where: B A - Scenaro - Asset 46 Mathematcal Modellng Methods for Tme Seres

47 7. Volatlty Brdge 7.1 Purpose The purpose of the Volatlty Brdge s generatng smaller tme nterval values (e.g. daly) between the bgger tme nterval values (e.g. monthly) of a tme seres n the Monte Carlo smulaton. 7.2 Input data TValues of the tme seres generated va the smulaton scenaros. 7.3 Output data After the volatlty brdge method s appled, the tme seres has addtonal values amongst the bgger tme nterval values. 7.4 Propertes The Monte Carlo smulaton, presented n the prevous secton, generates a smaller number of scenaros over longer horzons. The volatlty brdge generates scenaros for arbtrary horzons n between these values, as shown n Fg. 31 [1]. Fg. 31 Volatlty Brdge The volatlty brdge process generates values wth a normal dstrbuton n between two-tme ponts, where at each tme pont the value s known. Daly prce scenaros, generated n ths way, are normally dstrbuted. 47 Mathematcal Modellng Methods for Tme Seres

48 Fg.32 Volatlty Brdge, generatng values between several tme ponts Generated from Monte Carlo, monthly values represent level 1, whle volatlty brdge generated values are at level 2 of the smulaton. For each smulaton run for level 1, the values between tme ponts are recalculated. The level 2 smulaton performs a stochastc nterpolaton between the support ponts, usng the volatlty brdge. The followng propertes are taken nto account durng ths process. Expected values between the control ponts are lnearly nterpolated. Daly fluctuatons are generated by a second level Monte Carlo smulaton, whch generates the daly changes derved from the hstorcal volatlty and correlaton. The smulaton of scenaros s ndependent of the tme of smulated planned tems. Fg. 33 Multply smulated daly prce paths wth ther level 1 and level 2 ponts For b=1, 2,, τ1, τ1 + 1,, τr the daly prces are generated wth the followng propertes: At level 1 dates, that s b = τ1,,τr, level 2 scenaros match level 1 scenaros, so that q q dt st where r r q st are level 1 prces n scenaro q. r 48 Mathematcal Modellng Methods for Tme Seres