Intra-day Trading of the FTSE-100 Futures Contract Using Neural Networks With Wavelet Encodings

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1 Submtted to European Journal of Fnance Intra-day Tradng of the FTSE-00 Futures Contract Usng eural etworks Wth Wavelet Encodngs D L Toulson S P Toulson Intellgent Fnancal Systems Lmted Sute 4 Greener House Haymarket London SWY 4RF SWY 4RF Tel: (00) Emal: fs@f5com wwwf5com Please send correspondence and proofs to ths author

2 Intra-day Tradng of the FTSE-00 Futures Contract Usng eural etworks Wth Wavelet Encodngs ABSTRACT In ths paper, we shall examne the combned use of the Dscrete Wavelet Transform and regularsed neural networks to predct ntra-day returns of the FTSE-00 ndex future The Dscrete Wavelet Transform (DWT) has recently been used extensvely n a number of sgnal processng applcatons The manner n whch the DWT s most often appled to classfcaton / regresson problems s as a pre-processng step, transformng the orgnal sgnal to a (hopefully) more compact and meanngful representaton The choce of the partcular bass functons (or chld wavelets) to use n the transform s often based ether on some pre-set samplng strategy or on a pror heurstcs about the scale and poston of the nformaton lkely to be most relevant to the task beng performed In ths work, we propose the use of a specalsed neural network archtecture (WEAPO) that ncludes wthn t a layer of wavelet neurons These wavelet neurons serve to mplement an ntal wavelet transformaton of the nput sgnal, whch n ths case, wll be a set of lagged returns from the FTSE-00 future We derve a learnng rule for the WEAPO archtecture that allows the dlatons and postons of the wavelet nodes to be determned as part of the standard back-propagaton of error algorthm Ths ensures that the chld wavelets used n the transform are optmal n terms of provdng the best dscrmnatory nformaton for the predcton task We then focus on addtonal ssues related to use of the WEAPO archtecture Frst, we examne the ncluson of constrants for enforcng orthogonalty n the wavelet nodes durng tranng We then propose a method (M&M) for prunng excess wavelet nodes and weghts from the archtecture durng tranng to obtan a parsmonous fnal network We wll conclude by showng how the predctons obtaned from commttees of WEAPO networks may be exploted to establsh tradng rules for adoptng long, short or flat postons for the FTSE-00 ndex future usng a Sgnal Thresholded Tradng System (STTS) The STTS operates by combnng predctons of future returns over a varety of dfferent predcton horzons A set of tradng rules s then determned that act to optmse the Sharpe Rato of the tradng strategy usng realstc assumptons for bd/ask spread, slppage and transacton costs Keywords: Wavelets, neural networks, commttees, regularsaton, tradng system, FTSE- 00 future

3 Introducton Over the past decade, the use of neural networks for fnancal and econometrc applcatons has been wdely researched (Refenes et al [993], Wegend [996], Whte [988] and others) In partcular, neural networks have been appled to the task of provdng forecasts for varous fnancal markets rangng from spot currences to equty ndexes The mpled use of these forecasts s often to develop systems to provde proftable tradng recommendatons However, n practce, the success of neural network tradng systems has been somewhat poor Ths may be attrbuted to a number of factors In partcular, we can dentfy the followng weaknesses n many approaches: Data Pre-processng Inputs to the neural network are often smple lagged returns (or even prces!) The dmenson of ths nput nformaton s often much too hgh n the lght of the number of tranng samples lkely to be avalable Technques such as Prncpal Components Analyss (PCA) (Oa [989]) and Dscrmnant Analyss (Fukunaga [990]) can often help to reduce the dmenson of the nput data as n Toulson & Toulson (996a) and Toulson & Toulson (996b) (henceforth TT96a, TT96b) In ths paper, we present an alternate approach usng the Dscrete Wavelet Transform (DWT) Model Complexty eural networks are often traned for fnancal forecastng applcatons wthout sutable regularsaton technques Technques such as Bayesan Regularsaton MacKay (99a), MacKay (99b) (henceforth M9a, M9b) Buntme & Wegend(99) or smple weght decay help control the complexty of the mappng performed by the neural network and reduce the effect of overfttng of the tranng data Ths s partcularly mportant n the context of fnancal forecastng due to the hgh level of nose present wthn the data 3 Confuson Of Predcton And Tradng Performance Often researchers present results for fnancal forecastng n terms of root mean square predcton error or number of accurately forecasted turnng ponts Whlst these values contan useful nformaton about the performance of the predctor they do not necessarly mply that a successful tradng system may be based upon them The performance of a tradng system s usually dependent on the performance of the predctons at key ponts n the tme seres Ths performance s not usually adequately reflected n the overall performance of the predctor averaged over all ponts of a large testng perod We shall present a practcal tradng model n ths paper that attempts to address each of these ponts The Predcton Model In ths paper, we shall examne the use of commttees of neural networks to predct future returns of the FTSE-00 Index Future over 5, 30, 60 and 90 mnute predcton horzons We shall then comb ne these predctons and determne from them a set of tradng rules that wll optmse a rsk-adusted tradng performance (Sharpe rato) We shall use as nput to each of the neural networks the prevous 40 lagged mnutely returns of the FTSE-00 Future The requred output shall be the predcted return for the approprate predcton horzon Ths process s llustrated n Fgure

4 Predcton Horzon FTSE-00 Tme 40 lagged returns 5 mn 30 mn 60 mn 90 mn Fgure : Predctng FTSE-00 Index Futures: 40 lagged returns are extracted form the FTSE-00 future tme seres These returns are used as nput to (WEAPO) MLPs Dfferent MLPs are traned to predct the return of the FTSE-00 future 5, 30, 60 and 90 mnutes ahead A key consderaton concernng ths type of predcton strategy s how to encode the 40 avalable lagged returns as a neural network nput vector One possblty s to smply use all 40 raw nputs The problem wth ths approach s the hgh dmensonalty of the nput vectors Ths wll requre us to use an extremely large set of tranng examples to ensure that the parameters of the model (the weghts of the neural network) may be properly determned Due to computatonal complextes and the non-statonarty of fnancal tme seres, usng extremely large tranng sets s seldom practcal A preferable strategy s to reduce the dmenson of the nput nformaton to the neural network A popular approach to reducng the dmenson of nputs to neural networks s to use a Prncpal Components Analyss (PCA) transform to reduce redundancy n the nput vectors due to ntercomponent correlatons However, as we are workng wth lagged returns from a sngle fnancal tme seres we know, n advance, that there s lttle (auto) correlaton n the lagged returns In other work (TT96a), (TT96b), we have approached the problem of dmenson reducton through the use of Dscrmnant Analyss technques These technques were shown to lead to sgnfcantly mproved performance n terms of predcton ablty of the traned networks However, such technques do not, n general, take any advantage of our knowledge of the temporal structure of the nput components, whch wll be sequental lagged returns Such technques are also mplctly lnear n ther assumptons of separablty, whch may not be generally approprate when consderng nputs to (non-lnear) neural networks We shall consder, as an alternatve means of reducng the dmenson of the nput vectors, the use of the dscrete wavelet transform

5 Wavelets 03 Coeffcents Fgure : The dscrete wavelet transform The tme seres s convolved wth a number of chld wavelets charactersed by dfferent dlatons and translatons of a partcular mother wavelet 3 The Dscrete Wavelet Transform (DWT) 3 Background The Dscrete Wavelet Transform (Telfer et al [995], Meyer [995]) has recently receved much attenton as a technque for the pre-processng of data n applcatons nvolvng both the compact representaton of the orgnal data (e data compresson or factor analyss) or as a dscrmnatory bass for pattern recognton and regresson problems (Casasent & Smokeln [994], Szu & Telfer [99]) The transform functons by proectng the orgnal sgnal onto a sub-space spanned by a set of chld wavelets derved form a partcular Mother wavelet For example, let us select the Mother wavelet to be the Mexcan Hat functon t ( t ) e y( t) = π 4 () 3 The wavelet chldren of the Mexcan Hat Mother are the dlated and translated forms of (), e φ τ, ζ ( t) = ζ φ t τ ζ () ow, let us select a fnte subset C from the nfnte set of possble chld wavelets Let the members of the subset be dentfed by the dscrete values of poston τ and scale ζ, =,, K, where K s the number of chldren { τ, ζ,, } C = = K (3)

6 Suppose we have an dmensonal dscrete sgnal x ρ The th component of the proecton of the orgnal sgnal x ρ onto the K dmensonal space spanned by the chld wavelets s then y = x φ τ, ζ ( ) (4) = 3 Choce of Chld Wavelets The sgnfcant questons to be answered wth respect to usng the DWT to reduce the dmenson of the nput vectors to a neural network are: How many chld wavelets should be used and gven that, what values of τ and ζ, should be chosen? For representatonal problems, the chld wavelets are generally chosen such that together they consttute a wavelet frame There are a number of known Mother functons and choce of chldren that satsfy ths condton (Debauches [988]) Wth such a choce of mother and chldren, the proected sgnal wll retan all of ts orgnal nformaton (n the Shannon sense, Shannon [948]), and reconstructon of the orgnal sgnal from the proecton wll be possble There are a varety of condtons that must be fulflled for a dscrete set of chld wavelets to consttute a frame, the most ntutve beng that the number of chld wavelets must be at least as great as the dmenson of the orgnal dscrete sgnal However, the choce of the optmal set of chld wavelets becomes more complex n dscrmnaton or regresson problems In such cases, reconstructon of the orgnal sgnal s not relevant and the nformaton we wsh to preserve n the transformed space s the nformaton that dstngushes dfferent classes of sgnal In ths paper, we shall present a method of choosng a sutable set of chld wavelets such that the transformaton of the orgnal data (the 40 lagged returns of the FTSE-00 Future) wll enhance the nonlnear separablty of dfferent classes of sgnal whlst sgnfcantly reducng the dmenson of the data We show how ths may be acheved naturally by mplementng the wavelet transform as a set of wavelet neurons contaned n the frst layer of a mult-layer perceptron (Rummelhart et al [986]) (henceforth R86) The shfts and dlatons of the wavelet nodes are then found along wth the other network parameters through the mnmsaton of a penalsed least squares obectve functon We then extend ths concept to nclude automatc determnaton of a sutable number of wavelet nodes by applyng Bayesan prors on the chld wavelet parameters durng tranng of the neural network and enforcng orthogonalty between the wavelet nodes usng soft constrants 4 Wavelet Encodng A Pror Orthogonal etwork (WEAPO) In ths secton, we shall derve a neural network archtecture that ncludes wavelet neurons n ts frst hdden layer (WEAPO) We shall begn by defnng the wavelet neuron and ts use wthn the frst layer of the WEAPO archtecture We shall then derve a learnng rule whereby the parameters of each wavelet neuron (dlaton and poston) may be optmsed wth respect to the accuracy of the network's predctons Fnally, we shall consder ssues such as wavelet node orthogonalty and choce of the optmal number of wavelet nodes to use n the archtecture (skeletonsaton)

7 4 The Wavelet euron The most common actvaton functon used for neurons n the Mult-Layer Perceptron archtecture s the sgmodal actvaton functon ϕ( x) = x (5) x + e 0 The output of a neuron y s dependent on the actvatons of the nodes n the prevous layer x k, and the weghted connectons between the neuron and the prevous layer ω k,, e y I = x ϕ ω =, (6) otng the smlarty between Equatons (6) and (4), we can mplement the Dscrete Wavelet Transform as the frst layer of hdden nodes of a mult-layer perceptron (MLP) The weghts connectng each wavelet node to the nput layer ω, must be constraned to be dscrete samples of a partcular wavelet chld φ τ, ζ ( ) and the actvaton functon of the wavelet nodes should be the dentty transformaton ϕ( x) = x In fact, we may gnore the weghts connectng the wavelet node to the prevous layer and nstead characterse the wavelet node purely n terms of values of translaton and scale, τ and ζ The WEAPO archtecture s shown below n Fgure 3 We can note that, n effect, the WEAPO archtecture s a standard four-layer MLP wth a lnear set of nodes n the frst hdden layer and n whch the weghts connectng the nput layer to the frst hdden layer are constraned to be wavelets Ths constrant on the frst layer of weghts acts to enforce our a pror knowledge that the nput components are not presented n an arbtrary fashon, but n fact have a defned temporal orderng DWT Predcton Horzon FTSE-00 Tme 40 lagged returns 5 mn 30 mn 60 mn 90 mn Pseudo weghts τ ζ Wavelet nodes MLP Fgure 3: The WEAPO archtecture

8 4 Tranng the Wavelet eurons The MLP s usually traned usng error backpropagaton (backprop) [R86] on a set of tranng examples The most commonly used error functon s smply the sum of squared error over all tranng samples E D ED = y ρ ρ t = (7) Backprop requres the calculaton of the partal dervatves of the data error E D wth respect to each of the free parameters of the network (usually the weghts and bases of the neurons) For the case of the wavelet neurons suggested above, the weghts between the wavelet neurons and the nput pattern are not free but are constraned to assume dscrete values of a partcular chld wavelet The free parameters for the wavelet nodes are therefore not the weghts, but the values of translaton and dlaton τ and ζ To optmse these parameters durng tranng, we must obtan expressons for the partal dervatves of the error functon The usual form of the backprop algorthm s: E D wth respect to these two wavelet parameters E ω E y y ω = (8),, E The term y, often referred to asδ, s the standard backpropagaton of error term, whch may be y must be ω found n the usual way for the case of the wavelet nodes The partal dervatve substtuted wth the partal dervatves of the node output y wth respect to the wavelet parameters For a gven Mother wavelet φ (x), consder the output of the wavelet node, gven n Equaton (4) Takng partal dervatves wth respect to the translaton and dlaton yelds:, y τ y ζ = = = τ = ζ = x = = = x x ζ τ φ ' 3 ζ ζ ζ τ φ ζ τ φ ζ τ x φ 3 ζ ζ = x ( τ ζ 5 ) τ φ ' ζ (9)

9 Usng the above equatons, t s possble to optmse the wavelet dlatons and translatons For the case of the Mexcan Hat wavelet we note that φ' ( t) = π t( t ) e 3 t (0) Once we have derved sutable expressons for the above, the wavelet parameters may be optmsed n conuncton wth the other parameters of the neural network by tranng usng any of the standard gradent based optmsaton technques 43 Orthogonalsaton of the Wave let odes A potental problem that mght arse durng the optmsaton of the parameters assocated wth the wavelet neurons s that of duplcaton n the parameters of some of the wavelet nodes Ths wll lead to redundant correlatons n the outputs of the wavelet nodes and hence the producton of an overly complex model One way of avodng ths type of duplcaton would be to apply a soft constrant of orthogonalty on the wavelets of the hdden layer Ths could be done through use of the addton of an addtonal error term n the standard data msft functon, e φ E W = τ ζ τ ζ φ, φ =, () where denotes the proecton () f, g = f ( ) g( ) = In the prevous secton, backprop error gradents were derved n terms of the unregularsed sum of squares data error term, E D We now add n an addtonal term for the orthogonalty constrant to yeld a combned error functon M(W), gven by E W φ M( W) = αe + γe φ (3) ow, to mplement ths nto the backprop tranng rule, we must derve the two partal dervatves of wth respect to the dlaton and translaton wavelet parameters ζ and τ Expressons for the partal dervatves above are obtaned from (9) and are: D W E τ Φ W = K = t= φ τ, ξ ( t) τ φ τ, φ ( t) (4) E ξ Φ W = K = t= φ τ, ξ ( t) ξ φ τ, φ ( t)

10 These terms may then be ncluded wthn the standard backprop algorthm The rato α γ determnes the balance that wll be made between obtanng optmal tranng data errors aganst the penalty ncurred by havng overlappng or non-orthogonal nodes The rato may ether be ether estmated or optmsed usng the method of cross valdaton The effect of the orthogonalsaton terms durng tranng wll be to make the wavelet nodes compete wth each other to occupy the most relevant areas of the nput space wth respect to the mappng beng performed by the network In the case of havng an excessve number of wavelet nodes n the hdden layer ths generally leads to the margnalsaton of a number of wavelet nodes The margnalsed nodes are drven to areas of the nput space n whch lttle useful nformaton wth respect to the dscrmnatory task performed by the network s present 44 Weght and ode Elmnaton The a pror orthogonal constrants ntroduced n the prevous secton help to prevent sgnfcant overlap n the wavelets by encouragng orthogonalty However, redundant wavelet neurons wll stll reman n the hdden layer though they wll have been margnalsed to rrelevant (n terms of dscrmnaton) areas of the tme/frequency space At best, these nodes wll play no sgnfcant role n modellng the data At worst, the nodes wll be used to model nose n the output targets and wll lead to poor generalsaton performance It would be preferable f these redundant nodes could be elmnated A number of technques have been suggested n the lterature for node and/or weght elmnaton n neural networks We shall adopt the technque proposed by Wllams (993) and MacKay (99a, 99b) and use a Bayesan tranng technque, combned wth a Laplacan pror on the network weghts as a natural method of elmnatng redundant nodes from the WEAPO archtecture The Laplacan Pror on the network weghts mples an addtonal term n the prevously defned error functon (3), e M ( W ) = αe + γe βe (5) D φ W + W where E W s defned as E W =, ω, (6) A consequence of ths pror s that durng tranng, weghts are forced to adopt one of two postons A weght can ether adopt equal data error senstvty as all the other weghts or s forced to zero Ths leads to skeletonsaton of a network Durng ths process, weghts, hdden nodes or nput components may be removed from the archtecture The combned effects of the soft orthogonalty constrant on the wavelet nodes and the use of the Laplacan weght pror leads to what we term Margnalse and Murder (M&M) tranng At the begnnng of tranng process, the orthogonalty constrant forces certan wavelet nodes to nsgnfcant areas of the nput space wth regards to the dscrmnaton task beng performed by the network The weghts emergng from these redundant wavelet nodes wll then have lttle data error senstvty and are forced to zero and deleted due to the effect of the Laplacan weght pror

11 5 Predctng The FTSE-00 Future Usng WEAPO etworks 5 The Data We shall apply the network archtecture and tranng rules descrbed n the prevous secton to the task of predctng future returns of the FTSE-00 ndex future quoted on LIFFE The hstorcal data used was tck-by-tck quotes of actual trades suppled by LIFFE (see Fgure 4) The data was pre-processed to a -mnutely format by takng the average volume adusted traded prce durng each mnute Mssng values were flled n by nterpolaton but were marked as un-tradable Mnutely prces were obtaned n ths manner for 8 months, January 995-June 996, to yeld approxmately 00,000 dstnct prces The entre data set was then dvded nto three dstnct subsets, tranng/valdaton, optmsaton and test We traned and valdated the neural network models on the frst sx months of the 995 data The predcton performance results, quoted n ths secton, are the results of applyng the neural networks to the second sx months of the 995 data (optmsaton set) We reserved the frst 6 months of 996 for out-of-sample tradng performance test purposes Fgure 4: FTSE-00 Future January 995 to June Indvdual Predctor Performances Table to Table 4 show the performances of four dfferent neural network predctors for the four predcton horzons (5, 30, 60 and 90 mnutes) The predctors used were Smple early-stoppng [Hecht-elsen[989])] MLP traned usng all 40 lagged return nputs wth an optmsed number of hdden nodes found by exhaustve search (-3 nodes) A standard weght decay MLP (Hnton [987]) traned usng all 40 lagged returns wth the value of weght decay lambda optmsed by cross valdaton 3 An MLP traned wth Laplacan weght decay and weght/node elmnaton (as n Wllams [993]) 4 WEAPO archtecture usng wavelet nodes, soft orthogonalsaton constrants and Laplacan weght decay for weght/node elmnaton The performances of the archtectures are n terms of : RMSE predcton error n terms of desred and actual network outputs Turnng pont accuracy: Ths s the number of tmes the network correctly predcts the sgn of the future return

12 3 Large turnng pont accuracy: Ths s the number of tmes that the network correctly predcts the sgn of returns whose magntude s greater than one standard devaton from zero (ths measure s relevant n terms of expected tradng system performance) Predcton horzon % Accuracy Large % Accuracy RMSE 5 507% 5477% % 5955% % 546% % 508% Table : Results for MLP usng early stoppng Predcton horzon % Accuracy Large % Accuracy RMSE % 570% % 5608% % 5409% % 574% Table : Results for weght decay MLP Predcton horzon % Accuracy Large % Accuracy RMSE 5 55% 4855% % 5434% % 4348% % 508% Table 3: Results for Laplacan weght decay MLP Predcton horzon % Accuracy Large % Accuracy RMSE 5 537% 5743% % 569% % 5794% % 588% 0088 Table 4: Results for WEAPO We conclude that the WEAPO archtecture and the smple weght decay archtecture appear sgnfcantly better than the other two technques The WEAPO archtecture appears to be partcularly good at predctng the sgn of large market movements 53 Use of Commttees for Predcton In the prevous secton, we presented predcton performance results usng a sngle WEAPO archtecture appled to the four requred predcton horzons A number of authors (Hashem & Schmeser [993]) have suggested the use of lnear combnatons of neural networks as a means of mprovng the robustness of neural networks for forecastng and other tasks The basc dea of a commttee s to ndependently tran a number of neural networks and to then combne ther outputs Suppose we have traned neural networks and that the output of the ρ The commttee response s gven by gven by ( x ) y th net s

13 0 ρ ρ y( x) = α y ( x) + α 0 = (7) where α s the weghtng for the th network and α 0 s the bas of the commttee The weghtngs may ether be smple averages (Basc Ensemble Method) or may be optmsed usng an OLS procedure (Generalsed Ensemble Method) Specfcally, the OLS weghtngs may be determned by ρ ρ α = Ξ Γ (8) where Ξ and Γ are defned n terms of the outputs of the ndvdual traned networks and the tranng examples, e T ρ ρ [ ξ, ] y( xt ) y ( xt ) Ξ = = T ρ Γ = = T t= [ γ ] ( t ) T t= ρ y x t t (9) where x ρ s the tranng examples th nput vector, t s the correspondng th target response and T s the number of Below, we show the predcton performances of commttees composed of fve ndependently traned WEAPO MLPs, for each of the predcton horzons We conclude that the performances (n terms of RMSE) are superor to those obtaned usng a sngle WEAPO archtecture Turnng pont detecton accuracy, however, s broadly smlar Predcton horzon % Accuracy Large % Accuracy RMSE 5 535% 577% % 5698% % 577% % 5869% Table 5: Results for Commttees of fve ndependently traned WEAPO archtectures 6 The Sgnal Thresholded Tradng System 6 Background One mght thnk that f we have a neural network or other predcton model correctly predctng the future drecton of a market 60 percent of the tme, then t would be relatvely straghtforward to devse a proftable tradng strategy In fact, ths s not necessarly the case In partcular one must consder the followng:

14 What are the effectve transacton costs that are ncurred each tme we execute a round-trp trade? Over what horzon are we makng the predctons? If the horzon s partcularly short term (e 5- mnute ahead predctons on ntra-day futures markets) s t really possble to get n and out of the market quckly enough and more mportantly to get the quoted prces? In terms of buldng proftable tradng systems t may be more effectve to have a lower accuracy but longer predcton horzons What level of rsk s beng assumed by takng the ndcated postons? We may, for nstance, want to optmse not ust pure proft but perhaps some rsk-adusted measure of performance such as Sharpe Rato or Sterlng Rato An acceptable tradng system has to take account of some or all of the above consderatons 6 The Basc STTS Model Assume we have P predctors makng predctons about the expected FTSE-00 Futures returns Each of the predctors makes predctons for τ tme steps ahead Let the predcton of the th predctor at tme t be denoted by p ( t) We shall defne the normalsed tradng sgnal S( t) at tme t to be: P p( t) S( t) = ω τ = (0) where ω s the weghtng gven to the th predctor An llustraton of ths s gven n Fgure 5 5 mnutes 30 mnutes 60 mnutes 90 mnutes τ =5 τ 30 τ = P 60 τ = P 90 = ω ω ω P ω P S ( t) = P ( t) ω τ Fgure 5: Weghted summaton of predctons from four WEAPO commttee predctors to gve a sngle tradng sgnal We shall base the tradng strategy on the strength of the tradng sgnal S( t) at any gven tme t At tme t we compare the tradng sgnal S( t) wth two thresholds, denoted by α and β These two thresholds are used for the followng decsons: α s the threshold that controls when to open a long or short trade β s the threshold used to decde when to close out an open long or short trade

15 At any gven tme t, the tradng sgnal wll be compared wth the approprate threshold usng the current tradng poston In partcular, detals of the actons defned for each tradng poston are found n Table 6: Current poston Test Acton: Go Flat f S(t) > α Long Flat f S(t) < -α Short Long f S(t) < -β Flat Short f S(t) > β Flat Table 6: Usng the tradng thresholds to decde whch acton to take Fgure 6 demonstrates the concept of usng the two thresholds for tradng The two graphs shown n Fgure 6 are the tradng sgnals S( t) for each tme t (top) and the assocated prces p ( t) dsplayed n the bottom graph The prce graph s coded for the dfferent tradng poston that are recommended, thck blue and red lnes for beng n a long or short tradng poston, grey otherwse At the begnnng of tradng we are n a flat poston We shall open a trade f the tradng sgnal exceeds the absolute value of α At the tme marked ❶ ths s the case snce the tradng sgnal s greater than α We shall open a long trade Unless the tradng sgnal falls below - β, ths long trade wll stay open Ths condton s fulflled at the tme marked ❷, when we shall close out the long trade We are now agan n a flat poston At tme ❸ the tradng sgnal falls below -α, so we open a short tradng poston Ths poston s not closed out untl the tradng sgnal exceeds β, whch occurs at tme ❹ when the short trade s closed out s(t) Go long Go short Go short α β β α Prce ❶ ❷ ❸ ❹ Go flat Go flat Go flat Fgure 6: Tradng sgnals and prces 7 Results An STTS tradng system, as descrbed above, was formed usng as nput 4 WEAPO commttee predctors Each commttee contaned fve ndependently traned WEAPO networks and was traned to produce 5, 30, 60 and 90-mnute ahead predctons, respectvely A screenshot from the software used to perform ths smulaton (Amber) s shown below n Fgure 7

16 Fgure 7: The Tradng System for FTSE-00 futures 40 lagged returns are extracted from the FTSE-00 future tme seres and after standardsaton, nput to the 0 WEAPO predctors, arranged n four commttes Each commttee s responsble for a partcular predcton horzon The predctons are then combned for each commttee and passed onto the STTS tradng module The optmal values for the STTS thresholds α and β and the four STTS predctor weghtngs ω to use were found by assessng the performance of the STTS model on the optmsaton data (last 6 months of 995) usng partcular values for the parameters The parameters were optmsed usng smulated annealng (Krkpatrck et al [983]) wth the obectve functon beng the net tradng performance (ncludng transacton costs) on ths perod measured n terms of Sharpe rato In terms of tradng condtons, t was assumed that there would be a three mnute delay n openng or closng any trade and that the combned bd-ask spread / transacton charge for each round trp trade would be 8 ponts Both are consdered conservatve estmates It was also assumed that contracts of the FTSE-00 future are rolled over on the delvery month, where bass adustments are made and one extra trade s smulated After the optmal parameters for the STTS system were determned, the tradng system was appled to the prevously unseen data of the frst sx months of the 996 Table 7 summarses the tradng performance over the sx-month test perod n terms of net over-all proftablty, tradng frequency and Sharpe Rato Monthly net proftablty n tcks 53 Average monthly tradng frequency (roundtrp) 8 Sharpe rato daly (monthly) 036 (048) Table 7: Results of tradng system on the unseen test perod

17 8 Concluson We have presented a complete tradng model for adoptng postons n the LIFFE FTSE-00 future In partcular, we have developed a system that avods the three weaknesses that can be dentfed wth many fnancal tradng systems, namely Data Pre-Processng We have constraned the effectve dmenson of the 40 lagged returns by mposng a Dscrete Wavelet Transform on the nput data va the WEAPO neural network archtecture We have also, wthn the WEAPO archtecture devsed a method for automatcally dscoverng the optmal number of wavelets to use n the transform and also whch scales and dlatons should be used Regularsaton We have appled Bayesan regularsaton technques to constran the complexty of the neural network predcton models We have demonstrated the requrement for ths by comparng the predcton performances of regularsed and unregularsed (early-stoppng) neural network models 3 STTS Tradng Model The STTS model s desgned to transform predctons nto actual tradng strateges Its obectve crteron s therefore not RMS predcton error but the rsk adusted net proft of the tradng strategy The model has been shown to provde relatvely consstent profts n smulated out-of-sample hgh frequency tradng over a 6-month perod 9 Bblography [] Buntne WL & Wegend AS (99) Bayesan Back-Propagaton, Complex Systems 5, [] Casasent DP & Smokeln JS (994) eural et Desgn of Macro Gabor Wavelet Flters for Dstorton- Invarant Obect Detecton In Clutter, Optcal Engneerng, Vol 33, o7, pp [3] Debauches I (988) Orthonormal Bases of Compactly Supported Wavelets, Communcatons n Pure and Appled Mathematcs, Vol 6, o 7, pp [4] Fahlman SE (988) Faster Learnng Varatons On Back-Propagaton: An Emprcal Study, Proceedngs Of The 988 Connectonst Models Summer School, 38-5 Morgan Kaufmann [5] Fukunaga K (990), Statstcal Pattern Recognton ( nd Edton) Academc Press [6] Hashem S & Schmeser B (993), Approxmatng a functon and ts dervatves usng MSE-optmal lnear combnatons of traned feed-forward neural networks, Proc world congress on eural etworks, WC-93, I [7] Hecht-elsen R (989), eurocomputng, Addson-Wesley [8] Hnton, GE (987), Learnng translaton nvarant recognton n massvely parallel networks In JW de Bakker, AJ man and PC Treleaven (Eds), Proceedngs PARLE Conference on Parallel Archtectures and Languages Europe, -3, Sprnger-Verlag [9] Krkpatrck S, Gelatt CD & Vecch MP (983) Optmzaton by smulated annealng Scence 0 (4598), [0] MacKay DJC (99) Bayesan Interpolaton eural Comput 4(3), [] MacKay DJC (99) A Practcal Bayesan Framework For Backprop etworks, eural Comput 4(3), [] Meyer Y (995) Wavelets and Operators, Cambrdge Unversty Press [3] Oa E (989), eural etworks, Prncpal Components and Subspaces, Intl Journal On eural Systems,, 6-68

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