Transcription

1 Neural Neworks n Fnance Albero Suárez Compuer Scence Dp. Escuela Polécnca Superor Unversdad Auónoma de adrd (Span albero.suarez@uam.es ullayer percepron Inpu layer ( x ( x x (3 ( D x ( w ( w ( w JD Hdden layer(s ( w ( w ( w JK Oupu layer ( y y ( ( K y y ( k J D ( w jk f w j d ( jd x ( d ; θ j ; k,, K, K

2 Fnancal me seres Tme seres of asses are hghly rregular If marke effcency hypohess s correc hey are also unpredcable. Tme seres of asses are non-saonary They are usually ransformed n log-reurns, or, for shor perods of me, n relave reurns Asse reurns exhb devaons from normaly / arkov hypohess Lepokurc: Heavy als Heeroskedasc: Volaly cluserng 3 An example: IBE35 4

3 Daly reurns: IBE35 (5 years 5 Daly-reurns dsrbuon 6

4 7 Falure of normal model: Heavy als 8

5 9 emory effecs (IBE 35

6 Tme seres analyss Consder he me seres Tme seres analyss,, K,, K, T Forecasng Classfcaon Class odellng P d K F(,, ; θ ; F(,,K; θ,,k; θ ( d These problems are closely relaed o each oher: d F(,, K; θ ε P( ε,, K; θ d d ; Tme seres predcon: a Learnng vew Nework model for me-seres predcon Learnng devce p

7 Tasks n me seres analyss Obanng daa: Selecon of arbues: Choose relevan ndcaors Daa collecon Dscree daa: Groupng /averagng n me wndow Connuous daa: Imporance of samplng frequency Preprocessng daa Clean daa : ssng daa, oulers Normalzaon of daa µ Elmnae rends /seasonaly: Handle a-pror nfo explc / Saonary daa. σ ; ; ; log medan q ; ( max max mn 3 mn Paramerc / non-paramerc daa analyss Paramerc Formulae (resrcve hypohess dependen on a se of parameers Fnd parameers by daa-drven opmzaon ranng se Sensvy analyss Uncerany n esmaed parameers Robusness Valdaon of models es se Non-Paramerc Consder a famly of unversal approxmans Fx archecure / parameers by daa-drven opmzaon ranng se Sensvy analyss Robusness Uncerany Inellgbly Valdaon of models es se 4

8 Does NNes use lead o prof? Effcen marke hypohess: All publcly avalable nformaon s refleced n he prce of he asses. An effcen marke reflecs n an nsananeous manner he expecaons over he fuure evoluon of he asse prces. The only rend presen n he me evoluon of he prce of an asse raded freely n an effcen marke s a consan rend conssng of wo erms: a rskless neres rae and a rsk premum. arkes whou frcons should be more effcen. Predcable prof sraeges are deeced and exploed very rapdly. Emprcally suppored n mos sudes Whe, 988 Bu: Lmed nformaon processng capably of he marke. Predcable prof sraeges reman vald provded hey reman undeeced. Asymmerc nformaon / dvergen objecves 5 Use of neural neworks n fnance In order o reman compeve, a marke agen needs o mplemen sophscaed ools o deec rends n me seres of asse prces, neres and exchange raes, ec. Forecas /model of fnancal /economc quanes Economc nformaon: Inflaon, GNP growh, unemploymen rae, ec. Predcon n energy markes: Evoluon of demand, load, prce of elecrcy, ec. Fnancal producs Predcon (prce, radng volume, ec. Desgn of auomac radng sraeges Rsk analyss and managemen Volaly Probably of exreme evens Prcng of dervave producs 6

9 Classcal models n me-seres Consder he me seres,,,, K,, K The seres exhbs randomness. The process s covarance-saonary when: ean s me ndependen T E µ Auocovarance s ndependen of me-ranslaons E ( τ µ ( µ γ τ 7 Auorregressveovng average models Auorregressve model for a me-seres f (, u ; θ Vecors of delayed values: p q p q The sysemac erm f (, u ; θ reflecs rends. The nnovaons u are uncorrelaed nose. axmzaon of he lkelhood funcon yelds esmaes of he model parameers. u m L m u u u Lu m m 8

10 Condonal heeroskedascy odel he me-srucure of he volaly of he nnovaons u σ Z q σ h( u, σ p Z The quanes are assumed o be drv s. Eg. In GARCH(p,q he volaly follows he process σ q u β σ p κ α 9 Nework model AR( / ARCH(, odel φ u u σ Z; Z dvs. σ κ α u N(, Goal LL T 3 πσ ( exp φ σ Σ φ Σ Σ -φ Σ α Σ κ Σ σˆ

11 Nework model AR( / GARCH(, odel φ u u σ Z; Z dvs. σ κ α u β σ N(, Goal LL T 3 πσ ( exp φ σ σ Σ Σ Σ Σ -φ Σ β κ φ α Σ Σ σˆ Nne archecures for me seres Feedforward neworks (neural exensons of AR ullayer percepron. Radal bass funcons. Probablsc neworks. Recurren neworks (neural exenson of ARA Jordan nes Elman Neworks Fully recurren nes xure models xed Densy Neworks xures of expers

12 Auoregressve LP Inpu layer x ( θ Hdden layer(s Oupu layer ( w x x D ( w J ( θ ( w JD x θ ( w ( ( w J J D ( wj f j d w x ( jd x d θ ( j ; Sgmodal (logsc f ( x e x Hyperbolc angen: e f ( x e x x e e x x 3 Auoregressve RBF Inpu layer x x Hdden layer(s Oupu layer φ w x 3 φ φ J w w J x( x D x J j w j φ j D ( x w ; Gaussan kernel φ ( ( j ( x exp x µ j Σ j x µ j 4

13 ARA(p,q LP Inpu layer x x θ θ w AR Hdden layer(s Oupu layer ( w ( w ( w J x( x p delay delay delay xˆ xˆ xˆ q x q _ A w A wjd ( xd xˆ d θ j ; u d q x J f w j j d q p w AR jd x d 5 Jordan Recurren Nework (no self recurren loops Inpu layer x ( w ( w Hdden layer(s Oupu layer ( w x ( w ( w J x x p ( w Jp delay x delay x Self-recurren loop (nformaon sorage devce delay xq 6

14 Fully recurren Nework x x x 3 x D ( w J ( w ( w D ( w JD delay delay delay Hdden layer(s ( w JJ f f f g j( f D d w ( jd ( w ( w x d Oupu layer ( w J x J q l k J j w w ( jlk ( j x g ( j gl ( k θ j 7 xure model s σ ODEL ODEL GATING NETWORK g g g J Σ σ σ s ODEL J 8

15 Gang Nework Gang Nework h h -c a r k k c a b h exp 9 r h J a r- Probables ( ; J j j J J j j g g J,, h h g K Herarchcal mxures Herarchcal mxures µ µ exp exp µ µ µ c a b c a b r k k k r k k k exp c a b r Inpu Vecor of Delayed values 3 ODEL ODEL ODEL 3 3 odel ; odel ; odel µ µ µ µ µ µ µ µ µ exp exp µ µ µ c a b c a b r k k k k k k

16 Some gudelnes for consrucng NNes Daa: Carefully selec arbues: Too many rrelevan or redundan arbues may make learnng relevan paerns dffcul. Include all relevan arbues. Preprocess daa appropraely: Oulers, mssng values. Elmnae rends, seasonal paerns, ec. The learnng process may aemp o model hese broad feaures, mssng (more neresng fner srucure. odel: ore sophscaed models have he poenal of beer performance. However, flexble models are also prone o overfng fnancal me seres. When avalable, nclude a-pror nformaon n he model. Expermen wh dfferen models. Use he resuls of several models smulaneously (e.g. baggng, commees 3 Some gudelnes for consrucng NNes Archecure: Lm complexy of he model (number of hdden layers / number of nodes per layer. Use prunng, regularzaon erms n he cos funcon ha penalze complexy, ec. Consruc several models rangng from smple o more complcaed archecures. Sop when no furher performance mprovemen s deeced. Valuae a range of models wh a valdaon se ha s no been n selecng he model parameers. Oher echnques: Cascade correlaon, genec algorhms, wegh elmnaon E.g., - Inpu layer: D nodes - Sar wh a sngle hdden layer wh N D/, D/±, D/±, N (D - If needed, add anoher layer wh N N /, N /±, N /±, Learnng: On-lne backpropagaon (learnng rae, momenum Bach learnng: Quas-Newon, Conjugae gradens, Specalzed learnng echques: Ieraed Exended Kalman Fler. (* We may need early soppng 3

17 Desgn of radng sraeges wh NNes Objecve: Desgn an auomac radng sysem ha maxmzes prof. Tradonal sraegy Sep : Use Neural Nework o predc prces. Sep : Use he predced prces o desgn radng sraegy ha maxmzes he prof. Beer: Use Neural Nework o drecly desgn radng sraegy. Choey Wegend (* ; oody Wu (* axmze a measure rskadjused performance Eg. Sharpe rao mean(er / sdev(er, where ER s he excess reurn over a benchmark Bengo (* axmze performance of a (dervable sof decson module usng as npu a predcon module. (* These references can be found n Wegend e al IBE 35 daly-reurns dsrbuon Heavy als Heeroskedasc srucure 34

18 xure of Gaussans for -ndependen pdf Emprcal sample odel pdf P x Two seps:,, K N K ( k k k p k N( x; µ, σ Toss a K-sded loaded dce o choose componen. Exrac value from he seleced model. Advanages: Close o he normal world. Accouns for lepokuross of emprcal uncondonal dsrbuons n fnance

19 37 38

20 39 xure of Gaussans Inuon: Implcly marke forecass are made n erms of scenaros. Each of hese scenaros s characerzed by an expeced reurn and a volaly. arkes assgn a dfferen probably o each scenaro. Dynamcal pcure? Drec me aggregaon of he process yelds a normal model (by Cenral Lm Theorem. I s possble o consruc a dsconnuous jump process mananng he mxure form. No realsc. 4

21 xure of ARCH processes xure of ARCH processes xarch (, ( r m g probably wh u θ, φ 4 The model for he resduals s The quanes are assumed o be N(, ( ( ( ( u Z u q α κ σ σ Z xure of GARCH processes xure of GARCH processes xgarch, (, ( ˆ ˆ r m g probably wh u θ φ 4 The model for he resduals s The quanes are assumed o be N(, ( ( ( ( ( u Z u p q σ β α κ σ σ Z

22 AR( / ARCH( for IBE35 The maxmum-lkelhood f of he me-seres IBE35 yelds he model ˆ.9 ˆ σ Z σ ( ˆ.9ˆ The quanes Z are assumed o follow a N(, dsrbuon. 43 Resdual correlaons: ARCH( 44

23 Normaly hypohess: ARCH( Y Quanles Quanles KS Tes. 45 IARCH for IBE35 The mxure model s odel odel ˆ.559ˆ σ Z σ ˆ.38ˆ σ Z σ ( ˆ.559ˆ ( ˆ.38ˆ The probables for he mxure are g g ( ( exp g {.6839(.555 } ( ; 46

24 Resdual correlaons: IARCH 47 Normaly hypohess: xarch( Y Quanles Quanles KS Tes.83 48

25 IARCH odel f 49 AR( / GARCH(, for IBE35 The maxmum-lkelhood f of he me-seres IBE35 yelds he model ˆ.358ˆ σ Z σ σ Z ( ˆ.358ˆ The quanes are assumed o follow a N(, dsrbuon. 5

26 Resdual correlaons: GARCH agnude Auocorrelaons of resduals Auocorrelaons of abs(resduals agnude Delay 5 Normaly hypohess: GARCH(, Y Quanles Quanles KS Tes.56 5

27 Tes Daa Volaly Tme Y Quanles KS Quanles agnude agnude Auocorrelaons of resduals Auocorrelaons of abs(resduals Delay 53 IGARCH for IBE35 The mxure model s odel ˆ σ.55ˆ σ Z ( ˆ σ odel ˆ σ.334ˆ σ.63. Z ( ˆ.334ˆ.85σ The probables for he mxure are g g ( ( exp g {.548 ( ˆ 4.87 } ( ; 54

28 Resdual correlaons: IGARCH agnude Auocorrelaons of resduals Auocorrelaons of abs(resduals agnude Delay 55 Normaly hypohess: IGARCH Y Quanles Quanles KS es.95 56

29 IGARCH odel f Enropy Probables odel odel Volaly Tme 57 Tes Daa Volaly Tme Y Quanles KS Quanles agnude agnude Auocorrelaons of resduals Auocorrelaons of abs(resduals Delay 58

30 Conclusons Auoregressve mxure models can be used o mprove he modelng of fnancal me seres odel s pecewse-lnear. emory effecs. Exreme evens. Dffcules: Parsmony? (Convex lnear combnaon of smple base models Consraned opmzaon of LL funcon. E algorhm s no praccal. 59