ハイブリッドモンテカルロ法に よる実現確率的ボラティリティモデルのベイズ推定

Transcription

1 ハイブリッドモンテカルロ法に よる実現確率的ボラティリティモデルのベイズ推定 Tesuya Takas Hrosma Unversy of Economcs

2 Oulne of resenaon 1 Inroducon Realzed volaly 3 Realzed socasc volaly 4 Bayesan nference 5 Hybrd Mone Carlo 6 Mnmum Norm negraor 7 Smulaon sudy 8 Emrcal sudy 9 GPGPU alcaon 10 Summary

3 Inroducon Sylzed roeres of asse reurns Absence of auocorrelaons Slow decay of auocorrelaon n absolue reurns Fa-aled eavy al dsrbuons Volaly cluserng Leverage effec..

4 Gokrsnan e. al1999 Absence of long auocorrelaons Slow decay of auocorrelaon n absolue reurns

5 Fa-aled eavy al dsrbuons Gokrsnan e. al1999

6 Suden- dsrbuon Tsalls and Aneneodo003

7 volaly cluserng Gokrsnan e al.cond-ma/

8 How can we exlan e sylzed facs? Some of e sylzed facs are exlaned by e mxure of dsrbuons yoess MDH Reurns are descrbed by Gaussan w me-varyng volaly Clark1976 R ~N01 Conssen Absence of auocorrelaons Slow decay of auocorrelaon n absolue reurns Fa-aled eavy al dsrbuons # of nformaon Volume # of ransacons Volaly cluserng

9 Volaly model Deermnsc model GARCH-ye model GARCH EGARCH GJR-GARCH ec GARCH model 1 R 1 arameers Parameer esmaon: easy Model arameers can be deermned by e maxmum lkelood esmaon

10 Socasc model Basc socasc volaly model ln ln 1 ~ N0 Parameer esmaon: dffcul Model arameers are deermned by e Bayesan nference. Te Bayesan nference s erformed by Markov Can Mone Carlo meods Hybrd Mone Carlo meod s used for arameer esmaons of e realzed socasc volalyrsv model Molecular dynamcs smulaons Merools es Imroved negraor Perform Bayesan nference of e RSV model by Hybrd Mone Carlo algorm Invesgae effcency of Hybrd Mone Carlo algorm w e mroved negraor

11 Realzed Volaly Le us assume a e logarmc rce rocess follows a socasc dffuson as d ln dw Andersen Bollerslev 1998 :so volaly W: Sandard Brownan moon T T N RV r s ds Realzed volaly s defned by a sum of squared fnely samled reurns. 1 T * k Inegraed volaly IV for T erod IV N T k k 0 k: samlng erod r ln ln k reurns calculaed usng g-frequency daa

12 Daly reurn R 1 ln 1 ln 1 Realzed volaly N RV r 1 r * k * k

13 Non-radng ours ssue Le us consder daly volaly Usually sock excange markes are no oen for a wole day. Domesc sock rade a e Tokyo sock excange break sar end 09:00 11:00 1:30 break 15:00 break mornng sesson afernoon sesson How o deal w e nraday reurns durng e breaks? RV wou ncludng reurns n e breaks

14 Mcrosrucure nose r r 0 : WN rue nose Observed reurns are also conamnaed by nose Prce dscreeness bd-ask sreads ec. N N N N r r r r RV Nose erms ln ln P P Observed rces are conamnaed by mcrosrucure nose In e resence of nose RV s calculaed as follows N Zou1996

15 Bas correcon Hansen and Lunde 005 nroduced an adjusmen facor Correc RV so a e average of RV maces e varance of e daly reurns c: adjusmen facor c T 1 T R 1 R RV 0 0 RV crv T: radng days Varance of daly reurns Average of orgnal realzed volales Orgnal realzed volaly

16 Realzed Socasc volaly model Idea : daly reurn + realzed volaly Takaas e al.009 Informaon gven from fnancal markes 1 y ~ N01 1 ~ N0 1 Volaly varable ln 3 y Dynamcs of RV u u ~ N0 y ln RV HL facor Bas o RV 0 y ln RV 0 RV crv ln c ln c Model arameers u

17 d T d Y Y f L 1 1 Lkelood funcon of Realzed socasc volaly model T u u u T y e y e Y Y f 1 1/ 1/ 1 / 1/ 1 1 ex /1 ex ex ex T y y y Y T y y y Y 1

18 Bayesan nference Bayes eorem y f y y f y :oseror dsrbuon f y :lkelood funcon :ror dsrbuon f y d f y Probably dsrbuon of θ y f y Parameers are esmaed as execaon values: 1 E [ ] y d Z Esmae by Markov Can Mone Carlo

19 For Socasc volaly model f d d d E[ ] 1 T exln f dd We need T udaes of volaly varables a eac MCMC se. An advanage of e ybrd Mone Carlo algorm: All T volaly varables can be udaed smulaneously. De-correlae Mone Carlo samles of volaly varables

20 Hybrd Mone Carlo f d d d E[ ] 1 T exln f dd A local udae sceme s no effecve For RSV model u Mul-move samler WaanabeOmor004 Basc dea: HMC = molecular dynamcs smulaon + Merools es Inroduce momena 1 T conjugae o 1 T 1 Z ex 1 ln f ddd Z ex H ddd Z: normalzaon consan Defne Hamlonan H ln f T V Knec + oenal

21 H d d H d d / / / H Coose canddaes of by solvng Hamlon s equaons of moon 0 d dh Hamlonan s conserved Solve Hamlon s equaons of moon numercally nd order Leafrog negraor H H H O Hamlonan s No conserved 0 H H H Se sze

22 / elemenary se reea s se / 3: : / / 1: H / H H mn1 ex 1 ex mn H H H P Merools acce/rejec es Acceance deends on se sze H H H O Molecular dynamcs smulaon

23 Effcency of Hybrd Mone Carlo cos 1/ Effcency P P P Trade-off beween e acceance and se sze Effcency funcon akes a maxmum a o For nd order Leafrog negraor Omum acceance 1 P o ex Takas For n- order negraor P o ex 1 n

24 Mnmum Norm Inegraor Hger order negraors can be used. Effcency deends on e model we use. Omelyan e al. 003 found e nd order mnmum norm MN negraor a s more effcen an e leafrog negraor. H ln f T V Leafrog negraor ex H ex MN negraor ex H ex 3 T / exv ext / O Leafrog 3 T exv / ext 1 exv / ext O MN for O Leafrog 3 3 O MN CosMN=CosLF

25 Smulaon Sudy u Arfcal 4000 daa Bayesan nference H H H Hger order error erms

26 Effcency funcon P acc 5 mes effecve Comuaonal cos: CosMN ~ CosLeafrog In oal MN negraor s abou.5 mes effecve an e leafrog negraor.

27 Volaly samlng by MN negraor

28 Resuls u 10 Inu average S.D ACF sor

29 Emrcal Sudy Daa: Panasonc Co Inu daa: Daly reurns + RV1mn

30 Resuls Inu daa: Daly reurns + RV1mn u average S.D y u y ln crv ln RV ln c c:hl facor ln c

31 HL facor

32 GPGPU alcaon Hybrd Mone Carlo algorm can be arallelzed Tycally e number of volaly varables are ousands Volaly varables can be udaed n arallel Seed u e Hybrd Mone Carlo algorm NVIDIA Cuda + GT0

33 Devce code global vod cuda_hmc n nd floa *d_ floa *d_dy floa *d_ { j- volaly n j = blockdm.x*blockidx.x + readidx.x ; d_[j]=d_[j]+d_[j]*d*0.5; d_[j]= d_[j+1] + d_[j-1] } Leafrog negraor

34 Call GPU kernel funcon on Hos Memory on GPU cudamallocvod**&d_ szeoffloa*nd; Transfer daa o GPU memory cudamemcyd szeoffloa*nd cudamemcyhostodevce; Defne blocks and sleds dm3 Dgnd/5611Db5611; Call kernel funcon cuda_hmc<<<dgdb>>>ndd_d_dyd_ ; ransfer daa from GPU o Hos cudamemcy_d_ szeoffloa*nd cudamemcydevcetohos;

35 GPUGT0 vs CPUAMD 3.0GHz # of reads 56 # of blocks B # of daa 56*B GT0 48 Cuda cores Measure comuaonal me of molecular dynamcs smulaons

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37 Gan=TmeCPU/TmeGPU

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39 Summary Bayesan nference of realzed socasc volaly model as been erformed by e Hybrd Mone Carlo meod. We found a e MN negraor s more effecve an e convenonal leafrog negraor. Te Hybrd Mone Carlo meod can decorrelae volaly varables fas enoug. Te arameer ξ of e realzed volaly model exlans e bas smlar o e HL facor. GPGPU can be used o erform e Hybrd Mone Carlo algorm Furer alcaon of e Hybrd Mone Carlo meod for e Realzed socasc volaly model w leverage effec and Te Mulvarae realzed socasc volaly can be ossble.