4 ( ) NATURE SCIENCE [Battiston 16] 2008 ( ) 5 JPX [ 13] [ 15a, 15b] [ 15,Mizuta 16c] [ 15a, 15b] δt (δt =1) (δt > 1) 4 [ 09, 12] 5 [LeBaron 06,Chen 1

ਓ޻ࢢ৔γϛϡϨʔγϣϯΛ༻͍ͨόονΦʔΫγϣϯͷ෼ੳ

ਫా ޹৴

∗1 Takanobu Mizuta

࿨ઘ ܿ

∗2 Kiyoshi Izumi ∗1

_{εύʔΫεɾΞηοτɾϚωδϝϯτגࣜձࣾ}

SPARX Asset Management Co., Ltd.

∗2

_{౦ژେֶେֶӃ ޻ֶܥݚڀՊ}

School of Engineering, The University of Tokyo ۙ೥ɼऔҾࢢ৔ಉ࢜ͷڝ૪΍౤ࢿՈͷཁ๬ͳͲʹΑΓɼऔҾࢢ৔ͷγεςϜͷߴ଎Խ͕ਐΜͩɽऔҾγεςϜͷߴ଎ ԽʹΑΓɼྲྀಈੑΛڙڅͯ͠རӹΛಘΔϚʔέοτϝʔΧʔઓུͷ஫จྔ͕ҎલΑΓ૿͑ྲྀಈੑ͕޲্ͨ͠ͱ͍͏ධՁ ͕͋ΔҰํɼ౤ࢿՈಉ࢜ͷऔҾͷεϐʔυڝ૪Λট͖ɼͦͷڝ૪ͷͨΊʹඅ΍͞ΕͨγεςϜίετ͸ଞͷ౤ࢿՈʹస Խ͞ΕΔͱ͍͏൷൑͕͋Δɽߴ଎ͳऔҾͷ༏ҐੑΛແޮʹ͢ΔऔҾํࣜͱͯ͠όονΦʔΫγϣϯํ͕ࣜఏҊ͞Ε͍ͯ Δ͕ɼྲྀಈੑΛڙڅ͢ΔϚʔέοτϝʔΧʔઓུ͸ଛӹͷϦεΫ͕ߴ͘ͳΔͨΊܧଓ͢Δ͜ͱ͕೉͘͠ͳΓɼྲྀಈੑͷ ڙڅ͕ݮͬͯ͠·͍ɼΉ͠Ζ౤ࢿՈͷऔҾίετ্͕ঢ͢ΔՄೳੑ͕͋Δͱ͍͏൷൑΋͋ΔɽຊݚڀͰ͸ɼβϥόํࣜ ͱόονΦʔΫγϣϯํࣜΛൺֱՄೳͳਓ޻ࢢ৔γϛϡϨʔγϣϯΛ༻͍ͯɼϚʔέοτϝʔΧʔઓུͷଛӹͷϦεΫ Λ෼ੳ͢Δ͜ͱʹΑΓͦͷଘଓՄೳੑΛٞ࿦͠ɼϚʔέοτϝʔΧʔઓུ͕όονΦʔΫγϣϯํࣜʹ͓͍ͯ΋ྲྀಈੑ Λڙڅ͠ଓ͚Δ͜ͱ͕Ͱ͖Δ͔Ͳ͏͔Λௐ΂ͨɽͦͷ݁Ռɼ൘دִ͕ͤؒେ͖͘ͳΔͱɼऔҾ੒ཱ཰͕ݮগ͠ྲྀಈੑڙ څ͕ݮগ͢ΔՄೳੑ͕ࣔ͞Εͨɽ͞Βʹɼ൘دִ͕ͤؒେ͖͘ͳΔͱɼΦʔόʔφΠτͷϙδγϣϯΛθϩʹ͢Δ͜ͱ ͕೉্͍͠ʹϙδγϣϯ΋େ͖͘Ձ֨มಈϦεΫ͕େ͖͘ͳΓɼϦεΫΛ͓͑ͯ͞ϚʔέοτϝʔΧʔઓུΛܧଓ͢Δ ͜ͱ͕ࠔ೉ʹͳΔՄೳੑ͕ࣔ͞Εͨɽ͞Βʹɼβϥόํࣜͷͱ͖ͷΈɼϦεΫʹݟ߹ͬͨऩӹΛಘΔՄೳੑ͕͋ΔՄೳ ੑ͕ࣔ͞ΕɼόονΦʔΫγϣϯํࣜͷͱ͖͸ɼগͳ͘ͱ΋βϥόํࣜͰ͸ػೳͨ͠ϚʔέοτϝʔΧʔઓུͰ͸ɼϦ εΫʹݟ߹ͬͨऩӹΛ͋͛Δͷ͸೉͘͠ͳΔՄೳੑ͕ࣔ͞Εͨɽ͜ΕΒͷࣔࠦ͸ɼβϥόํࣜͰ͸ྲྀಈੑΛڙڅ͍ͯ͠ ͨϚʔέοτϝʔΧʔઓུ͕ɼόονΦʔΫγϣϯํࣜʹͳΔͱͦͷڙڅΛҡ࣋Ͱ͖ͳ͘ͳΓɼͦΕΒ͕ఫୀ͢Δ͜ͱ ʹΑΓɼྲྀಈੑ͕௿Լ͢ΔՄೳੑΛ҉͍ࣔͯ͠Δͱ΋ߟ͑Δ͜ͱ͕Ͱ͖Δɽ

1.

͸͡Ίʹ

2000೥୅͔Β2010೥୅ॳ಄ʹ͔͚ͯɼऔҾࢢ৔ಉ࢜ͷڝ ૪΍౤ࢿՈͷཁ๬ͳͲʹΑΓɼऔҾࢢ৔ͷγεςϜͷߴ଎Խ͕ ਐΜͩ∗1ɽऔҾγεςϜͷߴ଎ԽʹΑΓɼചΓങ͍ͱ΋ʹ஫ จΛৗʹఏࣔͯ͠ଞͷ౤ࢿՈʹऔҾػձ(ྲྀಈੑ)Λఏڙ͢Δ ͜ͱͰఏࣔͨ͠ചΓͱങ͍ͷՁ֨ࠩ(஫จεϓϨου)෼ͷར ӹΛಘΔϚʔέοτϝʔΧʔͱ͍͏౤ࢿઓུͷ஫จྔ͕ҎલΑ Γ૿͑ɼྲྀಈੑ͕޲্ͨ͠ͱ͍͏ධՁ͕͋ΔɽҰํͰɼա౓ͳ ߴ଎Խ͸औҾࢢ৔ͷγεςϜίετΛ૿େͤ͞Δ͚ͩͰͳ͘ɼ ౤ࢿՈಉ࢜ͷऔҾͷεϐʔυڝ૪Λট͖ɼͦͷڝ૪ͷͨΊʹඅ ΍͞ΕͨγεςϜίετ͸औҾίετͱͯ͠ଞͷ౤ࢿՈʹసԽ ͞Ε͍ͯΔͱ͍͏൷൑͕͋Δ[Farmer 12, Budish 15]ɽ [Budish 15]͸ɼͦͷΑ͏ͳ౤ࢿՈͷऔҾͷߴ଎Խڝ૪Λऴ ΘΒͤΔํ๏ͱͯ͠ɼߴ଎ͳऔҾͷ༏ҐੑΛແޮʹ͢ΔऔҾํ ࣜͱͯ͠όονΦʔΫγϣϯํࣜΛఏҊͨ͠ɽݱࡏଟ͘ͷऔҾ ࢢ৔ͰɼചΓखͱങ͍खͷ૒ํ͕Ձ֨Λఏࣔ͠ചΓखͱങ͍ खͷఏࣔՁ͕֨߹க͢ΔͱͦͷՁ֨Ͱ௚ͪʹऔҾ͕੒ཱ͢Δɼ βϥόํࣜ(࿈ଓμϒϧΦʔΫγϣϯํࣜ)͕࠾༻͞Ε͍ͯΔɽ ҰํɼόονΦʔΫγϣϯํࣜͰ͸ɼ͋Δఔ౓ͷظؒɼྫ͑͹ ਺ඵͱ͍ͬͨظؒɼ஫จΛ੒ཱͤͣ͞஫จͷड෇ͷΈߦ͍ɼͦ ͷظ͕ؒऴΘΔͱू·ͬͨ஫จΛ͖ͭ߹ΘͤͯऔҾՁ֨ͷܾఆ Λߦ͏ɽ͜͏͢Δ͜ͱʹΑΓɼߴ଎ͳऔҾͷ༏Ґੑ͕ͳ͘ͳΔ ͨΊɼ౤ࢿՈಉ࢜ͷεϐʔυڝ૪Λ͠ͳͯ͘Α͘ͳΔͱ͍͏ओ ுͰ͋Δɽ [Budish 15]͸͞Βʹɼ࣮ূ෼ੳͱ؆୯ͳϞσϧʹΑΔ෼ੳ ʹΑΓɼβϥόํࣜͰ͸ඇৗʹ୹͍ظؒͷΈࡋఆػձ͕ଘࡏ͢ Δ͕ɼόονΦʔΫγϣϯํࣜͰ͸ͦΕ͕ͳ͘ͳΓɼߴ଎Խڝ ࿈བྷઌ:ਫా ޹৴ɼ[email protected] http://www.mizutatakanobu.com/jindex.htm †_{ຊߘ͸JPXϫʔΩϯάϖʔύʔ[ਫా16a]Λ࠶ߏ੒͠} ͨ΋ͷͰ͋Δɽ ∗1 ༏ΕͨϨϏϡʔͱͯ͠ [ਗ਼ਫ 13] ͕͋Δɽ·ͨɼऔҾࢢ৔ؒͷڝ ૪ʹ͓͚Δߴ଎Խͷॏཁੑʹؔͯ͠ͷղઆʹ [ਫా 12] ͕͋Δɽ ૪Λ͠ͳͯ͘͢Ήͱओுͨ͠ɽ·ͨ[Fricke 15]͸؆୯ͳϞσ ϧΛ༻͍ͯɼϘϥςΟϦςΟ(Ձ֨มಈͷେ͖͞)͕΋ͬͱ΋ খ͘͞ͳΔͳͲͷ৚͔݅ΒɼόονΦʔΫγϣϯํࣜͷ࠷దͳ ஫จड෇ظؒΛٞ࿦ͨ͠ɽ͞Βʹ[Manahov 16]͸؆୯ͳϞσ ϧΛ༻͍࣮ͨূ෼ੳ͔ΒϨΠςϯγʔΞʔϏτϥʔδ∗2͕Մ ೳͳ࣌ؒεέʔϧΛࢉग़͠ɼͦΕΛ๷͙؍఺͔ΒόονΦʔΫ γϣϯํࣜͷ࠷దͳ஫จड෇ظؒΛٞ࿦ͨ͠ɽ ҰํɼόονΦʔΫγϣϯํࣜʹ͸൷൑΋ଟ͍ɽ[େ෿ 14] ͸ɼྲྀಈੑΛڙڅͯ͠རӹΛಘΔϚʔέοτϝʔΧʔઓུʹ ͓͍ͯɼόονΦʔΫγϣϯํࣜͩͱ͍͘ΒͰऔҾ͕੒ཱ͢Δ ͔ݟࠐΈͮΒ͘ͳΔͨΊ͜ͷઓུͷଛӹͷϦεΫ͕ߴ͘ͳΓɼ ܧଓ͢Δ͜ͱ͕೉͘͠ͳΓɼྲྀಈੑͷڙڅ͕ݮͬͯ͠·͏Մೳ ੑΛࢦఠ͍ͯ͠Δɽྲྀಈੑͷڙڅ͕ݮΔͱɼ࠷ྑͷങ͍஫จՁ ֨ͱചΓ஫จՁ֨ͷ͕ࠩେ͖͘ͳΔͳͲͯ͠ɼΉ͠Ζଞͷ౤ࢿ ՈͷऔҾίετ্͕ঢ͢Δ͜ͱ΍ɼͦ΋ͦ΋औҾͰ͖Δػձ͕ ݮͬͯ͠·͏͜ͱ͕ߟ͑ΒΕΔ∗3ɽ ࣮ࡍɼ[Budish 15, Fricke 15]ͷϞσϧͰ͸ɼྲྀಈੑΛڙڅ ͢Δ౤ࢿՈͷଛӹ͸ߟྀ͓ͯ͠Βͣɼ[େ෿14]͕ࢦఠ͢Δྲྀ ಈੑΛڙڅ͢Δ౤ࢿՈͷଘଓՄೳੑΛऔΓѻ͍ͬͯͳ͍ɽ ྲྀಈੑΛڙڅ͢Δ౤ࢿՈͷଘଓՄೳੑΛऔΓѻ࣮ͬͨূݚ ڀͱͯ͠[Bellia 15]͕͋Δɽ[Bellia 15]͸ࢀߟʹͳΔࣄྫͱ ͯ͠ɼ౦ژূ݊औҾॴʹ͓͍ͯɼऔҾ։࢝ɾऴྃ࣌ʹߦΘΕΔ ൘دͤํࣜͷͱ͖ͱͦͷؒʹߦΘΕΔβϥόํࣜͷͱ͖ͷ஫ จσʔλΛൺֱ͢Δ࣮ূ෼ੳΛߦ͍ɼϚʔέοτϝʔΧʔઓུ ∗2 େޱ஫จΛෳ਺ͷࢢ৔ʹ෼ׂͯ͠ग़ͨ͠ͱ͖ɼ֤஫จ͕֤ࢢ৔ʹ ౸ୡ͢Δ·Ͱʹ͔͔ΔΘ͔ͣͳ࣌ؒࠩΛར༻ͯ͠ɼଞͷ౤ࢿՈ͕ࡋ ఆऔҾΛߦ͏͜ͱɽಛʹถࠃͰ͸ɼଞͷࢢ৔ͷํ͕༗རͳՁ֨Ͱऔ ҾͰ͖Δ஫จΛड͚ͨࢢ৔͸ɼͦͷ༗རͳՁ֨ͰऔҾͰ͖Δࢢ৔ʹ ͦͷ஫จΛճૹ͢Δٛ຿͕͋Γɼ͜ͷճૹͷ଎౓͕஗͍ͨΊʹࡋఆ ػձ͕ൃੜ͢Δ͜ͱ͕͋Γɼͦͷੋඇ͕େ͖ͳٞ࿦ͱͳ͍ͬͯΔɽ ೔ຊͰ͸ถࠃ΄Ͳࢢ৔͸෼அ͞Ε͓ͯΒͣ͜ͷΑ͏ͳճૹٛ຿΋ͳ ͍ͨΊɼ͜ͷΑ͏ͳࡋఆػձ͸΄ͱΜͲͳ͍ͱߟ͑ΒΕ͍ͯΔɽৄ ͘͠͸ [େ෿ 14]ɽ ∗3 [େ෿ 14] ͸͞ΒʹɼόονΦʔΫγϣϯํࣜͰ͸஫จड෇ظؒ ͷ࠷ޙͷํʹ஫จ͕ूதͯ͠ैདྷҎ্ͷߴ଎Խڝ૪ʹͳΔՄೳੑ΋ ࢦఠ͍ͯ͠Δɽ

Λߦ͍ͬͯΔͱߟ͑ΒΕΔ౤ࢿՈ͸൘دͤํࣜͷͱ͖ΑΓ΋β ϥόํࣜͷͱ͖ͷ΄͏͕औҾ΁ͷࢀՃ཰͕ߴ͍͜ͱΛࣔͨ͠ɽ ͜ͷઓུ͸औҾͷ੒ཱ͕ଈ࠲ʹߦΘΕΔ͜ͱ͕ॏཁͰ͋ΓɼԾ ʹόονΦʔΫγϣϯํࣜʹͳͬͨ৔߹ɼྲྀಈੑΛڙڅ͢Δऔ ҾࢀՃ͕ݮΔՄೳੑΛࢦఠ͍ͯ͠Δɽ βϥόํ͔ࣜΒόονΦʔΫγϣϯํࣜ΁ͷมߋͱ͍͏ࣄྫ ͸ͳ͍ͨΊ࣮ূݚڀ͸ෆՄೳͰ͋Δɽ·ͨɼઌʹड़΂ͨΑ͏ͳ ࢀߟʹͳΔࣄྫ͸͋Δ΋ͷͷɼՁ֨ܗ੒΍ྲྀಈੑʹ͸͞·͟· ͳཁҼ͕ෳࡶʹؔΘ͍ͬͯΔͨΊɼ࣮ূݚڀͰ͸੍౓ͷҧ͍ͷ ޮՌ͚ͩΛऔΓग़͢͜ͱ͸ࠔ೉Ͱ͋Δɽ ͜ͷΑ͏ͳ࣮ࣾձͰಋೖ͞Εͨ͜ͱ͕ͳ͍γεςϜΛݕূ ͢Δํ๏ͱͯ͠ɼίϯϐϡʔλ্ͰԾ૝తʹͦͷঢ়گΛ࡞Γग़ ͠ݕূ͢ΔɼࣾձγϛϡϨʔγϣϯͱ͍͏ख๏͕͋Δɽࣾձγ ϛϡϨʔγϣϯ͸ɼྫ͑͹ɼࣗಈंಓͷ੔උ͕ަ௨ौ଺΁༩͑ ΔӨڹ෼ੳ΍ɼςϩ΍Րࡂɼ఻છප͕ൃੜͨ͠৔߹ͷආ೉ͷํ ๏΍͋Δ΂͖ରࡦͷ෼ੳͳͲͰɼେ͖ͳ੒ՌΛ͍͋͛ͯΔ∗4ɽ ۚ༥ࢢ৔ʹؔͯ͠ͷࣾձγϛϡϨʔγϣϯ͸ɼΤʔδΣϯ τϕʔευϞσϧͷҰछͰ͋Δਓ޻ࢢ৔ϞσϧΛ༻͍ͯߦΘ ΕΔɽਓ޻ࢢ৔ϞσϧΛ༻͍ͨγϛϡϨʔγϣϯ(ਓ޻ࢢ৔γ ϛϡϨʔγϣϯ)Λ༻͍Ε͹ɼ͜Ε·Ͱʹಋೖ͞Εͨ͜ͱ͕ͳ ͍ۚ༥ࢢ৔ͷن੍ɾ੍౓΋ٞ࿦͢Δ͜ͱ͕Ͱ͖Δ͏͑ɼͦͷ७ ਮͳӨڹΛநग़Ͱ͖Δɽ͜Ε͕ਓ޻ࢢ৔γϛϡϨʔγϣϯݚڀ ͷڧΈͰ͋Δɽ ۙ೥ɼֶज़քͷΈͳΒͣۚ༥ͷن੍౰ہ΍औҾॴؔ܎ऀ΋ɼ ۚ༥ࢢ৔ͷن੍΍औҾॴͷ੍౓ͳͲΛ෼ੳ͢Δਓ޻ࢢ৔γϛϡ ϨʔγϣϯʹڵຯΛࣔ࢝͠Ί͍ͯΔɽ࣮ࡍɼNATUREࢽͱฒ ΜͰ࠷΋ݖҖ͕͋Δֶज़ࡶࢽͰ͋ΔSCIENCEࢽʹܝࡌ͞Ε ͨ[Battiston 16]Ͱ͸ɼ“2008೥ͷۚ༥ةػҎ߱ɼܦࡁ΍ۚ༥ ࢢ৔Λཧղ͢ΔͨΊʹɼωοτϫʔΫཧ࿦΍(ਓ޻ࢢ৔Ϟσϧ ΛؚΉ)ΤʔδΣϯτϕʔευϞσϧͱ͍ͬͨෳࡶܥཧ࿦Λ༻ ͍ͨख๏ʹؔ৺͕ू·͖͍ͬͯͯΔ”ͱड़΂ɼ͞Βʹ“ۚ༥ͷ ෳࡶܥʹજΉਖ਼ͷϑΟʔυόοΫݱ৅Λऑۚ͘͠༥γεςϜͷ ҆ఆԽͤ͞ΔΑ͏ͳɼ੓ࡦ΍ن੍͸ͲͷΑ͏ͳ΋ͷ͔ͷ஌ݟΛ ಘΒΕΔɽ”ͱड़΂͍ͯΔɽ ͦͯ͠ɼଟ͘ͷਓ޻ࢢ৔γϛϡϨʔγϣϯݚڀ͕৽͍ۚ͠༥ ࢢ৔ͷن੍΍੍౓ɼ৽͍͠ํࣜͷࢢ৔Λ෼ੳ͠ɼͲͷΑ͏ͳن ੍΍੍౓͕ྑ͍͔ͱ͍͏ٞ࿦ʹߩݙͨ͠∗5ɽ JPXϫʔΩϯάϖʔύʔʹ͓͍ͯ΋ਓ޻ࢢ৔γϛϡϨʔγϣ ϯΛ༻͍ͨۚ༥ࢢ৔ͷ੍౓มߋͷӨڹΛ෼ੳͨ͠΋ͷ͕͍ͭ͘ ͔͋Δɽ[ਫా13]͸ݺͼ஋ͷࠁΈͷมߋ͕Ձ֨ܗ੒ʹ༩͑Δ ӨڹΛɼ[૲ా15a,૲ా15b]͸ϚʔέοτϝʔΧʔͷ༗ແ͕ ྲྀಈੑʹ༩͑ΔӨڹΛɼ[ਫా15, Mizuta 16c]͸औҾॴγες Ϝͷߴ଎Խ͕ࢢ৔ޮ཰ੑʹ༩͑ΔӨڹΛௐ΂ͨɽ ͔͠͠ͳ͕Βɼਓ޻ࢢ৔γϛϡϨʔγϣϯΛ༻͍ͯɼόον ΦʔΫγϣϯํࣜͷࢢ৔ʹ͓͍ͯϚʔέοτϝʔΧʔઓུͷଛ ӹͷϦεΫΛ෼ੳ͠ɼͦͷଘଓՄೳੑΛٞ࿦ͨ͠ݚڀ͸ͳ͍ɽ ͦ͜ͰຊݚڀͰ͸ɼ[૲ా15a,૲ా15b]ͷਓ޻ࢢ৔Ϟσϧ Λϕʔεʹɼ൘دִͤؒδtͱ͍͏ύϥϝʔλʔΛ৽ͨʹಋೖ ͢Δ͜ͱʹΑΓɼβϥόํࣜ(δt = 1)ͱόονΦʔΫγϣϯ ํࣜ(δt > 1)Λ࿈ଓతʹมߋͰ͖ΔՁܾ֨ఆϝΧχζϜΛ࣮ ૷͠ɼϚʔέοτϝʔΧʔઓུͷଛӹͷϦεΫΛ෼ੳ͢Δ͜ͱ ʹΑΓͦͷଘଓՄೳੑΛٞ࿦͠ɼϚʔέοτϝʔΧʔઓུ͕ྲྀ ಈੑΛڙڅ͠ଓ͚Δ͜ͱ͕Ͱ͖Δ͔Ͳ͏͔Λௐ΂Δɽ ∗4 ྫ͑͹ɼ[ग़ޱ 09, ࣮ੈ 12] ͳͲ͕ৄ͍͠ɽ

∗5 ༏ΕͨϨϏϡʔͱͯ͠ɼ[LeBaron 06,Chen 12,࿨ઘ 12,Cristelli 14, Mizuta 16b]ɽ·ͨɼਓ޻ࢢ৔γϛϡϨʔγϣϯͷ௕ॴ͓Αͼݶ քʹ͍ͭͯ͸ɼຊߘͷ෇࿥ “Ϟσϧߏஙͷجຊཧ೦” ΋ࢀরɽ

2.

ਓ޻ࢢ৔Ϟσϧ

[Chiarella 02]Ͱ͸ɼγϯϓϧͰ͋Γͳ͕Βɼ࣮ূ෼ੳͰಘ ΒΕͨ௕ظؒʹଘࡏ͢ΔՁ֨มಈͷ౷ܭతੑ࣭(stylized fact) Λ࠶ݱͰ͖ΔΤʔδΣϯτϞσϧͷߏஙʹ੒ޭ͍ͯ͠Δɽ[ਫ ా13]Ͱ͸ɼ[Chiarella 02]ͷϞσϧΛϕʔεʹϞσϧΛߏங ͠ɼ[Chiarella 02]ͷϞσϧͰ͸࠶ݱ͞Ε͍ͯͳ͔ͬͨ໿ఆ݅ ਺΍Ωϟϯηϧ཰ɼ1ςΟοΫ͝ͱͷಅམ཰ͷඪ४ภࠩͳͲɼ ߴස౓ͳ࣌ؒεέʔϧͰͷੑ࣭(ϚʔέοτɾϚΠΫϩɾετ ϥΫνϟʔ)΋࠶ݱͨ͠ɽ[૲ా15a,૲ా15b]Ͱ͸ɼ[ਫా13] ͷϞσϧʹɼ[Nakajima 04]ΛϕʔεʹϞσϧԽͨ͠Ϛʔέο τϝʔΧʔઓུΛߦ͏ΤʔδΣϯτΛ௥Ճͨ͠ɽ ຊݚڀͰ͸[૲ా15a,૲ా15b]ͷਓ޻ࢢ৔ϞσϧΛϕʔε ʹɼ൘دִͤؒδtͱ͍͏ύϥϝʔλʔΛ৽ͨʹಋೖ͢Δ͜ͱʹ ΑΓɼβϥόํࣜ(δt = 1)ͱόονΦʔΫγϣϯํࣜ(δt > 1) Λ࿈ଓతʹมߋͰ͖ΔՁܾ֨ఆϝΧχζϜΛ࣮૷ͨ͠ɽ ຊݚ ڀͷ໨తʹ͸Ϟσϧ͕γϯϓϧͰ͋Δ͜ͱ͸ͱͯ΋ॏཁͰ͋ ΔɽϞσϧߏஙͷجຊཧ೦͸ຊߘͷ෇࿥“Ϟσϧߏஙͷجຊཧ ೦”ʹͯઆ໌ͨ͠ɽ

2.1

Ձܾ֨ఆϝΧχζϜ

ຊϞσϧͰ͸ɼ൘دִͤؒδtΛಋೖ͢Δ͜ͱʹΑΓɼβϥ όํࣜ(δt = 1)ͱόονΦʔΫγϣϯํࣜ(δt > 1)Λ࿈ଓత ʹมߋͰ͖ΔՁܾ֨ఆϝΧχζϜΛߏஙͨ͠ɽ βϥόํࣜ͸ɼചΓखͱങ͍खͷ૒ํ͕Ձ֨Λఏࣔ͠ɼചΓ खͱങ͍खͷఏࣔՁ͕֨߹க͢ΔͱͦͷՁ֨Ͱ௚ͪʹऔҾ͕੒ ཱ͢ΔํࣜͰ͋ΔɽόονΦʔΫγϣϯํࣜͰ͸ɼ͋Δఔ౓ͷ ظؒ஫จΛ੒ཱͤͣ͞஫จͷड෇ͷΈߦ͏ɽͦͷظ͕ؒऴΘΔ ͱɼങ͍ख͸ߴ͍஫จՁ֨ͷ஫จ͔ΒചΓख͸͍҆஫จՁ֨ͷ ஫จ͔ΒॱʹऔҾΛ੒ཱ͍͖ͤͯ͞ɼ෇͖߹ΘͤΔ஫จ͕ͳ͘ ͳΔ·Ͱ܁Γฦͨ͠ͱ͜ΖͰऔҾՁ֨ͷܾఆΛߦ͏ɽ͜ͷ஫จ ͷ෇͚߹ͤͱऔҾՁ֨ͷܾఆΛߦ͏࡞ۀΛ൘دͤͱΑͿɽ ຊݚڀͰ͸ϊʔϚϧΤʔδΣϯτ(NA)͕஫จΛग़͢͝ͱʹ ࣌ࠁt͕1૿͑Δɽ൘دͤͷִ࣌ؒؒΛδtͱ͠ɼδt = 1ͷͱ ͖ɼ஫จ͕ೖΔͨͼʹ൘د͕ͤߦΘΕͯβϥόํࣜͱҰக͢Δ Α͏ʹϞσϧԽͨ͠ɽ࣌ࠁtʹ൘د͕ͤߦΘΕͨͱ͖ɼ࢒ͬͨ ஫จͷ஥஋(࠷΋ߴ͍ങ͍஫จͷՁ֨ͱ࠷΋͍҆ചΓ஫จͷՁ ֨ͷฏۉ)ΛऔҾՁ֨Pt_{ͱ͢Δɽ·ͨɼ൘د͕ͤߦΘΕͳ͍} ࣌ࠁtʹ͓͍ͯ΋ɼԾʹͦͷ࣌ࠁʹ൘دͤΛͨ͠৔߹ʹܾఆ͞ ΕΔऔҾՁ֨(औҾݟࠐΈՁ֨)ΛٻΊͦΕΛPtͱ͢Δɽ͜ ΕʹΑΓɼδt͕͍͔ͳΔ஋Ͱ͋ͬͯ΋ɼ͢΂ͯͷ࣌ࠁtʹ͓͍ ͯɼऔҾՁ֨(औҾݟࠐΈՁ֨)Pt͕࿈ଓతʹࢉग़͞ΕΔɽ ਤ1ʹຊݚڀͷՁܾ֨ఆϝΧχζϜͷྫΛࣔ͢ɽਤ1ͷ্ ͷஈ͸δt = 1(βϥόํࣜ)ͷ৔߹ɼԼͷஈ͸δt = 4(όον ΦʔΫγϣϯํࣜ)Ͱ͋Γɼࠨ͔Β࣌ࠁt = 0, 1, 2, 3, 4ͷͱ͖ ͷ஫จঢ়گΛ͍ࣔͯ͠Δɽt = 0Ͱ͸ɼδt = 4Ͱ΋൘د͕ͤى ͖ͨ௚ޙͰ͋Γɼಉ͡஫จঢ়گͰ͋ͬͨͱ͢Δɽt = 1ʹ͓͍ ͯɼՁ֨99ͷചΓ஫จ(৽ن஫จ)͕͋ͬͨɽδt = 1Ͱ͸ɼ͜ ͷ৽ن஫จ͸Ձ֨99ͷങ͍஫จͱऔҾ͕੒ཱ͜͠ͷങ͍஫จ ͸ফ͑ΔɽҰํɼδt = 4Ͱ͸൘دͤΛߦ͏࣌ࠁͰ͸ͳ͍ͨΊɼ ৽ن஫จΛ࢒͢ɽಉ༷ʹɼt = 2, 3ʹ͓͍ͯɼ৽ن஫จͱͯ͠ɼ Ձ֨100ͷങ͍ɼՁ֨101ͷങ͍ͷ஫จ͕དྷΔ͕ɼδt = 1ͷ ৔߹͸͍ͣΕ΋औҾ͕੒ཱ͠ɼδt = 4ͷ৔߹͸൘دͤΛߦ͏ ࣌ࠁͰ͸ͳ͍ͨΊɼ৽ن஫จΛ࢒͢ɽt = 4ʹ͓͍ͯɼ৽ن஫ จͱͯ͠Ձ֨98ͷചΓ஫จ͕དྷΔɽδt = 1ͷ৔߹͸ɼ΍͸Γ औҾ͕੒ཱ͢Δɽδt = 4ʹ͓͍ͯ͸ɼt = 4͸൘دͤΛߦ͏࣌ ࠁͷͨΊɼങ͍ख͸ߴ͍஫จՁ֨ͷ஫จ͔ΒചΓख͸͍҆஫จ Ձ֨ͷ஫จ͔ΒॱʹऔҾΛ੒ཱ͍͖ͤͯ͞ɼ෇͖߹ΘͤΔ஫จ ͕ͳ͘ͳΔ·Ͱ܁Γฦ͢ɽͦͷ݁ՌɼՁ֨101ͷങ͍ͱՁ֨

኎䜚 㻥㻥 ㈙䛔 㻝㻜㻜 ㈙䛔 㻝㻜㻝 ኎䜚 㻥㻤 ᫬้ 㼠㻩㻜 㼠㻩㻝 㼠㻩㻞 㼠㻩㻟 㼠㻩㻠 ኎䜚 ౯᱁ ㈙䛔 ኎䜚 ౯᱁ ㈙䛔 ኎䜚 ౯᱁ ㈙䛔 ኎䜚 ౯᱁ ㈙䛔 ኎䜚 ౯᱁ ㈙䛔 䝄䝷䝞᪉ᘧ 㻝 㻝㻜㻝 㻝 㻝㻜㻝 㻝 㻝㻜㻝 㻝 㻝㻜㻝 㻝 㻝㻜㻝 䃓㼠㻩㻝 㻝 㻝㻜㻜 㻝 㻝㻜㻜 㻝 㻝㻜㻜 㻝 㻝㻜㻜 㻝㻜㻜 㻥㻥 㻝 㻝 㻥㻥 㻝 㻥㻥 㻥㻥 㻥㻥 㻥㻤 㻝 㻥㻤 㻝 㻥㻤 㻝 㻥㻤 㻝 㻝 㻥㻤 㻝 ኎䜚 ౯᱁ ㈙䛔 ኎䜚 ౯᱁ ㈙䛔 ኎䜚 ౯᱁ ㈙䛔 ኎䜚 ౯᱁ ㈙䛔 ኎䜚 ౯᱁ ㈙䛔 㻝 㻝㻜㻝 㻝 㻝㻜㻝 㻝 㻝㻜㻝 㻝 㻝㻜㻝 㻝㻝 㻝 㻝㻜㻝 㻝 㻝 㻝㻜㻜 㻝 㻝㻜㻜 㻝 㻝㻜㻜 㻝㻝 㻝 㻝㻜㻜 㻝 㻝 㻝㻜㻜 㻝 䃓㼠㻩㻠 㻥㻥 㻝 㻝㻝 㻥㻥 㻝 㻝 㻥㻥 㻝 㻝 㻥㻥 㻝 㻝 㻥㻥 㻝 㻥㻤 㻝 㻥㻤 㻝 㻥㻤 㻝 㻥㻤 㻝 㻝㻝 㻥㻤 㻝 ≉ᐃ䛾᫬้䛜 ᮶䜛䛸ྲྀᘬᡂ❧ 䝞䝑䝏䜸䞊䜽 䝅䝵䞁᪉ᘧ ྲྀᘬᡂ❧ 䛥䛫䛺䛔 ᪂つὀᩥ ༶ᗙ䛻 ྲྀᘬᡂ❧ ༶ᗙ䛻 ྲྀᘬᡂ❧ ྲྀᘬᡂ❧ 䛥䛫䛺䛔 ྲྀᘬᡂ❧ 䛥䛫䛺䛔 ༶ᗙ䛻 ྲྀᘬᡂ❧ ༶ᗙ䛻 ྲྀᘬᡂ❧ ਤ1: Ձܾ֨ఆϝΧχζϜͷྫ

኎䜚㻌

౯᱁㻌

㈙䛔㻌

㻝㻜㻜㻝㻝㻌

㻝㻌㻝㻜㻜㻝㻜㻌

㻝㻜㻜㻜㻥㻌

㻝㻜㻜㻜㻤㻌

㻝㻜㻜㻜㻣㻌

㻝㻜㻜㻜㻢㻌

㻝㻜㻜㻜㻡㻌

㻝㻜㻜㻜㻠㻌㻌㻌㻌

㻝㻜㻜㻜㻟㻌

㻝㻜㻜㻜㻞㻌㻝㻌

㻝㻜㻜㻜㻝㻌

㻝㻜㻜㻜㻜㻌

኎䜚㻌

౯᱁㻌

㈙䛔㻌

㻝㻌

㻌 㻝㻜㻜㻝㻝㻌

㻝㻌㻌 㻝㻜㻜㻝㻜㻌

㻝㻜㻜㻜㻥㻌

㻝㻜㻜㻜㻤㻌

㻝㻜㻜㻜㻣㻌

㻝㻜㻜㻜㻢㻌

㻝㻜㻜㻜㻡㻌

㻝㻜㻜㻜㻠㻌

㻝㻜㻜㻜㻟㻌㻝㻌

㻝㻜㻜㻜㻞㻌 㻝㻌

㻝㻜㻜㻜㻝㻌

㻝㻜㻜㻜㻜㻌

኎䜚䞉㈙䛔䠎䛴䛾ὀᩥ䜢ྠ᫬䛻ฟ䛩

ᯈᐤ䛫䜎䛷ẖ᫬้ὀᩥ䜢ධ䜜䜛

䊹㻌 㻼

㼒㼍㼕㼞㻌

䊹㻌 㻼

㼒㼍㼕㼞㻌

䊹㻌 㻼

㼟㼜㼞㼑㼍㼐㻌

㻼

㼟㼜㼞㼑㼍㼐㻌

䊻㻌

ਤ2: ϚʔέοτϝʔΧʔΤʔδΣϯτ(MM)ͷ஫จ 98ͷചΓɼՁ֨100ͷങ͍ͱՁ֨99ͷചΓ͕ͦΕͧΕ෇͚ ߹Θ͞ΕɼऔҾ͸Ձ֨͸ͦΕΒͷ஫จΛ࡟আͨ͠ޙͷ஥஋Ͱ͋ Δ99.5Ͱܾఆ͢Δɽ ͜͜Ͱt = 4ʹ͓͍ͯ࢒ͬͨ஫จ͕ҟͳΔ͜ͱʹ஫໨͢΂ ͖Ͱ͋Δɽδt = 1ͷ৔߹͸ɼ͢΂ͯͷ஫จ͕ͳ͘ͳ͓ͬͯΓɼ δt = 4ͷ৔߹͸ɼ4ͭͷ஫จ͕࢒͍ͬͯΔɽͦͯ͠ɼδt = 1ͷ ৔߹͸ɼ਺ྔ4ͷങ͍஫จ͕੒ཱ͍ͯ͠Δͷʹରͯ͠ɼδt = 4 ͷ৔߹͸ɼ਺ྔ2ͷങ͍஫จͷΈ੒ཱ͍ͯ͠Δɽ͜ͷΑ͏ʹ δtʹԠͯ͡ɼՁ֨ܗ੒΍஫จঢ়گ͸ҟͳΔ΋ͷͱͳΔɽ

2.2

ΤʔδΣϯτ

ຊϞσϧ͸1ͭͷূ݊ͷΈΛऔҾର৅ͱ͢ΔɽnମͷϊʔϚ ϧΤʔδΣϯτ(NA)ͱ1ମͷϚʔέοτϝʔΧʔΤʔδΣϯ τ(MM)͕͍ΔɽNA͸ΤʔδΣϯτ൪߸j = 1͔Βॱ൪ʹ j = 2, 3, 4, ...ͱ஫จΛग़͢ɽMM͸NA͕஫จΛग़͢௚લʹ ങ͍ͱചΓͷ̎ͭͷ஫จΛग़͢ɽਤ2ʹࣔ͢Α͏ʹɼMM͸ ൘د͕ͤߦΘΕΔ·Ͱຖ࣌ࠁ஫จΛೖΕɼ൘دͤ௚ޙʹ͢΂ ͯͷ஫จΛΩϟϯηϧ͢ΔɽNAͷ஫จ͸ɼ஫จΛߦ͔ͬͯΒ Ωϟϯηϧ࣌ؒtc͚ͩܦաͨ͠৔߹Ωϟϯηϧ͞ΕΔɽ࠷ޙ ͷNAɼj = n͕஫จΛग़͢ͱɼ࣍ͷ࣌ࠁʹ͸·ͨॳΊͷNAɼ j = 1͔Β஫จΛग़͠܁Γฦ͞ΕΔɽ࣌ࠁt͸1ମͷNA͕஫ จΛग़͢͝ͱʹ1૿͑Δɽ ஫จ਺ྔ͸ৗʹ1ͱҰఆͱ͢Δɽ·ͨɼ͍ͣΕͷΤʔδΣϯ τ΋อ༗͢Δࢿ࢈ͷ਺ྔʹ੍ݶ͸ͳ͘(Ωϟογϡ͕ແݶେ)ɼ ϚΠφεͷอ༗਺ྔ(ۭചΓ)ʹ΋੍ݶ͸ͳ͍ɽՁ֨ͷมԽ෯ ͷ࠷খ୯Ґ(ݺ஋ͷࠁΈ)͸δP ͱ͠ɼͦΕΑΓখ͍͞୺਺͸ɼ ങ͍஫จͷ৔߹͸੾ΓࣺͯɼചΓ஫จͷ৔߹͸੾Γ্͛Δɽ 2.2.1 ϊʔϚϧΤʔδΣϯτ(NA) NA͸ɼ࣮ࡍͷࢢ৔ͷՁ֨ܗ੒ͷੑ࣭Λ࠶ݱ͢ΔͨΊʹಋೖ ͢Δ΋ͷͰ͋Γɼstylized fact΍ߴස౓औҾʹ͔͔ΘΔ౷ܭྔ Λ࠶ݱ͢ΔͳΔ΂͘γϯϓϧͳɼ͘͝Ұൠతͳ౤ࢿՈΛϞσϧ Խͨ͠΋ͷͱͨ͠ɽ NA͸஫จՁ֨Pt o,jɼചΓങ͍ͷผΛҎԼͷΑ͏ʹܾΊΔɽ ࣌ࠁtʹNAɼj͕༧૝͢ΔՁ֨ͷมԽ཰(༧૝Ϧλʔϯ)rte,j ͸ɼ re,jt = 1 w1,j+ w2,j+ uj  w1,jlogPf Pt + w2,jr t h,j+ ujtj  ɽ (1) ͜͜Ͱɼwi,j͸࣌ࠁtɼNAɼjͷi߲໨ͷॏΈͰ͋Γɼγϛϡ

䝫䝆䝅䝵䞁䛜ቑຍ䛩䜛ὀᩥ䜢ฟ䛥䛺䛔

᫬㛫㻌

䏓㼀ᮇ㛫㻌

㻝᪥䛾㻌

⤊䜟䜚㻌

䏓㼀㼑㼚㼐ᮇ㛫㻌

㻝᪥䛾㻌

⤊䜟䜚㻌

㻝᪥䛾㻌

⤊䜟䜚㻌

䝫䝆䝅䝵䞁㻌

䝊䝻䛻䛩䜛㻌

኎䜚㻌 ౯᱁㻌 ㈙䛔㻌 㻝㻌 㻝㻜㻜㻝㻝㻌 㻝㻜㻜㻝㻜㻌 㻝㻜㻜㻜㻥㻌 㻝㻜㻜㻜㻤㻌 㻝㻜㻜㻜㻣㻌㻝㻌 ᭱ᚋ䏓㼀㼑㼚㼐ᮇ㛫䚸኎䜚ᣢ䛱䛾ሙྜ㻌 ฟ䛥䛺䛔䊺㻌 䝫䝆䝅䝵䞁䛜ῶᑡ䛩䜛ὀᩥ䜢཯ᑐഃ䛾౯᱁䛷ฟ䛩 ኎䜚㻌 ౯᱁㻌 ㈙䛔㻌 㻝㻜㻜㻝㻝㻌㻝㻌 㻝㻜㻜㻝㻜㻌 㻝㻜㻜㻜㻥㻌 㻝㻜㻜㻜㻤㻌 㻝㻜㻜㻜㻣㻌 ᭱ᚋ䏓㼀㼑㼚㼐ᮇ㛫䚸኎䜚ᣢ䛱䛾ሙྜ㻌 䊹䛣䛣䛻ฟ䛩㻌

䏓㼀㼑㼚㼐ᮇ㛫㻌

䏓㼀㼑㼚㼐ᮇ㛫㻌

䏓㼀ᮇ㛫㻌

䏓㼀ᮇ㛫㻌

PMM4 PMM3 䊹㻌 㻼㼒㼍㼕㼞㻌 䊹㻌 㻼㼒㼍㼕㼞㻌 ਤ3: ϚʔέοτϝʔΧʔΤʔδΣϯτ(MM)͕ϙδγϣϯΛด͡Δ࣌ؒଳ

䝫䝆䝅䝵䞁䛜ቑຍ䛩䜛ὀᩥ䜢ฟ䛥䛺䛔

᫬㛫㻌

䏓㼀ᮇ㛫㻌

㻝᪥䛾㻌

⤊䜟䜚㻌

䏓㼀㼑㼚㼐ᮇ㛫㻌

㻝᪥䛾㻌

⤊䜟䜚㻌

㻝᪥䛾㻌

⤊䜟䜚㻌

䝫䝆䝅䝵䞁㻌

䝊䝻䛻䛩䜛㻌

኎䜚㻌

౯᱁㻌

㈙䛔㻌

㻝㻌 㻝㻜㻜㻝㻝㻌

㻝㻜㻜㻝㻜㻌

㻝㻜㻜㻜㻥㻌

㻝㻜㻜㻜㻤㻌

㻝㻜㻜㻜㻣㻌㻝㻌

᭱ᚋ䏓㼀㼑㼚㼐ᮇ㛫䚸኎䜚ᣢ䛱䛾ሙྜ㻌

ฟ䛥䛺䛔䊺㻌

䝫䝆䝅䝵䞁䛜ῶᑡ䛩䜛ὀᩥ䜢཯ᑐഃ䛾౯᱁䛷ฟ䛩

኎䜚㻌

౯᱁㻌

㈙䛔㻌

㻝㻜㻜㻝㻝㻌㻝㻌

㻝㻜㻜㻝㻜㻌

㻝㻜㻜㻜㻥㻌

㻝㻜㻜㻜㻤㻌

㻝㻜㻜㻜㻣㻌

᭱ᚋ䏓㼀㼑㼚㼐ᮇ㛫䚸኎䜚ᣢ䛱䛾ሙྜ㻌

䊹䛣䛣䛻ฟ䛩㻌

䏓㼀㼑㼚㼐ᮇ㛫㻌

䏓㼀㼑㼚㼐ᮇ㛫㻌

䏓㼀ᮇ㛫㻌

䏓㼀ᮇ㛫㻌

PMM4

PMM3

䊹㻌 㻼

㼒㼍㼕㼞㻌

䊹㻌 㻼

㼒㼍㼕㼞㻌 ਤ4: ϚʔέοτϝʔΧʔΤʔδΣϯτ(MM)͕ϙδγϣϯΛด͡Δ࣌ؒଳʹग़͢஫จ Ϩʔγϣϯ։࢝࣌ʹɼͦΕͧΕ0͔Βwi,max·ͰҰ༷ཚ਺Ͱ ܾΊΔɽuj͸NAɼjͷ3߲໨ͷॏΈͰ͋ΓɼγϛϡϨʔγϣ ϯ։࢝࣌ʹ0͔Βumax·ͰҰ༷ཚ਺ͰܾΊΔɽlog͸ࣗવର ਺Ͱ͋ΔɽPf ͸࣌ؒʹΑΒͣҰఆͷϑΝϯμϝϯλϧՁ֨ɼ Pt͸લઅͰఆٛ͞ΕͨऔҾՁ֨(औҾݟࠐΈՁ֨)ɼtj͸࣌ࠁ tɼΤʔδΣϯτjͷཚ਺߲Ͱ͋Γɼฏۉ0ɼඪ४ภࠩσͷਖ਼ ن෼෍ཚ਺Ͱ͋Δɽrh,jt ͸࣌ࠁtʹNAɼj͕ܭଌͨ͠աڈϦ λʔϯͰ͋Γɼrh,jt = log (Pt/Pt−τj)Ͱ͋Δ∗6ɽ͜͜Ͱτj͸ γϛϡϨʔγϣϯ։࢝࣌ʹ1͔Βτmax·ͰͷҰ༷ཚ਺ͰNA ͝ͱʹܾΊΔɽ ࣜ(1)ͷୈ1߲໨͸ϑΝϯμϝϯλϧՁ֨ͱൺֱ͚ͯ҆͠ Ε͹ϓϥεͷ༧૝ϦλʔϯΛߴ͚Ε͹ϚΠφεͷ༧૝Ϧλʔϯ Λࣔ͢ɼϑΝϯμϝϯλϧՁ஋Λࢀরͯ͠౤ࢿ൑அΛߦ͏ϑΝ ϯμϝϯλϧ౤ࢿՈͷ੒෼Ͱ͋Δɽୈ߲̎໨͸աڈͷϦλʔϯ ͕ϓϥε(ϚΠφε)ͳΒϓϥε(ϚΠφε)ͷ༧૝ϦλʔϯΛ ࣔ͢ɼաڈͷՁ֨ਪҠΛࢀরͯ͠౤ࢿ൑அΛߦ͏ςΫχΧϧ౤ ࢿՈͷ੒෼Ͱ͋Γɼୈ߲̏໨͸ϊΠζΛද͍ͯ͠Δɽ ༧૝Ϧλʔϯrt e,jΑΓ༧૝Ձ֨Pe,jt ͸ɼ

Pe,jt = Ptexp (re,jt ) (2) Ͱٻ·Δɽ஫จՁ֨Po,jt ͸ฏۉPe,jt ɼඪ४ภࠩPσͷਖ਼ن෼ ෍ཚ਺ͰܾΊΔɽ͜͜ͰɼPσ͸ఆ਺Ͱ͋Δɽͦͯ͠ɼചΓങ ͍ͷผ͸༧૝Ձ֨Pt e,jͱ஫จՁ֨Po,jt ͷେখؔ܎ͰܾΊΔɽ ͢ͳΘͪɼ Pe,jt > Po,jt ͳΒ਺ྔ1ͷങ͍ Pe,jt < Po,jt ͳΒ਺ྔ1ͷചΓ (3) ͱ͢Δ∗7ɽ ∗6 ͨͩ͠ɼt < τjͷͱ͖͸ɼrth,j= 0 ͱͨ͠ɽ ∗7 ͨͩ͠ɼt < tcͷͱ͖͸े෼ͳ൘ͷް͞Λ֬อ͢ΔͨΊɼPf > 2.2.2 ϚʔέοτϝʔΧʔΤʔδΣϯτ(MM) ຊݚڀͰ͸[૲ా15a,૲ా15b]ΛϕʔεʹϚʔέοτϝʔ ΧʔΤʔδΣϯτ(MM)ΛϞσϧԽͨ͠ɽਤ2ʹࣔ͢Α͏ʹ MM͸ɼϑΣΞόϦϡʔPf airΑΓ஫จεϓϨουPspreadͩ ͚ߴ͍஋ஈ(Pf air+ Pspread)ʹ਺ྔ1ͷചΓ஫จɼPspreadͩ ͚͍҆஋ஈ(Pf air− Pspread)ʹ਺ྔ1ͷങ͍஫จΛೖΕΔɽ MM͸NA͕஫จΛग़͢௚લʹ͜ΕΒͷ஫จΛग़͠ɼ൘دͤ ͕ߦΘΕΔ·Ͱຖ࣌ࠁ஫จΛೖΕΔɽ͜͏͢Δ͜ͱʹΑΓɼδt ͷ஋ʹΑΒͣMMͷ૯஫จ਺ྔ͸ಉ͡ͱͳΔɽͦͯ͠൘دͤ ௚ޙʹ͢΂ͯͷ஫จΛΩϟϯηϧ͢Δɽ MM ʹ͸ҎԼͷ 4 छྨΛ࣮૷ͨ͠ɽ1 ͭΊ͸γϯϓϧ MM(SMM)Ͱ͋ΓɼPf air = Pt ͱ͢Δɽ2ͭΊ͸ϙδγϣ ϯMM(PMM)Ͱ͋Γɼ Pf air= (1− kS3)Pt (4) ͱ͢Δɽ͜͜Ͱɼk͸ఆ਺ɼS͸MMͷอ༗਺ྔ(ϙδγϣ ϯ)Ͱ͋Δɽϙδγϣϯ͕ਖ਼(S > 0)ͷ৔߹ɼPf air͸PtΑ Γখ͘͞ͳΓɼങ͍஫จ΋ചΓ஫จ΋҆͘ͳΔɽങ͍஫จ͸औ Ҿ੒ཱͮ͠Β͘ͳΓɼചΓ஫จ͸औҾ੒ཱ͠΍͘͢ͳΔͨΊɼ ϙδγϣϯͷઈର஋S͕ݮগ͠΍͘͢ͳΔɽϙδγϣϯ͕ ෛ(S < 0)ͷ৔߹ɼPf air͸PtΑΓେ͖͘ͳΓɼങ͍஫จ΋ ചΓ஫จ΋ߴ͘ͳΔɽങ͍஫จ͸औҾ੒ཱ͠΍͘͢ͳΓɼചΓ ஫จ͸औҾ੒ཱͮ͠Β͘ͳΓɼϙδγϣϯͷઈର஋S͸ݮ গ͠΍͘͢ͳΔɽͭ·ΓɼPMM͸SMMʹൺ΂ɼϙδγϣϯ Λ͓͑͞ɼϙδγϣϯ͕େ͖͘ͳΔ͜ͱʹΑΔՁ֨มಈͷϦε ΫΛ͓͑͞Δ͜ͱ͕Ͱ͖Δɻ ࣮ࡍͷࢢ৔ʹ͓͍ͯ΋ɼϚʔέοτϝʔΧʔઓུΛͱΔ౤ ࢿՈ͸Ձ֨มಈͷϦεΫΛ͓͑͞ΔͨΊʹɼෆཁͳϙδγϣϯ Pt o,jͳΒ਺ྔ 1 ͷങ͍ɼPf < Po,jt ͳΒ਺ྔ 1 ͷചΓͱ͢Δɽ· ͨɼPt_=P f ͱ͢Δ

ΛͱΒͳ͍Α͏ʹ޻෉͍ͯ͠Δɽಛʹɼ̍೔ͷऔҾ͕࣌ؒऴྃ ͔ͯ͠ΒཌӦۀ೔ͷऔҾ͕࣌ؒ։࢝͢Δ·Ͱͷ࣌ؒ(Φʔόʔ φΠτ)ʹϙδγϣϯ͕͋ΔͱɼΦʔόʔφΠτʹେ͖ͳग़དྷ ࣄ͕͋ͬͨ৔߹ͳͲʹେ͖ͳՁ֨มಈϦεΫ͕ൃੜ͢ΔͨΊɼ ΦʔόʔφΠτ͸ϙδγϣϯΛθϩʹ͢Δ͜ͱ͕΄ͱΜͲͰ͋ ΔɽͦͷͨΊຊݚڀͰ͸ɼ[૲ా15a,૲ా15b]Ͱ͸ߟྀ͞Εͯ ͍ͳ͔ͬͨɼΦʔόʔφΠτͷϙδγϣϯΛθϩʹ͢ΔPMM Λ2छྨ࣮૷ͨ͠ɽ ਤ3͸̍೔ͷఆٛͱϙδγϣϯΛθϩʹ͢ΔظؒΛࣔͨ͠ɽ ̍೔ͷ௕͞͸ΔT ͱ͠ɼ͜ͷظؒͷ͏ͪ࠷ޙͷΔTendظؒ ʹɼPMM͸ϙδγϣϯΛθϩʹ͢ΔͨΊͷ޻෉Λߦ͏ɽͦͷ 1ͭΊ͸PMM3Ͱ͋Γɼਤ4ͷࠨʹࣔ͢Α͏ʹɼϙδγϣϯ ͕ਖ਼(S > 0)ͷͱ͖͸ങ͍஫จΛग़͞ͳ͍ɼϙδγϣϯ͕ෛ (S < 0)ͷͱ͖͸ചΓ஫จΛग़͞ͳ͍ͱ͍͏΋ͷͰ͋Δɽ΋͏ ͻͱͭ͸PMM4Ͱ͋Γɼਤ4ͷӈʹࣔ͢Α͏ʹɼϙδγϣϯ ͕ਖ਼(S > 0)ͷͱ͖͸ങ͍஫จΛग़͞ͳ͍ͷΈͳΒͣɼചΓ ஫จͷՁ֨ΛPf air− Pspread(ങ͍஫จͰ࢖༻͢Δ༧ఆͩͬͨ Ձ֨)ʹมߋ͠ɼϙδγϣϯ͕ෛ(S < 0)ͷͱ͖͸ചΓ஫จΛ ग़͞ͳ͍ͷΈͳΒͣɼങ͍஫จͷՁ֨ΛPf air+ Pspread(ചΓ ஫จͰ࢖༻͢Δ༧ఆͩͬͨՁ֨)ʹมߋ͢Δɽ PMM4͸PMM3ΑΓ΋ɼΑΓऔҾ͕੒ཱ͠΍͍͢Ձ֨Ͱ ஫จΛग़͢͜ͱʹΑΓɼΑΓੵۃతʹϙδγϣϯΛ0ʹ͢Δ͜ ͱΛ໨ࢦ͢ɽ͔͠͠ͳ͕ΒɼNA͕ڙڅͨ͠஫จʹࣗ਎ͷ஫จ Λର౰ͤ͞Δɼ͢ͳΘͪྲྀಈੑΛୣ͏஫จΛग़ͨ͢ΊɼS͕ େ͖͔ͬͨ৔߹ɼҰํ޲ʹՁ֨Λมಈͤͯ͞͠·͍ͳ͕ΒS ΛݮΒ͢͜ͱʹͳΓɼՁ֨มಈϦεΫ͕ߴ͘ͳͬͯ͠·͏ɽͦ ͷͨΊɼ΋ͬͱ΋ݱ࣮తͳϞσϧ͸PMM3Ͱ͋Δ͕ɼ࣍ষͰ ड़΂ΔΑ͏ʹɼδt͕େ͖͍৔߹͸PMM3Ͱ͸̍೔ͷऴΘΓʹ ϙδγϣϯΛ0ʹͰ͖ͳ͍ͨΊɼPMM4Λ༻ҙͨ͠ɽ

3.

γϛϡϨʔγϣϯ݁Ռ

ຊݚڀͰ͸ [ਫా 13]͓Αͼ [૲ా 15a,૲ా 15b]ͱಉ͡ Ͱ͋ΔҎԼͷύϥϝʔλΛ༻͍ͨɽ۩ମతʹ͸, n = 1, 000

ɼw1,max= 1ɼw2,max= 10ɼumax= 1ɼτmax = 10, 000ɼσ=

0.06ɼPσ = 30ɼtc = 20, 000ɼδP = 0.02, Pf = 10, 000ɼk = 0.00000005ɼΔT = 20, 000ɼΔTend = 2, 000ͱͨ͠ɽ·ͨγ ϛϡϨʔγϣϯ͸࣌ࠁt = te= 10, 000, 000·Ͱߦͬͨ∗8ɽ ·ͨɼ4छྨͷMMɼPspread/Pf =0.03%, 0.1%, 0.3%, 1% ͓Αͼδt =1, 2, 5, 10, 20, 50, 100, 200, 500, 1000ʹର͠ ͯ∗9ɼͦͷଞͷ৚݅Λཚ਺ද΋ؚΊશ͘ಉ͡ʹͯ͠ɼ֤छ౷ ܭ஋Λࢉग़͢Δɽ͜ΕΛཚ਺දΛมߋͯ͠100ճߦ͍ɼͦͷ ฏۉ஋ΛҎޙ༻͍Δɽ

3.1

஫จεϓϨου P

spread

ͱऔҾ੒ཱ཰

ද1͸ɼ൘دִͤؒδt, MMͷ஫จεϓϨουPspread/Pf ͝ͱͷMMͷऔҾ੒ཱ཰Λࣔͨ͠ɽऔҾ੒ཱ཰͸ɼऔҾ੒ཱ ਺ྔ/૯஫จ਺ྔͱͨ͠ɽMM͸PMM4Λ༻͍ͨɽ δt͕େ͖͘ͳΔͱɼMMͷऔҾ੒ཱ཰͕ݮগ͍ͯ͠Δͷ͕ ෼͔ΔɽMM͸஫จΛৗʹఏࣔͯ͠NAʹऔҾػձ(ྲྀಈੑ) Λఏڙ͍ͯ͠ΔͷͰɼMMͷऔҾ੒ཱ཰ͷݮগ͸ྲྀಈੑͷݮ গΛ͍ࣔࠦͯ͠ΔɽΑͬͯɼδt͸খ͍͞ํ͕MMʹΑΓྲྀಈ ੑ͕ڙڅ͞Εɼβϥόํ͕ࣜ࠷΋MMʹΑΔྲྀಈੑڙڅ͕ଟ ͍ՄೳੑΛ͍ࣔͯ͠Δɽ ∗8 ͜ΕΒͷύϥϝʔλͷଥ౰ੑݕূʹ͍ͭͯ͸ຊߘͷ෇࿥ “Ϟσϧ ͷଥ౰ੑݕূ” Ͱઆ໌ͨ͠ɽ͞ΒͳΔৄࡉ͸ [ਫా 14] ʹॻ͔Εͯ ͍Δɽ ∗9 ͭ·Γɼ4 छྨͷ MMɼ4 ௨Γͷ Pspreadɼ10 ௨Γͷδt ͷܭɼ 160(= 4× 4 × 10) ௨ΓͷέʔεΛࢼͨ͠ɽ Pspread/Pf͕େ͖͘ͳΔͱ౰વɼMMͷऔҾ੒ཱ཰͸খ͞ ͘ͳΔɽPspread/Pf = 0.1%ͷ৔߹ɼδt = 1ͳΒ·ͩݱ࣮త ͳऔҾ੒ཱ཰Ͱ͋Γ෼ੳՄೳͰ͋Δ͕ɼδt ≥ 50ͰऔҾ͕΄ͱ ΜͲͳ͘ͳΓɼҎԼͰௐ΂ΔMMͷϙδγϣϯ΍ଛӹͷ෼ੳ ͕ҙຯΛͳ͞ͳ͘ͳΔɽͦͷͨΊҎޙ͸ɼδt ≥ 50Ͱ΋ҙຯͷ ͋Δ෼ੳͰ͖ΔΑ͏ʹɼPspread/Pf = 0.03%Λ༻͍Δɽ

3.2

MM ͷछྨ͝ͱͷϙδγϣϯ

ද2͸ɼ൘دִͤؒδtɼMMͷछྨ͝ͱͷɼશظؒ·ͨ͸ ̍೔ͷऴΘΓͷΈͰܭଌͨ͠ϙδγϣϯͷઈର஋Sͷฏۉ Λࣔͨ͠ɽPspread/Pf = 0.03%ͱͨ͠ɽ ࣮ࡍͷࢢ৔ʹ͓͍ͯ͸ɼϚʔέοτϝʔΧʔઓུΛͱΔ౤ ࢿՈ͸Ձ֨มಈͷϦεΫΛ͓͑͞ΔͨΊʹɼෆཁͳϙδγϣϯ ΛͱΒͳ͍Α͏ʹ޻෉͍ͯ͠Δɽಛʹɼ̍೔ͷऔҾ͕࣌ؒऴྃ ͔ͯ͠ΒཌӦۀ೔ͷऔҾ͕࣌ؒ։࢝͢Δ·Ͱͷ࣌ؒ(Φʔόʔ φΠτ)ʹϙδγϣϯ͕͋ΔͱɼΦʔόʔφΠτʹେ͖ͳग़དྷ ࣄ͕͋ͬͨ৔߹ͳͲʹେ͖ͳՁ֨มಈϦεΫ͕ൃੜ͢ΔͨΊɼ ΦʔόʔφΠτ͸ϙδγϣϯΛθϩʹ͢Δ͜ͱ͕΄ͱΜͲͰ ͋Δɽ ͦͷͨΊɼগͳ͘ͱ΋̍೔ͷऴΘΓͷΈͰܭଌͨ͠Sͷฏ ۉ͕େ͖͍ͱɼϦεΫ͕ߴ͗ͯ͢ݱ࣮తͰͳ͍ͱ͍͑Δɽ1೔ ͷऴΘΓʹSΛখ͘͢͞Δ͜ͱΛϞσϧԽ͍ͯ͠ͳ͍SMM ͱPMM͸౰વɼ͜ΕΒͷ஋͕0ͱͳ͓ͬͯΒͣݱ࣮తͰͳ͍ɽ PMM3ʹ͓͍ͯ͸ɼδt͕খ͍͞৔߹͸ɼ͜ΕΒͷ஋͕0ͱ ͳ͓ͬͯΓɼΦʔόʔφΠτͷϦεΫΛͱΒͣʹ͢ΜͰ͍Δɽ Ұํɼδt ≥ 20ͷ৔߹͸ɼS = 0ʹग़དྷ͓ͯΒͣɼΦʔόʔ φΠτͷϦεΫʹ͞Β͞Ε͓ͯΓɼݱ࣮తͰͳ͍ɽ͞Βʹɼલ અͰड़΂ͨΑ͏ʹδt͕େ͖͍΄ͲऔҾ੒ཱ͕গͳ͍ʹ΋͔͔ ΘΒͣɼશظؒͰͷS͕େ͖͘ͳ͓ͬͯΓՁ֨มಈϦεΫ͕ ߴ·͍ͬͯΔɽ͢ͳΘͪɼδt͕େ͖͍৔߹ɼΦʔόʔφΠτͷ S = 0Λ࣮ݱ͢Δ͜ͱ͕೉্͍͠ʹϙδγϣϯ΋େ͖͘Ձ ֨มಈϦεΫ͕େ͖͘ͳΓɼϦεΫΛ͓͑ͯ͞Ϛʔέοτϝʔ ΧʔઓུΛܧଓ͢Δ͜ͱ͕ࠔ೉ʹͳΔՄೳੑΛ͍ࣔͯ͠Δɽ PMM4ͷΑ͏ʹੵۃతͳՁ֨ͰSΛݮΒ͢஫จΛग़ͯ͠ ࢝Ίͯɼδt ≥ 20ͷ৔߹Ͱ΋SΛ0ۙ͘ʹݮΒ͢͜ͱ͕Ͱ͖ ͍ͯΔɽPMM4Ͱ͸ɼNA͕ڙڅͨ͠஫จʹࣗ਎ͷ஫จΛର ౰ͤ͞Δɼ͢ͳΘͪྲྀಈੑΛୣ͏஫จΛग़ͨ͢ΊɼS͕େ͖ ͔ͬͨ৔߹ɼҰํ޲ʹՁ֨Λมಈͤͯ͞͠·͍ͳ͕ΒSΛ ݮΒ͢͜ͱʹͳΓɼՁ֨มಈϦεΫ͕ߴ͘ͳͬͯ͠·͏ɽͦͷ Α͏ͳऔҾΛ͠ͳ͚Ε͹δt ≥ 20ͷ৔߹͸ɼSΛ0ۙ͘ʹݮ Β͢͜ͱ͕Ͱ͖ͳ͍ɽ͜ͷ͜ͱ͔Β΋δt͕େ͖͍৔߹ɼΦʔ όʔφΠτͷS = 0Λ࣮ݱ͢Δʹ͸·ͨผͷϦεΫΛͱΔ ඞཁ͕͋ΓɼϦεΫΛ͓͑ͯ͞ϚʔέοτϝʔΧʔઓུΛܧଓ ͢Δ͜ͱ͕ࠔ೉ʹͳΔՄೳੑΛ͍ࣔͯ͠Δɽδt͕େ͖͍৔߹ ͸ɼϦεΫ͕͋Δͱ͸͍͑ɼPMM4͕།ҰɼΦʔόʔφΠτ ͷSΛ0ۙ͘ʹͰ͖ΔͷͰɼҎޙɼPMM4Λ༻͍Δɽ

3.3

࠷ऴଛӹ

ද3͸ɼ൘دִͤؒδt͝ͱͷɼMMͷ࠷ऴଛӹ/Pfɼϙδ γϣϯͷઈର஋SͷฏۉɼMMͱNAͷऔҾ੒ཱ཰Λࣔ͠ ͨɽMM͸PMM4༻͍ɼPspread/Pf = 0.03%ͱͨ͠ɽ δt = 2, 5, 10ͷ৔߹ɼMMͷଛӹ͸ෛͱͳ͓ͬͯΓଛΛ͠ ͍ͯΔɽຊݚڀͰ͸ɼࢢ৔͕ඇৗʹ҆ఆ͍ͯ͠Δͱ͖ɼ͢ͳΘ ͪMMʹͱͬͯ͸࠷΋ऩӹ͕͋͛΍͍͢؀ڥͷΈΛऔΓѻͬ ͍ͯΔɽͦΕʹ΋͔͔ΘΒͣऩӹΛ͋͛ΒΕ͍ͯͳ͍ͱ͍͏ ͜ͱ͸ɼݱ࣮ͷࢢ৔Ͱ͸͞ΒʹऩӹΛ্͛Δͷ͸ݫ͍͠ͱߟ ͑ΒΕΔɽδt = 20, 50, 100, 200ͷ৔߹Ͱ΋ऩӹ͸͋Δͱ͸͍ ͑ɼδt = 1ͷ৔߹ʹൺ΂খ͘͞ɼલઅ·Ͱʹड़΂ͨΑ͏ʹδt ͕େ͖͍΄ͲՁ֨มಈϦεΫ͕େ͖͍͜ͱΛߟ͑Δͱɼδt = 1

ද1: ൘دִͤؒδt, MMͷ஫จεϓϨουPspread/Pf ͝ͱͷMMͷऔҾ੒ཱ཰(PMM4) MMͷऔҾ੒ཱ཰ Pspread/Pf 0.03% 0.10% 0.30% 1.00% δt 1(βϥόํࣜ) 8.06% 1.53% 0.00% 0.00% 2 6.30% 0.88% 0.00% 0.00% 5 3.93% 0.37% 0.00% 0.00% 10 2.47% 0.14% 0.00% 0.00% 20 1.49% 0.02% 0.00% 0.00% 50 0.77% 0.00% 0.00% 0.00% 100 0.48% 0.00% 0.00% 0.00% 200 0.32% 0.00% 0.00% 0.00% 500 0.21% 0.00% 0.00% 0.00% 1000 0.22% 0.00% 0.00% 0.00% ද2: ൘دִͤؒδtɼMMͷछྨ͝ͱͷɼશظؒ·ͨ͸̍೔ͷऴΘΓͷΈͰܭଌͨ͠ϙδγϣϯͷઈର஋Sͷฏۉ(Pspread/Pf = 0.03%) Sͷฏۉ SMM PMM PMM3 PMM4 શظؒ ̍೔ͷ શظؒ ̍೔ͷ શظؒ ̍೔ͷ શظؒ ̍೔ͷ ऴΘΓ ऴΘΓ ऴΘΓ ऴΘΓ ͷΈ ͷΈ ͷΈ ͷΈ δt 1(βϥόํࣜ) 12,357 12,371 3.18 3.08 2.90 0.00 2.89 0.00 2 17,42 17,441 3.10 3.25 2.79 0.00 2.79 0.00 5 4,409 4,414 3.87 3.95 3.48 0.00 3.48 0.00 10 1,744 1,744 4.44 4.34 4.01 0.02 3.96 0.00 20 548 548 4.84 4.71 4.52 0.78 4.35 0.00 50 384 385 5.27 5.14 5.02 2.63 4.63 0.00 100 369 370 5.57 5.51 5.56 4.26 4.80 0.00 200 174 174 5.91 6.11 5.92 5.69 4.38 0.00 500 72 71 5.75 6.06 5.70 5.81 2.32 0.03 1000 290 290 5.94 6.11 5.61 5.80 1.76 0.06 ͷ৔߹ʹൺ΂ϚʔέοτϝʔΧʔઓུΛܧଓ͢Δ͜ͱ͕ࠔ೉ʹ ͳ͍ͬͯΔͱ͍͑Δɽগͳ͘ͱ΋ɼδt = 1ͷ৔߹ʹൺ΂ɼϦ εΫʹݟ߹ͬͨऩӹΛ͋͛ͮΒ͘ͳΔՄೳੑ͕ࣔ͞Ε͍ͯΔɽ

4.

·ͱΊͱࠓޙͷ՝୊

ຊݚڀͰ͸[૲ా15a,૲ా15b]ͷਓ޻ࢢ৔ϞσϧΛϕʔε ʹɼ൘دִͤؒδtͱ͍͏ύϥϝʔλʔΛ৽ͨʹಋೖ͢Δ͜ͱʹ ΑΓɼβϥόํࣜ(δt = 1)ͱόονΦʔΫγϣϯํࣜ(δt > 1) Λ࿈ଓతʹมߋͰ͖ΔՁܾ֨ఆϝΧχζϜΛ࣮૷͠ɼϚʔέο τϝʔΧʔઓུͷଛӹͷϦεΫΛ෼ੳ͢Δ͜ͱʹΑΓͦͷଘଓ ՄೳੑΛٞ࿦͠ɼϚʔέοτϝʔΧʔઓུ͕ྲྀಈੑΛڙڅ͠ଓ ͚Δ͜ͱ͕Ͱ͖Δ͔Ͳ͏͔Λௐ΂ͨɽ ͦͷ݁Ռɼδt͕େ͖͘ͳΔͱɼMMͷऔҾ੒ཱ཰͕ݮগ͠ MMʹΑΔྲྀಈੑڙڅ͕ݮগ͢ΔՄೳੑ͕ࣔ͞Εͨɽ͞Βʹɼ δt͕େ͖͘ͳΔͱɼΦʔόʔφΠτͷϙδγϣϯΛθϩʹ͢ Δ͜ͱ͕೉্͍͠ʹϙδγϣϯ΋େ͖͘Ձ֨มಈϦεΫ͕େ͖ ͘ͳΓɼϦεΫΛ͓͑ͯ͞ϚʔέοτϝʔΧʔઓུΛܧଓ͢Δ ͜ͱ͕ࠔ೉ʹͳΔՄೳੑ͕ࣔ͞Εͨɽ͞Βʹɼδt = 1ͷͱ͖ɼ ͢ͳΘͪβϥόํࣜͷͱ͖ͷΈɼϦεΫʹݟ߹ͬͨऩӹΛಘΔ Մೳੑ͕͋ΔՄೳੑ͕ࣔ͞Εɼδt > 1ͷͱ͖ɼ͢ͳΘͪόο νΦʔΫγϣϯํࣜͷͱ͖͸ɼগͳ͘ͱ΋βϥόํࣜͰ͸ػೳ ͨ͠ϚʔέοτϝʔΧʔઓུͰ͸ɼϦεΫʹݟ߹ͬͨऩӹΛ͋ ͛Δͷ͸೉͘͠ͳΔՄೳੑ͕ࣔ͞Εͨɽ ͜ΕΒͷࣔࠦ͸ɼβϥόํࣜͰ͸ྲྀಈੑΛڙڅ͍ͯͨ͠Ϛʔ έοτϝʔΧʔઓུ͕ɼόονΦʔΫγϣϯํࣜʹͳΔͱͦͷ ڙڅΛҡ࣋Ͱ͖ͳ͘ͳΓɼͦΕΒ͕ఫୀ͢Δ͜ͱʹΑΓɼྲྀಈ ੑ͕௿Լ͢ΔՄೳੑΛ҉͍ࣔͯ͠Δͱ΋ߟ͑Δ͜ͱ͕Ͱ͖Δɽ ͜Ε͸ɼ[େ෿14]ͷࢦఠͱ੔߹తͰ͋Δɽ ࠓޙͷ՝୊͸ɼόονΦʔΫγϣϯํࣜʹదͨ͠Ϛʔέοτ ϝʔΧʔઓུ͕͋Δ͔ݕ౼͢Δ͜ͱͰ͋ΔɽຊݚڀͰ͸ɼβϥ όํࣜͰػೳͨ͠ϚʔέοτϝʔΧʔઓུ͕όονΦʔΫγϣ ϯํࣜͰ͸ػೳ͠ͳ͍ՄೳੑΛࣔͨ͠ɽ͔͠͠ɼόονΦʔΫ γϣϯํࣜͰͷΈػೳ͢ΔϚʔέοτϝʔΧʔઓུ͕ଘࡏ͢ Δ͜ͱ͸൱ఆͰ͖ͣɼࠓޙͷ՝୊Ͱ͋Δɽ·ͨɼຊݚڀͰ͸଴ ػ͍ͯ͠Δ஫จ͕ඇৗʹଟ͍ʢ൘͕ް͍ʣঢ়گͷΈΛऔΓѻͬ ͨɽ൘͕ബ͍໏ฑͰ͸ɼόονΦʔΫγϣϯํࣜͷํ͕஫จΛ ग़͠΍͘͢ྲྀಈੑ͕ߴ·Δ͜ͱ΋ߟ͑ΒΕɼࠓޙͷ՝୊Ͱ͋ Δɽ͞Βʹɼͦ΋ͦ΋ྲྀಈੑΛڙڅ͢ΔϚʔέοτϝʔΧʔઓ ུ͕͋·ΓࢀՃ͍ͯ͠ͳ͍໏ฑͰ͸ɼຊݚڀͷٞ࿦͸੒Γཱͨ ͣࠓޙͷ՝୊Ͱ͋Δɽ ͞Βʹɼطʹड़΂ͨΑ͏ʹɼਓ޻ࢢ৔γϛϡϨʔγϣϯ͸ͦ ͷಋೖͷ७ਮͳޮՌΛݟΔ͜ͱ͕Ͱ͖Δ͏͑ɼաڈʹಋೖ͞Ε ͨ͜ͱ͕ͳ͍΋ͷ΋෼ੳ͢Δ͜ͱ͕Ͱ͖ΔɽͨͩͦͷޮՌ͸

ද3: ൘دִͤؒδt͝ͱͷɼMMͷ࠷ऴଛӹɼϙδγϣϯͷઈର஋SͷฏۉɼMMͱNAͷऔҾ੒ཱ཰(PMM4ɼPspread/Pf = 0.03%) Sͷฏۉ MMͷ 1೔ͷ MMͷ NAͷ ࠷ऴଛӹ/Pf શظؒ ऴΘΓ औҾ੒ཱ཰ औҾ੒ཱ཰ ͷΈ δt 1(βϥόํࣜ) 51.98 2.89 0.00 8.06% 39.1% 2 -29.42 2.79 0.00 6.30% 39.1% 5 -14.90 3.48 0.00 3.93% 37.6% 10 -4.08 3.96 0.00 2.47% 36.3% 20 1.51 4.35 0.00 1.49% 34.9% 50 3.68 4.63 0.00 0.77% 33.4% 100 2.53 4.80 0.00 0.48% 32.5% 200 0.93 4.38 0.00 0.32% 31.8% 500 -0.06 2.32 0.03 0.21% 31.0% 1000 -0.10 1.76 0.06 0.22% 30.5% ࣮֬ͳ༧૝Ͱ͸ͳ͍ɽ͞·͟·ͳέʔεͰͷγϛϡϨʔγϣϯ Λߦ͍ɼ͜Ε·Ͱ༧૝͞Ε͍ͯͳ͔ͬͨɼ“͋ΓಘΔ”ϝΧχ ζϜͰͷݱ৅Λݟ͚͓ͭͯ͘͜ͱ͕ɼਓ޻ࢢ৔γϛϡϨʔγϣ ϯͷେ͖ͳ໾ׂͰ͋Γɼਓ޻ࢢ৔γϛϡϨʔγϣϯͷݶքͰ͋ ΔɽͦͷͨΊɼ͞ΒͳΔৄࡉͳٞ࿦Ͱ͸ɼ࣮ূ෼ੳͳͲଞͷख ๏ͷ݁Ռͱൺֱݕ౼͢Δඞཁ͕͋Δɽ

෇࿥

Ϟσϧߏஙͷجຊཧ೦

ਓ޻ࢢ৔γϛϡϨʔγϣϯΛ༻͍Ε͹ɼ͜Ε·Ͱʹಋೖ͞Εͨ͜ͱ͕ͳ͍ۚ ༥ࢢ৔ͷن੍ɾ੍౓΋ٞ࿦͢Δ͜ͱ͕Ͱ͖Δ͏͑ɼͦͷ७ਮͳӨڹΛநग़Ͱ͖ Δɽ͜Ε͕ਓ޻ࢢ৔γϛϡϨʔγϣϯݚڀͷڧΈͰ͋Δɽ ͦͯ͠ɼଟ͘ͷਓ޻ ࢢ৔γϛϡϨʔγϣϯݚڀ͕ن੍΍੍౓ͷมߋ΍ɼ৽͍͠λΠϓͷࢢ৔Λ෼ੳ ͖ͯͨ͠ [LeBaron 06, Chen 12, ࿨ઘ 12, Cristelli 14, Mizuta 16b]ɽ

ͨͩͦͷޮՌ͸࣮֬ͳ༧૝Ͱ͸ͳ͍ɽ͞·͟·ͳέʔεͰͷγϛϡϨʔγϣ ϯΛߦ͍ɼ͜Ε·Ͱ༧૝͞Ε͍ͯͳ͔ͬͨɼ“͋ΓಘΔ” ϝΧχζϜͰͷݱ৅Λ ݟ͚͓ͭͯ͘͜ͱ͕ɼਓ޻ࢢ৔γϛϡϨʔγϣϯͷେ͖ͳ໾ׂͱͳΔɽۚ༥ࢢ৔ Ͱ͜Ε͔Β࣮ࡍʹ͓͜Δݱ৅Λఆྔతʹ΋஧࣮ʹ࠶ݱ͢Δ͜ͱ͕໨తͰ͸ͳ͘ɼ ن੍΍੍౓ͷมߋ͕ɼͲͷΑ͏ͳϝΧχζϜͰՁ֨ܗ੒ʹӨڹΛ༩͑ɼͲͷΑ ͏ͳ͜ͱ͕ى͜ΓಘΔͷ͔ͱ͍͏஌ࣝ֫ಘ͕໨తͰ͋Δɽ͜Ε͸ྫ͑͹࣮ূ෼ ੳͳͲଞͷख๏Ͱ͸Ͱ͖ͳ͍͜ͱͰ͋Δɽ ਓ޻ࢢ৔Ϟσϧ͸ීวతʹଘࡏ͢ΔϚΫϩݱ৅Λ࠶ݱ͢΂͖Ͱ͋Δͱߟ͑Β ΕΔɽਓ޻ࢢ৔γϛϡϨʔγϣϯͰ͸ɼϚΫϩݱ৅Ͱ͋Δࢢ৔Ձ֨ͷಅམ཰΍ ചങ਺ྔΛϞσϧԽ͠ͳ͍ɽ͋͘·Ͱɼ౤ࢿՈΛ໛ͨ͠ “ΤʔδΣϯτ” ͱऔҾ ॴΛ໛ͨ͠ “Ձܾ֨ఆϝΧχζϜ” ͱ͍ͬͨϛΫϩϝΧχζϜΛϞσϧԽ͠ɼͦ ͷϛΫϩϝΧχζϜͷ૬ޓ࡞༻ͷੵΈ্͛ͱͯ͠ϚΫϩݱ৅͕ग़ྗ͞ΕΔɽͦ ͷͨΊɼϛΫϩϝΧχζϜͷϞσϧԽ͸ݱ࣮ͷࢢ৔ʹଈͨ͠΋ͷͱ͠ɼ݁Ռͱ ͯ͠ग़ྗ͞ΕΔϚΫϩݱ৅͸ɼݱ࣮ͷࢢ৔Ͱීวతʹଘࡏ͢ΔϚΫϩతੑ࣭Λ ࠶ݱ͞ΕΔΑ͏ʹ࡞Δඞཁ͕͋Δɽ ͔͠͠ɼීวతͰ͸ͳ͘ಛఆͷ࣌ظ΍ࢿ࢈ɼ஍ҬͰग़ݱ͢ΔϚΫϩతੑ࣭͢ ΂ͯΛ࠶ݱ͢Δ͜ͱ͸ຊݚڀͷ໨తͰ͸ͳ͍ɽඞཁҎ্ʹଟ͘ͷϚΫϩతੑ࣭ ΛҰͭͷϞσϧͰ࠶ݱ͠Α͏ͱ͢Δͱɼա৒ʹෳࡶͳϞσϧΛ΋ͨΒ͠ɼؔ࿈ ͢Δཁૉ͕ଟ͘ͳΓ͗ͯ͢ɼൃੜϝΧχζϜͷཧղΛ๦͛ͯ͠·͏ɽ ࣮ࡍɼෳࡶͳਓ޻ࢢ৔Ϟσϧʹରͯ͠ɼϞσϧ͕ෳࡶʹͳΔͱύϥϝʔλ͕ ૿͑ϞσϧͷධՁ͕ࠔ೉ʹͳΔͱ͍͏൷൑͕͋Δ [Chen 12]ɽϞσϧ͕ෳࡶ͢ ͗Δͱؔ࿈͢Δཁૉ͕ଟ͘ͳΓ͗ͯ͢ɼൃੜϝΧχζϜͷཧղΛ๦͛ͯ͠·͏ɽ ·ͨ, ύϥϝʔλ͕૿͑Δ΄Ͳ͞·͟·ͳग़ྗ͕ͩͤΔΑ͏ʹͳΓɼϞσϧΛ ࡞ͬͨਓ͕ಋ͖͍ͨ݁Ռ΁ዞҙతʹಋͨ͘Ίͷύϥϝʔλઃఆ͕ߦΘΕΔڪΕ ͕͋ΔɽγϯϓϧͰύϥϝʔλ͕গͳ͍Ϟσϧ΄Ͳɼύϥϝʔλௐ੔ʹΑͬͯ ಛఆͷ݁Ռʹಋ͘͜ͱ͕ࠔ೉Ͱ͋ΔͨΊධՁ͕༰қͱͳΔɽ Ҏ্ʹΑΓɼຊݚڀͰ͸ɼ෼ੳ໨తΛՌͨͤΔൣғ಺ͰͳΔ΂͘γϯϓϧͳ ϞσϧͷߏஙΛߦ͍ͬͯΔɽ࣮ࡍͷࢢ৔Λ׬શʹ࠶ݱ͢Δ͜ͱΛ໨తͱ͓ͯ͠ ΒͣɼීวతͰ͸ͳ͘ಛఆͷ࣌ظ΍ࢿ࢈ɼ஍ҬͰग़ݱ͢ΔϚΫϩతੑ࣭͢΂ͯΛ ࠶ݱ͢Δ͜ͱ΍, ࣮ࡍʹ͸ଘࡏ͢ΔͰ͋Ζ͏౤ࢿՈΛ͢΂ͯ໢ཏ͢Δ͜ͱ͸͋͑ ͯߦ͍ͬͯͳ͍ɽ

Ϟσϧͷଥ౰ੑݕূ

ਓ޻ࢢ৔Ϟσϧͷଥ౰ੑ͸࣮ূ෼ੳͰಘΒΕ͍ͯΔ fat-tail ΍ volatility-clustering ͱ͍ͬͨ୅දతͳ stylized fact ͕࠶ݱͰ͖Δ͔Ͳ͏͔ͰධՁ͞Ε

ද4: βϥόํࣜ(δt = 1)ɼMMͳ͠ͷ৔߹ͷ֤छ౷ܭྔ ໿ఆ཰ 32.3% औҾ Ωϟϯηϧ཰ 26.1% ஫จ݅਺/ 1೔ 6467 ඪ४ 1ظؒ 0.0512% ภࠩ 1೔ (ΔT = 20000) 0.562% ઑ౓ 1.42 ϥά 1 0.225 ೋ৐Ϧλʔϯͷ 2 0.138 ࣗݾ૬ؔ܎਺ 3 0.106 4 0.087 5 0.075

Δ [LeBaron 06, Chen 12, ࿨ઘ 12, Cristelli 14, Mizuta 16b]ɽ ϑΝοτɾ ςʔϧ͸ɼࢢ৔Ձ֨ͷಅམ཰ͷ෼෍͕ਖ਼ن෼෍Ͱ͸ͳ͘੄͕ް͍ɼ͢ͳΘͪɼઑ ౓͕ਖ਼Ͱ͋Δ͜ͱͰ͋ΔɽϘϥςΟϦςΟɾΫϥελϦϯά͸ࢢ৔Ձ֨ͷಅམ཰ ͷ 2 ৐͕ɼେ͖ͳϥάͰ΋ࣗݾ૬ؔ܎਺͕༗ҙʹਖ਼Ͱ͋Δ͜ͱͰ͋Δɽ [Sewell 11] ͳͲଟ͘ͷݚڀͰड़΂ΒΕ͍ͯΔΑ͏ʹɼۚ༥ࢢ৔͸ෆ҆ఆͰ ͋Γɼ҆ఆతʹɼͲͷΑ͏ͳ࣌ظʹ΋༗ҙʹ؍ଌ͞ΕΔελΠϥΠζυɾϑΝΫ τ͸ϑΝοτɾςʔϧͱϘϥςΟϦςΟɾΫϥελϦϯάͷ 2 ͔ͭ͠ͳ͍ɽ ͔͠΋ɼ͜ΕΒ͸౷ܭྔͷ༗ҙʹਖ਼Ͱ͋Δ͜ͱ͚͕ͩ҆ఆͯ͠؍ଌ͞Εɼ஋ͦ ͷ΋ͷ͸ɼ࣌ظʹΑͬͯҟͳΔɽϑΝοτɾςʔϧʹ͍ͭͯ͸ɼ࣮ূ෼ੳͰΑ͘؍ ଌ͞ΕΔಅམ཰ͷ෼෍ͷઑ౓͸ 1∼ 100 ఔ౓Ͱ͋ΓɼϘϥςΟϦςΟɾΫϥελ Ϧϯάʹ͍ͭͯ͸ɼ࣮ূ෼ੳͰΑ͘؍ଌ͞ΕΔಅམ཰ͷࣗݾ૬ؔ͸ 0.01∼ 0.2 ఔ౓ͱɼ͔ͳΓ͹Β͖͕ͭ͋Δ [Sewell 11]ɽ ຊݚڀͷΑ͏ʹɼۚ༥ࢢ৔ʹڞ௨͢Δੑ࣭Λ෼ੳର৅ͱ͢Δਓ޻ࢢ৔͕࠶ݱ ͢΂͖͸ɼ͜ΕΒͷ౷ܭྔ͕༗ҙʹਖ਼Ͱ͋Γɼ໰୊ͳ͍ൣғʹ஋͕ऩ·͍ͬͯΔ ͜ͱͰ͋ͬͯɼಛఆͷ஋ʹ͚ۙͮΔ͜ͱ͸ຊ࣭తͰ͸ͳ͍ɽ ද 4 ͸ɼβϥόํࣜ (δt = 1)ɼMM ͳ͠ͷ৔߹ͷ౷ܭྔΛࣔͨ͠ɽ౷ܭྔ ͸ 100 ճͷࢼߦͷฏۉ஋Λ༻͍͍ͯΔɽ໿ఆ཰ɼΩϟϯηϧ཰ͱ΋ʹ͍ͣΕͷ ςΟοΫαΠζʹ͓͍ͯ΋࣮ࡍͷࢢ৔ͷ஋ʹ͍ۙ஋Λಘ͓ͯΓɼϞσϧͷଥ౰ੑ Λ͍ࣔࠦͯ͠Δ∗10ɽ1 ೔ (ΔT ) ͝ͱͷಅམ཰ͷඪ४ภࠩ∗11΋ɼ֓Ͷ࣮ࡍͷ ࢢ৔ʹ͍ۙ஋͕ಘΒΕɼ͜ͷଆ໘͔Β΋Ϟσϧͷଥ౰ੑΛ͍ࣔࠦͯ͠Δɽ ද 4 ͸ɼ10 ظؒ͝ͱ∗12ͷಅམ཰Λ༻͍ͯܭࢉͨ͠ઑ౓ͱಅམ཰ͷ 2 ৐ͷ ∗10 ໿ఆ཰ = ໿ఆ݅਺/஫จ݅਺ɼΩϟϯηϧ཰=Ωϟϯηϧ݅਺/(஫จ݅ ਺+Ωϟϯηϧ݅਺)ɽ ∗11 ຊݚڀͷγϛϡϨʔγϣϯͰ͸ΦʔόʔφΠτͷՁ֨มಈ͕ଘࡏ͠ͳ͍ͷ Ͱɼ͜͜Ͱͷ 1 ೔͝ͱͷಅམ཰ͷඪ४ภࠩ͸ɼ࣮ࡍͷࢢ৔ͷΠϯτϥσΠͷ ϘϥςΟϦςΟʹ૬౰͢Δɽ ∗12 ຊϞσϧͷ࣌ࠁ͸஫จΛ͚ͨͩ͠ͰऔҾ͕੒ཱ͠ͳ͍৔߹΋࣌ࠁ͕ਐΉͨ Ίɼ࣌ࠁ 1 ͝ͱͷશͯͷՁ֨Λ༻͍ͨελΠϥΠζυɾϑΝΫτ͸ଟ͘ͷՁ ֨มಈ͕ແ͍σʔλʹΑΓόΠΞε͕͔͔ͬͯ͠·͏ͨΊɼ10 εςοϓִؒ

ࣗݾ૬ؔ΋͍ࣔͯ͠Δɽಅམ཰ͷઑ౓͕ϓϥεͰɼ࣮ূ෼ੳͰΑ͘؍ଌ͞ΕΔ 1∼ 100 ఔ౓ͷൣғʹऩ·͍ͬͯΔɽΏ͑ʹɼϑΝοτɾςʔϧ͕࠶ݱ͞Εͯ ͍Δɽ·ͨɼಅམ཰ͷ 2 ৐ͷࣗݾ૬ؔ΋ϥά͕͋ͬͯ΋ϓϥεͰ࣮ূ෼ੳͰΑ ͘؍ଌ͞ΕΔ 0.01∼ 0.2 ఔ౓ʹऩ·͍ͬͯΔɽΏ͑ʹɼϘϥςΟϦςΟɾΫϥ ελϦϯά͕࠶ݱ͞Ε͍ͯΔͱߟ͑ΒΕΔɽ Ҏ্ʹΑΓɼຊݚڀͷϞσϧ͕ɼ໿ఆ݅਺΍Ωϟϯηϧ཰ɼ̍೔ͷಅམ཰ͷ ඪ४ภࠩͳͲ୹ظͷϚΠΫϩɾετϥΫνϟʔͷੑ࣭͓Αͼɼ௕ظʹ؍ଌ͞Ε ΔՁ֨มಈͷ౷ܭతͳੑ࣭΋࠶ݱ͍ͯ͠Δ͜ͱ͕ࣔ͞ΕͨɽͦΕΏ͑ຊݚڀͷ ໨తͰ͋ΔɼόονΦʔΫγϣϯํࣜʹ͓͍ͯϚʔέοτϝʔΧʔઓུͷଘଓ ՄೳੑΛٞ࿦͢Δͷʹଥ౰ͳϞσϧͰ͋Δ͜ͱ͕ࣔ͞Εͨɽ

ཹҙࣄ߲ͱँࣙ

ຊ࿦จ͸εύʔΫεɾΞηοτɾϚωδϝϯτגࣜձࣾͷެࣜݟղΛද͢΋ ͷͰ͸͋Γ·ͤΜɽ͢΂ͯ͸ݸਓతݟղͰ͋Γ·͢ɽຊݚڀͷҰ෦͸ɼJSTɼ CREST ͓Αͼ JSPS Պݚඅ 15H02745 ͷॿ੒Λड͚ͨ΋ͷͰ͢ɽ

ࢀߟจݙ

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