ࢢ৔΁ͷ਺ֶ͔ΒͷΞ ϓϩʔν (લฤ) ૣҴాେֶ ࠓҪ༔ਓ

֓ཁ લฤͱޙฤͷೋຊཱͯͰߦ͍·͢ ϑΝΠφϯεͱ͸Կ͔ɺԿΛѻ͏ͷ͔ ͳ͔Ͱ΋਺ཧϑΝΠφϯεͱ͸ زԿL´evyաఔͱ֤छϞσϧ ਺஋ղੳతख๏ Local risk-minimization strategy

਺ཧϑΝΠφϯεͱ͸ʁ גՁ΍ҝସϨʔτͳͲͷࢿ࢈Ձ֨Λ֬཰աఔΛ༻͍ ͯϞσϧԽͯ͠ɺͦΕΛجʹ࠷దઓུɺۚ༥೿ੜ঎ ඼(σϦόςΟϒ)ͷՁ֨ɺͦͷଞछʑͷ͜ͱΛղੳ͢ Δ෼໺Λ਺ཧϑΝΠφϯεͱ͍͏ (ؠ೾਺ֶࣙయ4൛) גՁʁɹϞσϧԽʁɹσϦόςΟϒʁ લ൒͸ɺۚ༥ͷجૅతͳ༻ޠͷఆٛͱߟ͑ํΛ঺հ

Ͱ͸ɺϑΝΠφϯεͱ͸ʁ

༨ஊ1 (NPV) key word : কདྷͷ100ສԁ < ࠓͷ100ສԁ NPV := ¯ I + T X t=1 Ct (1 + r)t ɾNPV(Net Present Value) : ׂҾݱࡏՁ஋ I : ॳظ౤ࢿࢿຊɺCt : ୈtظͷcash flow r : ׂҾ཰(ར཰)ɺT : ຬظ NPV>0ͳΒ͹ɺͦͷϓϩδΣΫτ͸࣮ߦ͢΂͖ ໰୊఺ɿׂҾ཰͸Ͳ͏΍ܾͬͯΊΔʁ

ෆ࣮֬ੑͱϦεΫ

ෆ࣮͔֬Β೺ѲՄೳͳʮϦεΫʯ΁

ɾ҆શࢿ࢈(ແϦεΫࢿ࢈)ɿݱۚ΋͘͠͸ݱۚಉ౳෺ ࣮຿తʹ͸ɺ10೥ࠃ࠴͕ࢦඪͱͯ͠༻͍ΒΕΔ ೥རrˋͷ࣌ɺ҆શࢿ࢈͸࿈ଓෳར ɾةݥࢿ࢈(ϦεΫࢿ࢈)ɿ҆શࢿ࢈Ҏ֎ͷࢿ࢈ ۩ମతʹ͸ɺגͳͲͷ҆શࢿ࢈Ҏ֎ͷ΋ͷ ҆શࢿ࢈ͱةݥࢿ࢈ ࢀߟɿ೔ຊࠃ࠴10೥෺ 0.213% ΞϝϦΧࠃ࠴10೥෺ 2.036% υΠπࠃ࠴10೥෺0.472% Ce rt

ϙʔτϑΥϦΦ key word : ཛ͸ҰͭͷόεέοτʹೖΕΔͳ Mean - Variance (MV) approach Markowitz (1927-) Miller, Sharpeͱڞʹ1990೥ϊʔϕϧܦࡁֶ৆ 1952೥ͷֶҐ࿦จʮϙʔτϑΥϦΦબ୒࿦ʯͰఏএ ౤ࢿՈ͸ɺ༩͑ΒΕͨฏۉऩӹΛ༗͢Δ͢΂ͯͷϙʔτ ϑΥϦΦͷதͰ෼ࢄ͕࠷খͱͳΔϙʔτϑΥϦΦΛ࡞੒ ͠อ༗͢΂͖Ͱ͋Δɹ͜ͷࡍɺ෼ࢄΛʮϦεΫʯͱΈΔ

MV approachͷਤΛೖΕΔ

Ծఆɿ(a)ࢿຊࢢ৔͸׬શࢢ৔Ͱ͋Δ (b)ظத͸ಉҰۚརͰແ੍ݶʹିआՄ (c)ظத͸ۚར͕มಈ͠ͳ͍ (d)ظத͸ݪࢿ࢈(underlying)ʹ഑౰͸ͳ͍ (e)ݪࢿ࢈Ձ֨͸ҏ౻աఔʹै͏ Black-Scholes ݱ୅ϑΝΠφϯεཧ࿦ͷத֩Λ઎ΊΔoptionཧ࿦ͷઌۦ͚ ʮۚ༥ࢢ৔ʹԙ͍ͯՁ͕֨ͲͷΑ͏ʹܾ·Δ͔ʯΛཧղ ͢ΔͨΊʹ֬཰ղੳΛಋೖͨ͠ dSt = µStdt + StdWt

ͱ͜ΖͰɾɾɾ ׬શࢢ৔ɿ೚ҙͷࢿ࢈ΛചങՄೳɺ೚ҙͷࢿ࢈͕ۭ ചΓՄೳɺຎࡲ͕ͳ͍(੫ۚ΍औҾඅ༻͕ଘࡏ͠ͳ ͍)ɺݸਓͷߦಈ͕Ձ֨ʹӨڹ͠ͳ͍ɺͲͷࢿ࢈΋ແ ݶʹ෼ׂՄೳ(ϚʔέοτϙʔτϑΥϦΦ͕1ԁͰ΋ߏ ੒Մೳ)ɺࢢ৔ʹ࠷௿ػձ͕ଘࡏ͠ͳ͍…

༨ஊ2 (Market Indexes) ֤ࠃͷࢢ৔Ͱܭࢉ͞Ε͍ͯΔ֤छͷࢢ৔ࢦඪ Nikkei225(೔ܦฏۉ)…౦ূ1෦ͷ͏ͪɺྲྀಈੑͱηΫλʔΛצ Ҋͨ͠255໏ฑͷ୯७ฏۉࢦ਺ TOPIX…౦ূ̍෦શ໏ฑͷ࣌Ձ૯ֹՃॏฏۉɻ68. 1. 4Λ100 NY Daw…ओཁۀछͷ୅දతͳ30໏ฑͰߏ੒͞ΕΔגՁͷ୯७ ฏۉࢦ਺ DAX…ϑϥϯΫϑϧτূ݊औҾॴ্৔ͷυΠπاۀͷ͏ͪ30 ໏ฑͷ࣌Ձ૯ֹՃॏฏۉɻ87 . 12. 31Λ1000

remark:(1)্ࣜΛຬͨ͞ͳ͍f͸ແࡋఆͰ͸ͳ͍ (2)optionՁ֨ͷಋग़͕Մೳ Black-Scholes-Merton equation ೚ҙͷΦϓγϣϯͷՁ֨f(S,t)͸ҎԼͷํఔࣜΛຬͨ͢ @f @t + @f @S rS + 1 2 @2f @S2 2S2 = rf European call optionͷ৔߹ʹ͸໌ࣔతͳղ͕ಘΒΕΔɿ C(S, t) = SN(d1) Ke r(T t)N(d2) d1 = log( S/K ) + ( r + 2/ 2)( T t ) p T t d2 = d1 p T t N͸ඪ४ྦྷੵਖ਼ن෼෍

̑ϲ݄෺ͷίʔϧΦϓγϣϯͰɺݪࢿ࢈ͷגࣜ͸ݱࡏͷ Ձ஋͕62υϧͰϘϥςΟϦςΟ͸೥ؒ20%Ͱ͋ͬͨɻݖ རߦ࢖Ձ֨(ετϥΠΫϓϥΠε)͸60υϧͰɺۚར͸ 10%ͱ͢Δɻ ͭ·ΓɺS_0=62, K=60, σ=0.2, r=0.1. e. g. BSΛ༻͍ͨoptionͷܭࢉ C = 62 ⇥ 0.739332 60 ⇥ 0.95918 ⇥ 0.654740 = 5.798

Φϓγϣϯɺͦͷલʹ σϦόςΟϒ(৚݅෇͖੥ٻݖ):ݪࢿ࢈͕ଘࡏ͠ɺͳΜΒ ͔ͷ৚݅Λ͚ͭͨ݁Ռ೿ੜ͢Δ΋ͷ ۚ༥σϦόςΟϒऔҾ:ઌ෺(ઌ౉)ɺεϫοϓɺΦϓγϣϯ ػೳ:ϦεΫͷίϯτϩʔϧखஈ ݱ෺ͷదਖ਼ͳՁ֨Λ଎΍͔ʹൃݟ ݱ෺ࢢ৔ͷग़དྷߴΛ૿Ճͤ͞ɺྲྀಈੑΛߴΊΔ

ઌ෺ͱ͸ ઌ෺औҾͱ͸ɺ(1) কདྷͷ༧ΊఆΊΒΕͨظ೔ʹ(2)ಛఆ ͷ঎඼ʢݪࢿ࢈ʣΛ(3)ݱ࣌఺ͰऔΓܾΊͨՁ֨Ͱചങ ͢ΔࣄΛ໿ଋ͢ΔऔҾͷ͜ͱ কདྷͷՁ֨มಈϦεΫΛճආͰ͖Δ ۃ୺ͳྫ:̒ϲ݄ઌ෺ݪ༉ͷઌ෺ܖ໿Λߟ͑Δɻݪ༉Ձ͕֨30.43($/bbl;1/ 19)ͳͷͰɺ͜ͷ··஋Լ͕Γ͢Δͱͯ͠30($/bbl)Ͱઌ෺ͷʮചΓʯܖ໿ ΛަΘ͢ ̒ϲ݄ޙ…25$·ͰԼམ͍ͯͨ͠ˠ25$Ͱݱ෺Λങͬͯ30$ͰചΕ͹5$ͷ Ṷ͚ɹ 35$ʹ্ঢ͍ͯͨ͠ˠ35$Ͱݱ෺Λങͬͯ30$ͰചΔͷͰ5$ͷଛ ઌ෺ܖ໿͸ɺܖ໿ͷཤߦ͕ٛ຿Ͱ͋Δ

Φϓγϣϯ ઌ෺औҾͱ͸ɺ(1) কདྷͷ༧ΊఆΊΒΕͨظ೔ʹ(2)ಛఆͷ ঎඼ʢݪࢿ࢈ʣΛ(3)ݱ࣌఺ͰऔΓܾΊͨՁ֨Ͱചങ͢Δ ݖརͷ͜ͱ ۃ୺ͳྫ:̒ϲ݄ઌ෺ݪ༉ͷΦϓγϣϯܖ໿Λߟ͑Δɻݪ༉Ձ͕֨ 30.43($/bbl;1/19)ͳͷͰɺ͜ͷ··஋Լ͕Γ͢Δͱͯ͠30($/bbl)Ͱϓοτ Φϓγϣϯ(ചΔݖར)Λܖ໿ ̒ϲ݄ޙ…25$·ͰԼམ͍ͯͨ͠ˠ25$Ͱݱ෺Λങͬͯ30$ͰചΕ͹5$ͷ Ṷ͚ɹ 35$ʹ্ঢ͍ͯͨ͠ˠΦϓγϣϯͷݖརΛߦ࢖͠ͳ͍ ઌ෺ͱҧ͍ɺٛ຿Ͱ͸ͳ͘ݖརͳͷͰߦ࢖͠ͳͯ͘΋ྑ ͍

ΦϓγϣϯͷՁ֨ΛͲͷΑ͏ʹܾΊͨΒྑ͍ͷͩΖ͏͔ ਤͰඳ͘ͱ ίʔϧ ϓοτ (St K)+ (K St)+ རಘ ݪࢿ࢈ͷকདྷՁ֨ C P + e r(T t)K = S

༷ʑͳΦϓγϣϯ ϤʔϩϐΞϯ:ຬظ೔ʹ͓͍ͯͷΈݖརߦ࢖Մೳ ΞϝϦΧϯ:ຬظ೔·Ͱͷ೚ҙͷ࣌఺Ͱݖརߦ࢖Մೳ όϦΞ:όϦΞՁ֨ʹҰ౓Ͱ΋ୡ͢Δͱɺݖརߦ࢖Մೳ(ϊοΫ Πϯ)ɺόϦΞՁ֨ʹҰ౓΋ୡ͠ͳ͚Ε͹ߦ࢖Մೳ(ϊοΫΞ΢ τ) ΞδΞϯ:ຬظ࣌ͷΦϓγϣϯՁ஋͕ɺຬظ೔·ͰͷฏۉՁ֨ʹ ج͍ͮͯࢉఆ͞ΕΔ σδλϧ:ຬظ೔ʹΠϯɾβɾϚωʔʹͳͬͨ৔߹ʹ͸ݻఆ͞Ε ͨՁ஋Λ࣋ͪɺຬظ೔ʹΦ΢τɾΦϒɾβɾϚωʔ·ͨ͸Ξο τɾβɾϚωʔͩͬͨ৔߹͸શ͘Ձ஋Λ࣋ͨͳ͍

Φϓγϣϯઓུͷྫ1

Φϓγϣϯઓུͷྫ2

Wall streetͰ࠷΋༗໊ͳ਺ֶऀ ͓ ஌ Β ͤ ɹຊֶ໊༪ڭतɼҏ౻ਗ਼ࡾઌੜʢཧֶ ෦਺ֶՊɼ౰࣌ʣ͸ ೥ ݄ ೔ʹ ࡀͰ੦ڈ͞Ε·ͨ͠ɻઌੜ͸ ೥ʹ໊ݹ԰େֶཧֶ෦਺ֶՊΛଔ ۀ໊͠ݹ԰େֶཧֶ෦ॿखɼߨࢣΛܦ ͯɼ ೥ʹຊֶཧֶ෦ߨࢣʹண೚͞ Ε ೥ʹఆ೥ୀ׭͞ΕΔ·Ͱ਺ֶڭ ࣨʹ͓͍ͯղੳֶͷڭҭݚڀʹ͝ਚྗ͞ Ε·ͨ͠ɻ ɹઌੜ͸ɼ֦ࢄํఔࣜͷڥք஋໰୊ɼφ ϏΟΤɾετʔΫεํఔࣜͷॳظ஋໰୊ ͳͲͷݹయղͷଘࡏɾҰҙੑͷূ໌ɼପ ԁܕภඍ෼ํఔࣜͷཧ૝ڥքͷݚڀͳͲ ภඍ෼ํఔࣜʹؔͯ͠ݚڀۀ੷Λڍ͛Β Ε·ͨ͠ɻڭҭͰ͸ɼઐ໳Ͱ͸ͳ͍֬཰ աఔ࿦ͷߨٛ΍ηϛφʔ΋୲౰͞ΕΔͳ Ͳɼղੳֶશൠʹ͓͚Δݚڀऀͷҭ੒ʹ ਚྗ͞Ε·ͨ͠ɻஶॻ΋ଟ͘ɼ ೥ ʹग़൛͞Εͨʮϧϕʔάੵ෼ೖ໳ʯ͸ܦ ࡁֶΛؚΉ͞·͟·ͳՊֶ෼໺ͷݚڀऀ ΍ֶੜʹࠓͳ͓ಡ·Εଓ͚ΒΕ͍ͯ·͢ɻ ɹઌੜ͸Ժ΍͔ͳํͰ͕ͨ͠ؤݻͳҰ໘ ΋͋ΓɼֶੜେձͷͨΊͷߨٛதࢭͷཁ ੥ֶ͕ੜ࣏ࣗձ͔Β͋ͬͨ࣌ɼ ʮͦͷ೔ ͷߨٛΛௌ͔ͳͯ͘΋ࢧো͕ͳ͍Α͏ʹɼ ҧ͏಺༰ͷߨٛΛҰճ͚ͩߦ͏ʯͱߨٛ ͸தࢭ͠ͳ͔ͬͨ͜ͱ͕͋Γ·ͨ͠ɻ· ͨɼීஈ͸ֶੜʹਅ໘໨ͳҰ໘͔͠ݟͤ ͳ͍ઌੜͰ͕ͨ͠ɼ࠙਌ձͰ౰࣌ͷ੓࣏ ՈΛൽ೑ͬͨࣗ࡞ͷସ͑ՎΛ൸࿐͞ΕΔ ೇԬɹ੒༤ʢ਺ཧՊֶݚڀՊɹڭतʣ ҏ౻ਗ਼ࡾઌੜͷ͝੦ڈΛౣΉ ҏ౻ਗ਼ࡾઌੜͷ͝੦ڈΛౣΉ ނɾҏ౻ਗ਼ࡾઌੜ ͜ͱ΋͋Γ·ͨ͠ɻ ɹֶੜࢥ͍Ͱڭҭ೤৺Ͱ͋ͬͨҏ౻ઌੜ ͷ້͝෱Λ৺ΑΓ͓فΓ͍ͨ͠·͢ɻ {fi (t)|t ∈ [0, ∞)}, i = 0, · · · , d Λ 2 ৐Մੵ෼ͳద߹ͨ͠࿈ଓ֬཰աఔɺx ∈ R ͱ͢Δɻ ·ͨɺ֬཰աఔΛ X(t) := x + d i=1 t 0 fi (s)dBi (s) + t 0 f0 (s)ds ͰఆΊΔɻࠓɺF : R → R ͸ 2 ճඍ෼Մೳͳؔ਺Ͱɺͦͷඍ෼ F′, F′′ ͸༗ք͔ͭ࿈ଓͰ ͋Δͱ͢Δɻ͜ͷ࣌ɺ͕࣍ࣜ੒ཱ͢Δ: F(X(t)) − F(x) = d i=1 t 0 (F′(X(s))fi (s))dBi (s) + t 0 F′(X(s))f0 (s)ds + 1 2 d i=1 t 0 F′′(X(s))fi (s)2ds ҏ౻ͷLemma

LTCMͷࣦഊ ݩιϩϞϯϒϥβʔζͷ࠴ݖτϨʔμʔɺϝϦ΢Σβʔ ͕ൃىਓͱͳΓઃཱ(94-00) M. γϣʔϧζ΍R. ϚʔτϯΒ͕ࢀը,12.5ԯUS$ͷ౰ॳࢿ ۚ γϯϓϨοΫε๏ͳͲΛ༻͍ͨ޿ٛͷࡋఆऔҾΛத৺ͱ ͨۚ͠༥޻ֶతख๏ ౰ॳͷ̐೥ؒ͸ฏۉ೥཰40%Λ௒͑ΔϦλʔϯ ΞδΞ௨՟ةػɺϩγΞةػʹΑΓ࣮࣭తʹഁ୼ ϩγΞͳͲͷ৽ڵࢢ৔Ͱ4.3ԯ$ ૬৔ͷ্Լʹ௚઀Ṍ͚ͨϙδγϣϯͰ3.71ԯ$ ϖΞτϨʔυͰ3.06ԯ$ גࣜࢦ਺ɾϘϥςΟϦςΟɾτϨʔυͰ13.12ԯ$ ࠴݊ࡋఆऔҾͷ16.28ԯ$(LTCMͷίΞऔҾʣ ܦࡁةػϦεΫͷݟੵ΋Γʮɹɹɹɹʯͩͬͨͱ͔(98.8Ͱ˛21ԯ$) 10 24

. . . . . . ਺ཧϑΝΠφϯεͰ͸ԿΛ࿦͡Δͷ͔?(Black-Scholes ϞσϧΛத৺ʹ) ੥ٻݖͷՁ֨෇͚͓Αͼϔοδͱ͸ Ұͭͷ҆શࢿ࢈ͱҰͭͷג͔ࣜΒ੒Δۚ༥ࢢ৔Λߟ͑Δɽ ݱ࣌఺Λ 0ɼຬظΛ T ͱ͢Δ࿈ଓ࣌ؒϞσϧΛߟ͑Δɽ ؆୯ͷͨΊɼۚར͸ߟ͑ͳ͍΋ͷͱ͢Δɽͭ·Γɼ҆શࢿ࢈ͷՁ֨͸ৗʹ 1 ͱ ͢Δɽ ͞ΒʹɼגࣜͷՁ֨͸ηϛϚϧνϯήʔϧ St Ͱ༩͑ΒΕΔ΋ͷͱ͢Δɽ ͜ͷͱ͖ɼ੥ٻݖ X Λ࣌ࠁ 0 Ͱച٫͠Α͏ͱ͢Δ౤ࢿՈ͸ɼ ͍͘ΒͰ X Λച٫͠ɼͲͷΑ͏ʹͯ͠কདྷͷϥϯμϜͳࢧ෷͍ʹඋ͑Δ΂͖͔?

. . . . . . ਺ཧϑΝΠφϯεͰ͸ԿΛ࿦͡Δͷ͔?(Black-Scholes ϞσϧΛத৺ʹ) ੥ٻݖͷՁ֨෇͚͓Αͼϔοδͱ͸ (cont’d) ΋͠ɼ X = c + T 0 ϑt dSt (1) Λຬͨ͢ c ∈ R ͱՄ༧ଌաఔ ϑ ͕ݟ͔ͭΕ͹ɼͦΕͧΕ͕ X ͷՁ֨ͱϔοδઓ ུʹͳΔɽ . ஫ҙ . . . . . . . . . . . 1 ۚརΛߟྀ͢Δ৔߹ɼ(1) ͸΋͏গ͠ෳࡶʹͳΔɽ ͜Εʹؔͯ͠͸ޙఔগ͚ͩ͠ݴٴ͢Δɽ . . . 2 S ͕ L´ evy աఔͷΑ͏ʹ jump ΛؚΉ৔߹͸ɼ(1) ͷΑ͏ͳදݱΛຬͨ͢ (c, ϑ) ͸ଘࡏ͠ͳ͍ɽ ͜ͷΑ͏ͳඇ׬උࢢ৔Ϟσϧͷ৔߹͸৭ʑͱ஫ҙͱ޻෉͕ඞཁͰ͋Δɽ Ұͭ͸ɼϚϧνϯήʔϧଌ౓͕ෳ਺ଘࡏ͢Δ͜ͱͰ͋Δɽ ΋͏Ұͭ͸ɼ࠷దϔοδઓུΛͲ͏ଊ͑Δ͔ͱ͍͏໰୊Ͱ͋Δɽ

. . . . . . ਺ཧϑΝΠφϯεͰ͸ԿΛ࿦͡Δͷ͔?(Black-Scholes ϞσϧΛத৺ʹ) Black-Scholes Ϟσϧͷ঺հ Black-Scholes Ϟσϧͱ͸ Ұͭͷ҆શࢿ࢈ͱҰͭͷג͔ࣜΒ੒Δۚ༥ࢢ৔Λߟ͑Δɽ ݱ࣌఺Λ 0ɼຬظΛ T ͱ͢Δ࿈ଓ࣌ؒϞσϧΛߟ͑Δɽ ҆શࢿ࢈ͷՁ֨Λ࣌ࠁ 0 Ͱ 1 ԁΛۜߦʹ༬͚ͨ࣌ͷ࢒ߴͱߟ͑Δɽ ͭ·Γɼۜߦ༬ۚͷۚརΛ r(≥ 0) ͱ͢Δ ࣌ࠁ t ʹ͓͚Δ҆શࢿ࢈ͷՁ֨͸ ert ͱͳΔɽ Ұํɼ࣌ࠁ t ʹ͓͚ΔגՁ St ͸ɼ࣍ͷ֬཰ඍ෼ํఔࣜͷղ: dSt = µSt dt + σSt dWt , S0 > 0. ͜͜ͰɼS0 > 0, µ ∈ R, σ > 0, W ͸ϒϥ΢ϯӡಈͱ͢Δɽ St = S0 exp µ − 1 2 σ2 t + σWt ͱ΋ॻ͚Δɽ

. . . . . . ਺ཧϑΝΠφϯεͰ͸ԿΛ࿦͡Δͷ͔?(Black-Scholes ϞσϧΛத৺ʹ) Black-Scholes Ϟσϧͷ঺հ Ձ֨ͱϔοδ ຬظ࣌ࠁ T Ͱͷࢧ෷͍͕ f(ST ) Ͱ͋Δ੥ٻݖΛߟ͑Δɽ ྫ͑͹ɼf(x) = (x − K)+ ͳΒ͹ίʔϧΦϓγϣϯͰ͋Δɽ ͜ͷ੥ٻݖʹର͢ΔՁ֨Λܭࢉ͢ΔͨΊʹɼϦεΫதཱଌ౓ (ಉ஋Ϛϧνϯήʔ ϧଌ౓) Λಋೖ͢Δɽ ͦΕ͸ɼҎԼͰఆٛ͞ΕΔ֬཰ଌ౓ Q Ͱ͋Δ: dQ dP = exp ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − 1 2 µ − r σ 2 T − µ − r σ WT ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ Q ͷԼͰɼׂҾגՁաఔ e−rt St ͸ Q-ϚϧνϯήʔϧʹͳΔɽ ੥ٻݖ f(ST ) ͷ࣌ࠁ t ʹ͓͚ΔՁ֨ Ct ͸ Ct = e−r(T−t)EQ [f(ST )|Ft ] Ͱ༩͑ΒΕΔɽ

. . . . . . ਺ཧϑΝΠφϯεͰ͸ԿΛ࿦͡Δͷ͔?(Black-Scholes ϞσϧΛத৺ʹ) Black-Scholes Ϟσϧͷ঺հ Ձ֨ͱϔοδ (cont’d) ͜ͷ੥ٻݖΛ࣌ࠁ 0 Ͱച٫ͨ͠౤ࢿՈͷϔοδઓུͰ͋ΔՄ༧ଌաఔ ϑ ͸ e−rT f(ST ) = C0 + T 0 ϑt d(e−rt St ) Λຬͨ͢ɽ ͜ͷ ϑ ͸σϧλϔοδͱݺ͹ΕΔ΋ͷͰ͋Γɼ ϑt = ∂Ct ∂St Ͱ༩͑ΒΕΔɽ

. . . . . . ਺ཧϑΝΠφϯεͰ͸ԿΛ࿦͡Δͷ͔?(Black-Scholes ϞσϧΛத৺ʹ) Volatility ͷਪఆ Implied volatility ຬظ Tɼߦ࢖Ձ֨ K ͷίʔϧΦϓγϣϯʹର͢Δ Black-Scholes ެࣜΛɼ volatility σ ͷؔ਺ͱݟͳͯ͠ C(σ; T, K) ͱॻ͘ɽ Implied volatility σIV ͸ɼ࣮ࡍʹ؍ଌ͞ΕͨՁ֨ ¯ C0 Λ༻͍ͯ ¯ C0 = C(σIV ; T, K) Ͱ༩͑ΒΕΔɽ σIV ͸ ( ¯ C0 ΋)T ͱ K ͷؔ਺Ͱ͋Δ͜ͱʹ஫ҙɽ ΋͠ Black-Scholes Ϟσϧ͕ຊ౰ʹਖ਼͍͠ͷͰ͋Ε͹ σIV ͸ఆ਺ʹͳΔ͸ͣͰ ͋Δɽ

. . . . . . ਺ཧϑΝΠφϯεͰ͸ԿΛ࿦͡Δͷ͔?(Black-Scholes ϞσϧΛத৺ʹ) Volatility ͷਪఆ 0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 1.6 0.2 0.22 0.24 0.26 0.28 0.3 0.32 Time to maturity Average profile of implied volatility surface Moneyness Implied volatility FIGURE 1.6: Implied volatilities of DAX index options, 2001.

. . . . . . ਺ཧϑΝΠφϯεͰ͸ԿΛ࿦͡Δͷ͔?(Black-Scholes ϞσϧΛத৺ʹ) Volatility ͷਪఆ 0 0.5 1 1.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 Time to maturity Average profile of implied volatility surface Moneyness Implied volatility 0 50 100 150 200 250 300 350 400 450 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 FIGURE 1.7: Left: The implied volatility surface for S&P 500 options. Right: At-the-money skew ∂Σ ∂K (K = S, T) as a function of the maturity T (in days).

. . . . . . ਺ཧϑΝΠφϯεͰ͸ԿΛ࿦͡Δͷ͔?(Black-Scholes ϞσϧΛத৺ʹ) ਺ཧϑΝΠφϯεʹ͓͚ΔϞσϧͷ֦ு ਺ཧϑΝΠφϯεʹ͓͚ΔϞσϧͷ֦ு ࣮σʔλΛ༻͍ͨ෼ੳ͔Βܦݧతʹ஌ΒΕ͍ͯΔגՁաఔͷಛ௃Λ stylized facts ͱ͍͏ɽ Volatility smile ΍ volatility skew ͸ͦ͏͍ͬͨ΋ͷͷҰͭͰ͋Δɽ ͜͏ͨ͠ stylized facts ͷଟ͘͸ɼBlack-Scholes ϞσϧͰ͸આ໌Ͱ͖ͳ͍ɽ ͦ͜Ͱɼదٓ Black-Scholes ϞσϧΛ֦ுͯ͠ɼΑΓݱ࣮తͱࢥΘΕΔϞσϧ ͷߏங͕ߦΘΕΔɽ ͜ͷ֦ுʹ͸༷ʑͳ΋ͷ͕͋Δ͕ɼେ͖ͳྲྀΕͱͯ͠ . . . 1 Stochastic volatility model . . . 2 Jump ΛؚΜͩϞσϧ ͷ 2 ͕ͭڍ͛ΒΕΔɽ Ҏޙɼ͜͜Ͱ͸ओʹޙऀʹয఺Λ͋ͯΔɽ

Numerical Approaches to Finance Monte Carlo ๏ quasi-Monde Carlo ๏ Fourier Transform Laplace Transform ࠷దԽ๏ (Nelder-Mead, Levenberg-Marquardt, SQP, · · · ) multinomial tree/lattice Explicit/Implicit Finite Difference Scheme Finite Elements Wavelet-Galerkin ͳͲͳͲ

࿦จ Numerical analysis on local risk-minimization for exponential L´ evy models (1) زԿ L´ evy Ϟσϧʹରͯ͠ɺCall option ͷ LRM Λ਺஋ܭࢉ Մೳͳදݱͱͯ͠ಋग़ (2) Merton model, Variance Gamma model Λྫʹ਺஋ܭࢉΛ ࣮ߦ Comparison of Local Risk Minimization and Delta Hedging for Exponential L´ evy Models (1) ্هͷ࿦จͷ࿮૊ΈͰ Delta hedge Λఆٛ (2) Merton model, VG model Λྫʹ (2-1) |LRM − ∆P∗ | ͷෆ౳ࣜධՁΛ༩͑ (2-2) |LRM − ∆P∗ | ʹ͍ͭͯ਺஋ܭࢉΛ࣮ߦ

༻͍ͨ਺஋ܭࢉख๏ Carr-Madan method ߴ଎ Fourier ม׵Λ༻͍ͯ call option ͷՁ֨Λಋग़͢Δख๏ model → Merton models, variance Gamma models VG Ͱ͸ࢢ৔σʔλ͔ΒϞσϧύϥϝʔλΛਪఆ ύϥϝʔλਪఆํ๏ → Ϟʔϝϯτ๏ٴͼ Levenberg-Marquardt ๏ ࢢ৔σʔλ → Nikkei225 2014 ೥ 3 ݄ऴ஋ (೔࣍)

લఏ ඇ׬උࢢ৔ʹ͓͚Δ৚݅෇͖੥ٻݖͷՁ֨෇͚ٴͼ࠷దϔοδઓུ ɾ׬උࢢ৔ →(1) ׬શෳ੡ઓུ͕ଘࡏ (2) औҾඅ༻ͳͲͷຎࡲ͕ଘ ࡏ͠ͳ͍ ແࡋఆ৚݅Λຬͨ͢׬උࢢ৔Ͱ͸ɺಉ஋Ϛϧνϯήʔϧଌ౓ (∗) ͕ ͨͩҰͭ ଘࡏͦ͠ͷԼͰͷ৚݅෇͖੥ٻݖͷظ଴஋͕Ձ֨Λ༩ ͑Δ ɾඇ׬උࢢ৔Ͱ͸ɺෳ੡ઓུ͕ଘࡏͤͣɺಉ஋Ϛϧνϯήʔϧଌ ౓͕ ແ਺ʹ ଘࡏ͢ΔͨΊɺແࡋఆ৚݅ԼͰͷՁ֨͸Ұҙʹ͸ఆ· Βͳ͍ → ԿΒ͔ͷҙຯͰʮ࠷దʯͳϔοδઓུΛߟ͑ɺͦͷॳظඅ༻Λ Ձ֨ͱΈͳ͢ (∗ ࢿ࢈աఔΛϚϧνϯήʔϧʹ͢Δଌ౓)

લఏ ඇ׬උࢢ৔ʹ͓͚Δ৚݅෇͖੥ٻݖͷՁ֨෇͚ٴͼ࠷దϔοδઓུ ຊ࿦จͰ͸ɺגՁ͕δϟϯϓΛؚΉΑ͏ͳ֬཰աఔɺͱΓΘ͚ L´ evy աఔʹΑͬͯۦಈ͞ΕΔ SDE ͷղͰهड़͞ΕΔඇ׬උࢢ৔Ϟ σϧΛର৅ͱ͢Δ ·ͨɺ ʮ࠷దʯͳϔοδઓུͱͯ͠ Local risk–minimization strategy Λߟ͑Δ

LRM (General case) ξt ͱ ηtɿͦΕͧΕגͱ҆શࢿ࢈ͷ࣌ࠁ t ʹ͓͚Δอ༗ྔ ϖΞ ϕ := (ξt, ηt ) ΛઓུͱݺͿ ౤ࢿՈͷ෋ (རಘ)ɿVt = ηt + ξt St ͜ͷ࣌ɺίετաఔ Ct (ϕ) ͸ Ct (ϕ) = Vt (ϕ) − t 0 ξs dSs Ͱ༩͑ΒΕΔ ϦεΫաఔ Rt (ϕ) ΛҎԼͰఆٛ͢Δɿ Rt (ϕ) := E (CT (ϕ) − Ct (ϕ))2 Ft ϕ ͕ risk-minimization Ͱ͋Δͱ͸ɺ͋ΒΏΔઓུͷதͰͦͷϦεΫ աఔ͕࠷খʹͳΔ΋ͷΛݴ͏ɻͭ·Γɺ೚ҙͷෳ੡ઓུ ϕ ʹରͯ͠ Rt (ϕ) ≤ Rt (ϕ) P-a.s. for every t ∈ [0, T] Λຬͨ͢ઓུͷ͜ͱͰ͋Δ

LRM (General case) Ҏ্͕ risk-minimization ͷఆٛͰ͋Δ͕ɺࢿ࢈Ձ֨աఔ St ͕ semi-martingale Ͱ͋Δ৔߹ʹ͸ risk-minimization ͕ ଘࡏ͢Δͱ͸ݶΒͳ͍ ͦ͜ͰɺSchweizer ͸ఆٛΛվྑ͠ semi-martingale ͷ৔߹Ͱ΋ར ༻Մೳͳ local risk-minimizing ͷߟ͑ํΛఏএͨ͠ ͔͜͠͠ͷ࿦จͷఆٛ͸ۃΊͯෳࡶͰ͋ΔͨΊɺ M. Schweizer (2008) Local Risk-Minimization for Multi-dimensional Assets and Payment Streams, Banach Center Publications 83, 213–229. ͷ Theorem 1.6 Λ LRM ͷఆٛͱΈͳ͢ (cf. ࣍ͷεϥΠυ) ͜ͷ Theorem ͸ɺLRM ͷඞཁे෼৚݅Λ༩͑Δ΋ͷͰ͋Δ:

LRM (General case) ఆٛ (LRM by Schweizer(2008)) 1 ΘS ͸ɼҎԼͷ৚݅Λຬͨ͢ R-஋Մ༧ଌաఔ ξ ͷू߹ͱ͢Δ: E T 0 ξ2 t d⟨M⟩t + ( T 0 |ξt dAt |)2 < ∞. 2 ξ ∈ ΘS ͱద߹աఔ η ͷ૊ ϕ = (ξ, η) ͕ L2-ઓུ Ͱ͋Δͱ͸ɼ V(ϕ) := ξS + η ͕ӈ࿈ଓաఔͰ͋Γɼ೚ҙͷ t ∈ [0, T] ʹର͠ ͯ E[V2 t (ϕ)] < ∞ Ͱ͋Δ࣌ʹݴ͏ɽ 3 F ∈ L2(P) ʹରͯ͠, ֬཰աఔ CF(ϕ) Λ CF t (ϕ) := F1{t=T} + Vt (ϕ) − t 0 ξs dSs ʹΑΓఆٛ͠ɼϕ = (ξ, η) ͷ F ʹର͢ΔίετաఔͱݺͿ. 4 L2-ઓུ ϕ ͕ F ʹର͢Δ LRM Ͱ͋Δͱ͸ɼVT (ϕ) = 0 Ͱ͋Γɼ CF(ϕ) ͕ M ͱ௚ަ͢Δ martingaleɼͭ·Γɼ[CF(ϕ), M] ͕Ұ ༷Մੵ෼ martingale Ͱ͋Δͱ͖ʹݴ͏ɽ

Fast Fourier Transform FFT : ࣍ͷܗͰ༩͑ΒΕͨ཭ࢄ Fourier ม׵Λߴ଎ʹܭࢉ͢Δख๏ (Cooley-Tukey, 1965) F(l) = N−1 j=0 e−i2π N jlxj (l = 0, · · · , N − 1, where {xj }j=0,··· ,N−1 ∈ R) N ͸ 2 ͷႈ৐ ܭࢉίετɿ ཭ࢄ Fourier ม׵: O(N2) FFT: O(N log 2 N)

Carr–Madan method Carr-Madan method: S ͕ P-martingale ͷͱ͖ɺE[(ST − K)+] Λߴ ଎ʹܭࢉ͢Δख๏. (Carr-Madan, 1999) k := log K, C(k) := E[(ST − ek)+] ͱ͢Δͱɺ C(k) = 1 π ∞ 0 e−i(v−iα)k φ(v − iα − i) i(v − iα)(i(v − iα) + 1) dv α > 0, φ: LT (= log ST ) ͷ ಛੑؔ਺ NOTE α ʹ͸ґΒͳ͍.

Carr–Madan method ୆ܗଇͱ Simpson ଇΛ༻͍ͯ C(k) ≈ e−αk π N−1 j=0 e−i2π N jleiπjψ(ηj − iα) η 3 (3 + (−1)j+1 − δj ), N : ֨ࢠ఺ͷ਺, η > 0 : ֨ࢠ఺ؒڑ཭, l := k + π η Nη 2π . ·ͨɺψ(z) := φ(z−i) iz(iz+1) ੵ෼۠ؒ −→ [0, Nη]. N ͱ η ͸ɺ࣍ࣜΛຬͨ͢Α͏ʹ্ख͘બͿ: 1 π ∞ Nη e−i(v−iα)kψ(v − iα)dv < ε ͜͜Ͱɺε > 0 ͸ڐ༰ޡࠩ.

ઃఆ S → exponential L´ evy process St := S0 exp µt + σWt + R0 xN([0, t], dx) , t ∈ [0, T] ͜͜Ͱɺ S0 > 0, µ ∈ R, σ > 0, R0 := R \ {0}. N ; Poisson random measure, N ; compensated measure of N id est N(dt, dx) := N(dt, dx) − ν(dx)dt. ͜͜Ͱ ν ͸ L´ evy measure ҰൠੑΛࣦΘͣʹɺ S0 = 1 ͱͯ͠Α͍. S ͸࣍ͷ SDE ͷղͰ΋͋Δ: dSt = St− µSdt + σdWt + R0 (ex − 1)N(dt, dx) ͜͜Ͱɺ µS := µ + σ2 2 + R0 (ex − 1 − x)ν(dx). ∀t ∈ [0, T] ʹ͍ͭͯ Lt := log St ͱॻ͘ͱ, L ͸ L´ evy process.

ઃఆ ͜͜Ͱ͸ɺMinimal Martingale Measure P∗ Λ࣍Ͱ༩͑Δ: dP∗ dP = exp − ξWT − ξ2 2 T + R0 log(1 − θx )N([0, T], dx) + T R0 θxν(dx) , ͜͜Ͱɺξ := µSσ σ2+ R0 (ex−1)2ν(dx) , θx := µS(ex−1) σ2+ R0 (ex−1)2ν(dx) (x ∈ R0 ). Lt ͸ɺ࣍ͷΑ͏ʹॻ͖׵͑ΒΕΔ: Lt = µ∗t + σWP∗ t + R0 xNP∗ ([0, t], dx), ͜͜Ͱɺ µ∗ := −σ2 2 + R0 (x − ex + 1)(1 − θx )ν(dx). Note L ͸ɺP∗ ͷԼͰ΋ L´ evy process. L´ evy ଌ౓͸ ∗

ઃఆ Ծఆ 1 R0 x2ν(dx) < ∞; R0 (ex − 1)nν(dx) < ∞ for n = 2, 4. ⇒ L´ evy process L ͸ L2 Ͱ༗ք µS, ξ, θx ͸ well-defined R0 (ex − 1)nν(dx) < ∞ for n = 1, 3 2 0 ≥ µS > −σ2 − R0 (ex − 1)2ν(dx). ⇒ θx < 1 for any x ∈ R0

ઃఆ LRMt := ξ(ST−K)+ t , I1 := EP∗ [1{ST>K} ST |Ft− ], I2 := R0 EP∗ [(ST ex − K)+ − (ST − K)+|Ft− ](ex − 1)ν(dx). ໋୊ A representation of LRM for call option (ST − K)+ ∀K > 0, ∀t ∈ [0, T] ʹରͯ͠ LRMt = σ2I1 + I2 St− σ2 + R0 (ex − 1)2ν(dx) . I , I ͷ۩ମతͳੵ෼දࣔʹ͍ͭͯ͸ޙ΄Ͳɾ ɾ ɾ

ੵ෼දࣔ Proposition 2 (Tankov, 2010) Λ༻͍ͯɺI1 (:= EP∗ [1{ST>K} ST |Ft− ]) ͷ ੵ෼ද͕ࣔٻ·Δ. I1 ͷੵ෼දࣔ φT−t (z) := EP∗ [eizLT−t ], ∀K > 0 ʹରͯ͠, EP∗ [1{ST>K} · ST |Ft− ] = 1 π ∞ 0 K−iv−α+1 α − 1 + iv φT−t (v − iα)Sα+iv t− dv. (∀t ∈ [0, T], ∀α ∈ (1, 2], z ∈ C) ͭ·Γɺk := log(K), ψ1 (z) := φT−t(z)Siz t− iz−1 ͱ͓͚͹ɺ I1 = EP∗ [1{ST>K} · ST |Ft− ] = ek π ∞ 0 e−i(v−iα)kψ1 (v − iα)dv. ⇒ FFT ͰܭࢉͰ͖Δ

ੵ෼දࣔ I2 (:= R0 EP∗ [(ST ex − K)+ − (ST − K)+|Ft− ](ex − 1)ν(dx)) ΋ಉ༷ EP∗ [(ST − K)+|Ft− ] ͷੵ෼දࣔ ∀K > 0 ʹରͯ͠, EP∗ [(ST − K)+|Ft− ] = 1 π ∞ 0 K−iv−α+1 φT−t (v − iα)Sα+iv t− (α − 1 + iv)(α + iv) dv, for any t ∈ [0, T] and any α ∈ (1, 2]. ψ2 (z) := φT−t(z)Siz t− (iz−1)iz , ζ := v − iα ͱ͓͚͹, EP∗ [(ST − K)+|Ft− ] = 1 π ∞ 0 K−iζ+1ψ2 (ζ)dv =: f(K). ⇒ f(K) ͸ FFT ͰܭࢉͰ͖Δ

I2 = R0 EP∗ [(ST ex − K)+ − (ST − K)+|Ft− ](ex − 1)ν(dx) = R0 exf(e−xK) − f(K) (ex − 1)ν(dx) = R0 ex π ∞ 0 (Ke−x)−iζ+1ψ2 (ζ)dv − 1 π ∞ 0 K−iζ+1ψ2 (ζ)dv (ex − 1)ν(dx) = R0 1 π ∞ 0 (eiζx − 1)K−iζ+1ψ2 (ζ)dv (ex − 1)ν(dx) = 1 π ∞ 0 K−iζ+1 R0 (eiζx − 1)(ex − 1)ν(dx)ψ2 (ζ)dv, (1) Remark ҎԼɺα ∈ (1, 2] Λ೚ҙͷఆ਺ͱͯ͠ݻఆ͢Δ ·ͨɺζ Λ v ͷؔ਺ͱͯ͠ΈΔ : ζ = v − iα

Merton models Black–Scholes model ʹɺͦͷ··δϟϯϓΛՃͨ୯७ͳϞσϧΛ ߟ͑Δ σ > 0 ͱͯ͠ɺ ν(dx) = γ √ 2πδ exp − (x − m)2 2δ2 dx. (ν = ν[γ, m, δ]) ͜ͷ࣌ɺSt Λ Merton jump diffusion ͱݺͿ ͜Ε͸ Merton(1976) Ͱఏএ͞Εͨ΋ͷͰ͋Γɺͦͷ࿦จͷதͰ͸ ΦϓγϣϯͷՁ֨෇͚΍ϔοδʹ͍ͭͯ΋ٞ࿦͞Ε͍ͯΔ Merton model ʹ͓͍ͯ St ͸ St = S0 exp{µt + σWt + Jt } Ͱ༩͑ΒΕɺJt ͸ jump part Ͱ͋Δ

Merton models ν Λ MMMɺͭ·Γ P∗ ʹม׵ͨ࣌͠ͷ νP∗ ͸࣍ͷΑ͏ʹͳΔ ໋୊ νP∗ (dx) ʹ͍ͭͯ νP∗ (dx) =ν[(1 + h)γ, m, δ2](dx) + ν −hγ exp 2m + δ2 2 , m + δ2, δ2 (dx) ͜͜Ͱɺ h = µS σ2 + R0 (ex − 1)2ν(dx) . ν(dx)= γ √ 2πδ exp −(x−m)2 2δ2 dx. (ν=ν[γ,m,δ])

Merton models Ծఆ (ຬͨ͢΂͖৚݅) Ϟσϧύϥϝʔλʔ͸ҎԼͷ৚݅Λຬͨ͞ͳ͚Ε͹ͳΒͳ͍ɿ m ∈ R , δ > 0 γ > 0 , 0 ≥ µ + σ2 2 + γ exp m + δ2 2 − 1 − m , µ + 3σ2 2 +γ exp(2m + 2δ2) − exp m + δ2 2 − m > 0 .

Merton models φT−t (z) := EP∗ [eizLT−t ] ʹ͍ͭͯ z ∈ C,t ∈ [0, T] ʹରͯ͠, φT−t (z) = exp (T − t) izµ∗ − σ2z2 2 + R0 (eizx − 1 − izx)νP∗ (dx) = exp (T − t) izµ∗ − σ2z2 2 + (1 + h)γ(eimz−z2 δ 2 2 − 1 − imz) − hγe2m+δ 2 2 [ei(m+δ2)z− z2 δ 2 2 − 1 − iz(m + δ2)] .

Merton models ˜ ψ(z) := ψ2 (z) exp −δ2 2 z2 , ˜ f(K) := 1 π ∞ 0 K−iζ+1 ˜ ψ(ζ)dv. ˜ f ΋ f ͱಉ༷ʹ FFT ͰܭࢉՄೳ. Expression of I2 ͷੵ෼දࣔ ೚ҙͷ t ∈ [0, T] ʹ͍ͭͯɺI2 ͸࣍ͷΑ͏ʹॻ͚Δ: R0 EP∗ [(ST ex − K)+ − (ST − K)+|Ft− ](ex − 1)ν(dx) = γe2m+ 3 2 δ2 ˜ f(Ke−m−δ2 ) − γem ˜ f(Ke−m) + γ(1 − em+ δ 2 2 ) f(K) Reminder: ψ2 (z) := φT−t (z)Siz t− (iz−1)iz , ζ := v − iα, EP∗ [(ST − K)+|Ft− ] = 1 π ∞ 0 K−iζ+1ψ2 (ζ)dv =: f(K).

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Merton model ͷ਺஋ܭࢉ model parameter: T = 1, µ = −0.7, σ = 0.2, γ = 1, m = 0, δ = 1. FFT parameter: N = 214, η = 0.025, α = 1.75 → Nη = 409.6 ࣍ͷೋͭͷ৔߹Λߟ͑Δ strike price K = 1 ͱͯ͠ɺ LRMt Λܭࢉ ͦͷࡍɺt = 0, 0.05, · · · , 0.95 ͱ͢Δ t = 0.5 ͱͯ͠ɺLRM0.5 Λܭࢉ ͦͷࡍɺK = 1, 1.25, 1.5, · · · , 8 ͱ͢Δ NOTE t ͷ஋ʹґΒͣɺLt− = 1 ͱ͍ͯ͠Δ

Merton models 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.986 0.988 0.99 0.992 0.994 0.996 0.998 t LRM t (a) Merton models, strike price K = 1, maturity T = 1 parameter: µ = −0.7, σ = 0.2, γ = 1, m = 0, δ = 1 .

Merton models 1 2 3 4 5 6 7 8 0.7 0.75 0.8 0.85 0.9 0.95 1 K LRM 0.5 (b) Merton models, maturity T = 1, at t = 0.5, parameter: µ = −0.7, σ = 0.2, γ = 1, m = 0, δ = 1 .

Variance Gamma models ࣍ʹɺσ = 0 ͷ৔߹Λߟ͑Δ ࠓɺGt Λ parameter (1/κ, 1/κ) Ͱ͋Δ gamma աఔͱ͢Δ ͜͜Ͱɺκ > 0 Ͱ͋Δ ͜ΕΛ subordinator ͱ͢Δ drift ෇͖ 1 ࣍ݩ Brown ӡಈͷ࣌ؒม ԽΛ Lt ͱ͢ΔϞσϧΛߟ͑Δ ͭ·Γɺ Lt = mGt + δBGt ͱ͢Δ Bt ͸ 1 ࣍ݩඪ४ Brown ӡಈɺm ∈ R, δ > 0 ͜ͷΑ͏ͳ֬཰աఔΛ variance Gamma (VG) աఔͱݺͿ

Variance Gamma models L´ evy measure : ν(dx) = C(1{x<0} e−G|x| + 1{x>0} e−M|x|) dx |x| = C(1{x<0} eGx + 1{x>0} e−Mx) dx |x| , ͨͩ͠ɺ C := 1 κ , G := m2 + 2δ2 κ δ2 + m δ2 , M := m2 + 2δ2 κ δ2 − m δ2 .

Variance Gamma models νP∗ (dx) ʹ͍ͭͯ νP∗ (dx) = ν(1+h)C,G,M (dx) + ν−hC,G+1,M−1 (dx), where h = µS R0 (ex−1)2ν(dx) . Ծఆ (ຬͨ͢΂͖৚݅) C > 0 , G > 0 , M > 4 , log (M − 1)(G + 1) (M − 2)(G + 2) > 0 ≥ log MG (M − 1)(G + 1) νC,G,M := C(1{x<0} eGx + 1{x>0} e−Mx) dx

Variance Gamma models φT−t (z) ʹ͍ͭͯ φT−t (z) = 1 + iz G 1 − iz M −(1+h)(T−t)C 1 + iz G + 1 1 − iz M − 1 h(T−t)C × exp (T − t)iz µ∗ + (1 + h)C M − G GM − hC M − G − 2 (G + 1)(M − 1) , where µ∗ = R0 (x − ex + 1)νP∗ (dx).

I2 = R0 EP∗ [(ST ex − K)+ − (ST − K)+|Ft− ](ex − 1)ν(dx) = 1 π ∞ 0 R0 (eiζx − 1)(ex − 1)ν(dx)K−iζ+1ψ2 (ζ)dv = 1 π ∞ 0 K−iζ+1ψVG (ζ)dv − 1 π ∞ 0 C log MG (M − 1)(G + 1) K−iζ+1ψ2 (ζ)dv. ͜͜Ͱɺ ψVG (ζ) := C log M − iζ M − 1 − iζ G + iζ G + 1 + iζ ψ2 (ζ). Recall : ψ2 (ζ) = φT−t(ζ)Siζ t− (iζ−1)iζ , σ = 0 −→ σI1 = 0.

Variance Gamma model ͷ਺஋ܭࢉ 1. model parameter : µ = −0.16, κ = 0.15, m = −0.5, δ = 0.45. LRMt Λ Merton model ͱಉ͘͡ܭࢉ͢Δ. 2. parameter ਪఆʹ͍ͭͯ ࠾༻ͨ͠ࢢ৔σʔλɿ̎̌̍̐೥݄̏ͷ೔ܦฏۉגՁ (ऴ஋) ਪఆํ๏ɿmoment ๏ͱ Levenberg–Marquardt ๏Λ༻͍ͨ C = 2.4693, G = 23.743, M = 24.903.

Variance Gamma model LRMt 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.98 0.985 0.99 0.995 1 1.005 t LRM t (c) variance Gamma model, parameter : µ = −0.16, κ = 0.15, m = −0.5, δ = 0.45.

Variance Gamma model LRM0.5 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K LRM 0.5 (d) variance Gamma model. t = 0.5 0.21 sec.

Variance Gamma model LRM0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 t LRM t (e) variance Gamma model. based on estimated data

Variance Gamma model LRM0.5 10000 11000 12000 13000 14000 15000 16000 17000 18000 19000 20000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K LRM 0.5 (f) variance Gamma model. based on estimated data

MMM (P∗) ͷԼͰͷ Delta Hedging Strategy ∆P∗ t ఆٛ For any K > 0 and any underlying asset pricing s > 0, delta hedging strategies under the minimal martingale measure is defined as ∆P∗ t := ∂EP∗ [(ST − K)+ | St− = s] ∂s . Black–Scholes model ͷ࣌͸ɺ௨ৗͷఆٛͱҰக͢Δ Theorem For any K > 0, t ∈ [0, T] and α ∈ (1, 2], we have delta hedging strategies as follows: ∆P∗ t = I1 St− . cf. Denkl et al., On the performance of delta hedging strategies in exponential

ෆ౳ࣜධՁ Theorem For any χt− > 0, t ∈ [0, T] and α ∈ (1, 2], there exists a constant C such that |LRMt − ∆P∗ t | ≤ Cχ1−α t− . We obtain, furthermore, lim χt−→0 |LRMt − ∆P∗ t | = 0 . ͜͜Ͱ χt− = K St− ͸ moneyness Λද͢ LRMt ΍ ∆P∗ t ͸ moneyness χ ͷؔ਺ͱͯ͠ॻ͚Δ C ͸ model ʹΑͬͯҟͳΔ

ෆ౳ࣜධՁ ఆཧ (Merton models) |LRMt − ∆P∗ t | ≤ C1 C(m,δ,α) σ 2π(T − t) χ1−α t− σ2 + R0 (ex − 1)2ν(dx) = C1 C(m,δ,α) σ(σ2 + C) 2π(T − t) χ1−α t− . ͜͜Ͱɺ C := R0 (ex − 1)2ν(dx) ,

ෆ౳ࣜධՁ C1 := exp (T − t) αµ∗ + σ2α2 2 + (1 + h)γ(emα+α 2 δ 2 2 − 1 − αm) − hγe2m+δ 2 2 e(m+δ2)α+α 2 δ 2 2 − 1 − α(m + δ2) , C(m,δ,α) :=γ em(α+1)+δ 2 2 (α+1)2 + em+δ 2 2 + emα+δ 2 2 α2 + 1 + γ m + δ2 √ 2 + 2α em(α+1)+δ 2 2 (α+1)2 + γ m + δ2 emα+δ 2 2 α2 .

ෆ౳ࣜධՁ ఆཧ (VG models) |LRMt − ∆P∗ t | ≤ 1 π C(C,G,M,α) C (ϵC0 )−1 2 (1+h)(T−t)C ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ˜ K1 2 h(T−t)Ca − C 1 2 h(T−t)C 4 2h(T − t)C + 1 a2h(T−t)C+1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ χ1−α t− . ͜͜Ͱɺະఆٛͳม਺ C(C,G,M,α), C, C0, ˜ K, C4, ϵ ͷఆٛ͸ࠐΈೖΓ͢ ͗ΔͷͰলུ

Merton Models Ͱ਺஋ܭࢉ model parameter (Assumption Λຬͨ͢ਓ޻తͳ஋): T = 1, µ = −0.7, σ = 0.2, γ = 1, m = 0, δ = 1. FFT parameter: N = 214, η = 0.025, α = 1.75 → Nη = 409.6 strike price K = 1 ͱͯ͠ɺ LRMtɺ∆P∗ t ɺ|LRMt − ∆P∗ t | Λܭࢉ ͦͷࡍɺt = 0, 0.05, · · · , 0.95 ͱ͢Δ t = 0.5 ͱͯ͠ɺLRM0.5ɺ∆P∗ 0.5 ɺ|LRM0.5 − ∆P∗ 0.5 | Λܭࢉ ͦͷࡍɺK = 1, 1.25, 1.5, · · · , 8 ͱ͢Δ

Merton Models, LRM0.5 and ∆P∗ 0.5 0 1 2 3 4 5 6 7 8 9 10 x 1016 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Moneyness χ LRM 0.5 LRM 0.5 ∆ (g) LRM0.5 and ∆P∗ 0.5 for moneyness χ

Merton Models, |LRM0.5 − ∆P∗ 0.5 | 0 1 2 3 4 5 6 7 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Moneyness χ |LRM 0.5 − ∆| (h) |LRM0.5 − ∆P∗ 0.5 | for moneyness χ

Variance Gamma Models ͷ਺஋ܭࢉ ࠾༻ͨ͠ࢢ৔σʔλ : 2014 ೥ 3 ݄ͷ೔ܦฏۉגՁ (ऴ஋) model parameter : C = 2.4693, G = 23.743, M = 24.903. model parameter ͷಋग़ʹ͸ Levenberg–Marquardt ๏Λ༻͍ͨ. strike price K = 1 ͱͯ͠ɺ LRMtɺ∆P∗ t ɺ|LRMt − ∆P∗ t | Λܭࢉɻ ͦͷࡍɺt = 0, 0.05, · · · , 0.95 ͱ͢Δ. t = 0.5 ͱͯ͠ɺLRM0.5ɺ∆P∗ 0.5 ɺ|LRM0.5 − ∆P∗ 0.5 | Λܭࢉɻ ͦͷࡍɺK = 1, 1.25, 1.5, · · · , 8 ͱ͢Δɻ

Variance Gamma Models, LRM0.5 and ∆P∗ 0.5 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Moneyness χ LRM 0.5 LRM t ∆ (i) LRM0.5 and ∆P∗ 0.5 for moneyness χ with S0 = 14841.07

Variance Gamma Models, |LRM0.5 − ∆P∗ 0.5 | 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Moneyness χ |LRM 0.5 − ∆| (j) |LRM0.5 − ∆P∗ 0.5 | for moneyness χ, with S0 = 14841.07

ँࣙ ͋Γ͕ͱ͏͍͟͝·ͨ͠

Appendix Tankov(2010) Proposition2 Suppose that there exists R 0 such that g(x)e−Rx ∈ L1(R) has finite variation onR , E[eRXT−t ] < ∞ and R |ΦT−t (u − iR)| 1 + |u| du < ∞ . Then the price at time t of the European call option with pay-off function G and characteristic function Φt of Xt satisfies P(t, St ) :=e−r(T−t)E[G(ST )|Ft ] = e−r(T−t) 2π R ˆ g(u + iR)ΦT−t (−u − iR)ˆ SR−iu t du , where ˆ g(u) := R eiuxg(x)dx .