Kimanta Farashin Sashin CDO na Musamman cikin Daidaito ba tare da Cin Ribar Ba: Hanyar Tsarin Ma'auni Mai Yawa

1. Gabatarwa

Wannan takarda tana magance kalubalen da suka shafi kimanta farashin sassan CDO na musamman cikin daidaito ba tare da cin riba ba. Kafin kuma yayin rikicin kudi na 2008, ma'aunin kasuwa shine hanyar taswirar asali. Duk da cewa tana taimakawa wajen sauƙaƙe ciniki da sarrafa haɗari, wanda ya haifar da haɓakar kasuwa, wannan hanyar tana da cikas a tushenta. Ba ta da daidaiton kimanta farashi, tana ba da damar cin riba, kuma tana iya samar da ma'auni na haɗari masu sabani, kamar yadda Morgan & Mortensen (2007) suka rubuta. Marubucin yana ba da shawarar wata sabuwar hanya, tare da fadada tsarin Li (2009), don kimanta tsoffin matsayi, sarrafa haɗari ga daidaitattun sassan fihirisa yayin da suke tsufa, da kuma ba da damar dabarun ciniki na ƙimar dangi.

2. Bita kan Taswirar Asali

Taswirar asali hanya ce da aka yi amfani da ita sosai amma tana da matsala a ka'ida. Babban iyakarta shine rashin iya samar da daidaitaccen rarraba lokutan kasawa (JDDT) ko alamomin kasawa ({JDDI(t)}). Wannan rashin daidaituwa yana iyakance amfaninta da farko ga tsaka-tsaki na rarraba asarar fayil - ma'auni mai mahimmanci amma wanda bai isa ba don ingantaccen kimanta farashi. Shaharar hanyar ta samo asali ne daga sauƙinta da sassauƙanta wajen gina waɗannan rarrabuwa, waɗanda aka ɗauka sun isa yayin haɓakar kasuwa. Duk da haka, kurakuranta sun sa ta zama mara dacewa don samar da ingantattun ma'auni na shinge ko don daidaitaccen kimanta farashi a cikin sassa da fayiloli daban-daban.

3. Hanyar Daidaitaccen Kimanta Farashi da Ake Shawarwari

Takardar tana ba da shawarar faɗaɗa ma'auni mai yawa ga tsarin Li (2009) don shawo kan gazawar taswirar asali.

3.1. Faɗaɗawar Tsarin Ma'auni Mai Yawa

Babban sabon abu shine sanya ma'auni na kasuwa daban ga kowane fihirisar bashi mai ruwa (misali, CDX, iTraxx). Ana ƙirƙira haɗin kai tsakanin waɗannan ma'auni na kasuwa a sarari. Wannan tsari yana ɗaukar dogaro na haɗarin tsarin tsakanin sassa ko yankuna daban-daban da fihirisoshi ke wakilta, yana ba da tsarin dogaro mafi gaskiya ga fayiloli na musamman waɗanda zasu iya shiga cikin ma'auni da yawa.

3.2. Tsarin Tsari da Ma'auni Mafi Muhimmanci

Tsarin ya nuna cewa lokacin kasawa $\tau_i$ na suna ɗaya yana gudana ne ta hanyar haɗakar ma'auni na tsarin kasuwa $M_k$ da ma'auni na musamman $\epsilon_i$. Ana ƙirƙira ƙimar kamfani $A_i(t)$ kamar haka: $$A_i(t) = \sum_{k} \beta_{i,k} M_k(t) + \sqrt{1 - \sum_{k} \beta_{i,k}^2} \, \epsilon_i(t)$$ inda $\beta_{i,k}$ ke wakiltar nauyin kamfani $i$ akan ma'auni na kasuwa $k$. Kasawa yana faruwa lokacin da $A_i(t)$ ya faɗi ƙasa da shinge da aka ƙayyade $B_i(t)$, wanda aka samo daga ƙimar haɗarin kamfanin. Don haka, ana ƙayyade rarraba kasawa ta hanyar tsarin haɗin kai na ma'auni na kasuwa $\rho_{k,l} = \text{Corr}(M_k, M_l)$ da nauyin kamfanoni ɗaya ɗaya.

4. Sakamakon Lissafi da Aiwartawa A aikace

4.1. Kwatancen Farashi da Taswirar TLP

Gwaje-gwajen lissafi sun nuna cewa tsarin da aka ba da shawara yana samar da farashin sassan musamman gabaɗaya daidai da waɗanda aka samo daga daidaitaccen hanyar taswirar asali ta amfani da Taswirar Kashi na Asara (TLP). Wannan sakamako ne mai amfani, yana nuna cewa tsarin zai iya zama madadin sauyawa ba tare da haifar da babban ɓarkewar ƙimar kasuwa ga littattafan da suke akwai ba.

4.2. Ma'auni na Haɗari: Sashin Deltas da Suna ɗaya Deltas

Babban fa'ida shine samar da ma'auni na haɗari masu karko da fahimta. Tsarin yana lissafta sashin deltas (hassada ga fihirisa) da suna ɗaya deltas (hassada ga faɗaɗawar bashi ɗaya ɗaya) a cikin tsari mai daidaituwa. Wannan yana ba da damar dabarun shinge mafi inganci idan aka kwatanta da rashin karko deltas wani lokacin taswirar asali ke samarwa.

4.3. Tattaunawa kan Gyaran Quanto

Takardar ta taɓa gyaran quanto, waɗanda suke da mahimmanci lokacin da aka ƙididdige kuɗin farko da biyan kuɗin kasawa na sashi a cikin kuɗi daban-daban. Tsarin tsari na ma'auni a sarari yana ba da tushe mafi haske don lissafin waɗannan gyare-gyare idan aka kwatanta da hanyoyin da aka saba amfani da su tare da taswirar asali.

5. Fahimta ta Asali & Ra'ayi na Manazarta

Fahimta ta Asali: Takardar Li wani hari ne mai hankali kan rashin kulawa da ya mamaye kasuwar CDO bayan rikicin. Ta gano daidai cewa ci gaba da dogaro da masana'antu akan taswirar asali - kayan aikin da aka sani ya lalace - shine bam mai ƙara tikitin don sarrafa haɗari, ba kawai sha'awar ka'ida ba. Babban fahimta ba kawai tsarin ma'auni mai yawa ba ne, amma yarda a sarari cewa tsarin kimanta farashi dole ne ya samar da daidaitaccen rarraba kasawa don ya zama mai amfani ga komai ban da ciniki mai sauƙi, wanda yarjejeniya ke tafiyar da shi. Wannan ya yi daidai da aikin tushe a ka'idar kimanta farashin kadari, kamar buƙatar yanayin rashin cin riba kamar yadda aka tsara a cikin ka'idar asali ta kimanta farashin kadari (Delbaen & Schachermayer, 1994). Tsarin da ya saba wa wannan, kamar taswirar asali, ba shi da dacewa a asali don lissafin ma'auni na shinge ko alamar littattafai masu sarƙaƙi zuwa tsari.

Kwararar Ma'ana: Hujja tana da gamsarwa kuma tana bin ma'ana mai tsabta, mai zuwa ga mai aiki: (1) Ga kayan aiki na yau da kullun (taswirar asali). (2) Ga dalilin da yasa yake da cikas a asali (babu daidaitaccen JDDT, cin riba). (3) Ga abin da muke buƙata don sarrafa haɗari na gaske (daidaitaccen JDDT, karko Greeks). (4) Ga maganina (faɗaɗawar ma'auni mai yawa na Li 2009). (5) Ga hujja cewa yana aiki kuma baya karya alamomin da suke akwai. Wannan kwararar tana kama da tsarin matsala-maganin-tabbatarwa da ake gani a cikin takardun kuɗi masu tasiri, kamar ainihin ƙirar Volatility na Gida ta Dupire (1994), wanda kuma ya nemi gyara aikin daidaitaccen kasuwa amma mara daidaituwa (amfani da ƙayyadaddun ƙima mai ƙima).

Ƙarfi & Kurakurai: Ƙarfin tsarin shine ƙirarsa mai amfani. Ta hanyar ɗaure ma'auni zuwa fihirisoshi masu ruwa, yana kafa tsarin a cikin masu canjin kasuwa da ake iya gani, yana haɓaka daidaitawa da shinge. Amfani da Monte Carlo na rabin-bincike shine cinikin inganci mai hankali. Duk da haka, babban aibin takardar shine lokacinta da iyakarta. An buga shi a cikin 2010, yana zuwa yayin da kasuwar CDO ta musamman ta lalace. "Nan gaba" shine sarrafa littafin gado a cikin gudu, aiki mai mahimmanci amma yana raguwa. Ya kewaye giwa a cikin daki: rashin al'ada na kasawa da rashin isasshen hanyoyin da suka dogara da Gaussian copula (ko da na ma'auni mai yawa) yayin rikice-rikicen tsarin, wani aibi da aka fallasa a 1994. Tsarin kamar na Hull da White (2004) ko kuma amfani da ƙirar ƙwaƙƙwaran gaba na baya-bayan nan sun yi jayayya don ƙarin hanyoyin da suka dogara da faɗaɗawa don ɗaukar haɗarin clustering mafi kyau.

Fahimta Mai Aiki: Ga masu ƙididdiga a bankunan da ke da littattafan bashi na tsari, wannan takarda ta zama dole ne. Aikin nan take shine gudanar da kwatancen tsari: sake kimanta samfurin sassan musamman a ƙarƙashin duka taswirar asali da wannan tsarin ma'auni mai yawa. Maɓalli ba bambancin PV ba ne, amma bambance-bambance a cikin deltas - a nan ne haɗarin ɓoye yake. Ga masu tsari, fahimta ita ce tilasta cewa lissafin babban jari don abubuwan da suka samo asali ya dogara ne akan tsarin da ke hana cin riba a sarari kuma yana samar da ma'auni na haɗari masu daidaituwa. Ga al'ummar ilimi, takardar tana nuna wani yanki mai albarka: haɓaka sauri, tsarin kimanta farashi mara cin riba don samfuran bashi na fayil wanda zai iya ɗaukar aikin kasawa mara layi, gungu na kasawa wanda sauƙaƙan tsarin ma'auni ya rasa. Nan gaba yana cikin tsarin haɗin gwiwa waɗanda suka haɗu da daidaiton wannan takarda tare da yanayin rikicin da ƙarin bincike na baya-bayan nan suka kama.

6. Cikakkun Bayanai na Fasaha da Tsarin Lissafi

Injin tsarin shine simintin Monte Carlo na rabin-bincike. Matakan sune:

1. Simintin Ma'auni: Ga kowane hanyar siminti $j$, samar da ƙimar dawowar ma'auni na kasuwa masu haɗin kai $M_k^j$ daga rarraba al'ada mai yawa: $\mathbf{M}^j \sim N(\mathbf{0}, \mathbf{\Sigma})$, inda $\mathbf{\Sigma}$ shine matrix haɗin kai na ma'auni.
2. Lissafin Ƙimar Kamfani: Ga kowane kamfani $i$, lissafa ƙimar kadarsa: $A_i^j = \sum_k \beta_{i,k} M_k^j + \sqrt{1 - \sum_k \beta_{i,k}^2} \, \epsilon_i^j$, tare da $\epsilon_i^j \sim N(0,1)$ i.i.d.
3. Binciken Kasawa: Ƙayyade ko kamfani $i$ ya kasa a cikin lokacin $[t, t+\Delta t]$ ta hanyar duba idan $A_i^j < \Phi^{-1}(PD_i(t))$, inda $PD_i(t)$ shine yuwuwar kasawa mai haɗari da aka samo daga faɗaɗawar CDS, kuma $\Phi$ shine daidaitaccen CDF na al'ada.
4. Haɗakar Asarar Fayil: Ƙara asarar daga ƙungiyoyin da suka kasa, tare da amfani da ƙimar dawo da dacewa, don samun hanyar asarar fayil $L^j(t)$.
5. Lissafin PV na Sashi: Ga sashi mai maƙallan haɗawa $A$ da maƙallan rabuwa $D$, asarar ita ce $L_{\text{tranche}}^j(t) = \min(\max(L^j(t)-A, 0), D-A)$. Ƙimar yanzu ita ce raguwar tsammanin ƙafa na farko da na asara a cikin dukkan hanyoyin.

Ribin inganci yana zuwa daga amfani da haɗakar bincike ko na lissafi don yuwuwar kasawa mai sharadi da aka ba da ma'auni na kasuwa, yana rage buƙatar simintin kowane suna na girgiza na musamman kai tsaye a yawancin lokuta.

7. Sakamakon Gwaji da Nazarin Chati

Takardar ta gabatar da misalan lambobi, ko da yake ba a sake yin takamaiman chatuna a cikin abin da aka samo ba. Dangane da bayanin, zamu iya ƙididdige sakamako mafi mahimmanci:

• Chati 1: Farashin Kwatancen Surface. Wannan zai iya zama shafi na 3D ko zafi mai zafi wanda ke nuna farashin (ko faɗaɗawa) na sassan musamman a cikin wuraren haɗawa daban-daban (x-axis) da matuƙa (y-axis), kwatanta tsarin da aka ba da shawara (Tsarin Z) da daidaitaccen Taswirar Asali tare da taswirar TLP (Market Std). Saman za su kasance masu daidaituwa sosai, tare da ƙananan bambance-bambance, musamman ga manyan sassa ko fayiloli marasa daidaito, suna nuna daidaiton kasuwar tsarin.
• Chati 2: Kwatancen Bayanin Delta. Chati na layi wanda ke zana sashin delta (hassada ga fihirisa) akan maƙallan haɗawa. Layin tsarin da aka ba da shawara zai kasance mai santsi kuma mai jujjuyawa. Layin don taswirar asali zai iya nuna halin da ba a saba gani ba "kaɗa-kaɗa" ko katsewa, musamman a kusa da wuraren rabuwa na daidaitaccen fihirisa (3%, 7%, 10%, 15%, 30%), yana nuna rashin karko sigina na shinge na tsohuwar hanya.
• Chati 3: Rarraba Delta na Suna ɗaya. Tarihin tarihi wanda ke nuna rarraba delta na suna ɗaya ga mambobin fayil na musamman. Tsarin da aka ba da shawara zai samar da ƙaƙƙarfan rarraba, mafi ma'ana wanda ya ta'allaka ne akan ƙimar fahimta dangane da ƙarƙashin ƙasa da haɗin kai. Taswirar asali na iya samar da rarraba bi-modal ko watsi da yawa, gami da deltas mara kyau ga wasu sunaye a cikin sassan daidaiton - sakamako mai sabani.

Waɗannan sakamakon sun tabbatar da ainihin alkawarin tsarin: daidaiton rashin cin riba ba tare da watsi da yarjejeniyar kasuwa akan matakan farashi ba.

8. Tsarin Nazari: Nazarin Shari'a Mai Aiki

Yanayi: Manajan haɗari yana riƙe da sashi na musamman na gado wanda ke nufin fayil na kamfanoni 100 na Arewacin Amurka. Sashin yana da ƙimar A, tare da haɗawa a 12% da rabuwa a 22%. Fayil yana da haɗuwa da fihirisar CDX.NA.IG amma bai yi daidai ba.

Aiwatar da Tsarin:

1. Daidaitawa: Daidaita tsarin ma'auni mai yawa. Babban ma'auni na kasuwa an taswireshi zuwa CDX.NA.IG. Nauyi ($\beta_{i,k}$) don sunaye a cikin fihirisa an daidaita su don dacewa da farashin sassan fihirisa. Don sunaye na musamman waɗanda ba a cikin fihirisa ba, ana ba da nauyi bisa wakilai na sashe/ƙima ko nazarin ƙididdiga.
2. Ƙima & Benchmarking: Kimanta sashin musamman ta amfani da tsarin da aka daidaita. A lokaci guda, kimanta shi ta amfani da kayan aikin taswirar asali/TLP na tebur. Kwatanta PVs. Yi zaton suna cikin faɗaɗawar nema-biyar (misali, Tsarin: 245 bps, BaseCorr: 250 bps).
3. Nazarin Haɗari (Mataki Mai Muhimmanci): Lissafa sashin delta na sashi zuwa sashin fihirisar CDX.NA.IG 12-22% a ƙarƙashin duka tsarin.
   • Delta Tsarin Taswirar Asali: 0.85 (amma yana da matukar mahimmanci ga ƙananan canje-canje a cikin haɗin kai na shigarwa, yana tsalle zuwa 1.1 ko 0.7 tare da ƙananan rikice-rikice).
   • Delta Tsarin da Aka Shawarwari: 0.88, tare da karko hassada ga canje-canjen shigarwa.
   Rashin karko a cikin delta taswirar asali yana nuna kuskuren ma'auni na shinge. Shinge dangane da shi zai iya haifar da kuskuren bin diddigin.
4. Aiki: Manajan haɗari ya yanke shawarar amfani da delta na tsarin da aka ba da shawara (0.88) don ƙayyade ƙimar sashin CDX.NA.IG 12-22% don siya/siyar don shinge. An sabunta tsarin sifa na P&L na tebur don saka idanu kan ingancin shinge dangane da wannan sabon ma'auni, mafi karko.

Wannan shari'ar tana nuna cewa babban ƙimar daidaitaccen tsarin ba a cikin canza alama ba, amma a samar da amintattun sigina don rage haɗari.

9. Aiwatarwa na Gaba da Hanyoyin Ci Gaba

Ka'idodin da aka zayyana suna da alaƙa fiye da CDOs na musamman na gado:

• Daidaituwar Haɗarin da ba na Daidaito ba: Za a iya amfani da hanyar ma'auni a sarari don kimanta farashi da sarrafa haɗari na sassan musamman akan sabbin nau'ikan kadari kamar CLOs (Ƙungiyoyin Lamuni na Haɗin gwiwa), inda za a iya amfani da ma'auni na fihirisa "daidaitacce" (misali, fihirisar lamuni mai ƙarfi).
• Haɗin Tsarin XVA: Daidaitattun rarraba kasawa suna da mahimmanci don lissafin Gyaran Ƙimar Bashi (CVA), Gyaran Ƙimar Bashi (DVA), da Gyaran Ƙimar Kuɗi (FVA). Wannan tsarin yana ba da tsari mai daidaituwa don simintin kasawar ɗayan abokin ciniki da kiran haɗin gwiwa a cikin mahallin bashi na fayil.
• Gwajin Matsala da Nazarin Yanayi: Masu tsari suna buƙatar yanayi mai tsanani amma mai yuwuwa. Tsarin ma'auni mai yawa yana ba da damar tsaftacewa, fahimtar girgiza ga takamaiman ma'auni na kasuwa (misali, "girgiza ma'auni na Turai da 3 daidaitattun karkatattu yayin da ake kiyaye ma'auni na Amurka") don tantance juriyar fayil.
• Haɓaka Koyon Injin: Aikin nan gaba zai iya haɗawa da amfani da dabarun koyon inji don daidaita nauyin ma'auni ($\beta_{i,k}$) da haɗin kai tsakanin ma'auni ($\mathbf{\Sigma}$) daga manyan bayanan faɗaɗawar CDS da ƙimar daidaiton, ƙaura fiye da wakilai na sashe/ƙima masu sauƙi.
• Haɗin kai tare da Tsarin Ƙungiyar Kasawa: Juyin halitta na gaba zai kasance maye gurbin tushen Gaussian copula tare da tsarin ƙwaƙƙwaran ƙarfi ko tsarin Hawkes wanda ke ɗaukar ƙungiyar kasawa a asali, yayin da yake riƙe da daidaitaccen, ma'auni mai yawa, tsarin kimanta farashi mara cin riba da aka ba da shawara a nan.

Hanyar ƙarshe ita ce zuwa ga tsarin haɗin kai, masu daidaituwa don duk samfuran bashi na fayil, daga fihirisoshi masu sauƙi na CDS zuwa sassan musamman masu sarƙaƙi, tabbatar da cewa ana auna haɗari kuma ana sarrafa shi akan kwatankwacin tushe a cikin cibiya.

10. Nassoshi

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5. Li, Y. (2009). [Nassoshi ga tsarin Li 2009].
6. Morgan, S., & Mortensen, A. (2007). Algorithms Taswirar CDO. Lehman Brothers.
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