Farashin Musayar Gaba da Sabon Abin Mamaki na Siegel: Hanyar Axiomatic zuwa Masu Haɗa Kaya Ba tare da Cin Ribar Ba

1. Gabatarwa

Sabon abin mamaki na Siegel, wanda ya samo asali daga Siegel (1972), yana gabatar da wata matsala ta asali a cikin kuɗaɗen duniya game da ƙayyadaddun farashin musayar gaba. Yana nuna rashin daidaituwa a fili lokacin da masu saka hannun jari marasa haɗari daga wuraren kuɗi daban biyu suka yi ƙoƙari su yarda da farashin gaba guda ɗaya bisa tsammaninsu na farashin lokaci na gaba. Sabon abin mamaki ya samo asali ne daga gaskiyar lissafi cewa matsakaicin lissafi da matsakaicin jituwa na jerin lambobi masu kyau gabaɗaya ba su daidaita ba, wanda ke haifar da rashin jituwa kan farashin gaba na "gaskiya". Wannan takarda ta Mallahi-Karai da Safari ta magance wannan matsalar da ta shafe shekaru da yawa ta hanyar gabatar da sabuwar hanyar axiomatic, suna neman aikin "mai haɗa kaya" wanda zai haifar da farashin gaba wanda zai yarda da bangarorin biyu a ƙarƙashin ƙayyadaddun tattalin arziki na halitta.

2. Sabon Abin Mamaki na Siegel da Mahallin Tarihi

Sabon abin mamaki ba wani abin sha'awa ne kawai na ka'ida ba amma yana da muhimman tasiri ga kasuwar musayar kuɗin waje ta yau da kullun da ta kai tiriliyoyin daloli, kamar yadda Obstfeld & Rogoff (1996) suka lura.

2.1 Bayanin Ƙa'idar Sabon Abin Mamaki

Yi la'akari da yanayin duniya guda biyu na gaba, $\omega_1$ da $\omega_2$, kowannensu yana da yuwuwar 50%. Bari farashin musayar lokaci na gaba (Yuro zuwa USD) a cikin waɗannan jihohin ya kasance $e_1$ da $e_2$, bi da bi. Mai saka hannun jari na Yuro, yana neman sayar da Yuro don USD a wani lokaci na gaba $T$, zai iya ba da shawarar matsakaicin lissafi a matsayin farashin gaba: $F_A = \frac{1}{2}(e_1 + e_2)$. Akasin haka, mai saka hannun jari na USD, yana yin ma'amalar ma'amala, zai yi la'akari da matsakaicin jituwa na farashin ma'amala: $F_H = \frac{2}{\frac{1}{e_1} + \frac{1}{e_2}}$. Tun da $F_A \geq F_H$ (tare da daidaito kawai idan $e_1 = e_2$), masu saka hannun jari biyu ba za su iya yarda da farashi ɗaya ba idan dukansu sun dage kan matsakaicin nasu. Wannan shine sabon abin mamaki na Siegel.

2.2 Ƙoƙarin Ka'idoji na Baya

Mafita na baya sau da yawa suna buƙatar gabatar da abubuwan waje kamar ƙin haɗari (Beenstock, 1985), ɗaukan ribar ana ɗaukar su a cikin kuɗin waje (Roper, 1975), ko kuma karɓar mai ƙima mai son kai (Siegel, 1972). Obstfeld & Rogoff (1996) sun ba da shawarar cewa farashin ma'auni zai yi shawarwari a wani wuri tsakanin $E(E_T)$ da $1/E(1/E_T)$. Marubutan wannan takarda sun soki waɗannan hanyoyin don ba su ba da takamaiman farashi, wanda za a iya yarda da shi a ƙarƙashin rashin haɗari.

3. Tsarin Axiomatic da Ma'anoni

Babban ƙirƙira na takardar shine tushen axiomatic. Maimakon farawa daga samfurin tattalin arziki na ɗabi'a, ya ayyana kaddarorin da aikin "mai haɗa kaya" na "gaskiya" $\phi$ dole ne ya bi.

3.1 Aikin Mai Haɗa Kaya

Bari $\mathbf{e} = (e_1, e_2, ..., e_n)$ ya zama vector na yuwuwar farashin lokaci na gaba (EUR/USD). Mai haɗa kaya $\phi(\mathbf{e})$ yana samar da farashin gaba guda ɗaya $F$.

3.2 Ka'idoji na Tsakiya

Ba Cin Riba (Babu Littafin Dutch): Dole ne ya zama ba zai yiwu a gina fayil na kwangilolin da aka ƙidaya akan $\phi(\mathbf{e})$ wanda ke ba da garantin riba mara haɗari.
Daidaituwa: Dole ne aikin $\phi$ ya zama mai daidaituwa a cikin hujjojinsa; lakabin jihohi ba shi da muhimmanci.
Rashin Canjin Ƙimar Kuɗi: Farashin gaba ya kamata ya kasance daidai ba tare da la'akari da wane kuɗi aka zaɓa a matsayin tushe ba. A bisa ƙa'ida, idan $\phi(\mathbf{e}) = F$ don EUR/USD, to don USD/EUR, farashin dole ne ya zama $1/F$. Wannan yana nuna $\phi(1/\mathbf{e}) = 1 / \phi(\mathbf{e})$.

Waɗannan ka'idoji suna da dabi'ar tattalin arziki kuma suna hana matsakaicin lissafi mai sauƙi (ya kasa rashin canjin ƙimar kuɗi) da matsakaicin jituwa (ya kasa lokacin da aka yi amfani da shi azaman babban mai haɗa kaya daga wani hangen nesa).

4. Ƙirƙirar Lissafi da Sakamako na Babba

4.1 Ƙirƙirar Mafita Gabaɗaya

Takardar ta nuna cewa ka'idojin daidaituwa da rashin canjin ƙimar kuɗi suna takura sosai ga nau'in $\phi$. Don lamarin jihohi biyu, sun nuna cewa mai haɗa kaya dole ne ya bi daidaitawar aiki kamar haka: $$\phi(e_1, e_2) = g^{-1}\left(\frac{g(e_1) + g(e_2)}{2}\right)$$ inda $g$ aiki ne mai ci gaba, mai tsauri. Yanayin rashin cin riba ya ƙara inganta wannan.

4.2 Aikin Ma'amala da Ka'idar Rarrabuwa

Mabuɗin biyayya ga rashin canjin ƙimar kuɗi shine ra'ayi na aikin ma'amala $\rho(x)$. Takardar ta tabbatar da cewa don mai haɗa kaya ya zama mara canzawa, dole ne a iya bayyana shi kamar haka: $$\phi(\mathbf{e}) = \rho^{-1}\left(\frac{1}{n} \sum_{i=1}^n \rho(e_i)\right)$$ inda aikin $\rho: \mathbb{R}^+ \to \mathbb{R}$ ya bi da sharadi $\rho(1/x) = -\rho(x)$ ko wani canji daidai. Wannan shine sakamako na fasaha na tsakiya.

Ka'idar Rarrabuwa: Duk masu haɗa kaya masu ci gaba, masu daidaituwa, marasa cin riba waɗanda ba su canzawa a ƙarƙashin canjin ƙimar kuɗi ana ba da su ta hanyar dabarar da ke sama, inda $\rho$ shine kowane aiki mai ci gaba, mai tsauri a ma'anar ninkawa (watau, $\rho(1/x) = -\rho(x)$).

Misali na al'ada shine matsakaicin geometric, wanda yayi daidai da zaɓin $\rho(x) = \log(x)$. Lalle ne, $\phi(e_1, e_2) = \sqrt{e_1 e_2}$, kuma $\log(1/x) = -\log(x)$.

5. Binciken Fasaha da Fahimta ta Tsakiya

Sharhin Manazarcin: Rarrabuwa ta Mataki Hudu

Fahimta ta Tsakiya

Takardar Mallahi-Karai da Safari ba wani ƙoƙari ne kawai na gyara sabon abin mamaki na Siegel ba; sake saiti ne na tushe. Sun gano daidai cewa tushen matsalar ba ilimin halayen mai saka hannun jari ba ne amma tambaya mara kyau. Neman farashin gaba na "gaskiya" ba tare da ayyana "gaskiya" ba ba shi da ma'ana. Hazakarsu ta ta'allaka ne a cikin sake ginawa na ma'anar: ana ayyana gaskiya ta hanyar rashin yiwuwar cin riba, daidaituwa tsakanin jihohi, da daidaito a cikin hangen nesa na kuɗi. Wannan hanyar axiomatic tana motsa muhawara daga tattalin arziki zuwa lissafi, inda za a iya magance shi sosai. Matsakaicin geometric ba wani matsakaici ne kawai na tsakiya ba; shi ne ɗaya (har zuwa canji) mafita wanda ke biyayya ga waɗannan buƙatun ma'ana marasa sasantawa ga ma'aikatan marasa haɗari. Wannan yana da muhimman tasiri ga ka'idar kuɗi ta asali, kamar yadda Black-Scholes PDE ke ayyana farashin zaɓi mara cin riba.

Kwararar Ma'ana

Kyawun gardama yana cikin sauƙinsa. 1) Ayyana Matsala ta Axiomatic: Lissafa kaddarorin (Babu Cin Riba, Daidaituwa, Rashin Canjin Ƙimar Kuɗi) waɗanda kowace mafita mai hankali dole ne ta samu. Wannan yana ketare shekaru da yawa na muhawara game da abubuwan da ake so na haɗari. 2) Fassara zuwa Lissafi: Waɗannan ka'idoji sun zama daidaitawar aiki don mai haɗa kaya $\phi$. 3) Warware Daidaitawar: Yanayin ma'amala $\phi(1/\mathbf{e}) = 1/\phi(\mathbf{e})$ shine ƙuntatawa mai kisa. Yana tilasta tsarin $\phi = \rho^{-1}(\mathbb{E}[\rho(e)])$, yana kwatanta nau'in tsammanin amfani amma a cikin ma'anar tsari mara yuwuwar, tsantsa. 4) Rarraba Duk Mafita: Ba su tsaya a gano misali ɗaya ba (matsakaicin logarithm/geometric). Sun ba da cikakken iyali na ayyuka, wanda aka siffanta ta hanyar baƙon dabi'ar $\rho$. Wannan cikakkiyar ka'idar ita ce ke ɗaga aikin daga wata dabara mai kyau zuwa babbar gudummawar ka'ida.

Ƙarfi & Kurakurai

Ƙarfi: Ƙaƙƙarfan takardar ba shi da aibi. Hanyar axiomatic tana da ƙarfi kuma tsafta ce. Ka'idar rarrabuwa ita ce cikakkiyar amsa ga takamaiman tambaya mai kyau. Ta yi bayani mai kyau game da dalilin da yasa matsakaicin geometric ya bayyana a zahiri a wasu fage kamar ƙimar girma na fayil (kwatanta da aikin Cover da Thomas akan fayiloli na duniya).

Kurakurai & Gibi: Tsaftar samfurin ita ma ita ce babban raunin aikace-aikace. Zato na sanannen, rarrabuwar jihohi na gaba $\{e_i\}$ tare da daidaiton yuwuwar yana da salo sosai. A cikin kasuwanni na gaske, wakilai suna da rarraba yuwuwar ci gaba da imani daban-daban. Takardar ta ɗan yi ishara da wannan amma ba ta haɗa yuwuwar son rai ko tsarin Bayesian ba, wata hanya da aka nuna ta aikin farko kan haɗa hasashen ƙwararru. Bugu da ƙari, yayin da yake magance sabon abin mamaki ga ma'aikatan marasa haɗari, ya kauce wa mamayar halayen ƙin haɗari na duniya na gaske. Tambayar tiriliyan daloli ta kasance: ta yaya wannan farashin gaba na axiomatic ke hulɗa tare da abubuwan rangwame na stochastic da farashin ruwa daban-daban? Samfurin, kamar yadda aka gabatar, yana wanzuwa a cikin sararin samaniya mara gogayya, mara riba.

Fahimta Mai Aiki

Ga masu ƙididdigewa da shugabannin tebur na ciniki, wannan takarda tana ba da ma'auni mai mahimmanci. Na farko, Tabbatar da Samfuri: Duk wani samfuri na ciki don samun farashin gaba na "ka'ida" daga abubuwan da ake tsammani na gaba ya kamata a duba shi da yanayin ma'amala. Idan aikin $\rho$ da aka nuna na samfurin ku ba shi da kyau, yana ƙunshe da son kai na kuɗi wanda za a iya amfani da shi. Na biyu, Ƙirar Algorithm: A cikin tsarin yin kasuwa ta atomatik don abubuwan haɓaka FX, yin amfani da mai haɗa kaya na tushen geometric a matsayin fifiko ko wurin tunani yana tabbatar da daidaiton ciki a cikin nau'ikan kuɗi da kuma karewa daga wasu nau'ikan cin riba na tsaye. Na uku, Fifikon Bincike: Mataki na gaba kai tsaye shine haɗa wannan tsarin tare da samfuran farashin ruwa na stochastic. Kalubalen shine nemo daidaiton "aikin ma'amala" a gaban ƙimar rangwame mara sifili, stochastic. Wannan haɗin kai zai iya haifar da haɗin kai, ka'idar farashin gaba na FX mara cin riba wanda a ƙarshe ya daidaita fahimtar Siegel da injiniyoyin farashin kadari na zamani.

6. Tsarin Bincike: Nazarin Lamari & Tasiri

Nazarin Lamari: Yin Shawarwari kan Kwangilar Gaba

Ka yi tunanin mai fitar da kayayyaki na Jamus da mai shigo da kayayyaki na Amurka sun yarda da biyan kuɗin Yuro miliyan ɗaya a cikin shekara guda. Suna son kulle farashin musayar EUR/USD na gaba a yau. Dukansu ba su da haɗari kuma suna da tsammani iri ɗaya: farashin lokaci na gaba zai kasance ko dai 1.05 ko 1.15 USD kowace Yuro, tare da daidaiton yuwuwar.

Hanyar Sauti (Lissafi): Bangaren Jamus zai iya ba da shawarar $F = (1.05 + 1.15)/2 = 1.10$.
Hanyar Ma'amala (Jituwa): Bangaren Amurka, yana tunani a cikin USD/EUR, yana ganin farashin gaba kamar ~0.9524 da ~0.8696. Matsakaicin lissafinsu shine ~0.9110, wanda yayi daidai da farashin EUR/USD na ~1.0977. Sun ba da shawarar $F \approx 1.0977$.
Mafita ta Axiomatic (Matsakaicin Geometric): Yin amfani da mai haɗa kaya na al'ada tare da $\rho=\log$, farashin gaba na gaskiya shine $F = \sqrt{1.05 \times 1.15} \approx 1.0997$.

Matsakaicin farashin geometric na ~1.0997 shine kawai farashi daga cikin iyali da aka rarraba wanda, idan aka yarda da shi, yana tabbatar da cewa babu ɓangaren da za a iya yin amfani da shi ta hanyar ɗayan ta hanyar jerin irin waɗannan kwangilolin, ba tare da la'akari da wane kuɗi aka keɓe a matsayin tushe ba. Wannan yana nuna tasirin aikace-aikacen mafita ta axiomatic: tana ba da takamaiman anga na shawarwari mai kariya.

7. Ayyuka na Gaba da Hanyoyin Bincike

Tsarin yana buɗe hanyoyi masu ban sha'awa da yawa:

Haɗin kai tare da Abubuwan Rangwame na Stochastic: Mafi mahimmancin ƙari shine haɗa ƙimar lokaci na kuɗi da ƙin haɗari. Mai haɗa kaya $\phi$ zai buƙaci yin aiki akan yuwuwar da aka daidaita haɗari ko farashin jiha, ba saƙon tsammani ba. Wannan zai iya haɗa tsarin zuwa samfuran abubuwan rangwame na stochastic (SDF) da suka yaɗu a cikin farashin kadari (duba Cochrane, 2005).
Kasuwanni marasa cikawa da Imani daban-daban: Ƙaddamar da samfurin zuwa rarraba ci gaba da wakilai tare da kimanta yuwuwar rarrabuwa. "Aikin ma'amala" $\rho$ zai iya zama kayan aiki don haɗa imani daban-daban ta hanya mai daidaito, mai alaƙa da wallafe-wallafen kan tafkin ra'ayi.
Cryptocurrency da Tsarin Kuɗi da yawa: A cikin kuɗaɗen rarraba (DeFi) tare da stablecoins da yawa da kadarori masu canzawa, ra'ayin daidaitaccen, farashin "matsakaici" mara cin riba a cikin kwandon yuwuwar farashin gaba yana da dacewa sosai don ƙirƙira masu yin kasuwa ta atomatik da tsarin oracle.
Gwajin Ƙididdiga: Duk da yake takardar ta kasance ta ka'ida, ana iya gwada hasashenta. Shin farashin gaba da aka yi shawarwari a cikin kasuwanni masu zurfi, masu ruwa (inda rashin haɗari ya fi dacewa) suna kama da matsakaicin geometric na abubuwan da ake tsammani na gaba fiye da matsakaicin lissafi? Wannan yana buƙatar auna tsammanin kasuwa a hankali.

8. Nassoshi

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Cover, T. M., & Thomas, J. A. (2006). Abubuwan Ka'idar Bayanai. Wiley-Interscience. (Don haɗin kai zuwa girma fayil da matsakaicin logarithm).
Edlin, A. S. (2002). Sabon Abin Mamaki na Siegel. A cikin Sabon Ƙamus na Tattalin Arziki da Doka.
Mallahi-Karai, K., & Safari, P. (2018). Farashin Musayar Gaba da Sabon Abin Mamaki na Siegel. Jaridar Kuɗi ta Duniya. https://doi.org/10.1016/j.gfj.2018.04.007
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Roper, D. E. (1975). Matsayin binciken ƙimar da ake tsammani don yanke shawara na hasashe a kasuwar kuɗin gaba. Jaridar Quarterly Journal of Economics.
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