Zaben Zuba Jari Mafi Kyau ga Mai Inshora a Kasuwannin Kuɗi Biyu: Nazarin Sarrafa Bazuwar

Table of Contents

1. Gabatarwa

Wannan takarda tana magance wani gibi mai mahimmanci a kimiyyar lissafin inshora da lissafin kuɗi: dabarar zuba jari mafi kyau ga kamfanin inshora da ke aiki a cikin kasuwannin kuɗi da yawa. Samfuran gargajiya, irin su na Browne (1995) da Schmidli (2002), sun fi mayar da hankali kan yanayin kuɗi ɗaya. Duk da haka, a cikin tattalin arzikin duniya da ke ƙara haɓaka, dole ne masu inshora su sarrafa kadarori da alawus da aka ƙididdige su a cikin kuɗi daban-daban, wanda ke fallasa su ga haɗarin musayar kuɗi na waje. Wannan binciken ya faɗaɗa samfurin rago na gargajiya na Cramér-Lundberg zuwa tsarin kuɗi biyu, yana haɗa farashin musayar kuɗi na bazuwar da aka ƙirƙira ta hanyar tsarin Ornstein-Uhlenbeck (OU). Manufar ita ce haɓaka tsammanin amfani mai ma'ana na arziƙin ƙarshe, ma'auni na gama gari na gujewa haɗari a cikin kuɗin inshora.

2. Tsarin Ƙirƙira Samfurin

2.1 Tsarin Rago

Tsarin ragon mai inshora $R(t)$ an ƙirƙira shi ta amfani da kusantar rarrabawa na samfurin gargajiya na Cramér-Lundberg: $$dR(t) = c dt - d\left(\sum_{i=1}^{N(t)} Y_i\right) \approx (c - \lambda \mu_Y) dt + \sigma_R dW_R(t)$$ inda $c$ shine ƙimar kuɗin inshora, $\lambda$ shine ƙarfin isowar da'awar, $\mu_Y$ shine matsakaicin girman da'awar, kuma $W_R(t)$ shine motsin Brownian na ma'auni. Wannan kusantarwa yana sauƙaƙa tsarin Poisson mai haɗaka don sauƙin nazari, dabarar gama gari a cikin wallafe-wallafen (duba, misali, Grandell, 1991).

2.2 Kasuwar Kuɗi

Mai inshora zai iya zuba jari a cikin:

1. Kadari Mai Tsaro na Cikin Gida: $dB(t) = r_d B(t) dt$, tare da ƙimar riba $r_d$.
2. Kadari Mai Haɗari na Waje: An ƙirƙira shi ta hanyar motsin Brownian na geometric: $dS_f(t) = \mu_f S_f(t) dt + \sigma_f S_f(t) dW_f(t)$.

Babban ƙirƙira shine ba da damar zuba jari a cikin kadarorin waje, wanda ke buƙatar ƙirar farashin musayar kuɗi.

2.3 Ƙa'idodin Canjin Kuɗi

Farashin musayar kuɗi $Q(t)$ (raka'a na kuɗin cikin gida a kowace raka'a na kuɗin waje) da motsinsa an ƙirƙira su kamar haka: $$dQ(t) = Q(t)[\theta(t) dt + \sigma_Q dW_Q(t)]$$ $$d\theta(t) = \kappa(\bar{\theta} - \theta(t)) dt + \sigma_\theta dW_\theta(t)$$ A nan, $\theta(t)$ shine matsakaicin ƙimar girma na nan take yana bin tsarin OU, yana ɗaukar halayen dawo da matsakaici na musamman na farashin musayar kuɗi da abubuwan tattalin arziki masu girma ke tasiri irin su bambance-bambancen hauhawar farashin kayayyaki da daidaiton ƙimar riba (Fama, 1984). $W_Q(t)$ da $W_\theta(t)$ su ne motsin Brownian masu alaƙa.

3. Matsalar Ingantawa

3.1 Aikin Manufa

Bari $X(t)$ ya zama jimillar arziƙi a cikin kuɗin cikin gida. Mai inshora yana sarrafa adadin $\pi(t)$ da aka zuba a cikin kadari mai haɗari na waje. Manufar ita ce haɓaka tsammanin amfani mai ma'ana na arziƙin ƙarshe a lokacin $T$: $$\sup_{\pi} \mathbb{E}[U(X(T))] = \sup_{\pi} \mathbb{E}\left[-\frac{1}{\gamma} e^{-\gamma X(T)}\right]$$ inda $\gamma > 0$ shine ma'aunin gujewa haɗari na cikakke mai dawwama. Amfani mai ma'ana yana sauƙaƙa daidaitun HJB yayin da yake kawar da dogaro da arziƙi a cikin dabarar mafi kyau a ƙarƙashin wasu sharuɗɗa.

3.2 Daidaitun Hamilton-Jacobi-Bellman

Bari $V(t, x, \theta)$ ya zama aikin ƙima. Daidaitun HJB mai alaƙa shine: $$\sup_{\pi} \left\{ V_t + \mathcal{L}^{\pi} V \right\} = 0$$ tare da sharuɗɗan ƙarshe $V(T, x, \theta) = U(x) = -\frac{1}{\gamma}e^{-\gamma x}$. Ma'aikacin bambance-bambance $\mathcal{L}^{\pi}$ ya haɗa da ƙa'idodin $X(t)$, $\theta(t)$, da alaƙar su. Warware wannan PDE shine babban ƙalubalen nazari.

4. Maganin Nazari

4.1 Dabarar Zuba Jari Mafi Kyau

Takardar ta samo zuba jari mafi kyau a cikin kadari mai haɗari na waje kamar haka: $$\pi^*(t) = \frac{\mu_f + \theta(t) - r_d}{\gamma (\sigma_f^2 + \sigma_Q^2 + 2\rho_{fQ}\sigma_f\sigma_Q)} + \text{Sharuddan Daidaitawa da suka haɗa da } \theta(t)$$ Wannan dabarar tana da fassarar fahimta: kalmar farko ita ce mafita irin ta Merton (Merton, 1969), inda zuba jari ya yi daidai da ribar da ta wuce ($\mu_f + \theta(t) - r_d$) kuma ya saba da haɗari ($\gamma$ da jimillar bambance-bambance). Sharuddan daidaitawa suna lissafin yanayin bazuwar na motsin farashin musayar kuɗi $\theta(t)$ da alaƙarsa da wasu matakai.

4.2 Aikin Ƙima

An gano aikin ƙima yana da siffar: $$V(t, x, \theta) = -\frac{1}{\gamma} \exp\left\{-\gamma x e^{r_d (T-t)} + A(t) + B(t)\theta + \frac{1}{2}C(t)\theta^2 \right\}$$ inda $A(t)$, $B(t)$, da $C(t)$ su ne ayyuka na ƙayyadaddun lokaci waɗanda suka gamsar da tsarin daidaitattun bambance-bambance (daidaitattun Riccati). Wannan tsarin ya zama gama gari a cikin matsalolin sarrafa murabba'i mai layi tare da amfani mai ma'ana.

5. Nazarin Lambobi

Takardar ta gabatar da nazarin lambobi don kwatanta halayyar dabarar mafi kyau. Abubuwan lura da suka haɗa da:

• Hankali ga $\theta(t)$: Zuba jari mafi kyau $\pi^*(t)$ yana ƙaruwa lokacin da tsammanin ƙimar farashin musayar kuɗi $\theta(t)$ ya yi yawa, yana ƙarfafa zuba jari a cikin kadarin waje.
• Tasirin Gujewa Haɗari ($\gamma$): Mafi girman gujewa haɗari yana rage matsayi a cikin kadari mai haɗari na waje sosai, kamar yadda ake tsammani.
• Tasirin Alaƙa: Alaƙar mara kyau tsakanin ribar kadarin waje da canjin farashin musayar kuɗi ($\rho_{fQ}$) na iya zama shinge na halitta, yana ba da damar mafi girman matsayi mafi kyau.

Nazarin yana iya haɗawa da kwaikwayon hanyoyi don $\theta(t)$ da zana $\pi^*(t)$ akan lokaci, yana nuna yanayinsa na ƙarfi da dogaro da yanayi.

6. Fahimtar Jiki & Ra'ayi na Manazarta

Fahimtar Jiki: Wannan takarda ba wani ƙarin gyara ne kawai ga samfurin zuba jari na mai inshora ba. Gudunmawar ta na asali ita ce haɗa haɗarin kuɗi na bazuwar a cikin tsarin sarrafa kadarori da alawus na mai inshora bisa ƙa'ida. Ta hanyar ƙirar motsin farashin musayar kuɗi a matsayin tsarin OU mai dawo da matsakaici, marubutan sun wuce samfuran ma'auni masu sauƙi kuma sun ɗauki wani mahimmin gaskiya ga masu inshora na duniya: haɗarin kuɗi abu ne mai dorewa, mai ƙarfi wanda dole ne a sarrafa shi da ƙarfi, ba kudin canji kawai ba.

Kwararar Hankali: Hankali yana da inganci kuma yana bin wasan kwaikwayon sarrafa bazuwar: (1) Faɗaɗa ragon Cramér-Lundberg zuwa rarrabawa, (2) Sanya Layer akan kasuwar kuɗi biyu tare da farashin musayar kuɗi na bazuwar, (3) Ayyana manufar amfani mai ma'ana, (4) Samar da daidaitun HJB, (5) Yin amfani da rabuwar amfani mai ma'ana don zato siffar mafita, da (6) Warware daidaitattun Riccati da suka biyo baya. Wannan hanya ce da aka yi amfani da ita sosai amma mai tasiri, mai kama da aikin ginshiƙi na Fleming da Soner (2006) akan sarrafa rarrabawa.

Ƙarfi & Kurakurai: Ƙarfi: Kyawun samfurin shine babban ƙarfinsa. Haɗin amfani mai ma'ana da ƙa'idodin affine don $\theta(t)$ yana haifar da mafita mai sauƙi, rufaffiyar fom - wanda ba kasafai ba a cikin matsalolin sarrafa bazuwar. Wannan yana ba da kwatankwacin lissafi bayyananne. Haɗa bayyananne na alaƙa tsakanin ribar kadari da na kuɗi shima yana da yabo, kamar yadda ya yarda cewa waɗannan haɗarin ba su keɓance ba. Kurakurai: Zato na samfurin shine ƙashin dugadugansa. Kusantar rarrabawa na ragon inshora yana cire haɗarin tsalle ( ainihin ainihin da'awar inshora), yana iya rage bayanin haɗarin wutsiya. Tsarin OU don $\theta(t)$, yayin dawo da matsakaici, bazai iya ɗaukar "canjin tsarin ƙulla" ko raguwar ƙima da aka gani a kasuwannin masu tasowa ba. Bugu da ƙari, samfurin ya yi watsi da farashin ma'amala da ƙuntatawa kamar rashin sayar da gajere, waɗanda ke da mahimmanci don aiwatarwa na aiki. Idan aka kwatanta da ƙarin ingantattun hanyoyi kamar zurfin ƙarfafa koyo don ingantaccen fayil (Theate & Ernst, 2021), wannan samfurin yana jin daɗin nazari amma yana iya zama mai rauni a duniyar gaske.

Fahimta Mai Aiki: Ga Shugabannin Zuba Jari a manyan masu inshora na duniya, wannan binciken ya jaddada cewa shinge na kuɗi ba zai iya zama tunani na baya ba. Dabarar mafi kyau tana da ƙarfi kuma ta dogara ne akan yanayin yanzu na motsin farashin musayar kuɗi ($\theta(t)$), wanda dole ne a ci gaba da ƙididdige shi. Masu aiki ya kamata su: 1. Gina Injunan Ƙididdiga: Haɓaka ingantattun tace Kalman ko hanyoyin MLE don ƙididdige yanayin ɓoyayye $\theta(t)$ da sigoginsa ($\kappa, \bar{\theta}, \sigma_\theta$) a ainihin lokaci. 2. Gwajin Damuwa Bayan OU: Yi amfani da tsarin samfurin amma maye gurbin tsarin OU tare da ƙarin samfuran rikitarwa (misali, canjin tsarin mulki) a cikin nazarin yanayi don auna juriyar dabarar. 3. Mayar da Hankali kan Alaƙa: Ci gaba da sa ido da ƙirar alaƙa ($\rho_{fQ}$) tsakanin ribar kadarin waje da motsin kuɗi, domin shi ne babban mai ƙayyadaddun ma'aunin shinge da mafi kyawun fallasa.

7. Cikakkun Bayanai na Fasaha & Tsarin Lissafi

Babban injin lissafi shine daidaitun Hamilton-Jacobi-Bellman (HJB) daga ka'idar sarrafa mafi kyau na bazuwar. Ƙa'idodin arziƙi a cikin kuɗin cikin gida, la'akari da zuba jari $\pi(t)$ a cikin kadarin waje, sune: $$dX(t) = \left[ r_d X(t) + \pi(t)(\mu_f + \theta(t) - r_d) + (c - \lambda\mu_Y) \right] dt + \pi(t)\sigma_f dW_f(t) + \pi(t)\sigma_Q dW_Q(t) + \sigma_R dW_R(t)$$ Daidaitun HJB don aikin ƙima $V(t,x,\theta)$ shine: $$ \begin{aligned} 0 = \sup_{\pi} \Bigg\{ & V_t + \left[ r_d x + \pi(\mu_f + \theta - r_d) + (c - \lambda\mu_Y) \right] V_x + \kappa(\bar{\theta} - \theta) V_\theta \\ & + \frac{1}{2}\left( \pi^2(\sigma_f^2 + \sigma_Q^2 + 2\rho_{fQ}\sigma_f\sigma_Q) + \sigma_R^2 + 2\pi(\rho_{fR}\sigma_f\sigma_R + \rho_{QR}\sigma_Q\sigma_R) \right) V_{xx} \\ & + \frac{1}{2}\sigma_\theta^2 V_{\theta\theta} + \pi \sigma_\theta (\rho_{f\theta}\sigma_f + \rho_{Q\theta}\sigma_Q) V_{x\theta} \Bigg\} \end{aligned} $$ Amfanin amfani mai ma'ana $V(t,x,\theta) = -\frac{1}{\gamma}\exp\{-\gamma x e^{r_d(T-t)} + \phi(t,\theta)\}$ yana sauƙaƙa wannan zuwa PDE don $\phi(t,\theta)$, wanda tare da zato murabba'i $\phi(t,\theta)=A(t)+B(t)\theta+\frac{1}{2}C(t)\theta^2$ yana haifar da daidaitattun Riccati don $A(t), B(t), C(t)$.

8. Tsarin Nazari: Wani Lamari na Aiki

Yanayi: Wani mai inshora na Japan wanda ba na rai ba (kuɗin cikin gida: JPY) yana riƙe da rago daga ayyukansa na cikin gida. Yana la'akari da zuba wani ɓangare na kadarorinsa a cikin hannun jarin fasaha na Amurka (kadari na waje, USD). Manufar ita ce tantance madaidaicin rabon ƙarfi ga wannan kadarin waje a tsawon shekaru 5.

Aikace-aikacen Tsarin:

1. Daidaituwar Sigogi:
   • Rago (JPY): Ƙididdige $c$, $\lambda$, $\mu_Y$ daga bayanan da'awar tarihi don samun motsi $(c-\lambda\mu_Y)$ da saurin girgiza $\sigma_R$.
   • Hannun Jarin Fasaha na Amurka (USD): Ƙididdige tsammanin dawowa $\mu_f$ da saurin girgiza $\sigma_f$ daga ma'aunin ma'auni (misali, Nasdaq-100).
   • Farashin Musayar USD/JPY: Yi amfani da bayanan tarihi don daidaita sigogin tsarin OU don $\theta(t)$: matsakaicin dogon lokaci $\bar{\theta}$, saurin dawo da matsakaici $\kappa$, da saurin girgiza $\sigma_\theta$. Ƙididdige alaƙa ($\rho_{fQ}, \rho_{fR},$ da sauransu).
   • Ƙimar Ribar Tsaro: Yi amfani da ribar Bond na Gwamnatin Japan (JGB) don $r_d$ da ribar Baitul Mali na Amurka (wanda aka canza zuwa tsarin samfurin).
   • Gujewa Haɗari: Saita $\gamma$ bisa isasshen jarin kamfanin da juriyar haɗari.
2. Lissafin Dabarar: Saka sigogin da aka daidaita cikin dabarar don $\pi^*(t)$. Wannan yana buƙatar ƙimar yanzu da aka ƙididdige na yanayin ɓoyayye $\theta(t)$, wanda za'a iya tace shi daga motsin farashin musayar kuɗi na kwanan nan.
3. Fitowa & Sa ido: Samfurin yana fitar da kaso mai canzawa da lokaci. Baitul malin mai inshora zai daidaita ma'aunin shinge na FX da rabon adalci daidai. Dole ne a sabunta ƙididdigar $\theta(t)$ lokaci-lokaci (misali, kowane wata), wanda ke haifar da sake daidaitawa mai ƙarfi.

Wannan tsarin yana ba da tsarin tsari, mai jagorar samfurin zuwa matsala mai rikitarwa na rabon kuɗi da yawa.

9. Ayyukan Gaba & Hanyoyin Bincike

Samfurin yana buɗe hanyoyi da yawa don faɗaɗawa da aikace-aikacen aiki:

• Fayilolin Kuɗi Da Yawa: Faɗaɗa samfurin zuwa fiye da kuɗin waje ɗaya da kadari, sarrafa hanyar sadarwar alaƙar kuɗi. Wannan ya yi daidai da bukatun masu inshora na ƙasashe daban-daban.
• Haɗa Haɗarin Tsalle: Maye gurbin kusantar rarrabawa tare da mafi dacewa na tsalle-tsalle ko tsarin Lévy don ragon inshora don ƙirar da'awar bala'i mafi kyau, ta amfani da dabarun daga Surya (2022) akan sarrafa mafi kyau ƙarƙashin tsarin tsalle.
• Samfuran Canjin Tsarin Mulki: Ƙirar $\theta(t)$ ko sigogin kasuwa tare da tsarin canjin tsarin mulki na Markov don ɗaukar manufofin kuɗi daban-daban ko zagayowar tattalin arziki, kamar yadda aka gani a cikin ayyukan Elliott da sauransu.
• Haɗa Koyon Injina: Yin amfani da hanyoyin sadarwar LSTM ko wakilan ƙarfafa koyo don ƙididdige yanayin ɓoyayye $\theta(t)$ da ƙa'idodinsa daga bayanan kasuwa masu yawan mitar, matsawa bayan zaton OU na sigogi.
• Haɗewar ALM: Shigar da wannan samfurin zuba jari cikin faɗaɗa tsarin Sarrafa Kadarori da Alawus (ALM) wanda kuma yana inganta farashin samfurin inshora da dabarun sake inshora.
• Kuɗi na Rarrabuwa (DeFi): Yin amfani da samfurin don sarrafa baitul malin tsarin inshora mai rarrabuwa (misali, Nexus Mutual) wanda ke riƙe da kadarorin crypto a cikin kuɗin asali na blockchain da yawa, inda saurin girgiza farashin musayar kuɗi ya yi tsanani.

10. Nassoshi

1. Browne, S. (1995). Manufofin Zuba Jari Mafi Kyau don Kamfani tare da Tsarin Haɗari na Bazuwar: Amfani Mai Ma'ana da Rage Yiwuwar Rushewa. Lissafin Ayyukan Bincike, 20(4), 937-958.
2. Fama, E. F. (1984). Farashin musayar kuɗi na gaba da tabo. Jaridar Tattalin Arziki ta Kuɗi, 14(3), 319-338.
3. Fleming, W. H., & Soner, H. M. (2006). Sarrafa Matakai na Markov da Maganganun Narkewa (bugu na 2). Springer.
4. Grandell, J. (1991). Bangarorin Ka'idar Haɗari. Springer-Verlag.
5. Merton, R. C. (1969). Zaɓin Fayil na Rayuwa Ƙarƙashin Rashin Tabbaci: Shari'ar Ci gaba da Lokaci. Bita na Tattalin Arziki da Ƙididdiga, 51(3), 247-257.
6. Schmidli, H. (2002). Kan rage yuwuwar lalata ta hanyar zuba jari da sake inshora. Takardun Aikace-aikacen Aikace-aikace, 12(3), 890-907.
7. Surya, B. A. (2022). Zuba jari mafi kyau da sake inshora ga mai inshora ƙarƙashin samfuran tsalle-tsalle. Jaridar Aikin Aktiya na Scandinavia, 2022(5), 401-429.
8. Theate, T., & Ernst, D. (2021). Aikace-aikacen Koyon Ƙarfafa Koyo mai zurfi zuwa Cinikin Algorithm. Ƙwararrun Tsarin tare da Aikace-aikace, 173, 114632.
9. Zhou, Q., & Guo, J. (2020). Sarrafa Zuba Jari Mafi Kyau ga Mai Inshora a Kasuwannin Kuɗi Biyu. arXiv preprint arXiv:2006.02857.