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

Table of Contents

1. Gabatarwa

Wannan takarda tana magance gibi mai mahimmanci a cikin wallafe-wallafen sarrafa haɗarin inshora: dabarun zuba jari mafi kyau ga masu inshora da ke aiki a kasuwannin kuɗi da yawa. Yayin da samfuran gargajiya suka mai da hankali kan yanayin kuɗi ɗaya, ayyukan inshora na duniya suna buƙatar fahimtar yanayin haɗarin kan iyaka. Binciken ya haɗu da kimiyyar lissafin inshora tare da lissafin kuɗi don haɓaka cikakken tsari don masu inshora su zuba jari a kasuwannin cikin gida da na waje.

Kalubalen asali yana cikin sarrafa haɗarai uku masu alaƙa: haɗarin da'awar inshora, haɗarin kasuwar kuɗi, da haɗarin canjin kuɗin waje. Ayyukan da suka gabata na Browne (1995), Yang da Zhang (2005), da Schmidli (2002) sun kafa tushe don matsalolin zuba jari na mai inshora amma sun yi watsi da girman kasuwannin kuɗi da yawa wanda ya zama mafi dacewa a cikin tattalin arzikin duniya na yau.

2. Tsarin Samfurin

2.1 Tsarin Rago

Tsarin ragon mai inshora yana bin kusantarwar bazuwa na samfurin Cramér-Lundberg na gargajiya:

$dX(t) = c dt - dS(t)$

inda $c$ ke wakiltar ƙimar kuɗin inshora kuma $S(t)$ shine tsarin tarin da'awar. Ƙarƙashin kusantarwar bazuwa, wannan ya zama:

$dX(t) = \mu dt + \sigma dW_1(t)$

inda $\mu$ shine motsi da aka daidaita nauyin aminci kuma $\sigma$ yana wakiltar sauyin da'awar.

2.2 Samfurin Canjin Kuɗin Waje

Canjin kuɗi tsakanin kuɗin cikin gida da na waje yana bin:

$dE(t) = E(t)[\theta(t)dt + \eta dW_2(t)]$

inda matsakaicin ƙimar girma nan take $\theta(t)$ yana bin tsarin Ornstein-Uhlenbeck:

$d\theta(t) = \kappa(\bar{\theta} - \theta(t))dt + \zeta dW_3(t)$

Wannan ƙayyadaddun ma'anar komawa tana ɗaukar halayen ƙididdiga na ƙimar canjin kuɗi da abubuwan tattalin arziki na asali ke tasiri kamar bambance-bambancen hauhawar farashin kayayyaki da fa'idodin riba.

2.3 Fayil na Zuba Jari

Mai inshora yana raba dukiyarsa a:

• Kayan cikin gida marasa haɗari tare da ƙimar $r_d$
• Kayan waje masu haɗari tare da yanayin dawowa a cikin kuɗin waje
• Canjin kuɗi ta hanyar canjin kuɗi $E(t)$

Tsarin dukiyar gabaɗaya $W(t)$ yana haɓaka bisa ga dabarar zuba jari $\pi(t)$, wanda ke wakiltar rabon da aka zuba a cikin kayan haɗari na waje.

3. Matsalar Haɓakawa

3.1 Manufar Amfani Mai Ma'ana

Mai inshora yana nufin haɓaka tsammanin amfani mai ma'ana na dukiyar ƙarshe:

$\sup_{\pi} \mathbb{E}[U(W(T))] = \sup_{\pi} \mathbb{E}[-\frac{1}{\gamma}e^{-\gamma W(T)}]$

inda $\gamma > 0$ shine madaidaicin ƙimar ƙin haɗari. Wannan aikin amfani yana da dacewa musamman ga masu inshora saboda daidaitaccen halin ƙin haɗari da sauƙin nazari.

3.2 Daidaitaccen Hamilton-Jacobi-Bellman

Aikin ƙima $V(t,w,\theta)$ yana gamsar da daidaitaccen HJB:

$\sup_{\pi} \{V_t + \mathcal{L}^\pi V\} = 0$

tare da yanayin ƙarshe $V(T,w,\theta) = -\frac{1}{\gamma}e^{-\gamma w}$, inda $\mathcal{L}^\pi$ shine janareta mara iyaka na tsarin dukiya ƙarƙashin dabarar $\pi$.

4. Maganin Nazari

4.1 Dabarar Zuba Jari Mafi Kyau

Dabarar zuba jari mafi kyau a cikin kayan haɗari na waje tana ɗaukar siffar:

$\pi^*(t) = \frac{\mu_F - r_f + \eta\rho\zeta\phi(t)}{\gamma\sigma_F^2} + \frac{\eta\rho}{\sigma_F}\frac{V_\theta}{V_w}$

inda $\mu_F$ da $\sigma_F$ su ne sigogin dawowar kayan waje, $r_f$ shine ƙimar riba mara haɗari ta waje, $\rho$ shine alaƙa tsakanin ƙimar canjin kuɗi da dawowar kayan waje, kuma $\phi(t)$ aiki ne na tsarin motsin canjin kuɗi.

4.2 Aikin Ƙima

Aikin ƙima ya yarda da siffar ma'ana mai ma'ana:

$V(t,w,\theta) = -\frac{1}{\gamma}\exp\{-\gamma w e^{r_d(T-t)} + A(t) + B(t)\theta + \frac{1}{2}C(t)\theta^2\}$

inda $A(t)$, $B(t)$, da $C(t)$ sun gamsar da tsarin daidaitattun bambance-bambance da aka samo daga daidaitaccen HJB.

5. Nazarin Lambobi

5.1 Hankan Sigogi

Gwaje-gwajen lambobi sun nuna:

• Tasirin Ƙin Haɗari: Babban $\gamma$ yana rage rabon zuba jari mafi kyau na waje daga kusan 60% zuwa 25% a cikin yanayin da aka gwada
• Sauyin Ƙimar Canjin Kuɗi: Dabarar mafi kyau ta ragu da 15-20% lokacin da $\eta$ ya karu daga 0.1 zuwa 0.3
• Saurin Komawar Ma'ana: Saurin komawa mafi sauri (babba $\kappa$) yana rage buƙatar kariya akan canje-canjen motsin canjin kuɗi

5.2 Aikin Dabarar

Nazarin kwatancen ya nuna dabarar kasuwannin kuɗi da yawa ta fi hanyoyin kuɗi ɗaya da 8-12% a cikin daidaitaccen dukiya a cikin sigogi daban-daban, musamman a lokutan ci gaba da yanayin canjin kuɗi.

6. Fahimta ta Tsakiya & Nazari

Fahimta ta Tsakiya: Wannan takarda tana ba da ci gaba mai mahimmanci amma mai mai da hankali sosai—ta yi nasarar faɗaɗa ka'idar zuba jari na mai inshora zuwa kasuwannin kuɗi biyu amma ta yi haka a cikin ƙayyadaddun zato waɗanda ke iyakance aikace-aikacen nan take. Ƙimar gaske ba ta cikin takamaiman maganin ba amma a cikin nuna cewa tsarin HJB zai iya sarrafa wannan rikitaccen, yana buɗe kofa don ƙarin faɗaɗa na gaskiya.

Kwararar Ma'ana: Marubutan suna bin samfurin sarrafa bazuwar na gargajiya: 1) Saita samfurin tare da kusantarwar bazuwa, 2) Tsarin HJB, 3) Maganin gwada-da-tabbatar tare da siffar ma'ana mai ma'ana, 4) Tabbatarwar lambobi. Wannan hanya tana da ƙwaƙƙwaran lissafi amma ana iya hasashenta ta hanyar koyarwa. Haɗa tsarin Ornstein-Uhlenbeck don motsin canjin kuɗi yana ƙara haɓakawa, tunawa da samfuran irin na Vasicek a cikin riba mai ƙayyadaddun kuɗi, amma maganin ya kasance mai tsabta a ka'idar maimakon tushen ƙididdiga.

Ƙarfi & Aibobi: Babban ƙarfi shine cikakkiyar fasaha—magani yana da kyau kuma dabarar raba masu canji ana amfani da ita da ƙware. Duk da haka, aibobi uku masu mahimmanci suna raunana dacewar aikace-aikace. Na farko, kusantarwar bazuwa na da'awar inshora tana kawar da haɗarin tsalle, wanda shine tushen inshora (kamar yadda aka jaddada a cikin aikin farko na Schmidli (2002, "On Minimizing the Ruin Probability by Investment and Reinsurance")). Na biyu, samfurin yana ɗauka ciniki ci gaba da kasuwanni masu cikakkiyar sassauci, yana yin watsi da ƙuntatawa na ruwa da ke addabar kasuwannin kuɗi a lokacin rikice-rikice. Na uku, nazarin lambobi yana jin kamar bayan baya—yana tabbatarwa maimakon bincika, rashin gwaje-gwajen ƙarfi da ake gani a cikin takardun kuɗi na zamani kamar waɗanda ke cikin Journal of Computational Finance.

Fahimta masu Aiki: Ga masu aiki, wannan takarda tana ba da ma'auni, ba tsari ba. Masu sarrafa haɗari yakamata su fitar da fahimta ta inganci—cewa hasashen motsin canjin kuɗi (ta hanyar tsarin OU) yana haifar da buƙatar kariya—amma yakamata su aiwatar da shi ta amfani da ƙarin dabarun ƙididdiga masu ƙarfi don sigogin OU. Ga masu bincike, matakai masu zuwa bayyananne sune: 1) Haɗa tsalle-tsalle na da'awar bin hanyar Kou (2002, "A Jump-Diffusion Model for Option Pricing"), 2) Ƙara sauyi mai bazuwa ga tsarin ƙimar canjin kuɗi, sanin cikakken rubutaccen tarin sauyi a cikin kasuwannin FX, da 3) Gabatar da farashin ciniki, mai yiwuwa ta amfani da hanyoyin sarrafa turawa. Fannin baya buƙatar ƙarin bambance-bambance akan wannan takamaiman samfurin; yana buƙatar kyawun wannan samfurin tare da gaskiyar ƙididdiga da aka samu a cikin mafi kyawun aikin Jarrow (2018, "A Practitioner's Guide to Stochastic Finance").

7. Cikakkun Bayanai na Fasaha

Sabon abu na lissafi mai mahimmanci ya haɗa da warware tsarin daidaitattun Riccati:

$\frac{dC}{dt} = 2\kappa C - \frac{(\eta\rho\zeta C + \zeta^2 B)^2}{\sigma_F^2} + \gamma\eta^2 C^2 e^{2r_d(T-t)}$

$\frac{dB}{dt} = \kappa\bar{\theta} C + (\kappa - \gamma\eta^2 e^{2r_d(T-t)} C) B - \frac{(\mu_F - r_f)(\eta\rho\zeta C + \zeta^2 B)}{\sigma_F^2}$

tare da yanayin ƙarshe $C(T)=B(T)=0$. Waɗannan daidaito suna sarrafa dogaro da aikin ƙima akan motsin canjin kuɗi mai bazuwa $\theta(t)$.

Dabarar mafi kyau ta rabu zuwa sassa uku:

1. Bukatar Myopic: $\frac{\mu_F - r_f}{\gamma\sigma_F^2}$ – ma'anar lokaci na bambance-bambance na al'ada
2. Kariyar Ƙimar Canjin Kuɗi: $\frac{\eta\rho}{\sigma_F}\frac{V_\theta}{V_w}$ – yana kare canje-canje a cikin saitin damar zuba jari
3. Daidaituwar Motsi: $\frac{\eta\rho\zeta\phi(t)}{\gamma\sigma_F^2}$ – yana lissafin hasashe a cikin motsin canjin kuɗi

8. Misalin Tsarin Nazari

Nazarin Shari'a: Mai Inshora na Duniya na P&C

Yi la'akari da mai inshora na kadarori & laifuka tare da alhaki a cikin USD da EUR. Ta amfani da tsarin takardar:

1. Ƙididdigar Sigogi:
   • Ƙididdige sigogin OU don motsin EUR/USD ta amfani da koma baya na shekaru 10
   • Daidaitu sigogin tsarin da'awar daga bayanan asarar tarihi
   • Ƙididdige ƙimar ƙin haɗari γ daga tsarin zuba jari na tarihin kamfani
2. Aiwatar da Dabarar:
   • Lissafa rabon zuba jari mafi kyau na EUR kowace rana
   • Lura da rabon kariya $\frac{V_\theta}{V_w}$ don siginonin sake daidaitawa
   • Aiwatar tare da bandeji na 5% don rage farashin ciniki
3. Siffanta Aiki:
   • Raba dawowa zuwa: (a) ɓangaren myopic, (b) kariyar canjin kuɗi, (c) lokacin motsi
   • Kwatanta da tsayayyen rabon cikin gida/waje na 60/40

Wannan tsarin, ko da yake an sauƙaƙa, yana ba da tsari mai tsari ga rabon kadarorin mai inshora na kasuwannin kuɗi da yawa wanda ya fi ƙwaƙƙwaran hanyoyin ad hoc na al'ada.

9. Ayyukan Gaba & Hanyoyi

Ayyukan Nan Take:

• Shirye-shiryen Rufin Kuɗi Mai Ƙarfi: Masu inshora za su iya aiwatar da dabarar a matsayin rufin kuɗi, daidaita rabon kariya bisa ga hasashen motsin canjin kuɗi
• Haɓakawar Solvency II: Haɗa tsarin cikin tsarin ORSA (Kima na Haɗarin Kai da Solvency) don masu inshora na Turai
• Baitul Mali na Kamfani Mai Yawan Ƙasashe: Ƙara zuwa sarrafa haɗarin kamfani fiye da inshora

Hanyoyin Bincike:

1. Faɗaɗawar Canjin Tsarin Mulki: Maye gurbin tsarin OU tare da samfurin canjin tsarin Mulki na Markov don ɗaukar karyewar tsari a cikin halayen ƙimar canjin kuɗi
2. Haɗakar Koyon Injina Amfani da hanyoyin sadarwar LSTM don ƙididdige tsarin motsin canjin kuɗi θ(t) maimakon ɗaukan ƙwaƙƙwaran sigogin OU
3. Ayyukan Kuɗi na Rarrabuwa: Daidaita tsarin don samfuran inshora na crypto tare da bayyanuwar cryptocurrency da yawa
4. Haɗakar Haɗarin Yanayi: Haɗa haɗarin canjin yanayi cikin yanayin canjin kuɗi don dogon lokacin zuba jari na mai inshora

10. Nassoshi

1. Browne, S. (1995). Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin. Mathematics of Operations Research, 20(4), 937-958.
2. Schmidli, H. (2002). On Minimizing the Ruin Probability by Investment and Reinsurance. The Annals of Applied Probability, 12(3), 890-907.
3. Yang, H., & Zhang, L. (2005). Optimal Investment for Insurer with Jump-Diffusion Risk Process. Insurance: Mathematics and Economics, 37(3), 615-634.
4. Kou, S. G. (2002). A Jump-Diffusion Model for Option Pricing. Management Science, 48(8), 1086-1101.
5. Jarrow, R. A. (2018). A Practitioner's Guide to Stochastic Finance. Annual Review of Financial Economics, 10, 1-20.
6. Zhou, Q., & Guo, J. (2020). Optimal Control of Investment for an Insurer in Two Currency Markets. arXiv:2006.02857.
7. Bank for International Settlements. (2019). Triennial Central Bank Survey of Foreign Exchange and OTC Derivatives Markets. BIS Quarterly Review.
8. European Insurance and Occupational Pensions Authority. (2020). Solvency II Statistical Report. EIOPA Reports.