Tsarin Ƙimar Musayar Yen-Dollar tare da Matsakaicin Motsi da Tasirin Sarrafa Kai

1. Gabatarwa

Wannan takarda ta gabatar da wani nau'in tsari mai kama da autoregressive mai tasirin sarrafa kai don ƙirar ƙimar musayar waje, musamman ma kan kasuwar Yen-Dollar. Binciken ya magance sanannen al'amuran "wutsiyoyi masu kiba" a cikin rarraba yuwuwar canjin ƙimar da kuma dogon lokacin daidaiton ƙarfin motsi (volatility), waɗanda suka bambanta da zato na rarraba al'ada (normal distribution). Marubutan sun gabatar da sabuwar fasaha ta raba ƙimar musayar zuwa wani ɓangare na matsakaicin motsi da kuma ragowar hayaniyar da ba ta da alaƙa. Binciken ya yi amfani da bayanan tick-by-tick na ƙimar musayar yen-dollar daga 1989 zuwa 2002, wanda CQG ta bayar.

2. Matsakaicin Motsi Mafi Kyau

Jigon hanyar ya ƙunshi ayyana ƙimar matsakaicin motsi "mafi kyau" $P(t)$ wanda ke raba hayaniyar da ba ta da alaƙa $ε(t)$ daga bayanan kasuwar da aka lura $P(t+1)$. An ayyana alaƙar kamar haka:

$P(t+1) = P(t) + ε(t)$

inda $P(t) = \sum_{k=1}^{K} w_P(k) \cdot P(t - k + 1)$. Ana daidaita ma'aunin nauyi $w_P(k)$ don rage alaƙar kai (autocorrelation) na kalmar ragowar $ε(t)$. Binciken ya gano cewa ma'aunin nauyi mafi kyau yana raguwa kusan bisa yanayin lissafi (exponentially) tare da halayyar lokaci na 'yan mintuna. Bugu da ƙari, ƙimar mutlaƙa na hayaniyar $|ε(t)|$ ita kanta tana nuna dogon lokacin daidaiton kai (long autocorrelation). Don ƙirar wannan, ana kuma rarraba lissafin logarithm na ƙimar mutlaƙa ta hayaniyar ta hanyar tsari mai kama da autoregressive:

$\log|ε(t+1)| = \log|\overline{ε}(t)| + b(t)$

inda $\log|\overline{ε}(t)| = \sum_{k=1}^{K'} w_\varepsilon(k) \cdot \log|ε(t - k + 1)|$. Muhimmanci, ma'aunin nauyi $w_\varepsilon(k)$ don ƙimar yen-dollar yana raguwa bisa ga dokar iko (power law) $w_\varepsilon(k) \propto k^{-1.1}$, kamar yadda aka nuna a cikin Fig.1 na takardar asali. Wannan yana nuna wani tsari na daban, mai dogon ƙwaƙwalwar ajiya (longer-memory) da ke tafiyar da ƙarfin motsi (volatility) idan aka kwatanta da farashin kansa.

3. Tsarin Sarrafa Kai don Ƙimar Musayar Waje

Dangane da binciken gabaɗaya, marubutan sun ba da shawarar cikakkiyar ƙirar sarrafa kai don ƙimar musayar waje:

$\begin{cases} P(t+1) = P(t) + ε(t) \\ ε(t+1) = \alpha(t) \cdot \overline{ε}(t) \cdot b(t) + f(t) \end{cases}$

A nan, $\alpha(t)$ alamar bazuwar ce (+1 ko -1), $b(t)$ kalmar hayaniyar da ba ta da alaƙa da aka zana daga rarraba da aka lura, kuma $f(t)$ tana wakiltar girgizar waje (misali, labarai, shisshigi). Matsakaicin motsi $P(t)$ da $\overline{ε}(t)$ an ayyana su kamar yadda yake a sashin da ya gabata. Simintin amfani da wannan ƙirar tare da aikin ma'aunin nauyi na yanayin lissafi $w_P(k) \propto e^{-0.35k}$ da hayaniyar waje ta Gaussian $f(t)$ sun yi nasarar sake fitar da mahimman gaskiyar gaskiya na kasuwa, kamar rarraba mai wutsiya mai kiba da tarin ƙarfin motsi (volatility clustering).

4. Fahimtar Jigo & Ra'ayin Mai Bincike

Fahimtar Jigo: Wannan takarda tana ba da fahimta mai ƙarfi, amma mai sauƙi da kyau: rawan rikice-rikicen ƙimar Yen-Dollar za a iya raba shi zuwa siginar yanayi mai gajeren ƙwaƙwalwar ajiya (matsakaicin motsi "mafi kyau") da tsarin ƙarfin motsi mai dogon ƙwaƙwalwar ajiya, wanda ke gudana ta hanyar dogaro gaɗaɗɗun 'yan kasuwa akan mayar da martani mai ma'auni na motsin farashi na kwanan nan. Gwanin gaske shine gano ma'auni biyu na daban na lokaci—raguwar yanayin lissafi (exponential decay) don farashi (~mintuna) da raguwar dokar iko (power-law decay) don ƙarfin motsi—waɗanda kai tsaye ke haifar da nau'ikan nau'ikan tsarin ƙananan kasuwa da tunanin 'yan kasuwa.

Kwararar Ma'ana: Hujjar tana da gamsarwa. Fara da wani wasan wasa na gabaɗaya (wutsiyoyi masu kiba, tarin ƙarfin motsi). Maimakon tsalle zuwa ƙirar ƙirar tushen wakili mai rikitarwa, sun yi tambaya mai tsabta: menene matsakaicin motsi mafi sauƙi wanda ke farfado da dawowar farashi? Amsar ta bayyana ingantaccen lokacin kasuwa. Sa'an nan, sun lura cewa girman hayaniyar da aka farfado ba farar fata ba ce—tana da ƙwaƙwalwar ajiya. Ƙirar wannan ƙwaƙwalwar ajiya ta bayyana tsarin dokar iko. Wannan rabe-raben mataki biyu ya tilasta yanke shawara na tsarin sarrafa kai inda ƙarfin motsi na baya yake daidaita ƙarfin motsi na gaba, wani ra'ayi mai kama da sauran tsarin rikitarwa da aka yi bincike a cikin ilimin kimiyyar lissafi.

Ƙarfi & Kurakurai: Ƙarfin ƙirar shine tushen gabaɗaya da ƙarancinsa. Ba ya dogaro sosai akan "nau'ikan wakili" da ba a iya gani. Duk da haka, babban aibinsa shine yanayinsa na zahiri. Ya bayyana "abin da" (ma'aunin nauyi na dokar iko) da kyau amma ya bar "dalilin" a bude. Me yasa 'yan kasuwa gabaɗaya suka haifar da ma'aunin nauyi na $k^{-1.1}$? Shin yana da kyau a ƙarƙashin wasu sharuɗɗa, ko kuma wani halitta, mai yuwuwar rashin inganci, halayyar garke? Bugu da ƙari, kula da girgizar waje $f(t)$ a matsayin hayaniyar Gaussian mai sauƙi aibi ne bayyananne; a zahiri, shisshigi da labarai suna da tasiri mai rikitarwa, mara daidaituwa, kamar yadda aka lura a cikin binciken daga Bankin don Haɗin Kan Ƙasashen Duniya (BIS) kan ingancin shisshigin bankin tsakiya.

Fahimta Mai Aiki: Ga masu ƙididdiga da masu kula da haɗari, wannan takarda ma'adinai ce. Na farko, tana tabbatar da amfani da matsakaicin motsi na gajeren lokaci sosai (ma'aunin mintuna) don cire siginar mitar girma. Na biyu, kuma mafi mahimmanci, tana ba da tsari don gina hasashen ƙarfin motsi mafi kyau. Maimakon ƙirar GARCH-dangi, mutum zai iya ƙididdige ma'aunin nauyi na dokar iko $w_\varepsilon(k)$ akan ƙarfin motsi kai tsaye don hasashen tashin hankalin kasuwa na gaba. Za a iya gwada dabarun ciniki da suka yi dogon lokaci (go long) akan ƙarfin motsi lokacin da ƙimar $\overline{ε}(t)$ ta ƙirar ta yi girma. Ƙirar kuma tana aiki azaman ma'auni mai ƙarfi; duk wani ƙirar AI/ML mai rikitarwa don hasashen FX dole ne aƙalla ta fi wannan rabe-raben mai sauƙi, mai ban sha'awa na kimiyyar lissafi don tabbatar da rikitarwarsa.

5. Cikakkun Bayanai na Fasaha & Tsarin Lissafi

Jigon lissafi na ƙirar shine rabe-raben biyu. Rabe-raben farashi na farko tsari ne mai kama da autoregressive (AR) akan matakin farashi da kansa, wanda aka tsara don farfado da dawowar mataki na farko:

$P(t+1) - P(t) = ε(t)$, tare da $\text{Corr}(ε(t), ε(t+\tau)) \approx 0$ don $\tau > 0$.

Na biyu, kuma mafi sabon abu, rabe-raben ya shafi tsarin AR akan log-volatility:

$\log|ε(t+1)| = \sum_{k=1}^{K'} w_\varepsilon(k) \cdot \log|ε(t - k + 1)| + b(t)$.

Binciken mai mahimmanci shine nau'in aikin kernels: $w_P(k)$ yana raguwa bisa yanayin lissafi (exponentially) (gajeren ƙwaƙwalwar ajiya), yayin da $w_\varepsilon(k)$ yana raguwa a matsayin dokar iko $k^{-\beta}$ tare da $\beta \approx 1.1$ (dogon ƙwaƙwalwar ajiya). Wannan alaƙar kai ta dokar iko a cikin ƙarfin motsi alama ce ta kasuwannin kuɗi, mai kama da al'amuran "Hurst exponent" da aka lura a yawancin jerin lokuta masu rikitarwa. Cikakkiyar ƙirar a cikin lissafin (5) da (6) ta haɗa waɗannan, tare da tsarin ninkawa $\alpha(t) \cdot \overline{ε}(t) \cdot b(t)$ yana tabbatar da ma'aunin ƙarfin motsi yana daidaita sabon abu na farashi mai bazuwar alama.

6. Sakamakon Gwaji & Binciken Chati

Takardar ta gabatar da adadi biyu masu mahimmanci dangane da bayanan tick na Yen-Dollar (1989-2002).

Fig.1: Ma'aunin nauyi $w_\varepsilon(k)$ na ƙimar mutlaƙa $|ε(t)|$. Wannan chati yana nuna a zahiri raguwar dokar iko na ma'aunin nauyi da aka yi amfani da su a cikin tsarin log-volatility autoregressive. Layin da aka zana yana nuna aikin $w_\varepsilon(k) \propto k^{-1.1}$, wanda ya dace da ma'aunin nauyi da aka ƙiyasta na gabaɗaya. Wannan shaida ce kai tsaye ta dogon ƙwaƙwalwar ajiya a cikin ƙarfin motsi, wanda ya bambanta da gajeren ƙwaƙwalwar ajiya a cikin farashi.

Fig.2: Alaƙar kai (Autocorrelations) na $|ε(t)|$ da $b(t)$. Wannan adadi yana aiki azaman chati na tabbatarwa. Yana nuna cewa ƙimar dawowar mutlaƙa danye $|ε(t)|$ suna da jinkirin raguwa, alaƙar kai mai kyau (tarin ƙarfin motsi). Akasin haka, kalmar ragowar $b(t)$ da aka cire bayan amfani da tsarin AR tare da ma'aunin nauyi na dokar iko ba ta nuna wata alaƙar kai mai mahimmanci ba, yana tabbatar da cewa ƙirar ta yi nasarar ɗaukar tsarin ƙwaƙwalwar ajiya a cikin ƙarfin motsi.

7. Tsarin Bincike: Wani Lamari Mai Aiki

Lamari: Bincika Haɗin Kuɗin Sirri (misali, BTC-USD). Yayin da takardar asali ta yi bincike kan Forex, wannan tsarin yana da amfani sosai ga kasuwannin kuɗin sirri, waɗanda aka sani da matsanancin ƙarfin motsi. Mai bincike zai iya maimaita binciken kamar haka:

1. Shirya Bayanai: Sami bayanan farashi na mitar girma (misali, minti 1) na BTC-USD daga musayar kamar Coinbase.
2. Mataki na 1 - Nemo $w_P(k)$: Gwada sigogi daban-daban na raguwar yanayin lissafi (exponential decay) don $w_P(k)$ don nemo saitin da zai rage alaƙar kai (autocorrelation) na sakamakon $ε(t)$. Sakamakon da ake tsammani shine halayyar lokaci mai yuwuwa a cikin kewayon mintuna 5-30 don kuɗin sirri.
3. Mataki na 2 - Bincika $|ε(t)|$: Dace da tsarin AR zuwa $\log|ε(t)|$. Ƙididdige ma'aunin nauyi $w_\varepsilon(k)$. Tambaya mai mahimmanci ita ce: shin suna bin dokar iko $k^{-\beta}$? Ma'auni $β$ na iya bambanta da 1.1, mai yiwuwa yana nuna ƙarin dagewar ƙwaƙwalwar ajiya na ƙarfin motsi a cikin kuɗin sirri.
4. Fahimta: Idan dokar iko ta kasance, yana nuna cewa 'yan kasuwar kuɗin sirri, kamar 'yan kasuwar Forex, suna amfani da dabarun tare da dogon ƙwaƙwalwar ajiya na mayar da martani akan ƙarfin motsi na baya. Wannan kamancen tsari yana da tasiri mai zurfi ga ƙirar haɗari da farashin abubuwan da aka samo dashi a cikin kuɗin sirri, wanda sau da yawa yana ɗaukarsa azaman sabon nau'in kadari gaba ɗaya.

8. Aikace-aikacen Gaba & Jagororin Bincike

Ƙirar ta buɗe hanyoyi masu ban sha'awa da yawa:

• Tabbatar da Haɗin Kadari: Amfani da hanyar guda ɗaya ga hannun jari, kayayyaki, da shaidu don ganin ko ma'auni $β \approx 1.1$ ya zama mai ɗorewa ko na musamman ga kasuwa.
• Haɗawa da Koyon Injin: Amfani da abubuwan da aka raba $P(t)$ da $\overline{ε}(t)$ a matsayin fasali masu tsabta, mafi tsayayye don ƙirar hasashen farashi na koyon zurfi, mai yuwuwar inganta aiki fiye da bayanan farashi danye.
• Tushen Ƙirar Tushen Wakili (ABM): Ayyukan ma'aunin nauyi na gabaɗaya $w_P(k)$ da $w_\varepsilon(k)$ suna ba da maƙasudin daidaitawa masu mahimmanci ga ABMs. Masu bincike za su iya ƙirƙirar ƙa'idodin wakili waɗanda gabaɗaya suka haifar da waɗannan kernels na mayar da martani daidai.
• Manufofi & Ƙa'ida: Fahimtar ma'auni na halayyar lokaci na martanin ɗan kasuwa (mintuna) na iya taimakawa wajen ƙirƙirar masu karya da'ira masu inganci ko kimanta tasirin ciniki mai mitar girma (HFT). Ƙirar za ta iya kwatanta tasirin kasuwa na canje-canjen ƙa'ida akan tsarin mayar da martani.
• Hasashen Girgizar Waje: Babban mataki na gaba shine matsawa fiye da ƙirar $f(t)$ a matsayin hayaniya mai sauƙi. Aikin gaba zai iya amfani da sarrafa harshe na halitta (NLP) akan ciyarwar labarai don ƙididdige $f(t)$, ƙirƙirar ƙirar haɗin kimiyyar lissafi-AI don abubuwan da ba kasafai ba amma masu tasiri.

9. Nassoshi

1. Mantegna, R. N., & Stanley, H. E. (2000). Gabatarwa ga Econophysics: Haɗin Kai da Rikitarwa a cikin Kuɗi. Cambridge University Press. (Don mahallin wutsiyoyi masu kiba da ma'auni a cikin kuɗi).
2. Mizuno, T., Takayasu, M., & Takayasu, H. (2003). Ƙirar ƙimar musayar waje ta amfani da matsakaicin motsi na bayanan kasuwar Yen-Dollar. (Takardar da aka bincika).
3. Bankin don Haɗin Kan Ƙasashen Duniya (BIS). (2019). Binciken Bankin Tsakiya na Shekara Uku kan musayar waje da kasuwannin abubuwan da aka samo dashi na OTC. (Don bayanai kan tsarin kasuwa da shisshigi).
4. Cont, R. (2001). Kaddarorin gabaɗaya na dawowar kadari: gaskiyar gaskiya da batutuwan ƙididdiga. Kuɗi mai ƙima, 1(2), 223-236. (Don cikakken jerin gaskiyar gaskiya na kuɗi).
5. Lux, T., & Marchesi, M. (2000). Tarin ƙarfin motsi a cikin kasuwannin kuɗi: ƙananan simintin hulɗar wakilai. Jaridar Ƙasashen Duniya na Ka'idar da Aikace-aikacen Kuɗi, 3(04), 675-702. (Don ra'ayoyin ƙirar tushen wakili akan tarin ƙarfin motsi).