Long Memory and Predictability of Financial Markets

Keynote for the presentation at JSAI 2017
https://kaigi.org/jsai/webprogram/2017/pdf/489.pdf

Transcript

1. LONG MEMORY AND PREDICTABILITY OF FINANCIAL MARKETS IMPLICATIONS FROM EMPIRICAL STUDIES AND APPLICATIONS OF THE LSTM MODEL ౦େɾ৽ྖҬɾ௠ݚڀࣨ ∁ڮफ़ଠ࿠, ௠₮ 2017೥5݄24೔ ਓ޻஌ೳֶձ
2. ໨࣍ ໨࣍ ▸ ݚڀഎܠɿۚ༥ࢢ৔ͷ౷ܭతಛ௃ ▸ ݚڀ಺༰ɿۚ༥ࢢ৔ͷग़དྷߴͱ௕ظهԱʹண໨ͯ͠ ▸ ࣮ূ෼ੳ ▸ ۚ༥ࢢ৔༧ଌ ▸ ·ͱΊ https://goo.gl/WLmTBP
3. ۚ༥ࢢ৔͸ͨͩϥϯμϜͰ͸ͳ͍ ▸ Stylized Facts ▸ ۚ༥ࢢ৔ʹ͓͍ͯීวతʹݟ͍ͩ͞ΕΔنଇੑ 1. Ϧλʔϯ෼෍ͷϑΝοτςʔϧ 2. ϘϥςΟϦςΟΫϥελϦϯά ݚڀഎܠ
4. Ϧλʔϯ෼෍ͷϑΝοτςʔϧ ▸ Ϧλʔϯ෼෍͕ਖ਼ن෼෍ͱҟͳΔ ▸ ϐʔΫ͕ਖ਼ن෼෍ΑΓઑ͍ͬͯΔ ▸ ςʔϧ෦෼͕ਖ਼ن෼෍ΑΓߴ͍஋ΛͱΓႈతʹͳ͍ͬͯΔ ʔਖ਼ن෼෍ ʔ࣮σʔλ ʔਖ਼ن෼෍ ʔ࣮σʔλ ݚڀഎܠ S&P500 Firms Price Return Distribution (linear-linear scale) S&P500 Firms Price Return Distribution (log-log scale, tail in positive region)
5. ϘϥςΟϦςΟΫϥελϦϯά ▸ Ձ֨มಈ͕େ͖͍࣌ظͱՁ֨มಈ͕খ͍࣌͞ظ͕ݻ·ͬͯݱΕΔ ▸ ࣗݾ૬ؔؔ਺͕ႈؔ਺ʹै͏ ACF X ( )= Corr X t ( ), X t + l ( ) ( )∝ l−α S&P500 Index Return Time-series ACF of S&P500 Index Absolute Return (Log-Log Scale) ݚڀഎܠ
6. ݚڀͷ໨త ▸ ۚ༥ࢢ৔Λग़དྷߴͱ௕ظهԱ͔Β෼ੳɾཧղ͢Δ ▸ ग़དྷߴʹؔ͢Δ Stylized Facts ͸গͳ͍ ▸ σʔλ࣮ূ෼ੳʹΑͬͯ࣍ͷೋ఺Λݕূ ▸ ग़དྷߴͱϦλʔϯ෼෍ͷؔ܎ ▸ ग़དྷߴͷࣗݾ૬ؔؔ਺ɺ௕ظهԱੑ ▸ ಘΒΕͨ஌ݟ͔Βۚ༥ࢢ৔ͷ༧ଌʹར༻͢Δ ▸ ۚ༥ࢢ৔ͷ௕ظهԱੑΛར༻ͨ͠ਂ૚ֶशϞσϧ Cont. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2), 223–236. Gallant, A. R., Rossi, P. E., & Tauchen, G. (1992). Stock Price and Volume. The Review of Financial Studies, 5(2), 199–242. Lobato, IN, & Velasco, C. (2000). Long memory in stock-market trading volume. Journal of Business & Economic Statistics. ݚڀ಺༰
7. ࢖༻ͨ͠σʔλ ▸ S&P500ΠϯσοΫε͓Αͼશͯͷݸผ໏ฑ ▸ ظؒɿڞʹ1962/01/03~2016/10/10 ▸ σʔλ਺ ▸ ΠϯσοΫεɿ13752݅ ▸ ݸผ໏ฑɿ3,350,831݅ ݚڀ಺༰-࣮ূ෼ੳ
8. ग़དྷߴͷτϨϯυআڈ S&P500 Index Before Detrending V t,k ^ = V t − µ t,k σ t,k k = 100 S&P500 Index After Detrending ▸ ௚ۙ kӦۀ೔ͷฏۉͱภࠩΛͱͬͯਖ਼نԽ͢Δ ݚڀ಺༰-࣮ূ෼ੳ
9. ग़དྷߴͱϦλʔϯ෼෍ͷؔ܎ʢ̍ʣ ݚڀ಺༰-࣮ূ෼ੳ ग़དྷߴ͕-1σ~0σͷͱ͖ͷ৚݅෇͖Ϧλʔϯ෼෍ S&P500 Firms S&P500 Firms
10. ग़དྷߴͱϦλʔϯ෼෍ͷؔ܎ʢ̎ʣ ग़དྷߴ͕-1σ~0σͷͱ͖ͷ৚݅෇͖Ϧλʔϯ෼෍ ग़དྷߴ͕8σ~9σͷͱ͖ͷ৚݅෇͖Ϧλʔϯ෼෍ ݚڀ಺༰-࣮ূ෼ੳ ग़དྷߴʹΑͬͯ෼෍ͷܗঢ়͕มΘΔ S&P500 Firms S&P500 Firms
11. ग़དྷߴͱϦλʔϯͷؔ܎ʢ̏ʣ ݚڀ಺༰-࣮ূ෼ੳ r normalized = r − µ V σ V µ V σ V ͋Δൣғͷग़དྷߴͷ Ϧλʔϯͷฏۉ ͋Δൣғͷग़དྷߴͷ Ϧλʔϯͷඪ४ภࠩ r ࣮ࡍͷϦλʔϯ
12. ग़དྷߴͷ௕ظهԱੑ ACF X ( )= Corr X t ( ), X t + l ( ) ( ) ACF X ( )∝ l−α α <1 α ≈ 0.51 ग़དྷߴͷࣗݾ૬ؔؔ਺͸௕ظʹΘͨͬͯႈʹै͏ ݚڀ಺༰-࣮ূ෼ੳ
13. ۚ༥ࢢ৔༧ଌͱ௕ظهԱੑ ▸ ϘϥςΟϦςΟͱग़དྷߴ͸௕ظهԱੑΛ࣋ͭ ▸ ۚ༥ࢢ৔͸௕ظهԱੑʹಛ௃͚ͮΒΕΔ ▸ ௕ظهԱੑΛ࠶ݱͰ͖ΔϞσϧͰۚ༥ࢢ৔ͷ༧ଌ ▸ ࠓճ͸ग़དྷߴͱϘϥςΟϦςΟ༧ଌ ݚڀ಺༰-ۚ༥ࢢ৔༧ଌ
14. LONG SHORT TERM MEMORY(LSTM) ▸ ݱ୅ͷσΟʔϓϥʔχϯάͷओཁٕज़ͷҰͭ ▸ ओʹܥྻσʔλ(ࣗવݴޠ,ಈըͳͲ)ʹରͯ͠ద༻͞ΕΔ ▸ ௕ظهԱΛ࠶ݱͰ͖Δ Hochreiter, S, & Schmidhuber, J. (1997). Long short-term memory. Neural computation. y t+1 Lin, H., & Tegmark, M. (2016). Critical Behavior from Deep Dynamics: A Hidden Dimension in Natural Language. arXiv. ݚڀ಺༰-ۚ༥ࢢ৔༧ଌ Image retrieved from Olah, C., (2015). Understanding LSTM Networks. http://colah.github.io/posts/2015-08-Understanding-LSTMs/img/LSTM3-chain.png
15. ύϥϝʔληοςΟϯάͱֶशσʔλ ▸ ϋΠύʔύϥϝʔλ ▸ ૚਺ 1 ▸ ೖྗ௕ 15 (ܭࢉྔͱͷ݉Ͷ߹͍) ▸ Ϣχοτ਺ 50 ▸ ίετ ೋ৐ޡࠩ ▸ ࠷దԽΞϧΰϦζϜ Adam ▸ ֶशσʔλɿϦλʔϯͱτϨϯυআڈ͞Εͨग़དྷߴ ▸ σʔλΛ80%:20%ʹ෼ׂ ▸ 80%Λ܇࿅༻ɹ1962~2003೥෼ʹ૬౰ ▸ 20%Λςετ༻ɹ2004~2016೥෼ʹ૬౰ ݚڀ಺༰-ۚ༥ࢢ৔༧ଌ Kingma, D., & Ba, J. (2014). Adam: A Method for Stochastic Optimization. arXiv. ֶशσʔλͷ࡞Γํ ೖྗ ग़ྗ ܇ ࿅ ༻ ε τ ༻ ς ࣌ؒ 1ϖΞ 2ϖΞ 3ϖΞ 1ϖΞ 2ϖΞ 3ϖΞ
16. ग़དྷߴͷ༧ଌ ▸ ग़དྷߴ͸ߴ͍ਫ਼౓Ͱ༧ଌͰ͖Δ ▸ ະདྷͷϦλʔϯ෼෍Λ༧ଌͰ͖Δ 2004~2016೥ͷS&P Indexͷग़དྷߴ༧ଌ ݚڀ಺༰-ۚ༥ࢢ৔༧ଌ ೖྗ:աڈ15εςοϓ ͷग़དྷߴ ग़ྗ:ݱࡏͷग़དྷߴ
17. ఏҊख๏ GARCH(1,1) RMSE (*10^-3) 7.725 7.871 2004~2016೥ͷS&P IndexͷϦλʔϯઈର஋ ೖྗ:աڈ15εςο ϓͷϦλʔϯઈର஋ ग़ྗ:ݱࡏͷ Ϧλʔϯઈର஋ ʔ༧ଌσʔλ ʔ࣮σʔλ RMSE…ฏํฏۉೋ৐ޡࠩ Ϧλʔϯઈର஋ͷ༧ଌ ݚڀ಺༰-ۚ༥ࢢ৔༧ଌ
18. ग़དྷߴΛೖྗʹՃ͑ͨ৔߹ͷ༧ଌਫ਼౓΁ͷӨڹ ϦλʔϯͷΈ Ϧλʔϯʴग़དྷߴ ݦஶͳਫ਼౓޲্͸ΈΒΕͳ͔ͬͨ Ϧλʔϯͷઈର஋ͱग़དྷߴ͸ಉ࣌త ݚڀ಺༰-ۚ༥ࢢ৔༧ଌ
19. ·ͱΊ ▸ ࣮ূݚڀ͔Β ▸ Ϧλʔϯ෼෍͸ग़དྷߴʹґଘ͍ͯ͠Δ ▸ ग़དྷߴͷࣗݾ૬ؔؔ਺͸௕ظʹΘͨͬͯႈతʔʼ௕ظهԱੑΛ࣋ͭ ▸ LSTM modelʹΑͬͯ ▸ ग़དྷߴΛߴ͍ਫ਼౓Ͱ༧ଌͰ͖Δ ▸ ओྲྀͰ͋ΔGARCH(1,1)ΑΓྑ͍ϘϥςΟϦςΟϞσϦϯά͕Ͱ͖Δ ·ͱΊ
20. จݙ ▸ Cont. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2), 223–236 ▸ Gallant, A. R., Rossi, P. E., & Tauchen, G. (1992). Stock Price and Volume. The Review of Financial Studies, 5(2), 199–242. The Review of Financial Studies. ▸ Hochreiter, S, & Schmidhuber, J. (1997). Long short-term memory. Neural computation. ▸ Kingma, D., & Ba, J. (2014). Adam: A Method for Stochastic Optimization. arXiv. ▸ Lin, H., & Tegmark, M. (2016). Critical Behavior from Deep Dynamics: A Hidden Dimension in Natural Language. arXiv. ▸ Lobato, IN, & Velasco, C. (2000). Long memory in stock-market trading volume. Journal of Business & Economic Statistics. จݙ