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Econophysics VI: Price Cross-Responses in Correlated Financial Markets - Thomas Guhr

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Let’s face complexity September 4-8, 2017

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Econophysics VI: Price Cross-Responses in Correlated Financial Markets - Thomas Guhr

  1. 1. Fakult¨at f¨ur Physik Econophysics VI: Price Cross–Responses in Correlated Financial Markets Thomas Guhr Let’s Face Complexity, Como, 2017
  2. 2. Outline Introduction price formation self–correlation of trade signs price self–response Empirical Analysis cross-responses cross-correlation of trade signs Model setup and construction comparision of simulated and empirical results impact functions
  3. 3. Introduction —price formation
  4. 4. Introduction —price formation log mi(t+ ) log mi(t) log mi(∞) Time Price in a liquid market, shares can be rapidly bought or sold with little impact on stock price. market liquidity measured by spread between best ask and best bid.
  5. 5. Introduction —price formation market orders take liquidity limit orders provide liquidity log mi(t+ ) log mi(t) log mi(∞) Time Price lack of short-run liquidity liquidity restoration in a liquid market, shares can be rapidly bought or sold with little impact on stock price. market liquidity measured by spread between best ask and best bid.
  6. 6. Introduction —correlations of trade signs How does liquidity influence the trades? liquidity volume price cost total liquidity ($) ($) cost ($) cost ($) high 10,000 2 20,000 20,000 0 5,000 2 10,000 low 2,000 2.2 4,400 21,500 1,500 3,000 2.5 7,500 order splitting correlations of trade signs in single stocks C0(l) = ⟨εn+l εn⟩ − ⟨εn⟩ 2 C1(l) = ⟨εn+l εn ln Vn⟩ C2(l) = ⟨εn+l ln Vn+l εn ln Vn⟩ fitted by C0(l) ≃ C0 lγ , (l > 1) , where γ = 1/5 for France-Telecom. Bouchaud, Gefen, Potters, Wyart, Quantitative Finance 4, 176 (2004) 10 0 10 1 10 2 10 3 10 4 Time (trades) 10 −2 10 −1 10 0 10 1 C(l) C2(l) C1(l) C0(l)
  7. 7. Introduction —price responses Price response measures how much price changes after time l, on average, conditioned on an initial buy or sell market order. Rii (l) = ⟨( Si (t + l) − Si (t) ) εi (t) ⟩ t Price reversion might seem to be at odds with long-memory sign correlation. Decaying quantity, i.e. an impact function, is required to reverse the price. 1 10 100 1000 Time (Trades) 0 0.05 0.1 R(l)(Arbitraryunits) Total TF1 Barclays Pechiney Bouchaud, Gefen, Potters, Wyart, Quantitative Finance 4, 176 (2004)
  8. 8. Introduction —questions What is the price response of one stock to the trades of the others? Is there a sign cross–correlation? —What kind of? so far: single stocks — but now: across stocks our papers: Shanshan Wang, Rudi Sch¨afer, Thomas Guhr, The European Physical Journal B89, 105 (2016) Shanshan Wang, Rudi Sch¨afer, Thomas Guhr, The European Physical Journal B89, 207 (2016) see also: M. Benzaquen, I. Mastromatteo, Z. Eisler, J.P. Bouchaud, arXiv:1609.02395
  9. 9. Empirical Analysis —data sets we used Trades and Quotes (TAQ) data set from NASDAQ stock market stocks from S&P 500 index in the year 2008 intraday data with trading time from 9:40 to 15:50 (New York time) for a given stock pair, we consider the common trading days resolution of 1 second
  10. 10. Empirical Analysis —accumulated trade signs in 1 second interval a time series of trades trade sign of n-th trade in time interval t is defined as ε(t; n) = { sgn ( S(t; n) − S(t; n − 1) ) , if S(t; n) ̸= S(t; n − 1) , ε(t; n − 1) , otherwise . accumulated trade sign in time interval t is ε(t) =    sgn ( N(t)∑ n=1 ε(t; n) ) , if N(t) > 0 , 0 , if N(t) = 0 . ε(t) =    +1, more buy market orders, 0, lack of trading OR a balance of buy and sell market orders −1, more sell market orders.
  11. 11. Empirical Analysis —time scale and sign accuracy Trading of different stocks is not synchronous ⇒ physical instead of event time scale. Wang, Sch¨afer, Guhr, Eur. Phys. J. B89, 105 (2016) -2 -1 0 1 2 (a) ε(t;n) empirical theoretical -2 -1 0 1 2 (b) ε(t) -2 -1 0 1 2 10:30:00 10:30:10 10:30:20 10:30:30 10:30:40 10:30:50 10:31:00 (c) ε(t) t/s empirical, Eq. (2) empirical, Eq. (3) theoretical, Eq. (2) for consecutive trades for stamp of one second for stamp of one second ε(t; n) =    sgn ( S(t; n) − S(t; n − 1) ) , if S(t; n) ̸= S(t; n − 1), ε(t; n − 1), otherwise. (1) ε(t) =    sgn   N(t)∑ n=1 ε(t; n)   , if N(t) > 0, 0, if N(t) = 0. (2) ε(t) =    sgn   N(t)∑ n=1 ε(t; n)v(t; n)   , if N(t) > 0, 0, if N(t) = 0. (3) for consecutive trades accuracy for emp. vs. theo. empirical signs theoretical signs fig.(a) AAPL 20080107 86% from TotalView– derived from average of 6 samples 85% ITCH data set Eq.(1) for stamp of one second accuracy for emp. Eq.(2) vs. theo. Eq.(2) accuracy for emp. Eq.(3) vs. theo. Eq.(2) accuracy difference fig.(b) AAPL 20080107 82% 77% 5% fig.(c) AAPL 20080602 87% 85% 2% average of 6 samples 82% 80% 2%
  12. 12. Empirical Analysis — cross–responses and sign cross–correlators midpoint price at time t is mi (t) = 1 2 (ai (t) + bi (t)) . logarithmic price change from t to t + τ is ri (t, τ) = log mi (t + τ) − log mi (t) = log mi (t + τ) mi (t) . price cross–response function of stock i to stock j is Rij (τ) = ⟨ ri (t, τ)εj (t) ⟩ t . cross–correlator of trade signs between stocks i and j is Θij (τ) = ⟨ εi (t + τ)εj (t) ⟩ t , where Θij (0) = Θji (0) and Θij (τ) = Θji (−τ) .
  13. 13. Empirical Analysis — cross–responses, sign cross–correlators Rij(τ) ×10-5 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 τ/s 100 101 102 Θij(τ) 10-7 10-6 10 -5 10-4 10-3 10-2 10-1 i=AAPL, j=MSFT i=MSFT, j=AAPL i=XOM, j=CVX i=GS, j=JPM τ/s 100 101 102 103 i=AAPL, j=GS i=GS, j=AAPL i=GS, j=XOM i=XOM, j=AAPL ×10-5 4 5 6 7 8 9 10 τ/s 100 101 102 103 10-7 10-6 10 -5 10-4 10-3 10-2 10-1 i=AAPL, j=AAPL i=GS, j=GS i=XOM, j=XOM Including εj (t) = 0 Θij (τ) = θij ( 1 + (τ/τ (0) ij )2 )γij /2 { 0 < γij < 1, a long–memory process γij > 1, a short–memory process Wang, Sch¨afer, Guhr, Eur. Phys. J. B89, 105 (2016) Sign Stock i Stock j γij correlators inc. 0 exc. 0 AAPL MSFT 1.00 1.35 MSFT AAPL 1.15 1.15 XOM CVX 1.04 1.16 Cross GS JPM 1.00 1.00 AAPL GS 1.00 0.91 GS AAPL 1.00 1.00 GS XOM 1.04 1.10 XOM AAPL 1.09 1.42 AAPL AAPL 1.27 1.27 Self GS GS 1.17 1.18 XOM XOM 1.12 1.14
  14. 14. Empirical Analysis — influence of zero trade signs τ/s 100 101 102 103 Rij(τ) ×10 -5 -3 -2 -1 0 1 2 3 4 5 6 7 τ/s 100 101 102 103 Θij(τ) -0.04 -0.02 0 0.02 0.04 0.06 0.08 for εj(t) = 0 included for εj(t) = 0 excluded for εj(t) = 0 stock i is MSFT, stock j is AAPL price cross–responses for εj (t) = 0: R (only 0) ij (τ) = R (inc. 0) ij − R (exc. 0) ij sign cross–correlators for εj (t) = 0: Θ (only 0) ij (τ) = Θ (inc. 0) ij − Θ (exc. 0) ij inclusion of zero trade signs weakens rather than strengthens cross–responses and sign cross–correlators.
  15. 15. Empirical Analysis — average cross–responses and average sign cross–correlators passive cross–response measures how price of stock i changes due to trades of all other stocks j, on average R (p) i (τ) = ⟨ Rij (τ) ⟩ j active cross–response quantifies which effect trades of stock j have on prices of all other stocks i, on average R (a) j (τ) = ⟨ Rij (τ) ⟩ i passive and active cross–correlators of trade signs Θ (p) i (τ) = ⟨ Θij (τ) ⟩ j Θ (a) i (τ) = ⟨ Θji (τ) ⟩ j in all averages i = j excluded
  16. 16. Empirical Analysis — average cross–responses and average sign cross–correlators passive active R (p) i(τ) 10-6 10 -5 10 -4 R (a) j(τ) 10-5 10-4 τ/s 100 101 102 103 104 Θ (p) i(τ) -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 i =AAPL, inc. 0 i =GS, inc. 0 i =XOM, inc. 0 i =AAPL, exc. 0 i =GS, exc. 0 i =XOM, exc. 0 τ/s 10 0 10 1 10 2 10 3 10 4 Θ (a) j(τ) -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 j =AAPL, inc. 0 j =GS, inc. 0 j =XOM, inc. 0 j =AAPL, exc. 0 j =GS, exc. 0 j =XOM, exc. 0 Passive cross–responses reverse faster than active ones. Sign cross–correlators are long memory after averaging due to noise reduction. Sign cross– stock γi or γj correlators i, j inc. 0 exc. 0 AAPL 0.68 0.73 Θ (p) i (τ) GS 0.92 0.90 XOM 1.32 1.33 AAPL 0.90 0.91 Θ (a) j (τ) GS 0.85 0.83 XOM 0.71 0.95
  17. 17. Empirical Analysis — market responses including zero trade signs τ = 1 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 2 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 60 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 300 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 1800 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 7200 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 excluding zero trade signs τ = 1 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 2 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 60 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 300 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 1800 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS τ = 7200 s stock j I HC CD IT U F M E CS TS stocki IHCCDITUFMECSTS -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Responses are normalized as Rij (τ)/max(|Rij (τ)|)
  18. 18. Empirical Analysis — market responses τ/s 10 0 10 1 10 2 10 3 10 4 R(τ) ×10-5 2 3 4 5 6 7 8 for εj(t) = 0 excluded for εj(t) = 0 included doubly averaged response for the market R(τ) = ⟨⟨Rij (τ)⟩j ⟩i excluding i = j. 99 stocks from 10 economic sectors in 2008 for each sector, first 9 or 10 stocks with largest average market capitalization Market efficiency is violated on short time scales, but restored on longer time scales. Wang, Sch¨afer, Guhr, Eur. Phys. J. B89, 105 (2016)
  19. 19. Empirical Analysis —identifying influencing and influenced stocks influencing stocks 300 COST FCX τ/s VZ 60 COP INTC PG AMZN VZ FCX T CSCO COP 2 FCX GS GS WMT JNJ AMZN T SLB CVX QCOM 1 INTC PG MSFT WMT JNJ CSCO INTC T FCX XOM AAPL COP CVX SLB QCOM AMZN PG MSFT INTC WMT CSCO AMZN XOM 16 AAPL CVX QCOM T COP 14 WMT MSFT CSCO XOM 12 AAPL CVX QCOM 10 MSFT stock j XOM 8 AAPL 6 4 2 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.1 R (a) j(τ) Industrials Health Care Consumer Discretionary Information Technology Utilities Financials Materials Energy Consumer Staples Telecommunications Services influenced stocks 300 MA CRM X τ/s 60 FLR ISRG FCX SHLD WIN WFR FLR SHLD 2 VNO CF WFR CME WYNN DVN MON CME Q ICE 1 APA ICE FLS OXY WYNN GS EOG X EOG S DVN Q FTR FLS ICE COP ISRG BEN APA WFR SLB 16 S HES CF MON FTR FLR HES 14 FCX WFR NUE OXY 12 CF FLR X 10 CF FCX stock i 8 NUE X 6 4 2 0.05 0.1 0.15 0.2 0.25 0.35 0.3 R (p) i(τ) Industrials Health Care Consumer Discretionary Information Technology Utilities Financials Materials Energy Consumer Staples Telecommunications Services 99 stocks from 10 economic sectors in 2008 for each sector, first 9 or 10 stocks with largest average market capitalization responses are normalized, Rij (τ)/max(|Rij (τ)|) zero trade signs included
  20. 20. Empirical Analysis —questions How to understand cross–responses between stocks? What is the relation between the cross–responses and sign cross–correlators? Why do active and passive average cross–responses differ? we need a model! our paper: Shanshan Wang, Thomas Guhr, arXiv:1609.04890 see also: M. Benzaquen, I. Mastromatteo, Z. Eisler, J.P. Bouchaud, arXiv:1609.02395
  21. 21. Price Impact Model —single stocks log mi (1) = log mi (0) + Gii (1)f ( vi (0) ) εi (0) + ηii (0) log mi (2) = Gii (1)f ( vi (1) ) εi (1) + ηii (1) +Gii (2)f ( vi (0) ) εi (0) + ηii (0) + log mi (0) log mi (t) = ∑ t′<t Gii (t − t ′ )f ( vi (t ′ ) ) εi (t ′ ) + ∑ t′<t ηii (t ′ ) + log mi (−∞) Bouchaud, Gefen, Potters, Wyart, Quantitative Finance 4, 176 (2004) log mi(t+ ) log mi(t) log mi(∞) Time Price f (vi (t)) Gii (τ) vi (t): traded volume ηii (t): random variable f (vi (t)): impact function of traded volumes Gii (τ): ‘bare’ impact function of time lags for a single trade
  22. 22. Price Impact Model —across stocks log mi (t) = ∑ t′<t [ Gii (t − t ′ )f ( vi (t ′ ) ) εi (t ′ ) + ηii (t ′ ) ] + ∑ t′<t [ Gij (t − t ′ )g ( vj (t ′ ) ) εj (t ′ ) + ηij (t ′ ) ] + log mi (−∞) log mi(t+ ) log mi(t) log mi(∞) Time Price liquidity→ f (vi (t)) information→ g(vj (t)) Gii (τ) → self–impact Gij (τ) → cross–impact
  23. 23. Price Impact Model —across stocks log mi (t) = ∑ t′<t [ Gii (t − t ′ )f ( vi (t ′ ) ) εi (t ′ ) + ηii (t ′ ) ] + ∑ t′<t [ Gij (t − t ′ )g ( vj (t ′ ) ) εj (t ′ ) + ηij (t ′ ) ] + log mi (−∞) for a buy market order for a sell market order Assume impact function G(τ) = Γ0 [ 1 + ( τ τ0 )2 ]β/2 + Γ Properties of impact function, e.g. positive or negative impact, temporary or permanent impact, are determined by data fits.
  24. 24. Price Impact Model —across stocks log mi (t) = ∑ t′<t [ Gii (t − t ′ )f ( vi (t ′ ) ) εi (t ′ ) + ηii (t ′ ) ] + ∑ t′<t [ Gij (t − t ′ )g ( vj (t ′ ) ) εj (t ′ ) + ηij (t ′ ) ] + log mi (−∞) log mi(t+ ) log mi(t) log mi(∞) Time Price liquidity→ f (vi (t)) information→ g(vj (t)) Gii (τ) → self–impact Gij (τ) → cross–impact
  25. 25. Price Impact Model —across stocks log mi (t) = ∑ t′<t [ Gii (t − t ′ )f ( vi (t ′ ) ) εi (t ′ ) + ηii (t ′ ) ] + ∑ t′<t [ Gij (t − t ′ )g ( vj (t ′ ) ) εj (t ′ ) + ηij (t ′ ) ] + log mi (−∞) R (C) ij (τ) = ⟨ r (L) ii (t, τ)εj (t) ⟩ t = ∑ t≤t′<t+τ Gii (t + τ − t ′ ) ⟨ f ( vi (t ′ ) )⟩ t Θij (t ′ − t) + ∑ t′<t [ Gii (t + τ − t ′ ) − Gii (t − t ′ ) ] ⟨ f ( vi (t ′ ) )⟩ t Θji (t − t ′ ) log mi(t+ ) log mi(t) log mi(∞) Time Price liquidity→ f (vi (t)) information→ g(vj (t)) Gii (τ) → self–impact Gij (τ) → cross–impact
  26. 26. Price Impact Model —across stocks log mi (t) = ∑ t′<t [ Gii (t − t ′ )f ( vi (t ′ ) ) εi (t ′ ) + ηii (t ′ ) ] + ∑ t′<t [ Gij (t − t ′ )g ( vj (t ′ ) ) εj (t ′ ) + ηij (t ′ ) ] + log mi (−∞) R (C) ij (τ) = ⟨ r (L) ii (t, τ)εj (t) ⟩ t = ∑ t≤t′<t+τ Gii (t + τ − t ′ ) ⟨ f ( vi (t ′ ) )⟩ t Θij (t ′ − t) + ∑ t′<t [ Gii (t + τ − t ′ ) − Gii (t − t ′ ) ] ⟨ f ( vi (t ′ ) )⟩ t Θji (t − t ′ ) R (S) ij (τ) = ⟨ r (I) ij (t, τ)εj (t) ⟩ t = ∑ t≤t′<t+τ Gij (t + τ − t ′ ) ⟨ g ( vj (t ′ ) )⟩ t Θjj (t ′ − t) + ∑ t′<t [ Gij (t + τ − t ′ ) − Gij (t − t ′ ) ] ⟨ g ( vj (t ′ ) )⟩ t Θjj (t − t ′ ) log mi(t+ ) log mi(t) log mi(∞) Time Price liquidity→ f (vi (t)) information→ g(vj (t)) Gii (τ) → self–impact Gij (τ) → cross–impact
  27. 27. Price Impact Model —across stocks cross–responses Rij (τ) = R (C) ij (τ) + R (S) ij (τ) . passive and active average cross–response functions R (p) i (τ) = ⟨ R (C) ij (τ) ⟩ j ⟨ f (p) i (vi ) ⟩ t ⟨ f (p) i (vi ) ⟩ t + ⟨ R (S) ij (τ) ⟩ j ⟨ g (p) i (vj ) ⟩ t,j ⟨ g (p) i (vj ) ⟩ t,j , R (a) i (τ) = ⟨ R (C) ji (τ) ⟩ j ⟨ f (a) i (vj ) ⟩ t,j ⟨ f (a) i (vj ) ⟩ t,j + ⟨ R (S) ji (τ) ⟩ j ⟨ g (a) i (vi ) ⟩ t ⟨ g (a) i (vi ) ⟩ t . average cross–responses: R (p) i (τ) = ⟨Rij (τ)⟩j R (a) i (τ) = ⟨Rji (τ)⟩j sign cross–correlators: Θ (p) i (τ) = ⟨Θij (τ)⟩j Θ (a) i (τ) = ⟨Θji (τ)⟩j according to empirical analysis ⟨ f (p) i (vi ) ⟩ t ∼ v δip i , ⟨ g (p) i (vj ) ⟩ t,j ∼ v δjp j ⟨ f (a) i (vj ) ⟩ t,j ∼ v δja j , ⟨ g (a) i (vi ) ⟩ t ∼ v δia i δip, δjp, δja, δia ∼ 0.5 ± 0.2 for small volumes of most stocks ⟨f (p) i (vi )⟩t , ⟨g (p) i (vj )⟩t,j , ⟨f (a) i (vj )⟩t,j , ⟨g (p) i (vi )⟩t → independent of time lag
  28. 28. Price Impact Model —simulations and data fits passive cross–responses R (p) i(τ) ×10 -5 3 4 5 6 7 8 9 10 11 R (p) i(τ) ×10 -5 3 4 5 6 7 8 9 10 11 R (p) i(τ) ×10 -5 3 4 5 6 7 8 9 10 11 τ/s 10 0 10 1 10 2 R (p) i(τ) ×10 -5 3 4 5 6 7 8 9 10 11 τ/s 100 101 102 R (p) i(τ) ×10 -5 3 4 5 6 7 8 9 10 11 empirical theoretical, Case (1) theoretical, Case (2) theoretical, Case (3) w = 0.10 w = 0.30 w = 0.50 w = 0.70 w = 0.90 active cross–responses R (a) i(τ) ×10 -5 4 6 8 10 12 14 R (a) i(τ) ×10-5 4 6 8 10 12 14 R (a) i(τ) ×10 -5 4 6 8 10 12 14 τ/s 10 0 10 1 10 2 R (a) i(τ) ×10 -5 4 6 8 10 12 14 τ/s 100 101 102 R (a) i(τ) ×10 -5 4 6 8 10 12 14 empirical theoretical, Case (1) theoretical, Case (2) theoretical, Case (3) w = 0.10 w = 0.30 w = 0.50 w = 0.70 w = 0.90 stock i is MSFT in 2008, and the pairwise stocks j are other 30 stocks with the largest average number of daily trades in S&P 500 index of 2008. Wang, Guhr, arXiv:1609.04890
  29. 29. Price Impact Model —impact functions sketch of price impacts after averaging, Gij (τ) → G (p) i (τ), G (a) i (τ) simulated impact function G(τ) = Γ0 [ 1 + ( τ τ0 )2 ]β/2 + Γ Wang, Guhr, arXiv:1609.04890 simulations of impact functions τ/s 10 0 10 1 10 2 10 3 10 4 G(τ) ×10-4 0 0.5 1 1.5 2 2.5 3 3.5 Gii(τ) G (p) i (τ) G (a) i (τ) 100 101 102 103 104 10-6 10-5 10 -4 10-3 MSFT in 2008 impact functions Γ (×10−10 ) Γ0 (×10−4 ) τ0 [s] β Gii (τ) 0.5 5.12 0.025 0.13 G (p) i (τ) 0 0.25 70.873 0.49 G (a) i (τ) 0 2.57 0.004 0.19
  30. 30. Summary price formation: interaction of market orders and limit orders role of liquidity empirical results: cross–responses of stock pairs trade sign cross–correlators average cross–responses average trade sign cross–correlators market responses influencing and influenced stocks price impact model: a self– and a cross–impact function two response components related to the cross– and the self–correlators, respectively comparison of empirical and simulated results self–, active and passive impact functions Rij(τ) ×10 -5 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 τ/s 100 101 102 Θij(τ) 10 -7 10 -6 10 -5 10 -4 10-3 10 -2 10 -1 i=AAPL, j=MSFT i=MSFT, j=AAPL i=XOM, j=CVX i=GS, j=JPM τ/s 100 101 102 103 i=AAPL, j=GS i=GS, j=AAPL i=GS, j=XOM i=XOM, j=AAPL ×10 -5 4 5 6 7 8 9 10 τ/s 100 101 102 103 10 -7 10 -6 10 -5 10 -4 10-3 10 -2 10 -1 i=AAPL, j=AAPL i=GS, j=GS i=XOM, j=XOM
  31. 31. Our papers [1] Shanshan Wang, Rudi Sch¨afer, Thomas Guhr, Cross-response in correlated financial markets: individual stocks, The European Physical Journal B89, 105 (2016) [2] Shanshan Wang, Rudi Sch¨afer, Thomas Guhr, Average cross-responses in correlated financial market, The European Physical Journal B89, 207 (2016) [3] Shanshan Wang, Thomas Guhr, Microscopic understanding of cross-responses between stocks: a two-component price impact model, arXiv:1609.04890

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