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Limit Order Market Modeling with Double Auction

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Young Researchers Workshop on Finance 2012 (The University of Tokyo)
http://www.comp.tmu.ac.jp/finance/Groupweb/workshop/Y2012/index.html

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Limit Order Market Modeling with Double Auction

  1. 1. Limit Order Market Modeling with Double Auction Mitsuru KIKKAWA (吉川満) (Graduate School of Advanced Mathematical Sciences, Meiji University) THIS FILE IS AVAILABLE AT http://kikkawa.cyber-ninja.jp/ Young Researchers Workshop on Finance 2012@The University of Tokyo
  2. 2. Aims: mathematical understanding of market mechanics for real market Market mechanics is one of complex phenomena in economics. Money game, Speculation (投機) Flash Crash, Shock, Bubble, … Detection of the fraud (不正取引の防止) : Insider trading, Manipulating quotations (相場操縦), Detection of bungled trade ( 誤発注) Mathematical model contributes the stability, efficiency and integrity of market 2
  3. 3. Today’s Talk • To formulate a financial market with the trader’s strategic behavior. • Focus on the order book (板情報) , which is the outcome of it. • Formulate a limit order market as a double auction. • Nonlinear strategy function (Kikkawa, 2009) • Micro-Econometrics (Multinominal Logit model) • Empirical analysis (Volume, Volatility, Price Discovery : the execution price, Walras equilibrium price)
  4. 4. 1. INTRODUCTION 4
  5. 5. Can we explain “real market” ? TOPIX at 1-day intervals (Jan. 6, 2011-Jan. 13, 2012) 700 750 800 850 900 950 1000 2011/1/6 2011/2/6 2011/3/6 2011/4/6 2011/5/6 2011/6/6 2011/7/6 2011/8/6 2011/9/6 2011/10/6 2011/11/6 2011/12/6 2012/1/6 5 pts day
  6. 6. Market treated by mechanism design theory Market can be treated as a double auction in mechanics design theory. [Results in double auction] 1. Hurwicz (1972) : in double auction, there is no institution satisfied with the following conditions : i) Individual Rationality (IR, 個人合理性) ii) Pareto Efficient (PE, パレート効率性) iii) Incentive Compatible (IC, 誘因両立性) Example: McAfee(1992), IR (○), IC (○), PE(×) 6 L. Hurwicz
  7. 7. Interpretation of Nash equilibrium (1950) 1st interpretation: Rationality (standard) 2nd interpretation: Mass-action(large populations, 統計的母集団) (for which he wanted to explain observable phenomena) Example: S1 S2 S1 a,b 0,0 S2 0,0 c,d player1 player 2 a,b,c,d ∊ R J.F. Nash 7
  8. 8. Limit Order Market : Order Book ( Bid (sell)) Price (Ask (buy)) - -------------------------------------------- 0 Market orders 0 --------------------------------------------- 492 9840 ----- --------------------------------------------- 506 9830 ----- ---------------------------------------------- 444 9820 ----- ------------------------------------------- 530 9810 ---- -------------------------------------------- 784 9800 ----- --------------------------------------------- ----- 9790 197 --------------------------------------------- ----- 9780 734 --------------------------------------------- ----- 9770 640 -------------------------------------------- ------ 9760 643 --------------------------------------------- ----- 9750 598 Center column : the prices, the second column from the left shows the volume of individual offers (sell). The right hand side of the table represents the bid side (buy). 8 8Nikkei Future Market(9:03, 5th, November, 2009)
  9. 9. 2. MODEL
  10. 10. Limit order market model as a double auction (Chatterjee and Samuelson (1983) ) • Players… large populations : seller and buyer (i=s, b) • Seller and buyer trade an asset. • Goods … one • Strategy … k (<∞) , ps, pb limit order price (how much does a player want to buy or sell an asset) • Payoff … Buyer : max[vb-pb] Prob(OB), Seller : max[ps-vs] Prob(OB), where vb, vs : reservation price respectively, Prob(OB) ∝ Prob (pb≧ ps(vs))×Prob(OA), Prob(OA) implies the market depth (市場の厚み). 10 10
  11. 11. One-Price Equilibrium A price is determined by looking at the prices at which the amount of aggregated bids and offers balance out. vb 1 vb=vs x Ex. Itayose Method O x 1 vs • This square is a turnover(出来高) 11
  12. 12. Zaraba method, linear equilibrium • Seller’s Strategy :ps(vs)=as+csvs、ps : uniform distribition on [as,as+cs] → ps=(ab+cb+vs)/2 • Buyer’s Strategy:pb(vb)=ab+cbvb、pb : uniform distribution on [ab,ab+cb] → pb=(vb+as)/2 ⇒ ps(vs)=2/3+vs/2, pb(vb)=1/3 + vb/2. vb vb=vs 1 vb = vs + 2/3 O 1 vs 12 Myerson and Satterthwaite (1983) →no Bayesian Nash equilibrium
  13. 13. Probability of choosing the strategy (related with logit model) Prop. 1. (Kikkawa 2009) Probability of choosing the strategy, πr, r=1,2…,k, Pi(πr)=Zi-1 exp(γi f(πr)), (i=1,2,…,n) πr: a group i’s strategy, γi: the optimal choice behavior for group i, f(πr): the player’s payoff from outcome πr, Zi: normalization parameter with ΣPi(πr)=1, for any i. This proposition is similar with quantal response equilibrium (質的応答均衡). (Mckelvey and Palfrey (1995, 1996) ) 13 13
  14. 14. Multinominal logit model • From Proposition 1, the probability of choosing the strategy for each group. + • Data (the probability of choosing the strategy for each player) • Regression analysis log Pi(πr):=Yi=α + γi f + u, where u : noise • We can estimate optimal parameters in this model with least squares method. 14
  15. 15. Limit Order Market : Order Book ( Bid (sell)) Price (Ask (buy)) - -------------------------------------------- 0 Market orders 0 --------------------------------------------- 492 9840 ----- --------------------------------------------- 506 9830 ----- ---------------------------------------------- 444 9820 ----- ------------------------------------------- 530 9810 ---- -------------------------------------------- 784 9800 ----- --------------------------------------------- ----- 9790 197 --------------------------------------------- ----- 9780 734 --------------------------------------------- ----- 9770 640 -------------------------------------------- ------ 9760 643 --------------------------------------------- ----- 9750 598 Center column : the prices, the second column from the left shows the volume of individual offers (sell). The right hand side of the table represents the bid side (buy). 15 15Nikkei Future Market(9:03, 5th, November, 2009)
  16. 16. Example (How to analyze the order book) Step 1) logit model (derive the probability of choosing the strategy (proposition 1) and transform this into log function.) Step 2) Regression analysis (回帰分析). OA: Ys=-0.65307+94079.26X1-9.59255X2, Yb=-0.66468+74928.44X1-7.6642X2. where X1 : valuation, X2 : order aggressiveness Step 3) Derive vs, vb, γs, γb: vs=9776, vb=9807.53, γs =0, γb =10.77. Step 4) Compute Walras equilibrium price (market clearing price), pw=9779.6. 【Movie】16 16
  17. 17. Dynamical framework Prop.2. We assume if an expected utility is greater, then the probability of playing the strategy will be higher in the next step. The following relationship about between the payoff and the population size is realized empirically : (i=s,b) where is the average payoff of the total population, is the group i ’s average payoff, Δ r is the whole population size variation, is the expected utilities’ variation by the population size changed. Proof. Price’s law + OLS 17     , ˆ ˆ' sp rpErp i ii i      pˆ  spiˆ  ii rpE 
  18. 18. 3. EMPIRICAL ANALYSIS
  19. 19. Dataset (8/6 ~ 8/10, 2007) • BNP Paribas shock (subprime lending) 19
  20. 20. Difference between the execution price (約定価 格) and Walras equilibrium price 20
  21. 21. Zoom UP 21 8/6 8/7
  22. 22. Expected utilities’ variance about sellers and buyers The variation in expected utility variance (Proposition 2) is large in the opening and closing auctions for the morning and afternoon sessions. →consistent with the classical microstructure research. 22
  23. 23. Price Discovery(価格発見) • Apply the standard method (Hasbrouck (1995)) • The (Phillips-Ouliaris) Cointegration (共和分)→ Yes • Information share : IS1(Walras) > IS2 (the execution) • Impulse response function (インパルス反応関数) 23 priceprice time time
  24. 24. 4. SUMMARY AND FUTURE WORK
  25. 25. Summary 1. The trading volume was proportional to the difference in reservation price between sellers and buyers, theoretically and empirically. 2. The volatility distribution in the model was consistent with classical market microstructure results. 3. In some cases, traders did not choose their strategy rationally. 4. Walras equilibrium price had a price discovery role, compared to the execution price. 25
  26. 26. Future work • Focus on “Information structure” • Bayes theorem (P(A|s)=Z P(s|A)P(A)) • (→ Proposition 1 has similar mathematical structure) • To visualize a “mood”, “feeling” in the market. • → text mining • Example: Bayesian estimator 26
  27. 27. Thank You For Your Attention Mitsuru KIKKAWA (mitsurukikkawa@hotmail.co.jp) This File is available at http://kikkawa.cyber-ninja.jp/
  28. 28. Acknowledgements • This research was supported in part by Meiji University Global COE Program (Formation and Development of Mathematical Sciences Based on Modeling and Analysis) of the Japan Society for the Promotion of Science.

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