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

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. 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. 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. 1. INTRODUCTION
4

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. 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. 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. 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. 2. MODEL

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. 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. 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. 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. 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. 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. 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. 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. 3. EMPIRICAL ANALYSIS

19. Dataset (8/6 ~ 8/10, 2007)
• BNP Paribas shock (subprime lending)
19

20. Difference between the execution price (約定価
格) and Walras equilibrium price
20

21. Zoom UP
21
8/6 8/7

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. 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. 4. SUMMARY AND FUTURE WORK

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. 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. Thank You For Your Attention
Mitsuru KIKKAWA
(mitsurukikkawa@hotmail.co.jp)
This File is available at
http://kikkawa.cyber-ninja.jp/

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.