1. The document describes several zero intelligence models for simulating financial markets, including the Daniels model and Mike-Farmer model. 2. The Daniels model uses a Poisson process to simulate the random placement and cancellation of limit and market orders. The Mike-Farmer model builds on this by incorporating empirical patterns in order placement, cancellation rates, and price distributions. 3. Both models are estimated using real order book data and their results are shown to closely match distributions of returns, spreads, and order lifetimes from the actual data.

1. Financial market simulation based on zero
intelligence models
Vyacheslav Arbuzov1,2
arbuzov@prognoz.ru
1Prognoz Risk Lab
2Perm State University
Perm 21.03.2014
Applied Economic Modeling Workshop
Vyacheslav Arbuzov Financial market simulation

2. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Basic knowledge about LOB
Continuous double auction scheme
Figure 1. Order book representation
Vyacheslav Arbuzov Financial market simulation

3. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Basic knowledge about LOB
Continuous double auction
Three fundamental processes specifying a LOB are:
1 Rate/size of market orders
2 Rate/placement/size of limit orders
3 Rate/placement/size of cancellations
Volume
Price
Figure 2. Different types of orders
Vyacheslav Arbuzov Financial market simulation

4. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Data
LSE data. Farmer, Patelli & Zovko
Data from Farmer, Patelli & Zovko (2005), The Predictive Power
of Zero Intelligence in Financial Markets
Only used data from electronic order book
01/08/1998 to 30/04/2000 (434 trading days)
Selected 11 stocks, each with over 80 events per day and over
300,000 in total
Vyacheslav Arbuzov Financial market simulation

5. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Data
LSE data. Farmer, Patelli & Zovko
stock num. events average limit market deletions # days
ticker (1000s) (per day) (1000s) (1000s) (1000s)
AZN 608 1405 292 128 188 429
BARC 571 1318 271 128 172 433
CW. 511 1184 244 134 134 432
GLXO 814 1885 390 200 225 434
LLOY 644 1485 302 184 159 434
ORA 314 884 153 57 104 432
PRU 422 978 201 94 127 354
RTR 408 951 195 100 112 431
SB. 665 1526 319 176 170 426
SHEL 592 1367 277 159 156 429
VOD 940 2161 437 296 207 434
Vyacheslav Arbuzov Financial market simulation

6. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Data
LSE data. Mike, Farmer
Data from Mike, Farmer (2008), An empirical behavioral model of
liquidity and volatility
Only used data from electronic order book
02/05/2000 to 31/12/2002
Selected 25 stocks
Trading day from 9:00 am to 16:00.
Vyacheslav Arbuzov Financial market simulation

7. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Data
LSE data. Mike, Farmer
Stock # of orders Stock # of orders Stock # of orders
SHEL050 3,560,756 BLT 984,251 III050 301,101
VOD 2,676,888 SBRY 927,874 TATE 243,348
REED 2,353,755 GUS 836,235 FGP 207,390
AZN 2,329,110 HAS 683,124 NFDS 200,654
LLOY 1,954,845 III050 602,416 DEB 182,666
SHEL025 1,708,596 BOC100 500,141 BSY100 177,286
PRU 1,413,085 BOC050 345,129 NEX 134,991
TSCO 1,180,244 BPB 314,414 AVE 109,963
BSY050 1,207,885
Vyacheslav Arbuzov Financial market simulation

8. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Data
MOEX data
Aeroflot JSC
Only used data from electronic order book
01/01/2012 to 31/01/2012 (21 trading days)
History of all orders and trades
2 765 074 orders
15 786 trades
Trading day from 10:00 am to 18:45.
Vyacheslav Arbuzov Financial market simulation

9. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Tool kit
Tools for market simulations
Data warehouse: Oracle
Statistical calculations and visualization: R-3.0.2
Market engine simulations: C++
R package (RODBC) for working with database
R package (Rcpp) for working with MinGW compilers (C++)
Vyacheslav Arbuzov Financial market simulation

10. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
ZI model of 2003
Daniels M.G., Farmer J.D., Gillemot L., Iori G., Smith E. (2003)
Quantitative model of price diffusion and market friction based on
trading as a mechanistic random process, Phys. Rev. Lett. 90
.
There is no established name of this model.
So in our research, we try to named this model as
The Daniels model
Vyacheslav Arbuzov Financial market simulation

11. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Theory
Basic knowledge
Standard settings and parameters of the zero-intelligence model.
Model works in the logarithm space.
ZI agents place and cancel orders randomly
The logarithm of the tick size is dp
The logarithm of the best (lowest) ask price is a(t)
The logarithm of the best (highest) bid price is b(t)
The spread at time t is s(t) = a(t) − b(t)
Each order/cancellation has characteristic size σ shares (the
sizes of limit orders and market orders are the same)
Vyacheslav Arbuzov Financial market simulation

12. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Theory
Poisson process
Impatient agents place market orders with Poisson rate µ
shares per unit time (buy and sell market orders equally likely
so effectively rate µ/2 for each).
Patient agents place buy limit orders with Poisson rate α
shares per price per time (uniformly in the semi-infinite
interval (−∞; a(t)) and sell limit orders with the same rate
in) (b(t); ∞)
Cancellations occur with probability δ per unit time (akin to
radioactive decay)
All processes are independent
Vyacheslav Arbuzov Financial market simulation

13. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Theory
Poisson process
Figure 3. Scheme of the Daniels model
Vyacheslav Arbuzov Financial market simulation

14. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Estimation of parameters
Estimation of α
We follow the methods of Farmer, Patteli, Zovko (2005). Given
real data of all orders/cancellations, can calibrate the parameters
σ, α, δ, µ
For buy orders calculate relative price ∆ = m − p and for sell
orders ∆ = p − m , where m - logarithm of midquote price
and p is the logarithm of order price
Rt = Qupper
t − Qlower
t , where Qlower
t is the 2 percentile of
density of ∆ and Qupper is the 60 percentile
α is calculated each day and then averaged. On day t,
αt = Lt/|Rt|, where Rt is the range of relative prices that
capture 58 % of day t’s limit orders and Lt is the total
number of shares of effective limit orders within this range.
Vyacheslav Arbuzov Financial market simulation

15. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Estimation of parameters
Estimation of σ, δ, µ
δt is calculated each day and then averaged. δt is calculated
using only cancelled limit orders in the price range Rt.
Measure δt as the inverse of the average lifetime of a
cancelled limit orders
σ is calculated simply as the average size of all limit orders.
The model assumes both averages equal and in practice the
average limit order size is only slightly larger than the average
market order size.
µ is calculated as the ratio of the number of shares of market
orders to the number of events during the trading day.
Vyacheslav Arbuzov Financial market simulation

16. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Practice
Estimation of α
Qupper = 12 tick size Qlower = −11 tick size
L = 1, 655, 646 α = 0.108
orders
perasecond · peraprice
Figure 4. Heavy tails of price distribution
(in this case ∆ = priceorder − pricebestaside)
Vyacheslav Arbuzov Financial market simulation

17. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Practice
Estimation of µ,δ,σ
Parameters Description Value
α Intensity of limit orders 0.108
µ Intensity of market orders 0.006
δ Intensity of cancellations 0.287
dp Tick size 0.01
σ Volume of orders 1184
Vyacheslav Arbuzov Financial market simulation

18. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Practice
Results of simulations
Figure 5. Distribution of spread
Vyacheslav Arbuzov Financial market simulation

19. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Practice
Results of simulations
Figure 6. Distribution of returns
Vyacheslav Arbuzov Financial market simulation

20. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Practice
Results of simulations
Figure 7. Orders lifetime distribution
Vyacheslav Arbuzov Financial market simulation

21. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Mike-Farmer model
Mike S., Farmer J. D. (2008) An empirical behavioral model of
liquidity and volatility, J. Econ. Dyn. Control 32
.
The Mike-Farmer model
Vyacheslav Arbuzov Financial market simulation

22. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Description
Basic knowledge
Important properties of the order flow for a future upgrade of the
model (from Farmer et al. (2006)):
Trending of order flow
Power placement of limit prices
Non-Poisson order cancellation process
Vyacheslav Arbuzov Financial market simulation

23. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Price distribution
Let’s x is logarithmic distance from the same best price. For buy
orders x = π − πb and for sell order x = πa − π.
-0.01 -0.005 0 0.005 0.01
x = relative limit price from same best
10
0
10
1
10
2
10
3
P(x)
Student distribution, alpha=1.3
S0 = 0
S0 = 0, BUY
S0 = 0, SELL
S0 = 0.003
AZN
MOEX data LSE data
Vyacheslav Arbuzov Financial market simulation

24. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Conditional cancellation process
Position in the order book
The distance of the price of the order i from the opposite best at
time t is:
∆i(t) = π − πb(t) - for sell orders
∆i(t) = πa(t) − π - for buy orders
∆i(0) - the distance to the opposite best when the order is placed
∆i(t) = 0 - when the order is executed
yi(t) = ∆i(t)
∆i(0)
yi = 1 - when order is placed
yi = 0 - when order is executed
Vyacheslav Arbuzov Financial market simulation

25. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Conditional cancellation process
Position in the order book
Bayes’ rule: P(Ci|yi) = P(yi|Ci)
P(yi) P(C)
P(Ci|yi) = K1(1 − D1e−yi ) P(Ci|yi) = K1(1 − e−yi )
0 1 2 3 4 5
y
10
-3
10
-2
10
-1
P(C|y)
real data
fitted curve
MOEX data LSE data
Vyacheslav Arbuzov Financial market simulation

26. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Conditional cancellation process
Order book imbalance
nimb = nbuy/(nbuy + nsell) for buy orders
nimb = nsell/(nbuy + nsell) for sell orders , where
nbuy - number of buy orders in order book
nsell - number of sell orders in order book
Bayes’ rule: P(Ci|nimb) = P (nimb|Ci)
P (nimb) P(C)
P(Ci|nimb) = K2(nimb + B)
0 0.2 0.4 0.6 0.8 1
nimb
0
0.004
0.008
0.01
P(C|nimb)
real data
linear fit
MOEX data LSE data
Vyacheslav Arbuzov Financial market simulation

27. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Conditional cancellation process
Number of orders in the order book
ntot = (nbuy + nsell)
Bayes’ rule: P(Ci|ntot) = P(ntot|Ci)
P(ntot) P(C)
P(Ci|ntot) = K3(1 − D2e−ntot ) P(Ci|ntot) = K3
ntot
MOEX data LSE data
Vyacheslav Arbuzov Financial market simulation

28. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Combined cancellation model
P(Ci|yi, nimb, ntot) = P(yi|Ci)P(nimb|Ci)P(ntot|Ci)
P(yi)P(nimb)P(ntot) P(C)
.
P(Ci|yi, nimb, ntot) = A(1 − D1e−yi )(nimb + B)(1 − D2e−ntot ) .
where
.
A = K1K2K3
P(C)2
Vyacheslav Arbuzov Financial market simulation

29. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Mike-Farmer results of simulations (LSE results)
10
-4
10
-3
10
-2
10
-1
R
10
-4
10
-2
10
0P(|r|>R)
real data
Simulation IV.
RETURN
Figure 8. Distribution of returns
Vyacheslav Arbuzov Financial market simulation

30. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Mike-Farmer results of simulations (LSE results)
10
-4
10
-3
10
-2
10
-1
S
10
-4
10
-2
10
0P(s>S)
real data
Simulation IV.
SPREAD
Figure 9. Spread distribution
Vyacheslav Arbuzov Financial market simulation

31. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Mike-Farmer results of simulations (LSE results)
10
0
10
1
10
2
10
3
tau
10
-6
10
-4
10
-2
P(tau)
Simulation, slope = -1.9
Real data, slope = -2.1
Figure 10. Lifetime distribution
Vyacheslav Arbuzov Financial market simulation

32. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Heavy tails in price distribution
Figure 11. Power Law of logarithmic distance
Vyacheslav Arbuzov Financial market simulation

33. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Fitting of price distribution
Figure 12. Price distribution fitting using Power Law and t-Student
Vyacheslav Arbuzov Financial market simulation

34. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Liquidity metric
Arbuzov V., Frolova M. Market liquidity measurement and econometric
modeling // Market Risk and Financial Markets Modeling, Springer, 2012
RTCI =
n
i=1
|pi−p|·ni
n
i=1
pini
where pi – price of order i,
ni - volume of order i,
p – best bid price for buy orders and best ask price for sell orders
Vyacheslav Arbuzov Financial market simulation

35. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Conditional cancellation process
Bayes’ rule: P(Ci|RTCI) = P (RT CI|Ci)
P (RT CI) P(C)
P(Ci|RTCI) = K4(RTCI + D3)
Figure 13. Conditional cancellation process
Vyacheslav Arbuzov Financial market simulation

36. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Results of simulations (MOEX)
Figure 14. Returns distribution
Vyacheslav Arbuzov Financial market simulation

37. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Results of simulations (MOEX)
Figure 15. Spread distribution
Vyacheslav Arbuzov Financial market simulation

38. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Empirical calculations
Results of simulations (MOEX)
tau
P(tau)
100
101
102
103
10−5
10−4
10−3
10−2
10−1
100
Empirical
Daniels
MF
Upgrade
Figure 16. Order lifetime distribution of analyzing models
Vyacheslav Arbuzov Financial market simulation

39. Intoduction Daniels model Mike-Farmer model Upgrading model Results of models
Answers and questions
References
Arbuzov V., Frolova M. (2012) Market liquidity measurement and econometric modeling. Market Risk and
Financial Markets Modeling, Springer.
Bouchaud J.-P., Gefen Y., Potters M., Wyart M., (2004) Fluctuations and response in financial markets:
the subtle nature of ‘random’ price changes. Quantitative Finance 4 (2), 176–190.
Daniels M.G., Farmer J.D., Gillemot L., Iori G., Smith E. (2003) Quantitative model of price diffusion and
market friction based on trading as a mechanistic random process, Phys. Rev. Lett. 90
Farmer J. D., Gillemot L., Iori G., Krishnamurthy S., Smith D. E., Daniels M. G. (2006) A Random Order
Placement Model of Price Formation in the Continuous Double Auction. The Economy as an Evolving
Complex System III, 133-173. New York: Oxford University Press.
Farmer J. D., Patelli P., Zovko I. I. (2005) The predictive power of zero intelligence in financial markets,
Proc. Natl. Acad. Sci. USA 102 2254–2259
Mike S., Farmer J. D. (2008) An empirical behavioral model of liquidity and volatility, J. Econ. Dyn. Control
32 200–234
R Core Team (2013) R: A language and environment for statistical computing. R Foundation for Statistical
Computing, Vienna, Austria.
Vyacheslav Arbuzov Financial market simulation