Financial market simulation based on zero intelligence models

Financial market simulation based on zerointelligence models

Vyacheslav Arbuzov1,2

[email protected]

1Prognoz Risk Lab

2Perm State University

Perm 21.03.2014Applied Economic Modeling Workshop

Vyacheslav Arbuzov Financial market simulation

[email protected]

Intoduction Daniels model Mike-Farmer model Upgrading model Results of models

Basic knowledge about LOB

Continuous double auction scheme

Figure 1. Order book representation



Basic knowledge about LOB

Continuous double auction

Three fundamental processes specifying a LOB are:1 Rate/size of market orders2 Rate/placement/size of limit orders3 Rate/placement/size of cancellations

Volume

Price

Figure 2. Different types of orders



Data

LSE data. Farmer, Patelli & Zovko

Data from Farmer, Patelli & Zovko (2005), The Predictive Powerof 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 over300,000 in total



Data

LSE data. Farmer, Patelli & Zovko

stock num. events average limit market deletions # daysticker (1000s) (per day) (1000s) (1000s) (1000s)

AZN 608 1405 292 128 188 429BARC 571 1318 271 128 172 433CW. 511 1184 244 134 134 432GLXO 814 1885 390 200 225 434LLOY 644 1485 302 184 159 434ORA 314 884 153 57 104 432PRU 422 978 201 94 127 354RTR 408 951 195 100 112 431SB. 665 1526 319 176 170 426SHEL 592 1367 277 159 156 429VOD 940 2161 437 296 207 434



Data

LSE data. Mike, Farmer

Data from Mike, Farmer (2008), An empirical behavioral model ofliquidity and volatility


02/05/2000 to 31/12/2002

Selected 25 stocks

Trading day from 9:00 am to 16:00.



Data

LSE data. Mike, Farmer

Stock # of orders Stock # of orders Stock # of orders

SHEL050 3,560,756 BLT 984,251 III050 301,101VOD 2,676,888 SBRY 927,874 TATE 243,348REED 2,353,755 GUS 836,235 FGP 207,390AZN 2,329,110 HAS 683,124 NFDS 200,654LLOY 1,954,845 III050 602,416 DEB 182,666

SHEL025 1,708,596 BOC100 500,141 BSY100 177,286PRU 1,413,085 BOC050 345,129 NEX 134,991TSCO 1,180,244 BPB 314,414 AVE 109,963BSY050 1,207,885



Data

MOEX data

Aeroflot JSC


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.



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++)



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 ontrading 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



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 (thesizes of limit orders and market orders are the same)



Theory

Poisson process

Impatient agents place market orders with Poisson rate µshares per unit time (buy and sell market orders equally likelyso effectively rate µ/2 for each).

Patient agents place buy limit orders with Poisson rate αshares per price per time (uniformly in the semi-infiniteinterval (−∞; a(t)) and sell limit orders with the same ratein) (b(t);∞)

Cancellations occur with probability δ per unit time (akin toradioactive decay)

All processes are independent



Theory

Poisson process

Figure 3. Scheme of the Daniels model



Estimation of parameters

Estimation of α

We follow the methods of Farmer, Patteli, Zovko (2005). Givenreal data of all orders/cancellations, can calibrate the parametersσ, α, δ, µ

For buy orders calculate relative price ∆ = m− p and for sellorders ∆ = p−m , where m - logarithm of midquote priceand p is the logarithm of order price

Rt = Quppert −Qlower

t , where Qlowert 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 thatcapture 58 % of day t’s limit orders and Lt is the totalnumber of shares of effective limit orders within this range.



Estimation of parameters

Estimation of σ, δ, µ

δt is calculated each day and then averaged. δt is calculatedusing only cancelled limit orders in the price range Rt.Measure δt as the inverse of the average lifetime of acancelled limit orders

σ is calculated simply as the average size of all limit orders.The model assumes both averages equal and in practice theaverage limit order size is only slightly larger than the averagemarket order size.

µ is calculated as the ratio of the number of shares of marketorders to the number of events during the trading day.



Practice

Estimation of α

Qupper = 12 tick size Qlower = −11 tick size

L = 1, 655, 646 α = 0.108orders

perasecond · peraprice

Figure 4. Heavy tails of price distribution(in this case ∆ = priceorder − pricebestaside)



Practice

Estimation of µ,δ,σ

Parameters Description Value

α Intensity of limit orders 0.108µ Intensity of market orders 0.006δ Intensity of cancellations 0.287dp Tick size 0.01σ Volume of orders 1184



Practice

Results of simulations

Figure 5. Distribution of spread



Practice


Figure 6. Distribution of returns



Practice


Figure 7. Orders lifetime distribution



Mike-Farmer model

Mike S., Farmer J. D. (2008) An empirical behavioral model ofliquidity and volatility, J. Econ. Dyn. Control 32.

The Mike-Farmer model



Description

Basic knowledge

Important properties of the order flow for a future upgrade of themodel (from Farmer et al. (2006)):

Trending of order flow

Power placement of limit prices

Non-Poisson order cancellation process



Empirical calculations

Price distribution

Let’s x is logarithmic distance from the same best price. For buyorders x = π − πb and for sell order x = πa − π.

-0.01 -0.005 0 0.005 0.01x = relative limit price from same best

100

101

102

103

P(x)

Student distribution, alpha=1.3

S0 = 0S0 = 0, BUYS0 = 0, SELLS0 = 0.003

AZN

MOEX data LSE data




Conditional cancellation processPosition in the order book

The distance of the price of the order i from the opposite best attime 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 placedyi = 0 - when order is executed




Conditional cancellation processPosition 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 5y

10-3

10-2

10-1

P(C

| y)

real datafitted curve

MOEX data LSE data




Conditional cancellation processOrder book imbalance

nimb = nbuy/(nbuy + nsell) for buy ordersnimb = nsell/(nbuy + nsell) for sell orders , wherenbuy - number of buy orders in order booknsell - 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 1nimb

0

0.004

0.008

0.01

P(C

| ni

mb)

real datalinear fit

MOEX data LSE data




Conditional cancellation processNumber 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




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




Mike-Farmer results of simulations (LSE results)

10-4

10-3

10-2

10-1

R

10-4

10-2

100

P(|r|

> R

)

real dataSimulation IV.

RETURN

Figure 8. Distribution of returns





10-4

10-3

10-2

10-1

S

10-4

10-2

100

P(s

> S

)

real dataSimulation IV.

SPREAD

Figure 9. Spread distribution





100

101

102

103

tau

10-6

10-4

10-2

P(ta

u)

Simulation, slope = -1.9Real data, slope = -2.1

Figure 10. Lifetime distribution




Heavy tails in price distribution

Figure 11. Power Law of logarithmic distance




Fitting of price distribution

Figure 12. Price distribution fitting using Power Law and t-Student




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




Conditional cancellation process

Bayes’ rule: P (Ci|RTCI) = P (RTCI|Ci)P (RTCI) P (C)

P (Ci|RTCI) = K4(RTCI +D3)

Figure 13. Conditional cancellation process




Results of simulations (MOEX)

Figure 14. Returns distribution





Figure 15. Spread distribution





tau

P(t

au)

100 101 102 103

10−

510

−4

10−

310

−2

10−

110

0

Empirical DanielsMF Upgrade

Figure 16. Order lifetime distribution of analyzing 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 EvolvingComplex 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.


Date post:	17-Aug-2014
Category:	Economy & Finance
Upload:	arbuzov1989
View:	563 times
Download:	2 times

Financial market simulation based on zero intelligence models

Economy & Finance