trading strategies via book imbalance2014/10/08 · trading strategies via book imbalance umberto...

Trading Strategies via Book Imbalance

Umberto Pesaventojoint work with Alexander Lipton and Michael G. Sotiropoulos

Algorithmic Trading Quantitative ResearchBank of America Merrill Lynch

Financial Engineering Workshops, Cass Business SchoolCity University London, 8 October 2014

U. Pesavento, Bank of America Merrill Lynch 1 of 26 8 October 2014

Bank of America Merrill Lynch

Contents

Limit order books and algorithmic trading

Empirical observations

Modeling the bid and ask queues

Adding trade arrival dynamics

Calibration

Conclusions and current work

Appendix

References



Limit order books and algorithmic trading: a top down approach

Sell trade127 800126 300

130 500129 1000

131 300132 800133 1500134 1000135 500

133 1500

131 100132 800

134 1000

128 1500127 800126 300

129 1000130 500

134 1500

132 100133 800

135 1000

129 1500128 800127 300

131 500130 1000

130 300131 800132 1500133 1000134 500

127 1500126 800125 300

129 500128 1000

t

t0 t t t t t t t t1 2 3 4 5 6 7 8

time

i

volu

me

ti+1

bid

ask

Buy trade

128 1500

A divide and conquer approach based on 3 phases:• calculating a time-volume schedule;

• limit order placement: optimal trading of the allocated shares within the given horizon;

• venue allocation.



Empirical observations: stopping times and book averaging

A separation of time scales:• queue length updates (≈ 3 s);

• best bid-ask updates (≈ 8 s);

• trade arrivals (≈ 12 s).

t0 pb0 pa

0 qb0 qa

0t1 pb

1 pa1 qb

1 qa1

t2 pb2 pa

2 qb2 qa

2t0 p0 q0 s0

t3 pb3 pa

3 qb3 qa

3t4 pb

4 pa4 qb

4 qa4

t1 p1 q1 s1

t5 pb5 pa

5 qb5 qa

5t6 pb

6 pa6 qb

6 qa6

t2 p2 q2 s2... ... ... ... ...



Empirical observations: mid price movements conditional on an book imbalance

Non-martingale properties of prices at small time scale:• future price variations can be predicted by using the imbalance in the bid and ask queues of

the order book I = (qb − qa)/(qb + qa);

• statistically significant (about 107data points for the plot above);

• the effect is not large enough to lead to a straightforward arbitrage but significant enough toyield savings in execution costs.



Empirical observations: trade arrivals and related stopping times

Changing the stopping time:• the overall trend in the expected price movements as a function of the book imbalance is the

same;

• conditioning on the arrival of trades on a particular side of the book breaks the symmetry inthe expected waiting time and price movements.



Modeling the bid and ask queues: replenishment processes and the constantspread approximation

130 8600

132 9000 133 6500135 3700

127 4500128 3000129 5000130 8600

128 3000129 5000130 8600131 7200

132 9000 133 6500

137 1500135 3700

initial queues

replenished queues

bid queue (up−tick)

ask

queu

e (d

own−

tick)

bid

ask

depletion replenishment

132 9000 133 6500135 3700

127 4500128 3000129 5000

131 7200

Simple models for queues and price dynamics:

• two correlated diffusion processes to represent the bid and ask queues (qb, qa) = (W b,W a);

• price moves are associated with queues depletion;

• queues replenishment, drawing from a stationary distribution;

• assume constant spreads (not a bad approximation for liquid stocks)

.



Modeling the bid and ask queues: time dependent dynamics

Pt +12

Pxx +12

Pyy + ρxy Pxy = 0, (1)

where ρxy is the correlation between the processes governing the depletion and replenishment ofthe bid and ask queues, which typically takes a negative value in a normal market.

α(x, y) = x

β(x, y) =−(ρxy x − y)√

1− ρ2xy

, (2)

yielding equation:

Pt +12

Pαα +12

Pββ = 0. (3)

And the second to cast the problem in polar coordinates:

α =− r sin(ϕ−$)

β =r cos(ϕ−$)←→

r =√α2 + β2

ϕ =$ + arctan(−α

β

).

(4)

where cos$ = −ρxy , so to yield the following equation for the hitting probabilities:

Pt +12

(Vrr +

1r

Pr +1r2

Pϕϕ)

= 0. (5)

with the final condition: P(T , T , r , ϕ) = 0 and boundary conditions:

P(t, T , 0, ϕ) = 0, P(t, T ,∞, ϕ) = 0, P (t, T , r , 0) = P0, P(t, T , r , $) = P1.



Modeling the bid and ask queues: Green’s function formulationWe seek the Green’s function to equation (5) by separating its radial and angular components:

G(τ, r ′, ϕ′) = g(τ, r ′)f (ϕ′), (6)This leads to two equations coupled by the positive constant Λ2 :

gτ =12

(gr′ r′ +

1r ′

gr′ −Λ2

r ′2g

), (7)

fϕ′ϕ′ =− Λ2f . (8)

The radial part is solved by:

g(τ, r ′) =e−

r′2+r20

2τ

τIΛ

(r ′r0

τ

), (9)

where IΛ(ξ) is the modified Bessel function of the first kind corresponding to Λ. After applying theboundary conditions on the angular part of the equation, the final formula for the Green’s function is:

G(τ, r0, r

′, ϕ0, ϕ

′) =2e−

r′2+r20

2τ

$τ

∞∑n=1

Iνn

(r ′r0

τ

)sin(νnϕ′) sin (νnϕ0). (10)

where νn = nπω . Finally, we integrate the equation above to obtain the hitting probability for the of

an up-tick (or down-tick) conditional on the initial condition of the queue.

P(t, T , r0, ϕ0) =−12

T

t

∞

0

Gϕ(t′− t, r , $)1r

drdt′. (11)



Modeling the bid and ask queues: infinite time limit

By writing out the explicit form for the Green’s function we obtain:

P (0, r0, φ0) =∞∑

n=1

ˆ T

0

ˆ ∞0

e−r2+r2

02t

$trIνn

(rr0

t

)dtdr

(−1)n+1νn sin (νnφ0) . (12)

We reverse the order of integration and evaluate the time integral using the following expression:

ˆ ∞0

e−r2+r2

02t

$trIνn

(rr0

t

)dt =

1

$νnr(√

s2 − 1 + s)νn (13)

where s = (r2 + r20 )/2rr0. We can then integrate along the radial component,

ˆ ∞0

1

$νnr(

max(

rr0,

r0r

))νn dr =1

$νnrνn0

ˆ r0

0rνn−1dr +

rνn0

$νn

ˆ ∞r0

r−νn−1dr =2

$ν2n. (14)

Finally, we sum the series to obtain:

P (0, r0, φ0) =2π

∞∑n=1

(−1)n+1

nsin(πn$φ0

)=φ0

$. (15)

As expected, the result depends only on the angular distance from the barrier.



Modeling the bid and ask queues: time independent formulation I

We now consider the time-independent problem from the onset:

12

Pxx +12

Pyy + ρxy Pxy = 0, (16)

P (x, 0) = 1, P (0, y) = 0. (17)

Again, we perform a change of coordinates to eliminate the correlation term,α(x, y) = x

β(x, y) =(−ρxy x + y)√

1− ρ2xy

, (18)

yielding equation:

Pαα + Pββ = 0. (19)



Modeling the bid and ask queues: time independent formulation II

We then perform a the second transformation to casts the modified problem in polar coordinates:

α =r sin(ϕ)

β =r cos(ϕ)←→

r =√α2 + β2

ϕ =arctan(α

β

),

(20)

where cos$ = −ρxy . Then the equation becomes

Pϕϕ(ϕ) = 0, (21)

with boundary conditions P(0) = 0 and P($) = 1. In this coordinate set the solution isstraightforward P(ϕ) = ϕ/$, which in the original set of coordinates has the form:

P(x, y) =12

1−arctan(

√1+ρxy1−ρxy

y−xy+x )

arctan(

√1+ρxy1−ρxy

)

.

(22)



Adding trade arrival dynamics: the trade arrival process

In analogy with the two Brownian processes representing the bid and ask queues, we add a third(unobservable) process to model trade arrival on the near side of the book:

(dqb, dqa

, dφ) = (dwb, dwa

, dwφ)



Adding trade arrival dynamics: handling correlation

12

Pxx +12

Pyy +12

Pzz + ρxy Pxy + ρxz Pxz + ρyz Pyz = 0 (23)

as in two dimensions, it is possible to eliminate the correlation terms,

α(x, y, z) =x

β(x, y, z) =(−ρxy x + y)√

1− ρ2xy

γ(x, y, z) =

[(ρxyρyz − ρxz ) x + (ρxyρxz − ρyz ) y + (1− ρ2

xy )z]

√1− ρ2

xy

√1− ρ2

xy − ρ2xz − ρ2

yz + 2ρxyρxzρyz

,

(24)

to obtain:

Pαα + Pββ + Pγγ = 0, (25)



Adding trade arrival dynamics: changing the domain

Again, we can write the exit probability problem in a simpler form by changing the computationaldomain Ω:

1sin2 θ

Pφφ (φ, θ) +1

sin θ∂

∂θ(sin θPθ (φ, θ)) = 0, (26)

P (0, θ) = 0, P ($, θ) = 0, P (φ,Θ (φ)) = 1. (27)

α =r sin θ sinϕ

β =r sin θ cosϕ

γ =r cos θ



Adding trade arrival dynamics: semi analytical solutions IWe introduce a new variable ζ = ln tan θ/2 and rewrite the exit problem again as [Lipton 2013]:

Pφφ (φ, ζ) + Pζζ (φ, ζ) = 0, (28)computational domain is now a semi-infinite strip with curvilinear boundary

ζ = Z (φ) = ln(

tan(

Θ (φ)

2

)). (29)

We look for the solution of the Dirichlet problem for the Laplace equation in the form

P(ϕ, ζ) =∞∑

n=1

cn sin(knϕ), kn =πn$

(30)

where the values of expansion coefficients cn can be determined by enforcing the boundarycondition

P(ϕ,Θ(ϕ)) = 1 (31)

ζ = ln tan θ/2

ϕ =ϕ



Adding trade arrival dynamics: semi analytical solutions IIIn order to compute the coefficients, we introduce the integrals

Jmn =

$

0

sin(kmϕ) sin(knϕ)e(kn+km)Z (ϕ)dϕ, (32)

Im =

$

0

sin(kmϕ)ekmZ (ϕ)dϕ. (33)

Then the boundary condition (31) becomes∑n

Jmncn = Im, (34)

and cn can be computed by matrix inversion as c = J−1I.



Calibration: putting everything together

Book event probabilities as a function of the bid-ask imbalance:• left region of the plot, price improvement is likely: get ready to reprice;

• central region of the plot, a trade on the near side is likely to anticipate an adverse price move:stay posted;

• right region of the plot: consider crossing the spread.



Calibration: the role of correlation

• Correlation is the main effect responsible for the symmetry breaking in the evolution of theprice expectation as a function of imbalance.

• It can also explain a big part of the adverse selection effect which we observe when postingorders in a limit order book.

• The model can capture the main features of symmetry breaking in the trade arrival process.



Conclusions and current work: from trade arrival rates to empirical fillprobabilities

Empirical fill probabilities, learning from our own execution data:• real data tends to be noisy, but it displays consistent trends• parametric forms of fill probabilities as a function of the limit order placement x can be

estimated, i.e. P(x) = 1− e−βx



Conclusions and current work: from empirical fill probabilities to optimizationschedules, a dynamic programming approach

Given an approximate functional form for the fill probability, we can solve the recursive optimizationproblem given by:

E [Pi ] = minx

((1− p(x))E [Pi+1] + p(x)x) (35)

where p(x) is the fill probability of a limit order with a limit price of x .• what is the optimal placement of a limit order? (blue line)

• what is the expected fill price? (red line)



Conclusions and current work: optimizing thresholds

Going on step further, parameter selection and price slippage estimation as a function of the slicesize:• as expected, larger slices will produce a larger slippage;• the optimal trade off between waiting and crossing the spread depends on the size of the slice

to be executed;• an optimal ridge in the parameter space can be calculated under certain assumptions.



Appendix: order flow and impact

We can also attempt to predict price movements and arrival times by conditioning on localmeasurements of prevailing order flows rather than book imbalance.



Appendix: a time dependent slice of the problem

Average time evolution of the mid-price across a trade event:• Before trade arrival prices tends to drift towards the near side of the book;

• At trade arrival impact dominates and prices moves towards the far side of the book.



Appendix: queues depletion and replenishment

Depletion of the bid and ask queues across bid up-ticks and down-tick price movements:• down-tick move, the initial queue size is thin while the next layer if fully formed;

• up-tick move, the previous layer is fully formed and the next queue distribution is thin;

• the ask queue is statistically unaffected.



References

[1] A. Lipton, U. Pesavento, M.Sotiropoulos, Risk, April, 2014.

[2] R. Almgren, C. Thum, H. L. Hauptmann, and H. Li. Equity market impact. Risk, 18:57, 2005.

[3] M. Avellaneda and S. Stoikov. High-frequency trading in a limit order book. QuantitativeFinance, 8:217–224, 2008.

[4] J.-P. Bouchaud, J. D. Farmer, and F. Lillo. How markets slowly digest changes in supply anddemand. In T. Hens and K Schenk-Hoppe, editors, Handbook of Financial Markets: Dynamicsand Evolution.

[5] J.-P. Bouchaud, D. Mezard, and M. Potters. Statistical properties of stock order books:empirical results and models. Quantitative Finance Finance, 2:251–256, 2002.

[6] R. Cont and A. de Larrard. Order book dynamics in liquid markets: limit theorems anddiffusion approximations. Working paper, 2012.

[7] R. F. Engle. The econometrics of ultra-high frequency data. Econometrica, 68:1–22, 2000.

[8] J. Hasbrouck. Measuring the information content of stock trades. Journal of Finance,46:179–207, 1991.

[9] A. Lipton and I. Savescu. CDSs, CVA and DVA - a structural approach. Risk, 26(4), 2013.

[10] S. Stoikov R. Cont and R. Talreja. A stochastic model for order book dynamics. Operationsresearch, 58(3):549–563, 2010.

[11] E. Smith, J.D. Farmer, L. Gillemot, and S. Krishnamurthy. Statistical theory of the continuousdouble auction. Quantitative Finance, 3:481– 514, 2003.


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