Numerical methods for an optimal order execution...Numerical methods for an optimal order execution...

IntroductionModel

DiscretizationAlgorithm

Numerical results

Numerical methods for an optimal order execution

Fabien Guilbaud

EXQIM, and LPMA, University Paris 7, [email protected]

June 18, 2010

Joint work with H.Pham1 and M.Mnif2

1LPMA, University Paris 7, CREST-ENSAE, and Institut Universitaire deFrance, [email protected]

2ENIT, Tunis, [email protected] Guilbaud Numerical methods for an optimal order execution

IntroductionModel


Numerical results

Contents

1 Introduction

2 Model

3 Discretization

4 Algorithm

5 Numerical results

Fabien Guilbaud Numerical methods for an optimal order execution

IntroductionModel


Numerical results

ProblemBibliographical review

Market impact and liquidation costs

Implementation shortfall: difference between a theoreticaltrading strategy and its implementation=⇒ Need of low-touch execution strategy

Market impact is key factor when executing large orders

One famous worst case example: Kerviel’s portfolio liquidation

Basic observation: market impact is reduced when theliquidation operation is extended in time

Idea is to split a big order into several small orders=⇒ trade-off between rapid execution (big impact but reducedrisk) and gradual trading (reduced impact but more risk)

=⇒ Our goal is to find optimal trading schedule and associatedquantities


IntroductionModel


Numerical results

ProblemBibliographical review

Bibliographical review

Almgren and Chriss (2001) [1] first provided a framework tomanage market impact: mean-variance criterion, staticstrategy

Several authors propose price impact models based on stylizeddynamics of order book: Schied et al. (2009)[10], Gatheral etal. (2010)[3] and Obizhaeva and Wang (2005) [6].

Some approaches using optimal control: Rogers and Singh(2008) [9], Forsyth (2009)[2], Ly Vath, Mnif and Pham(2007)[5], Predoiu, Shaikhet and Shreve (2010) [8], Kharroubiand Pham (2009)[4]

=⇒ In this talk we use the model investigated in this last paper:impulse control formulation.


IntroductionModel


Numerical results

The model of portfolio liquidationPDE characterization

The model

We consider a financial market where an investor has to liquidatean initial position of y > 0 shares of risky asset by time T .

We set a probability space (Ω,F ,P) equipped with a filtrationF = (Ft)0≤t≤T supporting a one-dimensional Brownianmotion W on a finite horizon [0,T ], T < ∞.

(Pt)t∈[0,T ] the market price of the risky asset

(Xt)t∈[0,T ] the cash holdings

(Yt)t∈[0,T ] the number of stock shares held by the investor

(Θt)t∈[0,T ] the time interval between t and the last tradebefore t


IntroductionModel


Numerical results


The model: trading strategies

We represent a trading strategy by:

α = (τn, ξn)n∈N

where (τn) are F-stopping times and (ξn) are Fτn -measurableR-valued variables.

Dynamics for the shares and lag processes are under α:

Θt = t − τn, τn ≤ t < τn+1

Θτn+1 = 0, n ≥ 0.

Ys = Yτn , τn ≤ s < τn+1

Yτn+1 = Yτn + ξn+1, n ≥ 0.


IntroductionModel


Numerical results


The model: price impact

Market price of risky asset process follows a geometric Brownian motion:

dPt = Pt(bdt + σdWt)

Suppose now that the investor decides to trade the quantity e. If thecurrent market price is p, and the time lag from the last order is θ, thenthe price he actually get for the order e is:

Q(e, p, θ) = pf (e, θ)

with

f (e, θ) = exp(λ|eθ|βsgn(e)

).(κa1e>0 + 1e=0 + κb1e<0

), (1)

Therefore cash holdings have the following dynamics

Xt = Xτn , τn ≤ t < τn+1, n ≥ 0.

Xτn+1 = Xτ−n+1− ξn+1Pτn+1 f (ξn+1,Θτ−n+1

)− ε, n ≥ 0.


IntroductionModel


Numerical results


Liquidation value and solvency constraints

Let us define the admissibility constraints on a trading strategy. We first definethe liquidative value of the position (x , y , p, θ) by:

L(x , y , p, θ) = max(x , x + ypf (−y , θ)− ε)

where ε > 0 is some fixed transaction fee. Constraints are:

No short sale constraint:

Yt ≥ 0 , ∀t ∈ [0,T ]

Solvency constraint:

L(Xt ,Yt ,Pt ,Θt) ≥ 0 , ∀t ∈ [0,T ]

We define the admissibility region:

S = (z , θ) = (x , y , p, θ) ∈ R× R+ × (0,∞)× [0,T ] : L(z , θ) ≥ 0 and y ≥ 0

Finally, we will call admissible the strategies α in the following set:

A = α = (τn, ξn)n : ∀t ∈ [0,T ] (Xαt ,Y

αt ,P

αt ,Θ

αt ) ∈ S


IntroductionModel


Numerical results


Optimal execution criterion

We choose a CRRA utility function U(x) = xγ with γ ∈ (0, 1)and denote UL(.) = U(L(.))

The value function is defined by (we denoted z = (x , y , p)):

v(t, z , θ) = supα∈A(t,z,θ)

E[UL(ZT )

], (t, z , θ) ∈ [0,T ]× S


IntroductionModel


Numerical results


From [4] v is a unique viscosity solution to a quasi-variational inequality(QVI) written as:

min

[− ∂

∂tv − Lv , v −Hv

]= 0, on [0,T )× S,

min [v − UL, v −Hv ] = 0, on T × S.

L is the infinitesimal generator associated to the process (X ,Y ,P,Θ) ina no trading period:

Lϕ =∂

∂θϕ+ bp

∂

∂pϕ+

1

2σ2p2 ∂

2

∂p2ϕ

H is the impulse operator:

Hϕ(t, z , θ) = supe∈C(t,z,θ)

ϕ(t, Γ(z , θ, e), 0)

with Γ(z , θ, e) = (x − epf (e, θ)− ε, y + e, p), z = (x , y , p) ∈ S, e ∈ R

From now, our goal is to solve numerically this HJBQVI.


IntroductionModel


Numerical results

Discrete schemeExplicit backward schemeConvergence analysis

Discrete scheme

We propose the following discretization scheme:

Sh(t, z , θ, vh(t, z , θ), vh) = 0, (t, z , θ) ∈ [0,T ]× S,

with

Sh(t, z , θ, r , ϕ)

:=

min

[r − E

[ϕ(t + h,Z 0,t,z

t+h ,Θ0,t,θt+h )

], r −Hϕ(t, z , θ)

]if t ∈ [0,T − h]

min[r − E

[ϕ(T ,Z 0,t,z

T ,Θ0,t,θT )

], r −Hϕ(t, z , θ)

]if t ∈ (T − h,T )

min[r − UL(z , θ) , r −Hϕ(t, z , θ)

]if t = T .

which can be formulated equivalently as an implicit backward scheme:

vh(T , z , θ) = max[UL(z , θ) , Hvh(T , z , θ)

],

vh(t, z , θ) = max[E[vh(t + h,Z 0,t,z

t+h , θ + h)],Hvh(t, z , θ)

], 0 ≤ t ≤ T − h,

and vh(t, z , θ) = vh(T − h, z , θ) for T − h < t < T .


IntroductionModel


Numerical results


Explicit backward scheme

The usual way to treat implicit backward scheme is to solve by iterations asequence of optimal stopping problems:

vh,n+1(T , z , θ) = max[UL(z , θ) , Hvh,n(T , z , θ)

],

vh,n+1(t, z , θ) = max[E[vh,n+1(t + h,Z 0,t,z

t+h , θ + h)],Hvh,n(t, z , θ)

],

starting from vh,0 = E[UL(Z 0,t,zT ,Θ0,t,θ

T )]. Due to the effect of the lag variableΘt in the market impact function, it is not optimal to trade immediately after atrade. Therefore we are able to write equivalently this scheme as an explicitbackward scheme:

vh(T , z , θ) = max[UL(z , θ) , HUL(z , θ)

],


t+h , θ + h)], sup

e∈Cε(z,θ)

E[vh(t + h,Z

0,t,zeθ

t+h , h)]],

where zeθ = Γ(z , θ, e)


IntroductionModel


Numerical results


Convergence analysis

Monotonicity

The numerical scheme Sh is monotone.

Stability

The numerical scheme Sh is stable.

Consistency

The numerical scheme Sh is consistent.

Theorem: convergence

The solution vh of the numerical scheme Sh converges locallyuniformly to v on [0,T )× S.


IntroductionModel


Numerical results

ImplementationAlgorithm

Implementation

We compute the conditional expectation arising in the numerical scheme Sh

using an optimal quantization method:

Eh(ti , z , θj) := E[vh(ti + h,Z 0,ti ,z

ti +h , θj + h)]

= E[vh(ti + h, x , y , p exp

((b − σ2

2)h + σ

√hU), θj + h)

],

that we approximate by

Eh(ti , z , θj) ' 1

N

N∑k=1

πk vh(ti + h, x , y , p exp((b − σ2

2)h + σ

√huk

), θj + h).

And we use weights (πk) and values (uk) from the optimal quantization of the

normal law. From a computational point of view, this is reduced to a inner

product computation, which is quite fast.


IntroductionModel


Numerical results


Algorithm


t+h , θ + h)],Hvh(t, z , θ)

]

t : TIME

Θ: LAG


IntroductionModel


Numerical results


Policy shape in plane (Price × Shares)

2500

2750

3000

3250

3500

3750

4000

4250

4500

4750

SH

AR

ES

Policy sliced in the (price,shares) plane

SELL

0

250

500

750

1000

1250

1500

1750

2000

2250

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

SH

AR

ES

PRICE

NT

BUY

Figure: Shape of policyFabien Guilbaud Numerical methods for an optimal order execution

IntroductionModel


Numerical results

Performance analysisBehavior on historical dataBibliography

Performance analysis

We provide some numerical results that we obtain from our implementation.We tested the optimal strategy against a benchmark of two other strategies.

Strategy Utility V Mean L Standard Dev. σ

Naive 0.99993 0.99986 0.00429

Uniform 0.99994 0.99988 0.00240

Optimal 1.00116 1.00233 0.00436

Quantity Formula Value

Winning percentage1

Q

Q∑i=1

1L(i)opt>max(L

(i)naive ,L

(i)uniform

) 58.8%

Utility Sharpe RatioVopt −max(Vnaive , Vuniform)

σopt0.28017

Performance Sharpe RatioLopt −max(Lnaive , Luniform)

σopt0.56140


IntroductionModel


Numerical results


Performance analysis

80

100

120

140

160

180

De

nsi

ty

Empirical distribution of performance

Naive

0

20

40

60

80

De

nsi

ty

Performance

Naive

Uniform

Optimal

Figure: Optimal strategy empirical distributionFabien Guilbaud Numerical methods for an optimal order execution

IntroductionModel


Numerical results


Number of trades in one day liquidation

0.15

0.2

0.25

De

nsi

ty

Empirical distribution of number of trades

0

0.05

0.1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

De

nsi

ty

Number of trades

Figure: Number of trades empirical distributionFabien Guilbaud Numerical methods for an optimal order execution

IntroductionModel


Numerical results


Behavior on historical data: price dependency

-268

802

230

-316

-433

-306

-600

-400

-200

0

200

400

600

800

1000

53.6

53.8

54

54.2

54.4

54.6

54.8

55

55.2

55.4

55.6

55.8

BNP.PA 19/04/2010

9H30 16H30 13H00

Figure: Strategy realization on the BNP.PA stock the 04/19/2010.Fabien Guilbaud Numerical methods for an optimal order execution

IntroductionModel


Numerical results


Behavior on historical data: price dependency

-1262

-298

-192

-1400

-1200

-1000

-800

-600

-400

-200

0

50.5

51

51.5

52

52.5

53

53.5

54

54.5

BNP.PA 22/04/2010

9H30 16H30 13H00

Figure: Strategy realization on the BNP.PA stock the 04/22/2010.Fabien Guilbaud Numerical methods for an optimal order execution

IntroductionModel


Numerical results


Bibliography

Almgren R. and N. Chriss (2001): “Optimal execution of portfolio transactions”, Journal of Risk, 3, 5-39.

Forsyth P. (2009): “A Hamilton-Jacobi-Bellman approach to trade execution”, preprint, University of

Waterloo.

Gatheral J., Schied A. and A. Slynko (2010): “Transient linear price impact and Fredholm integral

equations”, Preprint

Kharroubi I. and H. Pham (2009): “Optimal portfolio liquidation with execution cost and risk”, Preprint,

University Paris 7, LPMA.

Ly Vath V., Mnif M. and H. Pham (2007): “A model of optimal portfolio selection under liquidity risk and

price impact”, Finance and Stochastics, 11, 51-90.

Obizhaeva A. and J. Wang (2005): “Optimal trading strategy and supply/demand dynamics”, to appear in

Journal of Financial Markets.

Potters M. and J.P. Bouchaud (2003): “More statistical properties of order books and price impact”,

Physica A, 324, 133-140.

Predoiu S., Shaikhet G. and S.Shreve (2010): “Optimal Execution in a General One-Sided Limit-Order

Book”, Preprint.

Rogers L.C.G. and S. Singh (2008): “The cost of illiquidity and its effects on hedging”, to appear in

Mathematical Finance.

Schied A. and T. Schoneborn (2009): “Risk aversion and the dynamics of optimal liquidation strategies in

illiquid markets”, Finance and Stochastics, 13, 181-204.Fabien Guilbaud Numerical methods for an optimal order execution

IntroductionModel


Numerical results


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Numerical methods for an optimal order execution...Numerical methods for an optimal order execution...

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