Optimal portfolio liquidation - CREST · The optimal trajectories are very sensitive to the...

IntroductionAlmgren and Chriss model

Naive strategiesOptimal strategies

Optimal portfolio liquidation

Mathieu Rosenbaum

University Paris 6

February 2013

Mathieu Rosenbaum Optimal portfolio liquidation 1



Plan

1 Introduction

2 Almgren and Chriss model

3 Naive strategies

4 Optimal strategies




Portfolio liquidation

Financial problem

We want to sell a large quantity of a stock (or of severalstocks) in one day.

How to choose the transaction times ?




Strategies (1)

Naive strategies

2 extreme strategies :

Sell everything right now→ huge transaction cost since weneed to “eat” a lot in the order book. However this cost isknown.

Sell regularly in the day small amounts of assets→ smalltransaction costs (volumes are much smaller) but the finalprofit is unknown because of the daily price fluctuations :Volatility risk.




Strategies (2)

Optimization

We need to optimize between transaction costs and volatilityrisk.

To do so, we use the Almgren and Chriss framework whichtakes into account the market impact phenomenon andemphasizes the importance of having good statisticalestimators of market parameters.




Plan

1 Introduction


3 Naive strategies





Trading strategy

Setup

We consider we are selling one asset. We have X shares ofthis assets at t0 = 0.

We want everything to be sold at t = T .

We split [0,T ] into N intervals of length τ = T/N and settk = kτ , k = 0, . . . ,N.

A trading strategy is a vector (x0, . . . , xN), with xk thenumber of shares we still have at time tk .

x0 = X , xN = 0 and nk = xk−1 − xk is the number of assetssold between tk−1 and tk , decided at time tk−1.




Price decomposition

Price components

The price we have access to moves because of :

The drift → negligible at the intraday level.

The volatility.

The market impact.




Permanent market impact

Permanent impact component

Market participants see us selling large quantities.

Thus they revise their prices down.

Therefore, the “equilibrium price” of the asset is modified in apermanent way.

Let Sk be the equilibrium price at time tk :

Sk = Sk−1 + στ1/2ξk − τg(nk/τ),

with ξk iid standard Gaussian and nk/τ the average tradingrate between tk−1 and tk .




Temporary market impact

Temporary impact component

It is due to the transaction costs : we are liquidity taker sincewe “eat” the order book.

If we sell a large amount of shares, our price per share issignificantly worse than when selling only one share.

We assume this effect is temporary and the liquidity comesback after each period.

Let S̃k = (∑

nk,ipi )/nk , with nk,i the number of shares soldat price pi between tk−1 and tk . We set

S̃k = Sk−1 − h(nk/τ).

The term h(nk/τ) does not influence the next equilibriumprice Sk .




Profit and Loss

Cost of trading

The result of the sell of the asset is

N∑k=1

nk S̃k

= XS0 +N∑

k=1

(στ1/2ξk − τg(nk/τ)

)xk −

N∑k=1

nkh(nk/τ).

The trading cost C = XS0 −∑N

k=1 nk S̃k is equal to

Vol. cost + Perm. Impact cost + Temp. Impact cost.




Mean-Variance analysis

Moments

Consider a static strategy (fully known in t0), which is in factoptimal in this framework. We have

E[C] =N∑

k=1

τxkg(nk/τ)+N∑

k=1

nkh(nk/τ), Var[C] = σ2N∑

k=1

τx2k .

In order to build optimal trading trajectories, we will look forstrategies minimizing

E[C] + λVar[C],

with λ a risk aversion parameter.




Plan

1 Introduction


3 Naive strategies





Assumptions (1)

Permanent impact

Linear permanent impact : g(v) = γv .

If we sell n shares, the price per share decreases by γn. Thus

Sk = S0 + σk∑

j=1

τ1/2ξj − γ(X − xk).

and in E[C], the permanent impact component satisfies

N∑k=1

τxkg(nk/τ) = γ

N∑k=1

xk(xk−1 − xk) =1

2γX 2 − 1

2γ

N∑k=1

n2k .




Assumptions (2)

Temporary impact

Affine temporary impact : h(nk/τ) = ε+ η(nk/τ).

ε represents a fixed cost : fees + bid ask spread.

Let η̃ = η − 12γτ , we get

E[C] =1

2γX 2 + εX +

η̃

τ

N∑k=1

n2k .




Regular liquidation

Regular strategy

Take nk = X/N, xk = (N − k)X/N, k = 1, . . . ,N.

We easily get

E[C] =1

2γX 2 + εX + η̃

X 2

T,

Var[C] =σ2

3X 2T (1− 1

N)(1− 1

2N).

We can show this strategy has the smallest expectation.However the variance can be very big if T is large.




Immediate selling

Selling everything at t0

Take n1 = X , n2 = . . . = nN = 0, x1 = . . . = xN = 0.

We get

E[C] = εX +ηX 2

τ,

Var[C] = 0.

This strategy has the smallest variance. However, if τ is small,the expectation can be very large.




Plan

1 Introduction


3 Naive strategies





Optimization (1)

Optimization program

The trader wants to minimize

U(C) = E[C] + λVar[C].

U(C) is equal to

1

2γX 2 + εX +

η̃

τ

N∑k=1

(xk−1 − xk)2 + λσ2N∑

k=1

τx2k .




Optimization (2)

Derivation

For j = 1, . . . ,N − 1,

∂U

∂xj= 2τ

(λσ2xj − η̃

(xj−1 − 2xj + xj+1)

τ2).

Therefore

∂U

∂xj= 0⇔

(xj−1 − 2xj + xj+1)

τ2= K̃xj ,

with K̃ = λσ2/η̃.




Optimization (3)

Solution

It is shown that the solution can be written x0 = X and forj = 1, . . . ,N :

xj =sinh

(K (T − tj)

)sinh(KT )

X ,

nj =2sinh(Kτ/2)

sinh(KT )cosh

(K (T − jτ + τ/2)

),

where K satisfies 2τ2

(cosh(Kτ)− 1

)= K̃ .

If λ = 0, then K̃ = K = 0 and so nj = τ/T = X/N. Weretrieve the strategy with minimal expected cost.




Optimal strategies




Optimal strategies on real data

Rapport de Groupe de Travail - Dépendance Haute Fréquence entre Titres Année 2007-2008

Notons également que les stratégies d’écoulement illustrées par les graphiques ci-dessus sontplus resserrées autour du cas λ = 0 pour Saint-Gobain que pour Renault. Cela vient du fait que,sur notre historique, Saint-Gobain est moins volatile que Renault. Ceci conduit un market maker,même très averse au risque, à plus se rapprocher de la stratégie d’écoulement linéaire (nk = X/Npour tout k) pour Saint-Gobain que pour Renault puisque l’augmentation de variance associée serarelativement faible (du fait de la faible volatilité de Saint-Gobain par rapport à Renault) par rapportà la diminution de coût de market impact moyen associée.

L’estimation de la matrice de variance-covariance C des différents titres constituant la positioninitiale à écouler est cruciale, notamment l’estimation des corrélations. En effet, si le market makerutilise un estimateur naïf du coefficient de corrélation entre les rendements de deux titres surdes données à très haute fréquence, il va être victime du bruit de microstructure et de l’effet deEpps décrit dans les sections précédentes. Autrement dit il considérerait que la corrélation estnulle. Nous proposons d’analyser graphiquement l’impact d’un tel changement de corrélationsur les stratégies optimales d’écoulement. Rappelons que la corrélation estimée par la techniquedu previous tick vaut 96, 07%.

Stratégies optimales (Renault)

0

200

400

600

800

1000

1200

1 11 21 31 41 51 61 71 81 91

Période

Nom

bre

de ti

tres

en p

orte

feui

lle

lambda=0

lambda=1e-5

lambda=-1e-5

lambda=0 (correl=0)

lambda=1e-5 (correl=0)

lambda=-1e-5 (correl=0)

FIG. 6 – Stratégie optimale d’écoulement de la position en titres Renault selon λ pour deux estimations dela corrélation

ENSAE 28




Remarks on this approach

Remarks

It is easy to show that the solution is time homogenous : if wecompute the optimal strategy in tk , we obtain the valuebetween tk and T of the optimal strategy computed in t0.

In this approach, we obtain an efficient frontier of trading.

The optimal trajectories are very sensitive to the volatilityparameter. It is therefore important to obtain accuratevolatility estimates.

The Almgren and Chriss framework can be extended indimension n (if we sell several assets). In that case, correlationparameters come into the picture.




Optimal strategies in dimension 2


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