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IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
Optimal portfolio liquidation
Mathieu Rosenbaum
University Paris 6
February 2013
Mathieu Rosenbaum Optimal portfolio liquidation 1
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
Plan
1 Introduction
2 Almgren and Chriss model
3 Naive strategies
4 Optimal strategies
Mathieu Rosenbaum Optimal portfolio liquidation 2
IntroductionAlmgren and Chriss model
Naive strategiesOptimal 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 ?
Mathieu Rosenbaum Optimal portfolio liquidation 3
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 4
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 5
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
Plan
1 Introduction
2 Almgren and Chriss model
3 Naive strategies
4 Optimal strategies
Mathieu Rosenbaum Optimal portfolio liquidation 6
IntroductionAlmgren and Chriss model
Naive strategiesOptimal 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.
Mathieu Rosenbaum Optimal portfolio liquidation 7
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 8
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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 .
Mathieu Rosenbaum Optimal portfolio liquidation 9
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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 .
Mathieu Rosenbaum Optimal portfolio liquidation 10
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 11
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 12
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
Plan
1 Introduction
2 Almgren and Chriss model
3 Naive strategies
4 Optimal strategies
Mathieu Rosenbaum Optimal portfolio liquidation 13
IntroductionAlmgren and Chriss model
Naive strategiesOptimal 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 .
Mathieu Rosenbaum Optimal portfolio liquidation 14
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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 .
Mathieu Rosenbaum Optimal portfolio liquidation 15
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 16
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 17
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
Plan
1 Introduction
2 Almgren and Chriss model
3 Naive strategies
4 Optimal strategies
Mathieu Rosenbaum Optimal portfolio liquidation 18
IntroductionAlmgren and Chriss model
Naive strategiesOptimal 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 .
Mathieu Rosenbaum Optimal portfolio liquidation 19
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 20
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 21
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
Optimal strategies
Mathieu Rosenbaum Optimal portfolio liquidation 22
IntroductionAlmgren and Chriss model
Naive strategiesOptimal 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
Mathieu Rosenbaum Optimal portfolio liquidation 23
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
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.
Mathieu Rosenbaum Optimal portfolio liquidation 24
IntroductionAlmgren and Chriss model
Naive strategiesOptimal strategies
Optimal strategies in dimension 2
Mathieu Rosenbaum Optimal portfolio liquidation 25