Permanent market impact can be nonlinear
aa r X i v : . [ q -f i n . T R ] M a r Permanent market impact can be nonlinear ∗ Olivier Gu´eant † Abstract
There are two schools of thought regarding market impact modeling. On the onehand, seminal papers by Almgren and Chriss introduced a decomposition betweena permanent market impact and a temporary (or instantaneous) market impact.This decomposition is used by most practitioners in execution models. On the otherhand, recent research advocates for the use of a new modeling framework that goesdown to the resilient dynamics of order books: transient market impact. One of themain criticisms against permanent market impact is that it has to be linear to avoiddynamic arbitrage. This important discovery made by Huberman and Stanzl [17]and Gatheral [13] favors the transient market impact framework, as linear permanentmarket impact is at odds with reality. In this paper, we reconsider the point made byGatheral using a simple model for market impact and show that permanent marketimpact can be nonlinear. Also, and this is the most important part from a practicalpoint of view, we propose different statistics to estimate permanent market impactand execution costs that generalize the ones proposed in [10].
Introduction
Over the course of the last decade, following the seminal papers on optimal liqui-dation by Almgren and Chriss [7, 8, 11, 10], a new strand of literature has emergedregarding the execution of large blocks of shares. Researchers, along with practition-ers, have first adopted the modeling framework introduced by Almgren and Chriss,and developed or used variants of the initial models. At the heart of this frame-work, there is a decomposition of market impact into two separate parts. On theone hand, there is a permanent market impact, pushing prices up over the courseof a buy order – resp. pushing prices down over the course of a sell order –, hencemodifying permanently the price process. On the other hand, there is a temporary ∗ This research has been conducted with the support of the Research Initiative “Ex´ecutionoptimale et statistiques de la liquidit´e haute fr´equence” under the aegis of the Europlace Instituteof Finance. I would like to thank Robert Almgren (NYU and Quantitative Brokers), Jim Gatheral(CUNY Baruch), Nicolas Grandchamp des Raux (HSBC France), Charles-Albert Lehalle (CapitalFund Management), Guillaume Royer (Ecole Polytechnique) and Christopher Ulph (HSBC) forthe conversations we had on market impact modeling. The points of view developed in this paperare solely those of the author. † Universit´e Paris-Diderot, UFR de Math´ematiques, Laboratoire Jacques-Louis Lions. 175, ruedu Chevaleret, 75013 Paris, France. [email protected] For the first time, the authors decidedto go from a statistical model of market impact (as the one proposed by Almgrenand Chriss) to a descriptive model of order book resilience. This was the first modelinvolving what is now called transient market impact. This second generation of op-timal liquidation models, based on transient market impact, has developed in recentyears ([1], [2], [3] and [21]). It raised the theoretical question of the functional formsfor market impact that are compatible with the absence of price manipulation (see[4], [5], [13], [14], [15], [18]). This question of the admissible forms for market impacthad already been considered for permanent market impact by Huberman and Stanzl[17] but the most cited paper on the topic is the paper by Gatheral [13] who statesthat permanent market impact has to be linear to avoid dynamic arbitrage, or inother words, to avoid round trips that are profitable on average.No model using the Almgren-Chriss framework featured a nonlinear specification forthe permanent market impact. However, the fact that it was apparently impossibleturned out to be a problem. Indeed, in parallel to the study of market impact from atheoretical point of view, market impact estimates were carried out and they createda consensus toward a square root form for the impact – only Almgren and coauthorsfitted in [10] a close-to-linear functional form for permanent market impact.The goal of this paper is to reconciliate the Almgren-Chriss decomposition of marketimpact with reality, and to show that nonlinear permanent market impact is com-patible with the absence of dynamic arbitrage. We indeed propose a framework inwhich permanent market impact is not only a function of the traded volume but alsoof the cumulated volume executed so far. This framework allows to have at the sametime a nonlinear permanent market impact and the absence of dynamic arbitrage.We also show, under a power-law hypothesis for permanent market impact, that thecumulated instantaneous market impacts can be extracted from the slippage andboth the pre-trade and post-trade prices, leading therefore to a generalization of theformula proposed by Almgren et al. in [10].Section 1 is dedicated to the classical Almgren-Chriss framework and it recalls theresult of Gatheral. Section 2 presents the main idea of this paper to model permanentmarket impact. It shows that the modeling framework is compatible with both anonlinear permanent market impact and the absence of dynamic arbitrage. Section3 shows how the cumulated instantaneous market impacts can be extracted from This paper appeared for the first time on the Internet in 2005. It has been published far laterin 2012.
Let us fix a probability space (Ω , F , P ) equipped with a filtration ( F t ) t ∈ R + satisfy-ing the usual conditions. We assume that all stochastic processes are defined on(Ω , F , ( F t ) t ∈ R + , P ).We consider a trader who wants, over a time window [0 , T ] , T >
0, to go from a port-folio made of q shares of a given stock to a portfolio made of q T shares of the samestock. For that purpose, we consider absolutely continuous strategies q : [0 , T ] → R with q (0) = q and q ( T ) = q T .The execution process usually impacts stock price. The usual framework to modelmarket impact is the following. First, the price S of the stock has the followingdynamics: dS t = σdW t − k ( v t ) dt, where v t = − q ′ ( t ), and where k ( v ) has the same sign as v : a buy order pushes theprice up, while a sell order pushes the price down.Besides this permanent market impact on price, the price obtained for a trade attime t is not S t but rather S t − h ( t, v t ) where h ( t, v ) is assumed to have the samesign as v .The resulting dynamics for the cash account X is: dX t = v t S t dt − h ( t, v t ) v t dt. The first result we state is the value of the cash account at the end of the tradingperiod: X T . Proposition 1. X T = X + ( q − q T ) S + Z T ( q T − q t ) k ( v t ) dt − Z T h ( t, v t ) v t dt − Z T σ ( q T − q t ) dW t . Proof:
3e write the definition of X T and we integrate by parts to obtain: X T = X + Z T v t S t dt − Z T h ( t, v t ) v t dt = X + [( q T − q t ) S t ] T + Z T ( q T − q t ) k ( v t ) dt − Z T σ ( q T − q t ) dW t − Z T h ( t, v t ) v t dt = X + ( q − q T ) S + Z T ( q T − q t ) k ( v t ) dt − Z T h ( t, v t ) v t dt − Z T σ ( q T − q t ) dW t . Then, following Gatheral ([13]), we define a dynamic arbitrage as a round trip from q to q T = q that is profitable on average. In other words, there is a dynamicarbitrage if and only if there exists an absolutely continuous function q : [0 , T ] → R such that the two following assertions are true: • q (0) = q ( T ), • The associated cash process verifies E [ X T ] ≥ X .Gatheral shows in [13] that a necessary and sufficient condition for the absence ofdynamic arbitrage is that k is a linear function. More precisely: Theorem 1 (Gatheral’s Theorem) . We have: • If k ( v ) = kv , then for any absolutely continuous function q : [0 , T ] → R , with q (0) = q ( T ) , the associated cash process verifies E [ X T ] ≤ X . • If h = 0 and if k is not linear, there exists a dynamic arbitrage. Considering the case h = 0 is a conservative assumption. The linearity assumptionfor k ensures in fact that there is no dynamic arbitrage independently of h .This result has very important consequences. The most important one concerns theoverall price impact of a transaction when k ( v ) = kv . If we consider indeed a sellorder of q shares over a time window [0 , T ], then, for any T ′ ≥ T , the stock priceat time T ′ is given by: S T ′ − S = − kq + σW T ′ . This means that the permanent market impact at the macroscopic level is linear, atodds with the square root (or at least strictly concave) shape usually obtained inthe literature.The goal of the next section is to recover a nonlinear and concave permanent marketimpact at the macroscopic level while avoiding the existence of dynamic arbitrage.4
Nonlinearity and no-dynamic arbitrage
One of the limitations of the preceding framework is that the permanent marketimpact function at the microscopic level ( k ) was a function of v only. The concavityof price impact observed empirically at the macroscopic level can in fact be inter-preted in the following way. When a trading process starts, it adds new volume tothe market and market participants anticipate that there will be more volume in thenear future. Hence, they add volume to the market, decreasing therefore the impactof future trades. Mathematically, it means that we can replace the expression k ( v t )by an expression of the form f ( | q − q t | ) v t , where f is a positive and decreasingfunction: the impact is still linear at the microscopic level but it decreases as thequantity executed so far is increasing.The above remarks lead to the following model: dq t = − v t dt,dS t = σdW t − f ( | q − q t | ) v t dt,dX t = v t S t dt − h ( t, v t ) v t dt, where f is assumed to be a positive function in L loc ( R + ).Our first result concerns the value of the cash process at the end of the executionprocess: Lemma 1. X T = X +( q − q T ) S + Z T ( q T − q t ) f ( | q − q t | ) v t dt − Z T h ( t, v t ) v t dt − Z T σ ( q T − q t ) dW t . Proof:
The proof is the same as above: X T = X + Z T v t S t dt − Z T h ( t, v t ) v t dt = X + [( q T − q t ) S t ] T + Z T ( q T − q t ) f ( | q − q t | ) v t dt − Z T σ ( q T − q t ) dW t − Z T h ( t, v t ) v t dt = X + ( q − q T ) S + Z T ( q T − q t ) f ( | q − q t | ) v t dt − Z T h ( t, v t ) v t dt − Z T σ ( q T − q t ) dW t . Now, an important point of our paper is that there is no dynamic arbitrage in ourmodel. More precisely: 5 roposition 2 (No-dynamic arbitrage) . For any absolutely continuous function q : [0 , T ] → R , with q (0) = q ( T ) , the associated cash process verifies E [ X T ] ≤ X .Proof: Using Lemma 1 we have: X T = X + Z T ( q − q t ) f ( | q − q t | ) v t dt − Z T h ( t, v t ) v t dt − Z T σ ( q − q t ) dW t . Let us define G ( z ) = R z yf ( | y | ) dy . The above expression can be written as: X T = X + Z T G ′ ( q − q t ) v t dt − Z T h ( t, v t ) v t dt − Z T σ ( q − q t ) dW t = X + [ G ( q − q t )] T − Z T h ( t, v t ) v t dt − Z T σ ( q − q t ) dW t = X − Z T h ( t, v t ) v t dt − Z T σ ( q − q t ) dW t . Hence: E [ X T ] = X − Z T h ( t, v t ) v t dt ≤ X . In the model we propose, there is no-dynamic arbitrage. However, it does not leadto a linear macroscopic permanent market impact. If one considers indeed a tradeof size q over the time window [0 , T ], then the price move due to the transaction ischaracterized by the following Proposition: Proposition 3 (Nonlinear permanent market impact) . Let us define F ( z ) = R z f ( | y | ) dy .If q (0) = q and q ( T ) = 0 , then ∀ T ′ ≥ T : S T ′ − S = − F ( q ) + σW T ′ . Proof: S T ′ − S = σW T ′ − Z T v t f ( | q − q t | ) dt = σW T ′ − [ F ( q − q t )] T = σW T ′ − F ( q ) . F ofthe form F : q ksgn ( q ) | q | α , with k > α ≃ . f is of the form f : q ∈ R ∗ + kαq − α . In particular, when α ∈ (0 , f blows up at 0. This is not a real problem since the function remains in L loc ( R + ).Obviously, one can replace kαq − α by kα ( q + A ) − α where A >
We have shown that our model has two characteristics that were supposed to beincompatible: no dynamic arbitrage and nonlinear permanent market impact. Also,the model can be used in optimal execution models since the effect of permanentmarket impact is independent of the liquidation strategy. This can be seen in thefollowing Lemma that gives the expression of the cash account at the end of aliquidation process.
Lemma 2.
Let us consider the special case q T = 0 . Let us define F ( z ) = R z f ( | y | ) dy .Then: X T = X + q S − Z q F ( z ) dz − Z T h ( t, v t ) v t dt + Z T σq t dW t . In particular, the influence of the permanent market impact on X T is independentof the strategy q and only depends on q .Proof: We use Lemma 1 when q T = 0 to obtain: X T = X + q S + Z T − q t f ( | q − q t | ) v t dt − Z T h ( t, v t ) v t dt + Z T σq t dW t = X + q S + [ − q t F ( q − q t )] T − Z T v t F ( q − q t ) dt − Z T h ( t, v t ) v t dt + Z T σq t dW t = X + q S − Z q F ( y ) dy − Z T h ( t, v t ) v t dt + Z T σq t dW t If one considers the special case of a power function for permanent market impact,then we obtain the following straightforward corollary:7 orollary 1. If f : q ∈ R ∗ + kαq − α with α ∈ (0 , and q T = 0 then: X T = X + q S − k α | q | α − Z T h ( t, v t ) v t dt + Z T σq t dW t . This expression will allow to recover the cumulated instantaneous market impacts R T h ( t, v t ) v t dt : Proposition 4.
Under the assumptions of Corollary 1, we have: S T ′ + αS α − X T − X q = 1 q Z T h ( t, v t ) v t dt − σ Z T q t q dW t + σ α W T ′ Proof:
We have: X T − X − q S = − k α | q | α − Z T h ( t, v t ) v t dt + Z T σq t dW t and S T ′ − S = σW T ′ − ksgn ( q ) | q | α . Hence: X T − X − q S = − q α ( σW T ′ − ( S T ′ − S )) − Z T h ( t, v t ) v t dt + Z T σq t dW t . Reorganizing the terms, we get: S T ′ + αS α − X T − X q = 1 q Z T h ( t, v t ) v t dt − σS Z T q t q dW t + σ α W T ′ . This Proposition is important because it can be rewritten: q S − ( X T − X ) q S + 11 + α S T ′ − S S = 1 q S Z T h ( t, v t ) v t dt − σ Z T q t q dW t + σ (1 + α ) S W T ′ , or equivalently: Slippage (%) + 11 + α Price Return(%) = Cum. Inst. Market Impact(%) + Noise . Hence, both permanent market impact and instantaneous market impact can beestimated using three variables: (i) slippage, (ii) pre-trade price and (iii) post-tradeprice. This is more clearly stated in the following Theorem that generalizes theresult of Almgren et al. ([10]) – the case of Almgren et al. corresponding to α = 1: The quantities below are signed: slippage and cumulated instantaneous market impact arepositive when q is positive (sell order) and negative when q is negative (buy order). heorem 2. Under the assumptions of Corollary 1 and denoting T ′ = T + δ , wehave: S T ′ − S = − ksgn ( q ) | q | α + ǫ S T ′ + αS α − X T − X q = 1 q Z T h ( t, v t ) v t dt + ǫ where ( ǫ , ǫ ) = N , σ T + δ δ α − R T (cid:16) q t q − α (cid:17) dt δ α − R T (cid:16) q t q − α (cid:17) dt δ (1+ α ) + R T (cid:16) q t q − α (cid:17) dt . In particular, if we assume that q t = q (cid:0) − tT (cid:1) , then: ( ǫ , ǫ ) = N , σ T + δ δ α + T (cid:0) α − (cid:1) δ α + T (cid:0) α − (cid:1) δ (1+ α ) + T α (1+ α ) !! . Proof: ǫ = σW T ′ , ǫ = − σ Z T q t q dW t + σ α W T ′ . We can compute the variance of ǫ and ǫ and the associated covariance: V ( ǫ ) = σ T ′ V ( ǫ ) = σ δ (1 + α ) + Z T (cid:18) q t q −
11 + α (cid:19) dt ! Cov ( ǫ , ǫ ) = σ (cid:18) δ α − Z T (cid:18) q t q −
11 + α (cid:19) dt (cid:19) . The particular case where q t = q (cid:0) − tT (cid:1) is then obtained straightforwardly.This Theorem generalizes a result by Almgren and coauthors (in the case α = 1).This result was used to estimate the two components of market impact using adatabase of metaorders (assumed to be executed with a POV algorithm) and theirdecomposition into child orders. As permanent market impact is not linear, oneneeds to use the above Theorem to estimate the parameters using Almgren’s esti-mation methodology. Otherwise, the estimation of the instantaneous market impactfunction is biased by a term coming from permanent market impact.9 onclusion In this paper, we propose a new modeling framework compatible with both no dy-namic arbitrage and nonlinear permanent market impact at the macroscopic level.This framework is rooted into the classical decomposition between permanent andinstantaneous market impacts. It allows to replicate the very robust stylized factof a concave permanent market impact, in particular the square root shape usuallyobserved. Interestingly, it does not change anything to classical optimal executionmodels since the permanent component of market impact continues to have no in-fluence on optimal strategies. However, the classical estimation of the instantaneousmarket impact function needs to be reconsidered and we propose in Theorem 2 newequations that generalize [10] to estimate market impact functions.
References [1] A. Alfonsi, A. Fruth, and A. Schied. Constrained portfolio liquidation in a limitorder book model.
Banach Center Publ , 83:9–25, 2008.[2] A. Alfonsi, A. Fruth, and A. Schied. Optimal execution strategies in limit orderbooks with general shape functions.
Quantitative Finance , 10(2):143–157, 2010.[3] A. Alfonsi and A. Schied. Optimal trade execution and absence of price manipu-lations in limit order book models.
SIAM J. Financial Mathematics , 1:490–522,2010.[4] A. Alfonsi and A. Schied. Optimal trade execution and absence of price manip-ulations in limit order book models.
SIAM Journal on Financial Mathematics ,1(1):490–522, 2010.[5] A. Alfonsi, A. Schied, and A. Slynko. Order book resilience, price manipulation,and the positive portfolio problem. 2009.[6] R. Almgren. Optimal trading with stochastic liquidity and volatility. 2011.[7] R. Almgren and N. Chriss. Value under liquidation.
Risk , 12(12):61–63, 1999.[8] R. Almgren and N. Chriss. Optimal execution of portfolio transactions.
Journalof Risk , 3:5–40, 2001.[9] R. Almgren and J. Lorenz. Adaptive arrival price.
Journal of Trading ,2007(1):59–66, 2007.[10] R. Almgren, C. Thum, E. Hauptmann, and H. Li. Direct estimation of equitymarket impact.
Risk , 18(7):58–62, 2005.[11] R.F. Almgren. Optimal execution with nonlinear impact functions and trading-enhanced risk.
Applied Mathematical Finance , 10(1):1–18, 2003.1012] P. A. Forsyth, J. S. Kennedy, S. T. Tse, and H. Windcliff. Optimal tradeexecution: a mean quadratic variation approach.
Quantitative Finance , 2009.[13] J. Gatheral. No-dynamic-arbitrage and market impact.
Quantitative Finance ,10(7):749–759, 2010.[14] J. Gatheral, A. Schied, and A. Slynko. Transient linear price impact and fred-holm integral equations.
Mathematical Finance , 2010.[15] J. Gatheral, A. Schied, and A. Slynko. Transient linear price impact and fred-holm integral equations.
Mathematical Finance , 2011.[16] O. Gu´eant. Optimal execution and block trade pricing: the general case.
Work-ing Paper , 2012.[17] G. Huberman and W. Stanzl. Price manipulation and quasi-arbitrage.
Econo-metrica , 72(4):1247–1275, 2004.[18] F. Kl¨ock, A. Schied, and Y. Sun. Existence and absence of price manipulationin a market impact model with dark pool.
Available at SSRN 1785409 , 2011.[19] J. Lorenz and R. Almgren. Mean-Variance Optimal Adaptive Execution.
Ap-plied Mathematical Finance , To appear in 2011.[20] A. Obizhaeva and J. Wang. Optimal trading strategy and supply/demanddynamics. Journal of Financial Markets, 2012[21] S. Predoiu, G. Shaikhet, and S. Shreve. Optimal execution in a general one-sided limit-order book. 2010.[22] A. Schied and T. Sch¨oneborn. Risk aversion and the dynamics of optimalliquidation strategies in illiquid markets.
Finance and Stochastics , 13(2):181–204, 2009.[23] A. Schied, T. Sch¨oneborn, and M. Tehranchi. Optimal basket liquidation forcara investors is deterministic.