Liquidity Stress Testing using Optimal Portfolio Liquidation
aa r X i v : . [ q -f i n . R M ] F e b Liquidity Stress Testing using Optimal Portfolio Liquidation ∗Mike
Weber
BlueCove [email protected]
Manziuk
CMAP, Ecole [email protected] Bastien
Baldacci
CMAP, Ecole [email protected] 8, 2021
Abstract
We build an optimal portfolio liquidation model for OTC markets, aiming at minimizing thetrading costs via the choice of the liquidation time. We work in the Locally Linear Order Bookframework of [12] to obtain the market impact as a function of the traded volume. We find thatthe optimal terminal time for a linear execution of a small order is proportional to the square rootof the ratio between the amount being bought or sold and the average daily volume. Numericalexperiments on real market data illustrate the method on a portfolio of corporate bonds.
The European Securities and Markets Authority (ESMA) has set out guidance on liquidity stress test-ing supplementary to the existing requirements enshrined in the AIFMD and UCITS directives, withthe ESMA guidelines coming into force on 30th September 2020. The core of the liquidity stress testingframework is a model that can be used to estimate liquidation times and costs in a reasonably realisticway for a portfolio of investments, including funds that can take short positions such as hedge funds.The main components of a liquidity stress testing framework are predefined stress tests, a marketliquidity model that estimates liquidation cost and time, and a governance framework. This paperfocuses on the model of market liquidity applied to optimal portfolio liquidation for corporate bonds. ∗ This work benefits from the financial support of the Chaires Analytics and Models for Regulation, Financial Risk andFinance and Sustainable Development. Bastien Baldacci gratefully acknowledge the financial support of the ERC Grant679836 Staqamof. Bastien Baldacci and Iuliia Manziuk would like to thank Mathieu Rosenbaum (Ecole Polytechnique),Kaitong Hu (Squarepoint Capital) and Olivier Guéant (Université Paris 1 Panthéon-Sorbonne) for fruitful discussions.Mike Weber would like to thank Robert Almgren (Quantitative Brokers) and Jim Gatheral (Baruch College) for helpfulconversations. The opinions expressed in this paper belong solely to the authors. p, p +∆ p ], where p is a certain price level and ∆ p > In this section, we first recall the framework of the Locally Linear Order Book (LLOB for short) modelintroduced in [12]. We then show how to use it to assess the costs of liquidation, in particular onOTC markets, for example, the corporate bonds market. Even though there is no order book for thecorporate bond market, electronic trading platforms form a rough approximation of it. It is as if thereexists an unobservable order book, hereafter called a latent order book, and that one can observe blockprices as a function of the price-volume dynamics of this order book.
Initially the LLOB model emerged from an empirical fact that in limit order books the latent volumesaround the best ticks are linear in price deviation from the best price, even if it is not directly reflectedin the order book. In this section we describe the LLOB model in the initial context of order drivenmarkets. The general idea of the LLOB is that there exists a latent order book which, at any time t ,aggregates the total intended volume to be potentially sold at price p > V + ( t, p ) and thetotal intended volume to be potentially bought at price p or below V − ( t, p ). The latent volumes V + ( t, p )and V − ( t, p ) are not the volumes revealed in the observable order book but the volumes that would berevealed as limit or market orders if the price comes closer to p at some point (in short, as stated in[12], the latent volumes reflect intentions that do not necessarily materialize).Between t and t + dt , new buy and sell orders of unit volume may arrive at levels p t ∓ u where u > λ ( u ). At the same time the buyers and sellers who have already sentorders at p t ∓ u might want to change the price to p t ∓ u ′ , for u ′ >
0, at rate ν ( u, u ′ ), or even cancelan order temporarily in the case u ′ = + ∞ .Let us assume that the price process p t is a Brownian motion, which may not be well suited to orderbooks due to microstructural effects, but is suitable to approximate the price process on OTC markets.We assume that either u ′ = + ∞ with rate ν ∞ ( u ) or that the change of price is a Brownian motion.We define D ( u ) = R + ∞ ( u − u ′ ) ν ( u, u ′ ) du ′ interpretable as the squared volatility of intentions. Let usdenote by ρ ± ( t, u ) a latent volume averaged over price paths, the equation for which is ∂ t ρ ± ( t, u ) = ∂ uu (cid:16) D ( u ) ρ ± ( t, u ) (cid:17) − ν ∞ ( u ) ρ ± ( t, u ) + λ ( u ) ,ρ ± ( t, u ) = 0 for ( t, u ) ∈ R + × R − , where D ( u ) = D ( u ) + σ and σ > D ( u ) , λ ( u ) , ν ∞ ( u ) the explicit form of the stationary solution of the above PDE3s not known. However, in the case where new orders appear uniformly, i.e λ ( u ) = λ and D ( u ) = D independent of u , the exact stationary solution is ρ ( u ) = ρ ∞ (cid:18) − exp (cid:16) − uu ⋆ (cid:17)(cid:19) , ( ρ )where ρ ∞ = λν ∞ , u ⋆ = q D ν ∞ . The function ρ ( u ) is the density of order book trades, or more preciselyfor all u ′ > ρ ( u ′ ) = dVdu ( u ′ ), where V is the order book volume as a function of the distance fromthe mid-price. The meaning of u ⋆ is the width of the linear price change zone: for u ≪ u ⋆ , that isfor small deviations from the mid-price, the price depends linearly on the volume, whereas it staysconstant for u ≫ u ⋆ . The constant ρ ∞ is understood to be the density of an order book trade far awayfrom the mid-price p t . Precisely, it is the inverse of the asymptotic large size market elasticity. We de-fine the asymptotic market elasticity as ǫ asympt. = ρ ∞ , and the “naive” market elasticity as ǫ naive = σ ADV .Let us consider a buy order. We integrate Equation ( ρ ) over u from mid-price to ∆ p . The resultingequation gives the order book volume as a function of price change: V (∆ p ) = ρ ∞ (cid:18) ∆ p − u ⋆ (cid:16) − e − ∆ pu⋆ (cid:17)(cid:19) . In the case ∆ p ≪ u ⋆ , the price impact varies as the square root of the trade size. For large trade sizes,the price impact is a linear function of the trade size, imitating the increasing cost of trading when thetraded volume is bigger than one the market can digest.Note that the above model corresponds to trades that can be done in a single day, and is considered asa one day model. We consider a “linear” liquidation in which the block of assets is unwound in equalparts over a number of days T which needs to be determined. If we denote the total block size as N and a daily trade size of NT , the cost of trading each block is given by C block ( T ) = NT ∆ p (cid:18) NT (cid:19) , where ∆ p ( v ) is defined as the solution of V ( w ) = v for w ∈ R , v ∈ R + . The total direct costs are givenby the sum of C block over the number of days, which isDC( T ) = C block ( T ) T = N ∆ p (cid:18) NT (cid:19) . We introduce the following parametrization for the quantity ρ ∞ : ρ ∞ = α ∞ ADV σ , The asymptotic large size market elasticity is the incremental price needed to trade an incremental volume whentrading volume is large relative to ADV.
4o be compared with ρ ∞ = λν ∞ in [12]. In other words, we take the number of daily orders λ (of unitvolume) equal to the average daily number of unit volumes (equal to ADV), and the rate at whichbuy or sell orders are canceled equal to σα ∞ where α ∞ > α ∞ in terms of market elasticity so that α ∞ = σ ADV1 ρ ∞ = ǫ naive ǫ asympt. , So the physical meaning of and intuition behind this is the ratio of the “naive” market elasticity fortrades not large compared with ADV to the value of market elasticity for trades materially larger thanADV. We also assume that the width of the linear region is of the order of one day’s price move, sothat u ⋆ = σ . To account for a trading firm’s risk aversion, we consider a running penalty proportional to the standarddeviation of the PnL of the entire block liquidation, which is a measure of risk usually used in practice.As we assume a linear liquidation schedule, the penalty can be written φ ( T ) = γ q V ( P nL ) = γ √ P N σ √ T where γ is a number of standard deviations of the PnL representing the risk tolerance of the firm and P is taken to be the bond price (in units in which par is 1) and N is the face amount. The effect of thevolatility penalty is to incentivize the optimizer to not take too much time with liquidating the position.Let us consider the case ∆ p ≪ u ⋆ . By taking a Taylor expansion of Equation ( ρ ) at ∆ pu ⋆ = 0, we have∆ p ( V ) = s V u ⋆ ρ ∞ . This is the commonly assumed square root law for price penalty as a function of volume, see [2], forexample. Given an assumed linear liquidation schedule the cost for each block is therefore given by C block = NT ∆ p block = s u ⋆ ρ ∞ (cid:18) NT (cid:19) / . The total cost is a sum of costs of all blocks and the volatility penaltyTC( T ) = DC( T ) + φ ( T ) = s u ⋆ ρ ∞ N / √ T + γP N σ √ T √ . Let us express this money amount in terms of a cost per bond: c ( T ) = TC(T) N = s N u ⋆ T ρ ∞ + γP σ √ T √ .
5e now solve the optimal liquidation problem by setting the first derivative of c with respect to T equal to zero. Computations lead to the following optimal liquidation time: T ⋆ = √ NγP σ s u ⋆ ρ ∞ , which provides the cost per bond: c ( T ⋆ ) = 2 (cid:18) u ⋆ ρ ∞ (cid:19) / √ γP σN / / . Remark 2.1.
By setting u ⋆ = σ, α ∞ = 1 we obtain ˜ T ⋆ = √ γP s N ADV , ˜ c ( T ⋆ ) = 2 / √ γP σ / (cid:18) N ADV (cid:19) / . A similar analysis to the above may be done with the large trade size limit: lim N → + ∞ c ⋆ = 3 − / (2 − / + 2 / )( γP ) / (cid:18) N ADV (cid:19) / , So in between the small and large limits the cost dependence on trade size changes from N / to N / . In the following section, we show how to extend this framework to the multi-asset case.
When moving to the multi-asset version of the optimization, one needs to create multi-asset versionsof both the direct cost and the volatility penalty. The problem of cross-impact emerging when, forexample, trades of a certain amount of one asset influence the price of another asset, is not treated inour model. Optimal liquidation models taking into account cross-impact (see, for example,[10]) exist,however it is hard to estimate cross-impact matrices in OTC markets, notably due to fragmentation.For the sake of simplicity, we assume that the total direct cost is the sum of the individual direct costs.The multi-asset volatility penalty for the portfolio is a straightforward extension of the single assetversion. It is an integral over time of the covariances of the remaining positions. The position function N i ( t ) for the bond i that is linearly liquidated over time is given by N i ( t ) = N i (cid:18) − tT ⋆i (cid:19) + , where N i ∈ R is the initial position in the bond i , and T ⋆i refers to the final liquidation time, suchthat N i ( T ⋆i ) = 0. The total variance of the PnL can be expressed as a sum over covariance terms d X i,j =1 σ i σ j ρ i,j N i N j Z min( T ⋆i ,T ⋆j )0 (cid:18) − tT ⋆i (cid:19)(cid:18) − tT ⋆j (cid:19) dt = d X i,j =1 σ i σ j ρ i,j N i N j T ⋆i , T ⋆j ) (cid:18) − min( T ⋆i , T ⋆j )3 max( T ⋆i , T ⋆j ) (cid:19) . Remark 2.2 (Calibration of α ∞ ) . A simple and intuitive approach for fixing the value of α ∞ can be ound by looking at the small size asymptotic limit formula for the optimal liquidation time: T ⋆asympt = √ γP √ α ∞ s N ADV , where α ∞ has been reintroduced. Let us assume a bond priced at par, and impose the condition T ⋆asympt ( N = ADV ) = 1 , implying that it is reasonable to trade the ADV in one day, therefore α ∞ = 6 γ . In this section we present numerical examples of optimal liquidation using our methodology. We firstshow an application to a long-short portfolio of two correlated bonds sharing same characteristics exceptthat one is much more liquid than the other. Then, we present the results obtained on a long-shortportfolio of 20 bonds. In all the numerical results, we choose a risk aversion parameter γ = 0 . This test case demonstrates the disadvantages of a line by line liquidation of a long/short portfolio,typically used in vendor liquidity stress testing offerings. The test portfolio consists of two bond posi-tions of the same size (27 and −
27 respectively), where the bonds have same price of 141 .
49$ and 7%annualized volatility, but different ADVs: 30 for the first bond and 3 for the second, so that the firstone is more liquid.A liquidation strategy based on individual liquidation would result in the more liquid bond beingunwound rapidly and the less liquid one slower. But clearly the optimal way to liquidate this portfoliois to unwind these positions with the same liquidation strategy, especially the same timescale. Thiswould minimize total PnL variance thereby allowing for a longer liquidation time and less costs.
Correlation L i qu i da t i on C o s t s NaiveIndividualPortfolio
Figure 1: Optimal portfolio liquidation costs withrespect to correlation for different strategies.
Correlation L i qu i da t i on T i m e Bond 1Bond 2
Figure 2: Liquidation times for 2-bond case.7n Figure 1, we show the liquidation costs as a function of the correlation between the two bond pricereturns. We refer to individual optimization to be the line by line liquidation of the positions, eachbond is liquidated independently of the others. Such individual liquidation implies that the liquidationcost for the portfolio is the sum of the liquidation for each bond, including the standard deviationpenalty. By a naive strategy we refer to a strategy with T ⋆inaive = N i ADV i .The liquidation costs are decreasing functions of the correlation for all strategies. The difference be-tween portfolio optimization costs and individual optimization costs even for small correlations is theconsequence of the choice of the penalty function, and we have no intention to compare the values di-rectly. We are mostly interested to compare the costs dependence on the correlation level. Notably, inFigure 1, we see that the costs of portfolio liquidation are decreasing more steeply when the correlationlevel increases compared to the individual optimization. Before the 60% correlation level the decreasein costs is mostly linear for portfolio optimization, and for the correlation levels above 60% it becomesmore concave. In this specific case, the line by line liquidation provides lower costs than the naiveliquidation. However in general, this has no reason to be true.In Figure 2, we show the liquidation times for 2 bonds as a function of correlation in the case of portfoliooptimization. Below 45% correlation, the optimal liquidation times appear to be almost independentof correlation (3 days for the first bond and 6 days for the second bond). Then the liquidation time ofthe less liquid bond decreases so as to approach the liquidation time of the liquid bond. As correlationincreases, both times increase, converging to the same optimal time of 10 days. We have chosen a set of 20 random bonds from the USD Investment Grade and High Yield universes,with somewhat random position sizes assigned. In Table 1 we show the main characteristics of theportfolio and in Table 4 its correlation matrix.We can summarize the bonds’ parameters as (from 3rd April 2020 unless otherwise noted):• The gross value is about $40M: $25M long and $15M short.• ADV is estimated available daily volume calibrated on TRACE volume data and varies from $2Mto $21M per day across the 20 bond in this portfolio.• Volatility is 22 business days (one month) historical volatility and varies from 3 .
8% to 63% annual.• Bid-ask is set to 20bp to provide a minimum level for all bonds.For the particular portfolio chosen and for the time period chosen (three months to early June 2020) theaverage correlation between bonds was about 20%. However, this may not be representative of typicalbonds in typical time periods. It is therefore worth looking at the behavior of optimal liquidation costand time versus correlation. 8ond ADV$M/day minbid-ask Annual.vol Bond faceamount $M1 3.0 0.20 % 7.0 % 272 3.0 0.20 % 8.8 % -333 8.0 0.20 % 12.5 % -244 2.5 0.20 % 4.9 % -31.55 3.5 0.20 % 13.0 % 276 6.0 0.20 % 7.1 % -27 4.5 0.20 % 21.5 % -1.58 2.0 0.20 % 18.4 % -19 5.0 0.20 % 3.8 % -110 2.5 0.20 % 11.1 % -0.7111 5.0 0.20 % 32.9 % 4212 3.0 0.20 % 13.0 % -4213 4.5 0.20 % 11.3 % 4014 17.5 0.20 % 11.8 % -4015 21.5 0.20 % 10.8 % 37.516 20.5 0.20 % 63.4 % 217 2.5 0.20 % 60.7 % 1.518 9.5 0.20 % 11.7 % 119 3.0 0.20 % 26.4 % -120 2.0 0.20 % 12.9 % -0.77Table 1: Test portfolio characteristics.In Figure 3, we show the optimal portfolio liquidation cost - both the direct cost and the full costincluding volatility penalty – versus pairwise correlation, with all off-diagonal correlations set to thesame value.
Correlation L i qu i da t i on C o s t s NaiveIndividualPortfolio
Figure 3: Optimal liquidation costs with portfoliooptimizer.
Correlation L i qu i da t i on T i m e T-maxT-median
Figure 4: Optimal liquidation times with portfo-lio optimizer.As expected, this long-short portfolio has a decreasing liquidation cost as correlation increases. Asnone of the other models are sensitive to correlation, their direct cost values are constant, though theirvolatility penalty naturally decreases with increasing correlation when evaluated using the portfoliocost function. As in the 2-bond example we can notice that the decrease of the costs for the portfolio9ptimization is steeper for all correlation levels compared to other strategies considered. We can alsonotice that the costs of portfolio optimization becomes concave for the correlation levels above 50%.Note that, in this example, the line by line optimization provides higher liquidation costs compared toa simple naive liquidation strategy.It is also interesting to look at liquidation times for the optimal portfolio liquidation as correlationincreases, shown in Figure 4. The optimizer is taking advantage of the higher correlation causing areduced volatility penalty for slower liquidation. Both the median and maximum time are increasingmonotonously with respect to the correlation level.In all of these optimizations, there have been no constraints, apart from a very high time constraint of100 days which was never effective. However, this model can be used to calculate an optimal liquidationstrategy under time constraint, which is very useful for liquidity stress testing. Taking our examplecase portfolio, we can see that with a maximum liquidation deadline of 100 days, the median andmaximum liquidation times are about 9 and 32 days, with a direct cost of $0.805m. It is interesting tosee the impact on the cost function as we decrease the deadline.In Table 2, we compare the short deadline results where a time upper bound was used to constrain theoptimizer. In Figure 5, we show the excess cost above the optimal liquidation cost due to deadline fullliquidation shortening. Even though the median time to liquidate the portfolio underlyings was about9 days in the optimal case, shortening the liquidation time cutoff to a maximum of 10 days only causesa minor increase in cost, but as the deadline becomes shorter the costs increase drastically.Deadline Portfolioliq.cost Portfoliodirect cost PortfolioT-median PortfolioT-max100 1.605 0.805 9.4 32.520 1.607 0.812 9.4 2015 1.612 0.829 8.9 1510 1.635 0.903 8.2 107.5 1.677 1.004 7.5 7.55 1.780 1.176 5 53 1.999 1.503 3 32 2.282 1.875 2 21 3.061 2.772 1 1Table 2: Short deadline costs and times comparison.
Deadline P r e m i u m Figure 5: Short deadline premium.In Table 3, we compare liquidation costs for naive, individual and portfolio optimization strategies andpresent the median and the maximum liquidation times for the portfolio optimization across differentportfolios. The first portfolio corresponds to the test portfolio considered above with the correlationmatrix presented in Table 4, and other portfolios are the ones with all correlations set to a certain level.For every level of correlation and for the example of 20 bonds described in Table 1, we present theliquidation costs in the naive, individual and portfolio optimization case. For the last case, we also10resent the direct costs, the median and maximum liquidation time.Correlation Naiveliq. cost Individualliq. cost Portfolioliq. cost Portfoliodirect cost PortfolioT-median PortfolioT-maxTest 1.82 1.99 1.61 0.81 9.43 32.510% 1.90 2.05 1.73 0.86 7.19 23.4510% 1.87 2.03 1.69 0.84 7.66 24.9720% 1.84 2.01 1.65 0.82 8.22 25.6730% 1.81 1.99 1.61 0.80 8.06 27.8840% 1.78 1.97 1.57 0.78 8.36 29.2250% 1.74 1.94 1.52 0.76 7.99 27.3760% 1.71 1.92 1.46 0.74 8.73 29.8670% 1.67 1.90 1.39 0.70 9.28 30.7780% 1.63 1.87 1.30 0.67 10.45 36.2090% 1.59 1.84 1.20 0.64 11.76 40.62Table 3: Comparison between three type of liquidation.The methodology presented in this paper allows one to obtain the optimal liquidation strategy for aportfolio of bonds in time proportional to O ( d ) where d is the number of bonds. For the test portfolioexample, the method works in 5 seconds and for a portfolio of 1000 bonds, it takes less than 6 hours,which is reasonable in the context of liquidity stress testing. In this paper, we presented an optimal portfolio liquidation model based on the Locally Linear OrderBook framework with an application to liquidity stress testing on OTC markets. The model has onlyone free parameter to be calibrated. When the traded volume is small, the optimal liquidation time inthe single asset case is obtained analytically and is proportional to the square root of the ratio betweenthe volume being liquidated and the average daily volume. In the case of portfolio liquidation, oursimple and reasonably fast optimization procedure established in this paper can be applied.11 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 1.00 0.10 0.12 0.14 0.14 0.16 0.42 0.30 0.30 0.10 0.16 -0.11 0.32 0.35 0.48 0.24 -0.01 0.52 0.24 0.432 0.10 1.00 0.30 -0.23 0.16 0.26 -0.03 0.23 0.04 0.26 0.30 0.24 0.29 0.04 -0.09 0.22 0.19 0.11 0.36 0.253 0.12 0.30 1.00 -0.05 -0.17 0.59 0.28 0.29 0.22 -0.02 0.26 0.11 0.40 0.33 0.29 0.39 0.14 0.24 0.05 0.054 0.14 -0.23 -0.05 1.00 0.23 0.06 0.14 0.11 0.37 -0.13 0.05 0.31 0.22 0.30 0.31 0.25 0.36 0.25 0.25 0.195 0.14 0.16 -0.17 0.23 1.00 0.15 -0.31 0.34 0.09 0.12 -0.00 0.24 0.36 -0.03 0.07 0.13 -0.05 -0.02 0.23 -0.016 0.16 0.26 0.59 0.06 0.15 1.00 -0.00 0.40 0.46 0.22 0.15 0.13 0.35 0.25 0.13 0.23 -0.00 0.23 0.13 0.147 0.42 -0.03 0.28 0.14 -0.31 -0.00 1.00 0.08 0.39 0.13 0.03 0.01 0.17 0.08 0.08 0.09 0.09 0.27 0.03 0.288 0.30 0.23 0.29 0.11 0.34 0.40 0.08 1.00 0.48 0.36 0.10 -0.08 0.13 0.06 0.15 0.20 0.02 0.25 0.35 0.129 0.30 0.04 0.22 0.37 0.09 0.46 0.39 0.48 1.00 0.20 0.07 0.24 0.28 0.23 0.12 0.05 0.14 0.34 0.06 0.3410 0.10 0.26 -0.02 -0.13 0.12 0.22 0.13 0.36 0.20 1.00 -0.05 0.19 0.05 -0.04 -0.12 -0.08 0.05 0.13 0.14 0.3211 0.16 0.30 0.26 0.05 -0.00 0.15 0.03 0.10 0.07 -0.05 1.00 0.12 0.22 0.29 0.29 0.12 0.27 0.19 0.23 0.2312 -0.11 0.24 0.11 0.31 0.24 0.13 0.01 -0.08 0.24 0.19 0.12 1.00 0.39 0.18 0.05 0.11 0.52 0.03 0.08 0.2513 0.32 0.29 0.40 0.22 0.36 0.35 0.17 0.13 0.28 0.05 0.22 0.39 1.00 0.40 0.40 0.49 0.12 0.39 0.13 0.3814 0.35 0.04 0.33 0.30 -0.03 0.25 0.08 0.06 0.23 -0.04 0.29 0.18 0.40 1.00 0.82 0.48 0.25 0.69 0.38 0.5715 0.48 -0.09 0.29 0.31 0.07 0.13 0.08 0.15 0.12 -0.12 0.29 0.05 0.40 0.82 1.00 0.58 0.20 0.64 0.35 0.4216 0.24 0.22 0.39 0.25 0.13 0.23 0.09 0.20 0.05 -0.08 0.12 0.11 0.49 0.48 0.58 1.00 0.18 0.50 0.42 0.3417 -0.01 0.19 0.14 0.36 -0.05 -0.00 0.09 0.02 0.14 0.05 0.27 0.52 0.12 0.25 0.20 0.18 1.00 0.15 0.26 0.0618 0.52 0.11 0.24 0.25 -0.02 0.23 0.27 0.25 0.34 0.13 0.19 0.03 0.39 0.69 0.64 0.50 0.15 1.00 0.28 0.6119 0.24 0.36 0.05 0.25 0.23 0.13 0.03 0.35 0.06 0.14 0.23 0.08 0.13 0.38 0.35 0.42 0.26 0.28 1.00 0.2720 0.43 0.25 0.05 0.19 -0.01 0.14 0.28 0.12 0.34 0.32 0.23 0.25 0.38 0.57 0.42 0.34 0.06 0.61 0.27 1.00Table 4: Correlation matrix for the example set of bonds. eferences [1] R. Almgren and N. Chriss. Optimal execution of portfolio transactions. Journal of Risk , 3:5–40,2001.[2] J.-P. Bouchaud. Price impact.
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