The Square-Root Impace Law Also Holds for Option Markets

Tóth, Bence; Eisler, Zoltán; Bouchaud, Jean‐Philippe

doi:10.1002/wilm.10537

Cited by 26 publications

(21 citation statements)

References 10 publications

(16 reference statements)

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confidence: 65%

“…In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the squareroot law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013;Farmer et al, 2013;Donier et al, 2015;Tóth et al, 2016).Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.…”

supporting

confidence: 62%

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Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

2017

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We study optimal liquidation of a trading position (so-called block order or metaorder) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the squareroot law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013;Farmer et al., 2013;Donier et al., 2015;Tóth et al., 2016).Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.

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supporting

confidence: 65%

supporting

confidence: 62%

Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

2017

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“…A metaorder of total size Q impacts the price as ∼ √ Q and not proportionally to Q as naively expected and actually predicted by classical economics arguments [2]. The square-root law is surprisingly universal: it is found to be to a large degree independent of details such as the asset class, time period, execution style and market venues [3][4][5][6][7][8][9][10][11][12][13][14]. In particular, the advent of electronic markets and High Frequency Trading has not altered the square-root behaviour, in spite of radical changes in the microstructure of markets.…”

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confidence: 83%

Crossover from Linear to Square-Root Market Impact

et al. 2018

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Using a large database of 8 million institutional trades executed in the U.S. equity market, we establish a clear crossover between a linear market impact regime and a square-root regime as a function of the volume of the order. Our empirical results are remarkably well explained by a recently proposed dynamical theory of liquidity that makes specific predictions about the scaling function describing this crossover. Allowing at least two characteristic time scales for the liquidity ("fast" and "slow") enables one to reach quantitative agreement with the data.Financial markets sputter enormous amounts of data that can now be used to test scientific theories at levels of precision comparable to those achieved in physical sciences (see, e.g. [1] for a recent example). Among the most remarkable empirical findings in the last decades is the "square-root impact law", which quantifies how much prices are affected, on average, by large buy or sell orders, usually executed as a succession of smaller trades. Such a succession of small trades, all executed in the same direction (either buys or sells) and originating from the same market participant, is called a metaorder. A metaorder of total size Q impacts the price as ∼ √ Q and not proportionally to Q as naively expected and actually predicted by classical economics arguments [2]. The square-root law is surprisingly universal: it is found to be to a large degree independent of details such as the asset class, time period, execution style and market venues [3][4][5][6][7][8][9][10][11][12][13][14]. In particular, the advent of electronic markets and High Frequency Trading has not altered the square-root behaviour, in spite of radical changes in the microstructure of markets. The universality of this square-root law, together with its insensitivity to the high frequency dynamics of prices, suggests that its interpretation should lie in some general properties of the low frequency, large scale dynamics of liquidity [15]. In fact, the publicly displayed liquidity at any given time is usually very small -typically on the order of 10 −2 of the total daily transaction volume in stock markets. Financial markets are the arena of a collective hide-and-seek game between buyers and sellers, resulting in a somewhat paradoxical situation where the total quantity that markets participants intend to trade is very large (0.5% of the total market capitalisation changes hands every day in stock markets) while most of this liquidity remains hidden, or "latent". These observa-tions have lead to the development of a physics inspired, "locally linear order book" (LLOB) model for the coarsegrained dynamics of latent liquidity [7,[15][16][17], which naturally explains why the impact of metaorders grows like the square-root of its size in a certain regime of parameters [17]. But this LLOB model also suggests that for a given execution time T , the very small Q regime should revert to a linear behaviour. The model in fact predicts the detailed shape of the crossover between linear and square-root i...

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“…A now commonly accepted result is the so-called square-root law (see e.g. [2][3][4][5][6][7][8][9][10][11][12][13]) which states that in normal trading conditions…”

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confidence: 99%

Impact is not just volatility

Bucci

Mastromatteo

Benzaquen

et al. 2019

Quantitative Finance

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The notion of market impact is subtle and sometimes misinterpreted. Here we argue that impact should not be misconstrued as volatility. In particular, the so-called "square-root impact law", which states that impact grows as the square-root of traded volume, has nothing to do with price diffusion, i.e. that typical price changes grow as the square-root of time. We rationalise empirical findings on impact and volatility by introducing a simple scaling argument and confronting it to data.

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The Square-Root Impace Law Also Holds for Option Markets

Cited by 26 publications

References 10 publications

Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

Crossover from Linear to Square-Root Market Impact

Impact is not just volatility

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