Cross-impact and no-dynamic-arbitrage

Abstract

We extend the ‘No-dynamic-arbitrage and market impact’-framework of Gatheral [Quant. Finance, 2010, 10(7), 749–759] to the multi-dimensional case where trading in one asset has a cross-impact on the price of other assets. From the condition of absence of dynamical arbitrage we derive theoretical limits for the size and form of cross-impact that can be directly verified on data. For bounded decay kernels we find that cross-impact must be an odd and linear function of trading intensity and cross-impact from asset i to asset j must be equal to the one from j to i. To test these constraints we estimate cross-impact among sovereign bonds traded on the electronic platform MOT. While we find significant violations of the above symmetry condition of cross-impact, we show that these are not arbitrageable with simple strategies because of the presence of the bid-ask spread.

Keywords:

• Market microstructure
• Market impact
• Price manipulation
• Cross-impact
• Fixed-income markets
• MOT

JEL Classifications:

• C20
• C81
• D44
• G12

Acknowledgements

We thank Borsa Italiana for providing us with access to their datasets and we thank Michael Benzaquen, Jean-Philippe Bouchaud, Katia Colaneri, Thomas Guhr, Florian Klöck, Enrico Melchioni, Loriana Pelizzon, Damian Taranto, two anonymous referees and participants of Market Microstructure—Confronting Many Viewpoints #4, Paris, and the $\mathit{XVIII}$ Workshop on Quantitative Finance, Milan, for helpful comments. We are responsible for all remaining errors.

Notes

No potential conflict of interest was reported by the authors.

This paper represents the authors’ personal opinions and does not necessarily reflect the views of the Deutsche Bundesbank, its staff, or the Eurosystem.

1 Conventionally buyer (seller) initiated trades have positive (negative) volume and order sign.

2 http://www.lseg.com/areas-expertise/our-markets/borsa-italiana/fixed-income-markets/mot

3 Alfonsi et al. (2016) consider a slightly different case where instead of a round-trip strategy, they consider the liquidation of an existing portfolio. This is equivalent in the limit of building up the portfolio infinitely slowly and when impact is purely transient.

4 The non-increasing assumption is necessary already in the single-asset case to avoid arbitrage opportunities from simple buy-hold-sell strategies. In the multi-asset case we require it e.g. for our symmetry result in lemma 3.6.

5 While keeping the product of ${\rho }^{\mathit{ij}}T$ small for all ij.

6 CONSOB, Bollettino Statistico Nr. 8, March 2016, available at http://www.consob.it/web/area-pubblica/bollettino-statistico

7 The conclusion of contracts from the opening auction happens at a random time between 09:00:00 and 09:00:59.

8 In phases of heavy trading multiple updates of the LOB may be recorded as one update in our data. However there is at least one update per second whenever there are changes to the LOB and in the vast majority of our sample updates are more frequent.

9 The remaining $\sim 8\%$ can either be due to orders that were executed across more than one millisecond (so that they are recorded as two or more orders), missed LOB updates or exotic order types.

10 We suppose that this is related to the fact that many of the bonds considered here are easily substitutable for one another.

11 In principle this observation suggests the presence of arbitrage opportunities due to the violation of lemma 3.7. However we should remember that what is shown in figure 3 is the observed impact, which might be different from the virtual impact, since the former does not take into account the selection bias due to the fact that traders condition the market order volume to what is present at the opposite best. For a discussion of this point in the self-impact case, see Bouchaud et al. (2008).

12 In other words, each trade advances time by one step, unless when there are two or more trades (in the same or different assets) recorded at exactly the same timestamp (at millisecond resolution). In such a case our combined trade time advances only by 1. In our sample ca. $3\%$ of trades happen at the same time-stamp as another trade in a different bond.

13 $\mathit{H}$ corresponds to the elementwise product of $\mathit{f}$ and $\mathit{G}$ as defined in equation (2(2) $$\begin{array}{c}\hfill S_t^i=S_0^i+\sum _j{\int }_0^t{f}^{\mathit{ij}}({\dot{x}}_s^j)G^{\mathit{ij}}(t-s)\mathrm{d}s+{\int }_0^t{\sigma }^i\mathrm{d}{Z}_s^i\end{array}$$(2) ), given the assumption of indifference to trade size.

14 Note that even though we refer to it as such, $\stackrel{\mathbf{~}}{\mathit{C}}$ is not strictly speaking a correlation matrix, as we do not de-mean nor normalize ${ϵ}_t^i{I}_t^i$.

15 We have been unable to make out any obvious patterns which pairs are significantly asymmetric when ordering by various measures of liquidity and trading activity (time-to-maturity, maturity, bid-ask spread, average number of trades per day, average trade volume, turnover, tick size). This suggests that the asymmetry we observe is not just a mere artifact of any of those measures.

16 To see this, denote the average trade size in asset i as ${\overline{x}}^i$ shares and take the case of executing the strategy in three trades corresponding to 3 units of trade time. The first phase, i.e. the first trade, lasts T / 3 and is of size ${\overline{x}}^i$ shares, therefore $v^{i}T=3{\overline{x}}^i$ shares.

17 Considering different sets leads to similar results.

19 In the case that $f(v)<-f(-v)$ all we need is to change the sign in equation (A2(A2) $$\begin{array}{c}\hfill \epsilon :=\frac{1}{4}\frac{f(v)+f(-v)}{f(v)}>0\end{array}$$(A2) ) to ensure that $\epsilon$ is positive. In the cases that either $f(v)\le 0\wedge f(-v)\le 0$ or $f(v)\ge 0\wedge f(-v)\ge 0$ the price manipulation arises in a simple in-out or out-in strategy as above respectively. Finally if assuming $vf(v)\le 0$ the proof is analogous to the one above.

20 We can choose an equivalent $\epsilon >0$ in all other cases, i.e. for $v_b{f}^{\mathit{ba}}(v_a)>v_a{f}^{ab(v_b)}>0$ by interchanging $a\leftrightarrow b$ in equation (A11(A11) $$\begin{array}{c}\hfill \epsilon :=\frac{1}{2}\frac{v_a{f}^{\mathit{ab}}(v_b)-v_b{f}^{\mathit{ba}}(v_a)}{2{\eta }^{\mathit{aa}}{v}_a^2+2{\eta }^{\mathit{bb}}{v}_b^2+3v_a{f}^{\mathit{ab}}(v_b)+v_b{f}^{\mathit{ba}}(v_a)}\end{array}$$(A11) ) and below. In the case that the denominator in (A11(A11) $$\begin{array}{c}\hfill \epsilon :=\frac{1}{2}\frac{v_a{f}^{\mathit{ab}}(v_b)-v_b{f}^{\mathit{ba}}(v_a)}{2{\eta }^{\mathit{aa}}{v}_a^2+2{\eta }^{\mathit{bb}}{v}_b^2+3v_a{f}^{\mathit{ab}}(v_b)+v_b{f}^{\mathit{ba}}(v_a)}\end{array}$$(A11) ) is negative we interchange $a\leftrightarrow b$ in order to ensure $\epsilon >0$ while the case of $v_a$ and $v_b$ such that the denominator is exactly zero is resolved by a slight modification of the turnaround points in the strategy.

21 If $v{f}^{\mathit{ij}}(v)\le 0$ then arbitrage arises from a strategy with $\lambda >0$, i.e. trading in the same direction.

22 We make the choice $v_1,v_2>0$ for simplicity of our arguments, this is without loss of generality since by lemma 3.2 $f^{\mathit{ij}}(v)$ needs to be an odd function of v.

23 The choice is either $v_b=v_1$ and $-\kappa v_b=v_2>0$ or $v_b=v_2$ and $-\kappa v_b=v_1>0$ (with $v_1\leftrightarrow v_2$), chosen such that the first inequality in equation (A15(A15) $$\begin{array}{c}\hfill -\kappa v_a{f}^{\mathit{ab}}(v_b)>v_a{f}^{\mathit{ab}}(-\kappa v_b)>0\end{array}$$(A15) ) is fulfilled.

24 In the one-dimensional case Gatheral (2010) shows that for absence of dynamic arbitrage it is also necessary that $\gamma \ge {\gamma }^{\ast }=2-\frac{log3}{log2}\approx 0.415$ and $\gamma +\delta \ge 1$.

Additional information

Funding

Michael Schneider acknowledges financial support from the Association of Foundations of Banking Origin [Young Investigator Training Program].