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Figure 1.
The market impact of an hypothetical currency pair, which is even when expressed in one currency (blue line) is not even when expressed in the opposite currency (red line)
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Figure 2.
Graphical interpretation of supply–demand symmetry. A regular LS is symmetrical if and only if the plot of $ y = L(x)/m $ is symmetrical with respect to $ y = -x $, concave, increasing and passing through the origin. The orange line is such an example. These curves are confined to stay in the white area of the plane. Two extreme cases exist: the green line, which represents a perfectly liquid market, where quotes at all sizes coincide with fair price, ($ m(s) = m $, $ \forall s\in\mathbb{R} $); and the red line, which is the most illiquid possible market, where there aren't any quotes either on the bid or on the ask side ($ m(s) = +\infty $ if $ s<0 $, $ m(s) = 0 $ if $ s>0 $)
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Figure 3.
Graphical interpretation of the conjugation relationship $ s \leftrightarrow \tilde{s} $
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Figure 4.
Illustration of Proposition 3.16. $ L_+ $ represents the bid wing of a LS. The plot compares the ask wing $ L_- $ obtained assuming that impact is even and the ask wing $ \widetilde{L}_+ $ assuming a symmetrical LS. The former, is always more liquid
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Figure 5.
A symmetrical Liquidation Operator $ L(s) $, in the case of exponentially decaying MSDC bids and no bid–offer spread. We set $ m = 1 $, as in all the following similar plots
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Figure 6.
In the top plot the MSDC and SDC of the example in Figure 5. In the bottom plot, the corresponding marginal and average impacts. In this example, quotes decay to zero on the bid wing and diverge to infinity at some finite ask size
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Figure 7.
A symmetrical Liquidation Operator $ L(s) $, in the case of exponentially decaying MSDC bids with finite bid-offer spread
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Figure 8.
In the top plot the MSDC and SDC of the example in Figure 7. In the bottom plot, the corresponding marginal and average impacts
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Figure 9.
An asymptotically linear symmetrical Liquidation Operator $ L(s) $ with zero bid-offer spread. The corresponding MSDC is asymptotically flat as illustrated in Figure 12
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Figure 10.
In the top plot the MSDC and SDC of the example in Figure 9. In the bottom plot, the corresponding marginal and average impacts. MSDC and SDC are asymptotically flat in this example. These plots, at small size scales, are close to even functions in the sense of equation (1)
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Figure 11.
An asymptotically linear symmetrical Liquidation Operator $ L(s) $ with finite bid-offer spread. The corresponding MSDC is asymptotically flat as illustrated in Figure 12
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Figure 12.
In the top plot the MSDC and SDC of the example in Figure 11. In the bottom plot, the corresponding marginal and average impacts
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Figure 13.
A symmetrical Liquidation Operator $ L(s) $ built in such a way to have linear ask MSDC and no bid-offer spread
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Figure 14.
In the top plot the MSDC and SDC of the example in Figure 13. Both the MSDC and the SDC are linear on the ask side. The bid side is clearly not linear
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Figure 15.
A symmetrical Liquidation Operator $ L(s) $ built in such a way to have linear ask MSDC and finite bid–offer spread
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Figure 16.
In the top plot the MSDC and SDC of the example in Figure 5. Both the MSDC and the SDC are linear on the ask side. The bid side is clearly not linear
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Figure 17.
A symmetrical Liquidation Operator $ L(s) $ in the case where there are four finite bid quotes and four symmetrical finite ask quotes. The corresponding MSDC is piecewise constant, as illustrated in Figure 18
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Figure 18.
In the top plot the MSDC and SDC of the example in Figure 17. In the bottom plot, the corresponding marginal and average impacts. The visualization of the symmetry relation between corresponding bid and ask quotes is all but intuitive. Symmetrically placed bid and ask quotes have size and price in correspondence with each others as described by Proposition 3.6
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Figure 19.
Another symmetrical Liquidation Operator $ L(s) $ with symmetrical finite bid and ask quotes. The example is made with finer quotes with a lesser degree of illiquidity, giving rise to a more regular pattern
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Figure 20.
In the top plot the MSDC and SDC of example in Figure 19. In the bottom plot, the corresponding marginal and average impacts. This example, as compared with the one in Figure 18, is made of a strip of finer quotes and gives rise to a more regular pattern, with a behavior at small size scales which is closer to an even function
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Figure 21.
In the top plot a symmetrical LS obtained from a power–law (square–root) MSDC on the bid side and no bid–offer spread. The marginal impact is shown in the bottom plot to display that for small degree of illiquidity and/or small size, impact can be approximated by an even function, as proved in Corollary 3.9
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Figure 22.
A similar example to the one in Figure 21 but with a finite bid–offer spread. The bottom plot shows impact as a difference from mid–price. It can be noticed that in this case, impact is not an even function even at small size scales, as described in Corollary 3.9
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