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Jonathan Donier added Quite_strikingly_the_MSD_curves__.tex
almost 9 years ago
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Quite strikingly, the MSD curves are indeed \emph{linear} in the vicinity of the price that corresponds to about $5\%$ range, in perfect agreement with our dynamical theory of supply and demand in the limit of frequent auctions
(note in particular that $\partial_y S(p^*) = \partial_y D(p^*) \approx 0$!). Correspondingly, we do expect that impact of meta-orders should be well accounted by a square-root law in this region, which is indeed also found empirically
(see \cite{donier2014million} for the special case of Bitcoin case, and \cite{Barra:1997,grinold2000active,Almgren:2005,Moro:2009,Toth:2011,Iacopo:2013,Gomes:2013,Bershova2013,Brokmann:2014,bacry2014market,zarinelli2014beyond} for more mainstream markets). Further away from the price, the non-universal region clearly appears, where the shape of the MSD (here, approximately saturating to some constant value) depends on the detailed characteristics of the order flow, modelled in this paper
by the functions $\nu(y)$ and $\omega(y)$.
\subsection{Summary}
The above section presented our story in mathematical terms. The punchline is however quite simple, and well summarized by the graphs plotted in Figure \ref{fig:scheme}, where we show (a) the standard Walrasian
supply and demand curves just before the auction, from which the equilibrium price $p^*$ can be deduced; (b) the supply and demand curves just after an auction, when the inter-auction time $\tau$ is large enough -- in
which case the marginal supply and demand are both finite at $p^*$; and (c) the supply and demand curves in the continuous time limit $\tau \to 0$, for which the marginal supply and demand curves vanish linearly around the
current price, as found in the Bitcoin market.
