# Meeting Details

Title: Optimal Bidding in a Limit Order Book Probability and Financial Mathematics Seminar Giancarlo Facchi, PSU, Mathematics http:// An external buyer asks for a random amount $X>0$ of a certain asset. This agent will buy the amount $X$ at the lowest available price, as long as this price does not exceed a given upper bound $P$. One or more sellers are competing to fulfill the incoming order, by offering various quantities of the same asset for sale at different "limit" prices. The collection of all these sell orders at different prices is the "Limit Order Book". Having observed the prices asked by his competitors, each seller must determine an optimal pricing strategy, maximizing his expected payoff. Clearly, when other sellers are present, asking a higher price for the asset reduces the probability of selling it. In our model we assume that the $i$-th seller owns an amount $\kappa_i$ of stocks. He can put all of it on sale at a given price, or offer different portions at different prices. If the selling prices are allowed to be any real number in $[0,P]$, then a general pricing strategy is described by a measure on $[0,P]$. We analyze in detail two different scenarios. Let $\psi(s) = {\rm Prob.}[X>s]$ denote the tail distribution function of $X$. \begin{itemize} \item If $(\ln\psi(s))^{\prime\prime} \geq 0~ \forall s$, then a unique Nash equilibrium exists and can be explicitly determined in the special case where every player has the same payoff function. We show that the all the optimal strategies (except at most one) consist of measures which are absolutely continuous with respect to the Lebesgue measure. \item If $(\ln\psi(s))^{\prime\prime} < 0~ \forall s$, a Nash equilibrium does not exist, and the competition between sellers does not settle near any equilibrium state. \end{itemize} We also consider a different model where there is a positive tick size, which means that the only admissible pricing strategies are purely atomic, supported on a finite set of prices. In this case, we can prove the existence of Nash equilibria also in the more general case of heterogeneous players. Moreover, as the tick size goes to 0, any weak limit of discrete Nash equilibria provides a Nash equilibrium for the continuum model.