For more information about this meeting, contact Anna Mazzucato, Victor Nistor, Manfred Denker.

Title: | Optimal Bidding in a Limit Order Book |

Seminar: | Probability and Financial Mathematics Seminar |

Speaker: | Giancarlo Facchi, PSU, Mathematics |

Abstract Link: | http:// |

Abstract: |

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. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 04 / 30 / 2013 |

Time: | 01:30pm - 02:30pm |