In this thesis we study optimal brokerage problems in different scenarios. The thesisis structured in two parts:... Show moreIn this thesis we study optimal brokerage problems in different scenarios. The thesisis structured in two parts:
In the first part of this thesis, corresponding to Chapter 2 and 3, we construct optimal
brokerage contracts for multiple (heterogeneous) clients trading a single asset whose price
follows the Almgren-Chriss model. The distinctive features of this work are as follows:
(i) the reservation values of the clients are determined endogenously, and (ii) the broker is
allowed to not offer a contract to some of the potential clients, thus choosing her portfolio
of clients strategically. We find a computationally tractable characterization of the optimal
portfolios of clients (up to a digital optimization problem, which can be solved efficiently if
the number of potential clients is small) and conduct numerical experiments which illustrate
how these portfolios, as well as the equilibrium profits of all market participants, depend
on the price impact coefficients.
In the second part of this thesis, corresponding to Chapter 4, we establish existence
of a solution to the optimal contract problem in models where the state process is given by
a multidimensional diffusion with linearly controlled drift. Then, under certain concavity
assumptions, we show that the optimal contracts in the relaxed formulation also solve the
associated strong optimal contract problem. The main advantages of this approach, relative
to the existing methods, are due to the fact that it allows (i) to obtain the existence of
an optimal contract (as a limit point of epsilon-optimal ones), and (ii) to include various
additional constraints on the associated control problems (e.g., state constraints, difference
in filtrations of the agent and of the principal, etc.). Finally, we apply our results to the
problem of brokerage fees when the agent has access to a larger filtration. Show less
