This paper provides the optimal position management strategy for a market maker who has to face uncertain customer orders in an "illiquid " market, where the market maker's continuous trading through a traditional exchange incurs stochastic linear price impacts. In addition, it is supposed that the market participants can partially infer the position size held by the market maker and their aggregate reactions affect the security prices. Although the market maker can ask its OTC counterparties to transact a block trade without causing a direct price impact in the exchange, its timing is assumed to be uncertain. Another important way for the market maker to reduce its position is to match an incoming customer order to the outstanding position being warehoused in its balance sheet .  The solution of the problem is represented by a stochastic Hamilton-Jacobi-Bellman equation, which can be decomposed into three (one non-linear and two linear) backward stochastic differential equations (BSDEs). We provide the verification using the standard BSDE techniques for a single security case. For a multiple-security case, we use an interesting connection of the non-linear BSDE to a special type of backward stochastic Riccati differential equation (BSRDE) whose properties have been studied by Bismut(1976)  .
This paper deals with an optimal position management problem for a market maker who has to face uncertain customer order flows in an illiquid market, where the market maker's continuous trading incurs a stochastic linear price impact. Although the execution timing is uncertain, the market maker can also ask its OTC counterparties to transact a block trade without causing a direct price impact . We adopt quite generic stochastic processes of the securities, order flows, price impacts, quadratic penalties as well as security borrowing/lending rates. The solution of the market maker's optimal position-management strategy is represented by a stochastic Hamilton-Jacobi-Bellman equation, which can be decomposed into three (one non-linear and two linear) backward stochastic differential equations (BSDEs). We provide the verification using the standard BSDE techniques for a single security case. For a multiple-security case, we make use of the connection of the non-linear BSDE to a special type of backward stochastic Riccati differential equation (BSRDE) whose properties were studied by Bismut(1976) . We also propose a perturbative approximation scheme for the resultant BSRDE, which only requires a system of linear ODEs to be solved at each expansion order. Its justification and the convergence rate are also given .
R
provides the
deals with an
R
strategy
problem
R
orders in an "illiquid "
order flows in an illiquid
R
through a traditional exchange incurs
incurs a
R
impacts. In addition, it is supposed that the market participants can partially infer the position size held by the market maker and their aggregate reactions affect the security prices. Although the market maker can
impact. Although the execution timing is uncertain, the market maker can also
D
in the exchange, its timing is assumed to be uncertain. Another important way for the market maker to reduce its position is to match an incoming customer order to the outstanding position being warehoused in its balance sheet
A
We adopt quite generic stochastic processes of the securities, order flows, price impacts, quadratic penalties as well as security borrowing/lending rates.
R
problem
market maker's optimal position-management strategy
R
use an interesting
make use of the
R
have been
were
A
. We also propose a perturbative approximation scheme for the resultant BSRDE, which only requires a system of linear ODEs to be solved at each expansion order. Its justification and the convergence rate are also given