Abstract:
This dissertation thesis focuses on two applications of optimization in financial
mathematics. The first application is in optimal liquidation in presence of an adverse
temporary price impact. The novel aspect of our formulation of the problem
is that we give the pressure to liquidate endogenously and we use a stochastic time
horizon. This leads to a severely singular initial value problem for which standard
numerical methods fail. We propose a procedure to overcome the singularity by
solving related finite horizon boundary value problems obtained by introducing a
time dimension into the time-homogenous problem. The convergence of the solutions
of the finite horizon problems to the solution of the original problem is analyzed
analytically and, subsequently, confirmed numerically. We find that the model
is consistent with the square root law known from empirical literature.
The second examined application of optimization in financial mathematics is
quadratic hedging of options. We focus on studying the mean squared hedging
error (MSHE) of a discretely implemented delta hedging strategy for an arithmetic
Asian option. We heuristically derive an approximation of the MSHE which is consistent
with known approximations for European options. We propose a method of
evaluating the approximation by solving a system of two partial differential equations
and use this method to numerically confirm that the approximation produces
reasonable estimates of the MSHE.
