Jonathan Goodman
Courant Institute of Mathematical Sciences
New York University
New York City
New York, USA
Many optimal trading problems in finance lead to free boundary problems
for elliptic or parabolic partial differential equations. The boundaries
represent market conditions that trigger trading. One such problem arises
from a Merton optimal trading problem in the presence of transaction
costs. We generalize an asymptotic analysis, valid in the limit of small
transaction costs, that was given by Whalley and Wilmott. The scaling
they used follows from an informal analysis that balances trading costs
against the error in tracking the Merton optimal allocation. This
informal analysis is dual to the original problem, in the sense of
duality that is used to derive adjoint methods in PDE optimization.
This duality allows us to study a more complicated problem in which
the investor is allowed to purchase (with transaction costs) exchange
traded stock options as well as the underlying asset. This analysis
shows that using options and higher order hedging reduces the overal
transaction costs by a small power of the small parameter.