  We investigate the optimal strategy over a finite time horizon for a
portfolio of stock and bond and a derivative in an multiplicative Markovian
market model with transaction costs (friction). The optimization problem is
solved by a Hamilton-Bellman-Jacobi equation, which by the verification theorem
has well-behaved solutions if certain conditions on a potential are satisfied.
In the case at hand, these conditions simply imply arbitrage-free
("Black-Scholes") pricing of the derivative. While pricing is hence not changed
by friction allow a portfolio to fluctuate around a delta hedge. In the limit
of weak friction, we determine the optimal control to essentially be of two
parts: a strong control, which tries to bring the stock-and-derivative
portfolio towards a Black-Scholes delta hedge; and a weak control, which moves
the portfolio by adding or subtracting a Black-Scholes hedge. For simplicity we
assume growth-optimal investment criteria and quadratic friction.
