Colloquium
12:30 p.m., Thursday (January 18, 2007)
MATH 104
Joern Sass
Radon Institute for Computational and Applied Mathematics
Austrian Academy of Sciences, Linz
Optimal portfolio policies under transaction costs
In a Black Scholes market  consisting of one stock whose prices evolve like
a geometric Brownian motion and one risk free asset  an investor wants to
maximize the asymptotic growth rate of his wealth (portfolio value). Without
transaction costs the optimal policy would be given by the constant Merton
fraction which is the fraction of the wealth to be invested in the stock.
Facing transaction costs it is no longer adequate to keep this risky
fraction constant.
We consider a combination of fixed (proportional to wealth) and proportional
costs which punish the trading frequency as well as the magnitude of the
transactions. Then an optimal trading strategy consists of a sequence of
stopping times and the optimal transactions at those times. So we have to
deal with impulse control strategies which can be described as solutions of
quasivariational inequalities.
Motivated by various structural results we first look at a restricted class
of
trading strategies which can be described by four parameters, two for the
stopping boundaries and two for the new risky fractions. In this class the
problem can be simplified by renewal arguments to one period between two
trading times, where we have to weight the new risky fractions by their
invariant distribution. This yields an explicit functional that has only to
be maximized in these four parameters. So the computation of the best
strategy in this class is very simple. Then we use the corresponding
quasivariational inequalities to prove that an optimal solution exists and
that it can be found in this class. The approach also works for short
selling and borrowing.
Refreshments will be served at 12:15 p.m. (Lounge, MATX 1115).
