Claudio Albanese and Stathis Tompaidis
Transaction costs and non-Markovian delta hedging
(157K, PostScript)
ABSTRACT. We consider the problem of
hedging and pricing European and American
derivatives in the continuous time formalism.
The underlying
security is a stock whose trading involves a
small relative transaction cost $k$. If $k=0$,
the Black and Scholes optimal trading strategy is
Markovian, satisfies the self-financing condition
and assures a perfect portfolio replication.
If $k>0$, transactions occur at random but discrete
times. We find an optimal trading strategy
that minimizes
total transaction costs for a given degree
of risk aversion. Since the calculation of rehedging
times is part of the problem in the continuous
time setting, optimal strategies
are non-Markovian. They also break the self-financing
constraint because hedge slippages are risky.
We compute the leading term in $\sqrt k$ in
an asymptotic expansion in the limit of
small transaction costs. We express the rehedging thresholds
in terms of the Black and Scholes solution and
evaluate the total transaction cost
by solving a final value problem for a
parabolic equation of the Black and Scholes type.