 95224 Claudio Albanese and Stathis Tompaidis
 Transaction costs and nonMarkovian delta hedging
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May 17, 95

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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 selffinancing 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 nonMarkovian. They also break the selffinancing
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.
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