95-224 Claudio Albanese and Stathis Tompaidis

Transaction costs and non-Markovian delta hedging (157K, PostScript) 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 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.

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