Optimal trading strategy for European options with transaction costs

The European option with transaction costs is studied. The cost of making a transaction is taken to be proportional by a factor λ to the value (in dollars) of stock traded. When there are no transaction costs (i.e. when λ=0) the well-known Black-Scholes strategy tells how to hedge the option. Since no non-trivial perfect hedging strategy exists when λ>0 (see (Ann. Appl. Probab. 5(2) (1995) 327)), we instead try to maximize the expected utility attainable. We seek to understand the effect transaction costs have on the maximum attainable expected utility over all strategies, when λ is small but non-zero. It turns out that transaction costs diminish the expected utility by an amount which has the order of magnitude λ^2/3. We will compute that correction explicitly modulo an error which is small compared to λ^2/3. We will exhibit an explicit strategy whose expected utility differs from the maximum attainable expected utility by an error small in comparison to λ^2/3.