Asymptotics and duality for the Davis and Norman problem

We revisit the problem of maximizing expected logarithmic utility from consumption over an infinite horizon in the Black–Scholes model with proportional transaction costs, as studied in the seminal paper of Davis and Norman [Math. Operation Research, 15, 1990]. Similar to Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20, 2010], we tackle this problem by determining a shadow price, that is a frictionless price process with values in the bid-ask spread which leads to the same optimization problem. However, we use a different parametrization that facilitates computation and verification. Moreover, for small transaction costs, we determine fractional Taylor expansions of arbitrary order for the boundaries of the no-trade region and the value function. This extends work of Janeček and Shreve [Finance Stoch., 8, 2004], who determined the leading terms of these power series.

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