Asymptotics and duality for the Davis and Norman problem

作者：Gerhold Stefan; Muhle Karbe Johannes^*; Schachermayer Walter

来源：Stochastics: An International Journal of Probability and Stochastic Processes , 2012, 84(5-6): 625-641.

DOI：10.1080/17442508.2011.619699

• transaction costs
• optimal consumption
• shadow price
• asymptotics

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 Janecek and Shreve [Finance Stoch., 8, 2004], who determined the leading terms of these power series.

• 出版日期2012