Asymptotics and duality for the Davis and Norman problem

Abstract

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.

Keywords:

• transaction costs
• optimal consumption
• shadow price
• asymptotics

2000 Mathematics Subject Classification::

• 91B28
• 91B16
• 60H10

Acknowledgements

We thank Paolo Guasoni, Mete Soner, and, in particular, Steve Shreve for valuable discussions and comments. We are also grateful to an anonymous referee for his/her careful reading of the manuscript. The first author was partially supported by the Austrian Federal Financing Agency and the Christian-Doppler-Gesellschaft (CDG). The second author gratefully acknowledges partial support by the National Centre of Competence in Research ‘Financial Valuation and Risk Management’ (NCCR FINRISK), Project D1 (Mathematical Methods in Financial Risk Management), of the Swiss National Science Foundation (SNF). The third author was partially supported by the Austrian Science Fund (FWF) under grant P19456, the European Research Council (ERC) under grant FA506041, the Vienna Science and Technology Fund (WWTF) under grant MA09-003, and by the Christian-Doppler-Gesellschaft (CDG).

Notes

^1. [email protected]

^2. [email protected]

^3. This notation, also used in [25], turns out to be convenient in the sequel. It is equivalent to the usual set-up with the same constant proportional transaction costs for purchases and sales (compare, e.g. [4,12,14,23]). Indeed, set and . Then coincides with . Conversely, any bid-ask process with equals for and .