1010.0627.pdf - arXiv:1010.0627v2[q-fin.PM 26 Aug 2011...

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arXiv:1010.0627v2 [q-fin.PM] 26 Aug 2011 ASYMPTOTICS AND DUALITY FOR THE DAVIS AND NORMAN PROBLEM STEFAN GERHOLD, JOHANNES MUHLE-KARBE, AND WALTER SCHACHERMAYER Abstract. We revisit the problem of maximizing expected logarithmic utility from consumption over an infinite horizon in the Black-Scholes model with pro- portional transaction costs, as studied in the seminal paper of Davis and Nor- man [ Math. Operation Research , 15, 1990]. Similarly 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 dif- ferent parametrization, which facilitates computation and verification. More- over, for small transaction costs, we determine fractional Taylor expansions of arbitrary order for the boundaries of the no-trade region and the value func- tion. This extends work of Janeˇ cek and Shreve [ Finance Stoch. , 8, 2004], who determined the leading terms of these power series. 1. Introduction It is a classical problem of financial theory to maximize expected utility from consumption (cf., e.g., [15, 20] and the references therein). This is often called the Merton problem , because it was first formulated and solved in a continuous- time setting by Merton [18, 19]. More specifically, he found that – for logarithmic or power utility and one risky asset following geometric Brownian motion – it is optimal to keep the fraction of wealth invested into stocks equal to a constant θ , which is known explicitly in terms of the model parameters. Magill and Constantinides [17] extended Merton’s setting to incorporate propor- tional transaction costs. In particular, they showed that – again for logarithmic or power utility – it is optimal to engage in the minimal amount of trading necessary to keep the fraction of wealth in stocks inside some no-trade region [ θ , θ ] around θ . Their somewhat heuristic derivation was made rigorous in the seminal paper of Davis and Norman [4], who also showed how to compute θ , θ by solving a free boundary problem. Date : November 9, 2018. 2000 Mathematics Subject Classification. 91B28, 91B16, 60H10. Key words and phrases. Transaction costs, optimal consumption, shadow price, asymptotics. We thank Paolo Guasoni, Mete Soner, and, in particular, Steve Shreve for valuable discus- sions 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 acknowl- edges 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)..
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Christopher Reinemann
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