On using shadow prices in portfolio optimization with transaction costs

In frictionless markets, utility maximization problems are typically solved either by stochastic control or by martingale methods. Beginning with the seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676--713], stochastic control theory has also been used to solve various problems of this type in the presence of proportional transaction costs. Martingale methods, on the other hand, have so far only been used to derive general structural results. These apply the duality theory for frictionless markets typically to a fictitious shadow price process lying within the bid-ask bounds of the real price process. In this paper, we show that this dual approach can actually be used for both deriving a candidate solution and verification in Merton's problem with logarithmic utility and proportional transaction costs. In particular, we determine the shadow price process.Comment: Published in at http://dx.doi.org/10.1214/09-AAP648 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

An Equilibrium Model of Lumpy Housing Investment

We formulate and solve a dynamic general equilibrium model with heterogeneous agents and lumpy housing adjustment at the household level. We use the model to ask a simple question: how does the microeconomic lumpiness of housing adjustment affect the equilibrium dynamic properties of aggregate consumption and investment? Our main conclusion is that lumpiness matters: in particular, lumpiness in housing adjustment (1) reduces the volatility of both housing and business investment; (2) increases the volatility of aggregate consumption; (3) increases the correlation of housing investment with business investment and with GDP. We also show that lumpiness of investment activity at the household level has small but significant aggregate implications, in contrast with the literature that shows that the aggregate effects of lumpy investment at the firm level are negligible.

Option Pricing and Hedging with Small Transaction Costs

An investor with constant absolute risk aversion trades a risky asset with general It\^o-dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading-order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.Comment: 20 pages, to appear in "Mathematical Finance