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

The dual optimizer for the growth-optimal portfolio under transaction costs

We consider the maximization of the long-term growth rate in the Black-Scholes model under proportional transaction costs as in Taksar et al.(Math. Oper. Res. 13:277-294, 1988). Similarly as in Kallsen and Muhle-Karbe (Ann. Appl. Probab. 20:1341-1358, 2010) for optimal consumption over an infinite horizon, we tackle this problem by determining a shadow price, which is the solution of the dual problem. It can be calculated explicitly up to determining the root of a deterministic function. This in turn allows one to explicitly compute fractional Taylor expansions, both for the no-trade region of the optimal strategy and for the optimal growth rat

Equilibrium asset pricing with transaction costs

We study risk-sharing economies where heterogeneous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully coupled forward–backward stochastic differential equations. We show that a unique solution exists provided that the agents’ preferences are sufficiently similar. In a benchmark specification with linear state dynamics, the empirically observed illiquidity discounts and liquidity premia correspond to a positive relationship between transaction costs and volatility

The role of the dominant versus the non-dominant hemisphere: an fMRI study of Aphasia recovery following stroke

Background: Speech production is one of the most frequently affected cognitive functions following stroke; however, the neural mechanisms underlying the recovery of speech function are still incompletely understood. Aims: The current study aims to address the differential contributions of the dominant and non-dominant hemispheres in recovery from aphasia following stroke by comparing data from four stroke patients and 12 control participants to assess the patterns of activation during speech production tasks during functional magnetic resonance imaging (fMRI) scanning. Methods & Procedures: Four chronic stroke patients (three left-hemisphere lesion and one right-hemisphere lesion) diagnosed with Broca’s aphasia at the acute phase, but now recovered to near normal speech ability, were tested on speech production tasks (phonemic fluency, categorical fluency and picture naming) whilst undergoing fMRI. These patients were compared with 12 healthy controls undergoing the same procedure. Outcomes & Results: Individual subject analysis showed activation peaks in perilesional areas in three out of four patients. This included one patient with right-hemisphere lesion, who also showed predominant perilesional activation. Group analysis of control participants showed predominately left-hemisphere activation, but not exclusively so. Laterality indexes were calculated and showed predominant left-hemisphere lateralisation in the control group (LI = 0.4). Three out of the four patients showed speech lateralised to the same hemisphere as their lesion and the fourth patient showed speech lateralised to the opposite hemisphere to their lesion. Different speech production tasks resulted in varying lateralisation indices (LIs) within participants. Conclusions: The data suggest that perilesional areas support recovery of speech in the chronic phase post-stroke regardless of the site of the lesion. The study has implications for the understanding of functional recovery as well as for the paradigms used in fMRI to localise speech production areas. Specifically, a variety of speech tasks are required to elicit activation that is representative of the range of cortical involvement in speech in healthy adults and that also allows for accurate reporting of the extent of recovery experienced in patients

Goal-seeking compresses neural codes for space in the human hippocampus and orbitofrontal cortex

Humans can navigate flexibly to meet their goals. Here, we asked how the neural representation of allocentric space is distorted by goal-directed behavior. Participants navigated an agent to two successive goal locations in a grid world environment comprising four interlinked rooms, with a contextual cue indicating the conditional dependence of one goal location on another. Examining the neural geometry by which room and context were encoded in fMRI signals, we found that map-like representations of the environment emerged in both hippocampus and neocortex. Cognitive maps in hippocampus and orbitofrontal cortices were compressed so that locations cued as goals were coded together in neural state space, and these distortions predicted successful learning. This effect was captured by a computational model in which current and prospective locations are jointly encoded in a place code, providing a theory of how goals warp the neural representation of space in macroscopic neural signals

Goal-seeking compresses neural codes for space in the human hippocampus and orbitofrontal cortex

Humans can navigate flexibly to meet their goals. Here, we asked how the neural representation of allocentric space is distorted by goal-directed behavior. Participants navigated an agent to two successive goal locations in a grid world environment comprising four interlinked rooms, with a contextual cue indicating the conditional dependence of one goal location on another. Examining the neural geometry by which room and context were encoded in fMRI signals, we found that map-like representations of the environment emerged in both hippocampus and neocortex. Cognitive maps in hippocampus and orbitofrontal cortices were compressed so that locations cued as goals were coded together in neural state space, and these distortions predicted successful learning. This effect was captured by a computational model in which current and prospective locations are jointly encoded in a place code, providing a theory of how goals warp the neural representation of space in macroscopic neural signals.</p

On the Existence of Shadow Prices

For utility maximization problems under proportional transaction costs, it has been observed that the original market with transaction costs can sometimes be replaced by a frictionless "shadow market" that yields the same optimal strategy and utility. However, the question of whether or not this indeed holds in generality has remained elusive so far. In this paper we present a counterexample which shows that shadow prices may fail to exist. On the other hand, we prove that short selling constraints are a sufficient condition to warrant their existence, even in very general multi-currency market models with possibly discontinuous bid-ask-spreads.Comment: 14 pages, 1 figure, to appear in "Finance and Stochastics