Equilibrium returns with transaction costs

Abstract

We study how trading costs are reflected in equilibrium returns. To this end, we develop a tractable continuous-time risk-sharing model, where heterogeneous mean-variance investors trade subject to a quadratic transaction cost. The corresponding equilibrium is characterized as the unique solution of a system of coupled but linear forward-backward stochastic differential equations. Explicit solutions are obtained in a number of concrete settings. The sluggishness of the frictional portfolios makes the corresponding equilibrium returns mean-reverting. Compared to the frictionless case, expected returns are higher if the more risk-averse agents are net sellers or if the asset supply expands over time.Mathematics Subject Classification (2010): 91G10, 91G80.JEL Classification: C68, D52, G11, G12. We are grateful to Michalis Anthropelos, Peter Bank, Paolo Guasoni, and Felix Kübler for stimulating discussions and detailed comments. Moreover, we thank an anonymous referee for his or her careful reading and pertinent remarks.

Equilibrium Asset Pricing with Transaction Costs

“…In contrast, the corresponding trading rates controlling these positions need to be determined from their zero terminal values -near the terminal time, trading stops since additional trades can no longer earn back the costs that would need to be paid to implement them. If a constant volatility is given exogenously as in [8], then these forward-backward dynamics suffice to pin down the equilibrium returns. In this case, the FBSDEs are linear, and therefore can be solved explicitly in terms of Riccati equations and conditional expectations of the endowment processes [24,6].…”

Asset Pricing with General Transaction Costs: Theory and Numerics

“…Note that for the quadratic costs G(x) = λx 2 /2 considered in [29], the generator of the backward component does not depend on its volatility and itself. If the volatility is constant, the backward equation then becomes linear and can in turn be solved by reducing it to some standard Riccati equations [7,9]. For stochastic volatilities, these are replaced by a backward stochastic Riccati equation, compare [35,6].…”

“…To this end, definẽ . 9 As shown in Lemma A.6,gq is in fact the solution to the first-order equation (17) in [27,Theorem 6], with q = α + 1. 10 Notice that for quadratic costs q = 2, this is the standard normal distribution.…”

“…Accordingly, tractable models often focus on settings where asset prices [47,38,49] or trading volume [46] are deterministic. Models where asset prices and trading volume both fluctuate randomly have recently been analyzed by focusing on quadratic costs on the agents' trading rates [22,43,9,29]. The analysis of these models crucially exploits the linearity of the corresponding first-order conditions, thereby naturally raising the question how delicately the qualitative and quantitative predictions depend on the specific choice of the trading costs.…”

Asset Pricing with Heterogeneous Beliefs and Illiquidity

“…As a preparation for the equilibrium result, we first fix agent i and solve her individual optimization problem in the face of an exogenous price process. Similar linear-quadratic optimization problems have been considered, e.g., in [5,6,11,13,22,23,30]; we provide a self-contained derivation for the convenience of the reader.…”