Asset Pricing with General Transaction Costs: Theory and Numerics

Abstract

We study risk-sharing equilibria with general convex costs on the agents' trading rates. For an infinite-horizon model with linear state dynamics and exogenous volatilities, the equilibrium returns mean-revert around their frictionless counterparts -the deviation has Ornstein-Uhlenbeck dynamics for quadratic costs whereas it follows a doubly-reflected Brownian motion if costs are proportional. More general models with arbitrary state dynamics and endogenous volatilities lead to multidimensional systems of nonlinear, fully-coupled forward-backward SDEs. These fall outside the scope of known wellposedness results, but can be solved numerically using the simulation-based deep-learning approach of [28]. In a calibration to time series of returns, bidask spreads, and trading volume, transaction costs substantially affect equilibrium asset prices. In contrast, the effects of different cost specifications are rather similar, justifying the use of quadratic costs as a proxy for other less tractable specifications.Mathematics Subject Classification: (2010) 91G10, 91G80, 60H10.JEL Classification: C68, D52, G11, G12.

Equilibrium asset pricing with transaction costs

“…The assumption of quadratic rather than proportional costs also simplifies the analysis by ensuring that the FBSDE system describing the equilibrium remains linear in the agents' positions. Subquadratic trading costs lead to FBSDEs that are nonlinear in the agents' positions, see Gonon et al [27]; the limiting case of proportional costs corresponds to an FBSDE system with reflection, as is typical for such singular control problems (compare e.g. Élie et al [23]).…”

“…While these assumptions are made for tractability, numerical results reported in [27] suggest that the qualitative and quantitative properties of the equilibrium asset prices are surprisingly robust across different specifications of the trading costs (given that their absolute magnitudes are matched appropriately). This is in line with partialequilibrium results for models with small trading costs, see Moreau et al [50], where the fluctuations of frictional positions around their frictionless counterparts and the welfare effects of small trading costs are governed by the same drivers for different specifications of both preferences and trading costs.…”

Asset pricing with heterogeneous beliefs and illiquidity

“…Nevertheless, to obtain tractable first‐order conditions, we use a quadratic rather than linear cost. This simplification is motivated by recent results (Gonon, Muhle‐Karbe, & Shi, 2019) for risk‐sharing equilibria which show that equilibrium prices are robust with respect to the specification of the trading cost. Second, we assume as in Sannikov and Skrzypacz (2016) that all investors penalize inventories through a quadratic holding cost on positions.…”

Introduction to Solving Quant Finance Problems with Time-Stepped FBSDE and Deep Learning

“…Transaction cost can be included if the generator f is allowed to depend on the timederivativeπ i (t) of π i (t) as well, with G(x) = λ|x| q /q with q ∈ (1, 2] and additional terms −G(π i (t)) for all the components with transaction costs added to the f generator function in the BSDE, as in Gonon, Muhle-Karbe, and Shi [GMKS19]. (Alternatively, the transaction costs can be included in the running cost in the stochastic control problem introduced below.)…”