Abstract
We study risk-sharing equilibria with general convex costs on the agents’ trading rates. This nests quadratic costs as one special case, but also covers proportional costs as another limiting case. For an infinite-horizon model with linear state dynamics and exogenous volatilities, the equilibrium returns can be characterized explicitly up to the solution of a single nonlinear ODE.
Summary
This paper generalises the quadratic transaction cost equilibrium model of Herdegen et al. (2018) to general smooth convex cost functions G(x) = lambda|x|^q/q for q in (1,2], as well as proportional costs (the limiting case q → 1). Two agents with heterogeneous risk aversions trade a single risky asset with mean-variance preferences.
For smooth convex costs, the equilibrium is characterised by the solution of a nonlinear ODE (Lemma 3.4, equation 3.5): the function g determines the ergodic state variable X via a mean-reverting diffusion dX_t = (G’)^{-1}(g(X_t))dt + drift * dW_t. The equilibrium return is then mu_t = frictionless + (gamma^1 - gamma^2)sigma^2 / 2 * X_t (Theorem 3.7). For proportional costs (Section 4), the state variable becomes a doubly-reflected Brownian motion on [-l, l] with l = (3lambda(gamma^1 beta^1 - gamma^2 beta^2)^2 / ((gamma^1 + gamma^2)^3 sigma^4))^{1/3}.
The paper also develops a deep learning algorithm (Section 7) to solve the fully-coupled nonlinear FBSDEs that arise with endogenous volatilities and general costs, demonstrating that different cost specifications yield remarkably similar equilibrium returns when costs are calibrated to match trading volume.
Key Contributions
- Extension from quadratic to general convex costs G(x) = lambda|x|^q/q, q in (1,2]
- Proportional costs as singular control limit (Section 4, Theorem 4.2)
- Nonlinear ODE characterisation of equilibrium (Lemma 3.4, equation 3.5)
- No-trade region width l = (3lambda * (risk-aversion-weighted endowment)^2 / ((gamma_1 + gamma_2)^3 sigma^4))^{1/3} for proportional costs
- Deep learning algorithm for nonlinear FBSDEs with endogenous volatility (Section 7)
- Calibration to S&P500 data: different cost specifications yield similar equilibria when matched to trading volume
Methodology
For smooth costs, the equilibrium ODE (3.5) is derived from Lemma 3.4 (existence of g with xg(x) ⇐ 0) using properties of the Legendre transform G*. The mean-reverting SDE for X has nonlinear drift (G’)^{-1}(g(X_t)) ensuring ergodicity. For proportional costs, singular stochastic control replaces the absolutely continuous trading rates, with positions described by Jordan-Hahn decompositions into cumulative purchases and sales.
For endogenous volatilities (Section 6), the model leads to fully-coupled nonlinear FBSDEs (equations 6.11-6.13) that cannot be reduced to Riccati equations for non-quadratic costs. The deep learning algorithm approximates the decoupling field by neural networks, training via stochastic gradient descent to match terminal conditions.
Key Findings
- Different cost specifications (quadratic, 3/2-power, proportional) yield qualitatively similar equilibrium returns
- Proportional costs: doubly-reflected Brownian motion state variable with explicit no-trade region width
- The no-trade region width scales as lambda^{1/3} for proportional costs (consistent with single-agent literature)
- Calibration: quadratic cost equivalent lambda_2 = 1.08 x 10^{-10} matches the proportional cost estimate lambda_1 = 0.31
- Realistic transaction costs produce substantial fluctuations in equilibrium returns around frictionless values
- Endogenous volatility models lead to challenging open problems in wellposedness of nonlinear FBSDEs
Important References
- Equilibrium Returns with Transaction Costs — Herdegen, Muhle-Karbe (2018), the quadratic cost baseline
- Investing with Liquid and Illiquid Assets — Garleanu and Pedersen, single-agent quadratic costs
- Ergodic Control of Brownian Motion — portfolio choice with proportional costs and ergodic control