Alethograph

Asset Pricing with Heterogeneous Beliefs and Illiquidity

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Summary

This paper studies how heterogeneous beliefs interact with illiquidity to determine equilibrium asset prices. N agent types disagree about the drift of the state process X driving the asset’s terminal payoff f(X_T). Each agent maximises expected returns penalised by quadratic holding costs (gamma * phi^2 / 2) and quadratic transaction costs (lambda * (phi-dot)^2 / 2). The unique Markovian equilibrium is characterised by a weakly coupled system of linear parabolic PDEs (Theorem 4.1), with the equilibrium price function v determined by v = (1/N) sum v_i + (lambda G’(t))/(N G(t)) * a_0.

The key insight is the dual role of holding costs and transaction costs: they have opposite effects on equilibrium volatility and the agents’ forward-looking behaviour. Small transaction costs (lambda → 0) increase volatility via a sqrt(lambda) perturbation expansion (Theorem 5.2), while small holding costs (gamma → 0) decrease volatility via a linear gamma expansion. The two costs enter through their ratio gamma/lambda when net supply is zero.

Key Contributions

  • Multi-agent equilibrium with heterogeneous beliefs AND both holding and transaction costs
  • PDE characterisation via weakly coupled linear parabolic system (Theorem 4.1)
  • Dual role of costs: transaction costs increase equilibrium volatility, holding costs decrease it
  • Small-cost asymptotics: sqrt(lambda) scaling for transaction costs (Theorem 5.2), linear gamma for holding costs
  • Calibration to USD/EUR exchange rate data showing transaction costs increase volatility
  • Existence and uniqueness via fixed-point argument for reaction-diffusion equations with novel gradient estimates

Methodology

Individual optimality reduces to linear-quadratic control problems with FBSDEs (Lemma 3.1). The optimal portfolio tracks a moving target via G(t) = cosh(sqrt(gamma/lambda)(T-t)), with tracking speed -G’(t)/G(t) and discount kernel G(s)/G(t). Market clearing yields the parabolic PDE system. Existence uses a fixed-point iteration on C_b^N with Picard-type contraction, combined with Schauder estimates for Holder regularity.

Key Findings

  • With homogeneous beliefs, equilibrium prices are independent of liquidity costs (matching Herdegen et al.)
  • Heterogeneous beliefs generate nontrivial price effects through illiquidity
  • Transaction costs force agents to consider future trading opportunities, increasing equilibrium volatility
  • Holding costs reduce the importance of future opportunities, decreasing volatility
  • The exact equilibrium volatility interpolates between risk-neutral (highest, with transaction costs) and frictionless (lowest)
  • For zero net supply, costs enter only through gamma/lambda: small transaction costs = large holding costs

Important References

  1. Equilibrium Returns with Transaction Costs — Herdegen, Muhle-Karbe (2018), the homogeneous-beliefs precursor
  2. Asset Pricing with General Transaction Costs — Gonon, Muhle-Karbe, Shi (2020), general cost functions
  3. Heterogeneous Beliefs under Frictionless Markets — Scheinkman and Xiong survey

Atomic Notes

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