Abstract
We consider an Ito-financial market at which the risky assets’ returns are derived endogenously through a market-clearing condition amongst heterogeneous risk-averse investors with quadratic preferences and random endowments. Investors act strategically by taking into account the impact that their orders have on the assets’ drift. Under price impact and transaction costs, we characterize the Nash equilibrium through the (unique) solution of a system of FBSDEs and derive its closed-form expression.
Summary
This paper extends the Herdegen-Muhle-Karbe (2018) competitive equilibrium framework by introducing strategic behaviour: each agent takes into account the price impact of their trades. This creates a Nash equilibrium rather than a competitive (Radner) equilibrium. The model supports N agents with heterogeneous risk tolerances, d risky assets, random endowments, and noise traders, all in continuous time with quadratic transaction costs.
The key insight is the “revealed risk exposure” concept (Definition 3.3, equation 16): under price impact, agent n submits a demand that reveals a different risk exposure than their true one. The revealed exposure is a weighted average of the agent’s true exposure and the aggregate residual exposure of the other agents — agent n hides part of their true exposure while exploiting others’ aggregate demand.
For the frictionless case, the Nash equilibrium return mu-breve differs from the competitive equilibrium return mu by a “liquidity premium” driven by price impact (equation 20). Under common risk tolerances, this premium equals -(Sigma/(delta N(N-1))) * sum lambda_m * phi-hat_m — which vanishes only when there are no noise traders and investors have the same risk tolerance.
With transaction costs (Section 4), the competitive equilibrium FBSDE (equation 24) has the same structure as Herdegen et al. (2018), and the Nash equilibrium (Section 4.3) is characterised by a system of coupled linear FBSDEs (equation 30). Under common risk tolerances (Assumption 4.1), closed-form solutions are obtained: the Nash equilibrium return with transaction costs is mu-breve_Lambda = mu-breve + 2Lambda/(N(N-1)) * (a^psi - r*psi-dot), showing that price impact amplifies the effect of transaction costs when N > 2 strategic investors participate.
Key Contributions
- First continuous-time model combining endogenous price impact AND exogenous transaction costs
- “Revealed risk exposure” concept: strategic agents hide their true hedging needs under price impact
- Nash equilibrium returns with transaction costs in closed form (Theorem 4.10, equation 31)
- Price impact amplifies transaction cost effects when N > 2 (Remark 4.11)
- For N = 2, price impact and competitive equilibria yield identical transaction cost effects (Theorem 4.12)
- Utility surplus analysis: risk-neutral investors always benefit from price impact (Proposition 3.9)
Methodology
Best-response functions derived via calculus of variations lead to a fixed-point problem. In the frictionless case, the linear system has a unique solution (Theorem 3.8). With transaction costs, the competitive equilibrium uses the FBSDE from Herdegen et al. (Lemma 4.1 / equation 24). For Nash equilibrium, revealed exposures are substituted into the FBSDE, yielding the coupled system (30) with a modified target TP-breve_m that includes a price impact correction 2delta-barSigma^{-1}Lambda/(N+1) * (a^psi_n - rpsi).
Key Findings
- Strategic agents always reveal different hedging needs than their true ones (Corollary 3.6)
- Nash equilibrium returns differ from competitive returns whenever agents have non-zero demand
- Under common risk tolerance and no noise traders, frictional and frictionless Nash equilibria coincide (matching the competitive result from Herdegen et al.)
- Price impact creates liquidity premium even with common risk tolerance, if noise traders are present
- The transaction cost effect on Nash equilibrium returns is strictly larger than on competitive returns when N > 2
- As Lambda → 0, the frictional Nash equilibrium converges to the frictionless Nash equilibrium
Important References
- Equilibrium Returns with Transaction Costs — Herdegen, Muhle-Karbe (2018), the competitive equilibrium baseline
- Equilibrium Asset Pricing with Transaction Costs — Herdegen, Muhle-Karbe, Possamai (2020), endogenous volatility
- Asset Pricing with Heterogeneous Beliefs and Illiquidity — Muhle-Karbe, Nutz, Tan (2020), heterogeneous beliefs
Atomic Notes
- quadratic transaction costs
- FBSDE equilibrium characterisation
- liquidity premium
- heterogeneous beliefs