Abstract
Studies the effects of a quadratic transaction tax levied against agents in a continuous time, risk-sharing equilibrium model where agents have heterogeneous beliefs about the dynamics of the traded risky asset. The goal of each agent is to choose a trading strategy according to a mean-variance criterion, for which an optimal strategy exists in closed form as the solution to an FBSDE. This tractable setup allows analysis of the utility loss incurred from taxation. Also develops the theory of vague convergence for signed measures, extending a classical continuity theorem and Karamata’s Tauberian theorem to signed measures.
Summary
This thesis extends the transaction cost equilibrium framework of Bouchard et al. and Muhle-Karbe et al. to incorporate heterogeneous beliefs about the risky asset’s dynamics. Agents hold local mean-variance preferences and are penalised by a quadratic transaction tax on their trading rates. The unique equilibrium return is characterised via a coupled system of linear FBSDEs, with explicit solutions obtained for the N=2 agent case.
The utility loss analysis (Chapters 3-4) is the key normative contribution. Under homogeneous beliefs, the planner always finds the tax detrimental. Under heterogeneous beliefs, a small transaction tax can be beneficial when the damage from false beliefs outweighs the hedging needs — matching Keynes’s intuition that transaction taxes curb speculation. The condition is characterised by the sample covariance between agents’ belief deviations and their portfolio deviations from frictionless targets.
Chapters 5-6 develop the mathematical theory of vague convergence for signed (real-valued) measures, establishing the relationship between vague convergence and distribution function convergence, extending the continuity theorem for Laplace transforms, and proving Karamata’s Tauberian theorem for signed measures — motivated by the long-run asymptotics of the equilibrium model.
Key Contributions
- Extension of Bouchard et al. (2018) equilibrium model to heterogeneous beliefs with N agents
- Explicit FBSDE characterisation of equilibrium returns and portfolios (Theorem 2.2.7)
- Utility loss decomposition into direct, return, and portfolio components (Definition 2.4.1)
- Condition for when transaction tax is beneficial: sample covariance of beliefs and portfolio deviations must be positive (Section 2.4.1)
- Comprehensive theory of vague convergence for signed measures (Chapter 5, published as Herdegen-Liang-Shelley 2022)
- Extension of Karamata’s Tauberian theorem to signed measures (Chapter 6)
Methodology
The thesis uses coupled linear FBSDEs solved via matrix exponentials and power series representations. The equilibrium is derived by combining individual optimality (calculus of variations / Gateaux derivatives) with market clearing. For N=2 agents with arithmetic Brownian motion endowments, explicit formulas are obtained. The vague convergence theory uses the Riesz-Markov-Kakutani representation theorem and the Stone-Weierstrass theorem.
Key Findings
- Under homogeneous beliefs and risk aversions, the frictionless equilibrium clears the market with transaction costs (Corollary 2.2.8 analogue)
- Under heterogeneous beliefs, the liquidity premium depends on the sample covariance of risk aversions and portfolio deviations
- The transaction tax is beneficial precisely when it punishes false beliefs — confirming Keynes’s intuition
- The state variable Delta^n (deviation from frictionless target) follows an Ornstein-Uhlenbeck process, with mean-reversion speed determined by transaction cost parameter
- Small-cost asymptotics: the tracking speed scales as 1/sqrt(lambda), the equilibrium correction scales as sqrt(lambda)
Important References
- Equilibrium Returns with Transaction Costs — Bouchard, Fukasawa, Herdegen, Muhle-Karbe (2018), the direct precursor
- Asset Pricing with Heterogeneous Beliefs and Illiquidity — Muhle-Karbe, Nutz, Tan (2020), heterogeneous beliefs with holding costs
- Asset Pricing with General Transaction Costs — Gonon, Muhle-Karbe, Shi (2020), general convex costs
Atomic Notes
- quadratic transaction costs
- liquidity premium
- utility loss from transaction costs
- FBSDE equilibrium characterisation
- heterogeneous beliefs
- no-trade region