Holding costs (or inventory costs) are a penalty on the size of an agent’s risky asset position, modelled as gamma * phi_t^2 / 2 in the mean-variance goal functional. In Muhle-Karbe, Nutz, Tan (2020), holding costs serve as a proxy for risk aversion and create a preference for smaller positions.
The key insight from Muhle-Karbe et al. is that holding costs and quadratic transaction costs play dual and inverse roles in equilibrium:
- Transaction costs force agents to consider future trading opportunities, making them more forward-looking and increasing equilibrium volatility
- Holding costs discount the importance of future positions, making agents more present-focused and decreasing equilibrium volatility
When the asset is in zero net supply (a_0 = 0), the equilibrium price depends on the two costs only through their ratio gamma/lambda. Small transaction costs (lambda → 0) are thus equivalent to large holding costs (gamma → infinity) in terms of equilibrium outcomes.
Key Details
- Cost functional: gamma * integral phi_t^2 dt / 2 in the goal functional
- Dual role with transaction costs: opposite effects on volatility; enter through ratio gamma/lambda for zero supply
- Small holding costs: regular perturbation expansion (linear in gamma) around risk-neutral price
- Small transaction costs: singular perturbation expansion (sqrt(lambda) scaling) around frictionless price
- Interpretation: can represent aversion to carrying inventory, risk-based position limits, or capital requirements