Minimizing Volatility: Optimal Adjustment with Evolving Feasibility Constraints

Abstract

Minimizing volatility and adjustment costs is of central importance in many economic environments, yet it is often complicated by evolving feasibility constraints. We study a decision maker who repeatedly selects an action from a stochastically evolving interval of feasible actions in order to minimize either average adjustment costs or variance. We show that for strictly convex adjustment costs (such as quadratic variation), the optimal decision rule is a reference rule in which the decision maker minimizes the distance to a target action. In general, the optimal target depends both on the previous action and the expectation of future constraints; but for the special case where the constraints follow a random walk, the optimal mechanism is to simply target the previous action. If the decision maker minimizes variance, the optimal policy is also a reference rule, but the target is a constant, which is not necessarily equal to the long-term average action. Compared to mid-point heuristics, these optimal rules may substantially reduce quadratic variation and variance, in natural environments by $50\%$ or more. Applied to stock market auctions, our results provide an explanation for the wide-spread use of reference price rules. We also apply our results to bilateral trade in over-the-counter markets, capacity planning in supply chains, and positioning in political agenda setting.

Figures (7)

• Figure 1: Comparison of Benchmark Policies. The graph traces the actions selected by the Midpoint Rule (blue), the Status Quo Reference Rule (red), and the Anchoring Rule rule (green). The gray bars indicate the feasible interval in a given period.
• Figure 2: Evolution of the Feasible Interval. The figure illustrates the formation of the constraint at time $t$. First, the agent selects an action $P_{t-1}$ (blue dot) from the vertical interval $I_{t-1}$. This choice determines the location of the new stochastic anchor $C_t(P_{t-1})$ (red dot) via a stochastic transformation. Finally, the new feasible interval $I_t$ (green bar) is realized by adding a stochastic base interval $J_t$ to the anchor.
• Figure 3: The Global Optimal Policy in the Uniform Environment for Quadratic Variation. The solid blue line ($r^\ast(s)$) depicts an approximation of the optimal target function, solved via an Average Cost Bellman Equation. It interpolates between the Status Quo Reference Rule ($45^\circ$, dashed) and the Anchoring Rule (horizontal), pulling the action toward $s=0.5$ with a slope of $\approx 0.88$. However, the efficiency gain is marginal compared to the Status Quo Reference Rule, which captures over 99% of the theoretical limit.
• Figure 4: Dynamics of the Market Clearing Interval in Repeated Double Auctions. An example of repeated double auctions with seven buy and seven sell orders. The intersection of demand (blue) and supply (red) step functions defines a closed interval of market-clearing prices $I$. The market clearing-price chosen in the first period influences the distribution of demand and supply in the second period. The graphic illustrates the evolution of the market across two periods. In period $t-1$ (left), given realized demand and supply shocks, the clearing interval is $I_{t-1}=[L_{t-1}, R_{t-1}]$. A specific trade price $P_{t-1}$ is selected (here, the upper bound). In period $t$, trader valuations have changed, shifting the expected center of demand and supply.
• Figure 5: Optimal Bidding Strategy under Uniform Distributions ($R=1/2$). This example assumes costs and valuations are uniformly distributed on $[0,1]$ with a fixed reference price $R = 1/2$. The buyer compares the optimal monopsony utility $v^2/4$ (achieved by shading $b = v/2$) against the truthful price-taking utility $U_{stat}(v) = \frac{1}{2}(v^2 - 1/4)$. Equating these payoffs identifies a unique switching threshold at $\hat{v} = \sqrt{2}/2 \approx 0.707$. The figure plots the resulting optimal bid $b^*(v)$ against valuation $v$. Buyers below $\hat{v}$ aggressively shade their bids (reaching a maximum bid of $\approx 0.35$), while those just above $\hat{v}$ jump immediately to truthful bidding ($b=v$) to secure the reference price. This strategic discontinuity creates a distinct "hole" in the bid distribution between $0.35$ and $0.707$.

Theorems & Definitions (20)

• Definition 1: Admissible Policy
• Definition 2: Deterministic Stationary Markov Policy
• Definition 3: Translation Invariant Costs
• Theorem 1: Optimality of Reference Rules
• Theorem 2: Non-Expansiveness of Optimal Reference Rules
• Theorem 3: Optimality of the Status Quo Reference Rule
• Theorem 4: Optimality of Constant Anchoring
• Theorem 5: Characterization of the Optimal Anchoring Rule
• Proposition 1: Asymptotic Incentive Compatibility, Jantschgi25
• Proposition 2: Best Response Structure