Optimal Tick Size Selection

• Optimal Tick Size Selection defines the smallest price increment in limit order markets, influencing liquidity provision, bid-ask spreads, and trading costs.
• Analytic and simulation-based models calibrate tick sizes by matching spread dynamics, order flow characteristics, and execution costs across large-tick and small-tick regimes.
• Empirical studies and regulatory experiments uncover trade-offs between improved liquidity and risks such as volatility, guiding tailored tick size optimization strategies.

The optimal tick size is a central design parameter in limit order markets, delineating the finest price granularity at which participants can quote or transact. A suitably chosen tick size promotes liquidity by encouraging limit order provision and maintaining reasonable bid-ask spreads, but excessive coarseness or fineness can result in widened effective spreads, liquidity fragmentation, increased volatility, or adverse selection. This entry synthesizes recent advances in optimal tick-size selection, emphasizing microstructure models, empirical criteria, and exchange-level optimization, with focus on large-tick and small-tick regimes, side-specific tick sizes, and regulatory implications as established in multiple empirical and theoretical studies (Baldacci et al., 2020, Jain et al., 2024, Huang et al., 2015, Dayri et al., 2012, Kashyap, 2016).

1. Microstructure Foundations: Tick Size and Liquidity Metrics

The tick size, denoted $\delta$ or $\alpha$, determines the discrete set of allowable price levels. Microstructure models categorize stocks by the interaction between the tick and endogenous microstructure metrics:

• Quoted Spread ($\Delta_q$): $\Delta_{q}=\text{Ask}_{t}-\text{Bid}_{t}$
• Effective Spread ($\Delta_e$): $$\Delta_{e}=2(P_{\mathrm{trade}} - \frac{1}{2}(\text{Bid}_{t} + \text{Ask}_{t}))$$
• Implicit Spread ($2\eta\alpha$): “Uncertainty zone” width for large-tick assets, where $\eta$ is the microstructure parameter estimated by $\widehat{\eta} = N^{(c)}/(2N^{(a)})$—i.e., the ratio of continuation to alternation one-tick moves (Dayri et al., 2012).
• Relative Tick Size ($\epsilon$): $$\epsilon = \delta/\langle P_{\mathrm{mid}}\rangle \times 10^4$$ (bps) (Jain et al., 2024)
• Volume and Trade Count ($M$): Number of daily trades and total volume, entering the volatility-per-trade $\sigma/\sqrt{M}$ relation.

In large-tick assets, the spread is often fixed at a single tick, and microstructure equilibrium is established between market-makers' profit and takers’ cost, where tick size acts as a lower bound for the spread and effective spread dynamics are captured by the implicit spread metric (Dayri et al., 2012).

2. Theoretical Models for Optimal Tick Selection

Optimal tick-size selection is framed by analytic and simulation-based methodologies:

2.1 Model with Uncertainty Zones

• Efficient price process: $dY_t = a_t\,dt + \sigma_{t^-}\,dW_t$ with $a_t \equiv 0$ in calibration.
• Discrete price grids: Separate bid $(\alpha^b)$ and ask $(\alpha^a)$ ticks allow for asymmetric granularity.
• Tick-size-dependent order arrival: $\lambda^i = \lambda/(1 + (\kappa\,\alpha^i)^2)$, where $\kappa$ modulates sensitivity (Baldacci et al., 2020).
• Exchange revenue maximization: Maximizing $$v(\alpha^a, \alpha^b) = E^{\ell^*}[c(N_T^a + N_T^b)]$$ over bid/ask tick sizes, with optimal pairs $(\alpha^{a*}, \alpha^{b*})$ determined numerically due to non-smooth control boundaries.

2.2 Implicit Spread and Volatility Per Trade

• Large-tick equilibrium: $\eta\alpha \simeq \sigma/\sqrt{M}$, reflecting the efficient price’s volatility per trade (Dayri et al., 2012, Huang et al., 2015).
• Market order cost: $C_{\mathrm{mkt}}(\alpha) = \alpha/2 - \eta \alpha$
• Limit order profit: $G_{\mathrm{lim}}(\alpha) = \alpha/2 - \eta\alpha$
• Forecasting parameter shift: After a tick change $\alpha_0 \to \alpha$, $$\eta(\alpha) \simeq (\eta_0+\delta)(\alpha_0/\alpha)^{1/2} - \delta$$, $\delta \approx 0.1$ empirically (Huang et al., 2015).
• Criterion for optimal tick: Choose $\alpha^*$ such that $\eta(\alpha^*) = 1/2$, enforcing zero expected market-order cost and balanced liquidity provision (Dayri et al., 2012).

2.3 Simulation-Based Approach for Arbitrary Tick Regimes

• Stylized fact metrics: Average spread in ticks ($\overline{s}$), coefficient of variation of spread ($CV_s$), average mid-price move $\langle |r_{\mathrm{mid}}|\rangle$, and book shape features (mode $x^*$) are synthesized to diagnose current regime (“large-tick,” “medium-tick,” or “small-tick”) and guide target selection (Jain et al., 2024).
• Wide-Queue Hawkes Model: Multi-type Hawkes processes drive LOB events, with parameters $(\alpha, \beta, \hat{\eta})$ controlling in-spread activity and sparsity, supporting transition and calibration across tick regimes using empirical data.

3. Practical Procedures for Tick Size Calibration

3.1 Closed-Form Procedure for Large-Tick Assets

For assets with a one-tick spread:

1. Estimate pre-change $\eta_0$ and $\alpha_0$ from high-frequency data; compute $(N^{(c)}, N^{(a)})$, $\sigma$, $M$ (Dayri et al., 2012, Huang et al., 2015).
2. Compute optimal tick:
   • $$\alpha^* = \left(\frac{\eta_0 + 0.1}{0.6}\right)^2 \alpha_0$$ (Huang et al., 2015), or
   • $\delta^* \simeq \delta_0 (2\eta_0)^{1/(1-\beta/2)}$ with latent liquidity exponent $\beta$ (Dayri et al., 2012).
3. Forecast post-change microstructure: Predict $\eta(\alpha^*)$, confirm $\eta(\alpha^*) \approx 1/2$ ensures optimality.

3.2 Simulation-Based Multi-Regime Approach

For both large- and small-tick stocks (Jain et al., 2024):

1. Empirical diagnosis: Compute all stylized-fact metrics for current $\delta_0$.
2. Set target regime: E.g., $\overline{s}_{\text{target}} = 2\!\sim\!3$ ticks, $CV_s \approx 0.5$–1, $x^*\approx2$–5 ticks.
3. Grid search: Use the Hawkes-LOB model to simulate order-flow and book shape as $\delta$ varies; adjust $(\alpha, \beta, \hat{\eta}, \delta)$ to achieve target metrics.
4. Validation: Cross-check execution costs, price move tails, depth, and event endogeneity against empirical benchmarks.

3.3 Empirical Trade-Off Guidelines

Empirical studies of real-world tick-size changes, e.g., the Tokyo Stock Exchange pilot (Kashyap, 2016), establish actionable guidelines:

• Require quoted spread reduction of at least 20% post-change to justify reduction.
• Impose a maximum acceptable drop in average trade size (e.g., $\leq 15\%$) to preserve large-order liquidity.
• Use regression-based rules to ensure changes balance benefits to small traders (spread improvements) against potential costs to large traders (impact, depth decreases).

Approach	Regime focus	Required inputs
Uncertainty zone model	Large-tick assets	$\eta_0$, $\alpha_0$, volatility/trades
Hawkes simulation	All regimes	LOB event time series, calibrable model parameters
Empirical trade-off	Ex-post/pre-change	Order-level data, time series of spreads/depth

4. Side-Specific Tick Sizes and Asymmetric Optimization

Permitting distinct tick sizes on bid and ask sides, as analyzed in the uncertainty-zone/HJB framework (Baldacci et al., 2020), enhances exchange control over liquidity and market-maker incentives:

• Bid/ask-specific grids: Price levels defined by $\{k\alpha^b : k\}$ for bids, $\{k\alpha^a : k\}$ for asks.
• Optimization problem: Maximize expected fee flow $v(\alpha^a, \alpha^b)$ across $(\alpha^a, \alpha^b)$ space; numerical grid search preferred due to non-smooth optimal controls.
• Inventory penalty asymmetry: High penalty for short positions ($\phi_-\gg0$) justifies $\alpha^b > \alpha^a$, aligning maker incentives with exchange revenue.
• Arbitrage window dynamics: With $\alpha^a\ne\alpha^b$, temporary windows where both buy and sell are profitable for the market maker increase alternations, reduce inventory drift, and facilitate spread compression.

Practical implementation requires real-time monitoring to ensure continued “large-tick” status ($\eta^i\leq1/2$) on both sides, and may encounter challenges from non-Poisson order flow, strategic adaptation of liquidity takers, and inter-venue effects (Baldacci et al., 2020).

5. Empirical Findings and Exchange Policy Implications

Analysis of major tick-size experiments substantiates both intended and unintended microstructural outcomes (Kashyap, 2016):

• Key empirical results:
  • Tighter spreads and increased trade counts observed post-reduction.
  • Displayed depth and average trade size decrease, raising costs/uncertainty for large orders.
  • Volatility of execution costs and message traffic increase.
• Optimal tick policy recommendations:
  • Phase-in via pilots, calibrate by monitoring intraday depth and execution-size distribution.
  • Employ tiered tick schedules: finer ticks for liquid/high-priced issues; coarser for illiquid stocks.
  • Enforce revertive criteria (e.g., threshold breaches in depth or cost-volatility).
  • Weight welfare of small- versus large-order traders in optimization rules.

A plausible implication is that exchanges seeking to optimize tick size must continuously trade off small-order execution gains against potential harm to institutional liquidity and operational stability.

6. Limitations and Future Research Directions

Closed-form optimal tick rules are applicable primarily to large-tick assets with persistent one-tick spreads. Models based on Poisson arrivals and linear latent liquidity may not capture feedback loops, strategic order-splitting, or volatility-tail dependencies. Simulation-based and hybrid approaches are required for assets transitioning between small- and large-tick regimes, or where tick-induced regime boundaries are approached (Jain et al., 2024).

Challenges remain in universalizing “optimal” tick prescriptions across heterogeneous assets, in modeling cross-venue interactions, and in operationalizing dynamic tick adaptation frameworks responsive to evolving volume, volatility, and liquidity conditions.

7. Summary Table of Optimal Tick-Size Approaches

Methodology	Primary Scope	Key Formula/Rule	Reference
Uncertainty zone model	Large-tick assets	$\alpha^* = [(\eta_0+0.1)/0.6]^2 \alpha_0$	(Huang et al., 2015, Dayri et al., 2012)
Hawkes simulation	All regimes	Grid search for $\delta$ with target metrics	(Jain et al., 2024)
Empirical regression	Ex-post impact	Spread and trade-size thresholds ($>20\%$, $<15\%$)	(Kashyap, 2016)
Side-specific optimization	Order-book asymmetry	Maximize $v(\alpha^a, \alpha^b)$ numerically	(Baldacci et al., 2020)

The selection of an optimal tick size is thus both a theoretically tractable and empirically contingent problem, requiring the interplay of high-frequency data calibration, analytic microstructure models, and simulation-based robustness checks to address the diversity of market microstructure landscapes.