Transaction Costs & Market Impact: Models & Analysis › Estimating Transaction Cost Parameters

Modeling transaction costs requires estimating the relevant parameters from market data. These parameters vary across assets, time periods, and market conditions. Accurate estimation is essential for realistic backtesting and strategy evaluation. The challenge lies in the fact that we can only observe transaction costs for trades we execute, and these observations are inherently noisy due to the many factors influencing each individual trade. Nevertheless, systematic approaches to estimation can yield useful parameter values that significantly improve our cost models.

The bid-ask spread can be estimated directly from quote data or inferred from trade data when quotes are unavailable. The choice of method depends on the data available and the precision required.

From quote data: The quoted spread at time $t$ is simply:

$$s_t = \text{ask}_t - \text{bid}_t$$

where:

• $s_t$: quoted spread at time $t$
• $\text{ask}_t$: lowest sell price
• $\text{bid}_t$: highest buy price

This direct measurement is straightforward when high-frequency quote data is available. However, the quoted spread may overstate actual trading costs if orders execute at prices better than the posted quotes, a phenomenon called price improvement.

The effective spread, which accounts for orders that execute at prices better than the quoted spread, is calculated from trade data:

$$s_{\text{eff}} = 2 \cdot D_t \cdot (P_t - M_t)$$

where:

• $s_{\text{eff}}$: effective spread
• $D_t$: direction indicator (+1 for buys, -1 for sells)
• $P_t$: trade price
• $M_t$: quote midpoint at trade time

The factor of 2 appears because we measure the distance from the midpoint to the execution price, which represents half the round-trip cost. Multiplying by the direction indicator $D_t$ ensures that the spread is positive regardless of whether the trade was a buy or a sell. A buy trade should execute above the midpoint (positive deviation), while a sell trade should execute below (negative deviation), so the direction indicator aligns these cases.

Roll's spread estimator: When quote data is unavailable, Roll (1984) showed that the spread can be estimated from the autocovariance of price changes:

$$\hat{s} = 2\sqrt{-\text{Cov}(\Delta P_t, \Delta P_{t-1})}$$

where:

• $\hat{s}$: estimated effective spread
• $\text{Cov}$: covariance operator
• $\Delta P_t$: price change ($P_t - P_{t-1}$)

This estimator relies on the fact that bid-ask bounce creates negative serial correlation in observed price changes. If the true value is stable, a buy trade executed at the ask followed by a sell trade at the bid results in a negative price change, while a sell followed by a buy results in a positive change. This oscillation creates a negative covariance proportional to the square of the spread.

When prices bounce between bid and ask without any change in fundamental value, consecutive returns tend to have opposite signs: an uptick to the ask is often followed by a downtick to the bid, and vice versa. This bid-ask bounce creates a predictable negative correlation. Roll showed that under idealized conditions, the covariance equals $-s^2/4$, where $s$ is the spread, which leads directly to the estimator above.

The Roll estimator closely approximates the true spread using only close-to-close price data. This demonstrates how transaction costs can be inferred from price dynamics even when high-frequency quote data is unavailable, though the estimate becomes noisy in the absence of a strong bid-ask bounce signal.

Estimating market impact parameters requires data on trade sizes, volumes, and subsequent price movements. The typical approach is to regress observed price changes on participation rates. This regression framework allows us to estimate both the impact coefficient $\eta$ and the impact exponent $\gamma$ simultaneously.

For the square-root model, taking logs turns the relationship into a linear regression:

$$\log(\text{Impact}) = \log(\sigma \eta) + \gamma \log\left(\frac{Q}{V}\right)$$

where:

• $\text{Impact}$: market impact cost
• $\sigma$: asset volatility
• $\eta$: impact coefficient
• $\gamma$: impact exponent
• $Q$: trade quantity
• $V$: daily trading volume

This allows estimation of both $\eta$ and $\gamma$ using standard regression techniques. The slope of the regression in log-log space directly estimates $\gamma$, while the intercept allows us to back out $\eta$ once we know the volatility. This linearization is useful because it turns a potentially complex nonlinear estimation problem into a simple ordinary least squares regression.