Transaction Cost Modeling: Commissions, Slippage, and Market Impact

Layer 1: The Intuition

1.1 Costs in a Wet Market

Imagine you are buying vegetables at a wet market:

The first two remain roughly constant as a proportion regardless of how much you buy; the third one becomes more expensive the more you buy.

1.2 Why Backtests Often Underestimate Costs

• Backtests often execute at the "closing price," but in reality, the price has already changed by the time you place your order;
• They ignore the spread (especially for small-cap stocks or small-cap cryptocurrencies, where spreads can be massive);
• They ignore impact: In a backtest, buying 10 million and buying 10,000 results in the same execution price;
• High turnover strategies: Rebalancing every day means you pay the cost every single day.

Layer 2: Cost Model

2.1 Components of One-Sided Cost

$$\text{cost} = \underbrace{\text{fee}}_{\text{transaction fee}} + \underbrace{\tfrac{1}{2}\,\text{spread}}_{\text{half-spread}} + \underbrace{Y \cdot \sigma \cdot \sqrt{\frac{Q}{V}}}_{\text{market impact}}$$

2.2 The Square Root Law

"Impact ∝ $\sqrt{Q/V}$" is one of the most robust empirical laws in market microstructure research, observed repeatedly across different markets such as stocks, futures, options, and cryptocurrencies.

Intuitively: If trading volume quadruples, the unit impact only doubles, but the total impact cost (unit impact × volume) increases eightfold. Therefore:

• With small capital, impact is almost negligible;
• As capital scale increases, impact costs grow faster than returns, eventually consuming all profits—this is the strategy capacity.

2.3 Methods to Reduce Costs

Layer 3: Code: How Costs Eat a Strategy

3.1 A High-Turnover Strategy

Simulating a daily-frequency strategy that changes direction almost every day. It has some real predictive power (Gross Sharpe ≈ 3.5, which is extremely rare in reality; this is used here to clearly demonstrate the effect of costs):

import numpy as np, pandas as pd
rng = np.random.default_rng(5)
n = 252 * 4
ret = pd.Series(rng.normal(0, 0.015, n))
# A daily signal with true predictive power
signal = pd.Series(0.25 * ret.shift(-1).fillna(0) / 0.015 + rng.normal(0, 1, n))
pos = np.sign(signal).shift(1).fillna(0)          # Yesterday's signal, today's position (+1 / -1)
gross = pos * ret

def cost_per_trade(trade_value, adv, sigma, fee_bps=1, half_spread_bps=2, Y=1.0):
    """One-sided cost (as a % of trade value) = commission + half-spread + square root impact"""
    impact = Y * sigma * np.sqrt(trade_value / adv)
    return fee_bps / 1e4 + half_spread_bps / 1e4 + impact

turnover = pos.diff().abs().fillna(0)               # Daily turnover (change in position)
def net_returns(capital, adv=5e8, sigma=0.015):
    c = cost_per_trade(capital * turnover.clip(lower=1e-12), adv, sigma)
    return gross - turnover * c

def sharpe(x): return x.mean() / x.std() * np.sqrt(252)
print("Average daily turnover:", round(turnover.mean(), 2))
print("Gross Sharpe:", round(sharpe(gross), 2))
for cap in [0, 1e5, 1e6, 1e7, 1e8]:
    print(f"Capital {cap:>12,.0f}: Net Sharpe {sharpe(net_returns(cap)):.2f}  Annualized Net Return {net_returns(cap).mean()*252*100:.1f}%")

Output:

Avg Daily Turnover: 0.99
Gross Sharpe: 3.49
Capital            0: Net Sharpe 3.17  Annualized Net Return 73.3%
Capital      100,000: Net Sharpe 2.84  Annualized Net Return 65.8%
Capital    1,000,000: Net Sharpe 2.14  Annualized Net Return 49.7%
Capital   10,000,000: Net Sharpe -0.06  Annualized Net Return -1.3%
Capital  100,000,000: Net Sharpe -5.75  Annualized Net Return -162.6%

3.2 Interpretation

(The annualized return of $-162\%$ is because simple returns are summed for the annualization here, and it assumes a full position reversal every day; in reality, trading would have stopped long before this.)

Key conclusion: Strategy returns are not a single number, but a function of capital scale. Discussing returns without considering capital scale is meaningless.

3.3 A Simple Way to Estimate Capacity

Set the expected return of a single trade equal to the cost:

$$\text{Expected alpha} = \text{fee} + \tfrac{1}{2}\text{spread} + Y\sigma\sqrt{Q/V} \quad\Rightarrow\quad Q_{\max} = V\left(\frac{\alpha - \text{fee} - \tfrac{1}{2}\text{spread}}{Y\sigma}\right)^2.$$

This gives the upper limit for a single trade size; multiplying this by the number of assets and the turnover frequency provides a rough estimate of the total strategy capacity.

3.4 Practical Key points

Summary