Optimal Execution Cost Function in Market Making

Understanding optimal execution cost functions

The optimal execution cost function quantifies the total cost of executing orders while market making, incorporating various components:

1. Direct transaction costs (spreads, fees)
2. Price impact costs
3. Inventory holding costs
4. Opportunity costs

Mathematically, a basic form of the execution cost function can be expressed as:

$C(v,t) = \alpha v + \gamma \sigma^2 v^2 t + \eta \frac{v^2}{t}$

Where:

• $v$ is the trading volume
• $t$ is the execution time
• $\alpha$ represents direct costs
• $\gamma$ captures price impact
• $\sigma$ is price volatility
• $\eta$ represents urgency costs

Components of market making costs

Spread costs and fee structure

Market makers incur direct costs through the bid-ask spread and various exchange fees. These costs form the baseline of the execution cost function:

$C_{spread} = S \cdot v$

Where $S$ is the effective spread and $v$ is the volume traded.

Price impact and market depth

Market impact grows non-linearly with order size and affects the execution cost function through:

$C_{impact} = \gamma \sigma \sqrt{\frac{v}{V}}$

Where:

• $V$ is the average daily volume
• $\gamma$ is the market impact coefficient

Optimization techniques

Dynamic programming approach

Market makers optimize their execution strategy using dynamic programming, solving the Hamilton-Jacobi-Bellman equation:

$\frac{\partial V}{\partial t} + \min_v \left\{ C(v,t) + \frac{\partial V}{\partial x}v + \frac{1}{2}\frac{\partial^2 V}{\partial x^2}\sigma^2 \right\} = 0$

This helps determine optimal trading rates while considering:

• Current inventory position
• Time horizon
• Market conditions
• Risk parameters

Risk-adjusted optimization

The execution cost function incorporates risk adjustments through:

$C_{adjusted}(v,t) = C(v,t) + \lambda \cdot \text{Var}(C)$

Where $\lambda$ represents the risk aversion parameter.

Implementation considerations

Market microstructure effects

Market microstructure influences the execution cost function through:

• Order book depth
• Quote fade probability
• Adverse selection risk

Real-time adaptation

Modern market making systems continuously update their execution cost functions based on:

• Market conditions
• Inventory positions
• Risk limits
• Competitor behavior

This requires sophisticated adaptive trading algorithms that can respond to changing market conditions.

Applications in practice

Quote optimization

Market makers use the execution cost function to optimize their quotes by:

1. Determining optimal bid-ask spreads
2. Adjusting quote sizes
3. Managing skew based on inventory

Risk management integration

The execution cost function helps in:

• Setting position limits
• Calculating risk charges
• Determining capital requirements
• Evaluating trading performance

The model integrates with broader algorithmic risk controls to ensure safe and efficient market making operations.

Model limitations and challenges

Key challenges in implementing optimal execution cost functions include:

1. Parameter estimation uncertainty
2. Market regime changes
3. Competition effects
4. Regulatory constraints

Market makers must regularly validate and calibrate their models to maintain effectiveness while complying with regulatory requirements.

Future developments

The evolution of optimal execution cost functions continues with:

1. Machine learning integration for parameter estimation
2. Real-time adaptation mechanisms
3. Multi-asset class optimization
4. Improved risk modeling

These developments help market makers maintain competitiveness while managing risks effectively in modern electronic markets.