Latency Arbitrage Formula for HFT Strategies

Core components of latency arbitrage modeling

The fundamental latency arbitrage formula calculates the expected profit ($E[P]$) from exploiting temporal price discrepancies:

$E[P] = P(S) \times V \times \Delta P - C$

Where:

• $P(S)$ = Probability of successful execution
• $V$ = Trading volume
• $\Delta P$ = Price differential
• $C$ = Trading costs

The probability of successful execution $P(S)$ is further defined by the latency differential:

$P(S) = F(\Delta t_{own} - \Delta t_{other})$

Where:

• $\Delta t_{own}$ = Own latency
• $\Delta t_{other}$ = Competitor latency
• $F()$ = Probability distribution function

Speed advantage quantification

The effective speed advantage ($\Delta L$) in a latency arbitrage strategy can be expressed as:

$\Delta L = t_{slow} - t_{fast}$

Where:

• $t_{slow}$ = Slower participant's round-trip time
• $t_{fast}$ = Faster participant's round-trip time

This advantage must exceed the minimum threshold ($\theta$) required for profitable arbitrage:

$\Delta L > \theta = \frac{C}{V \times \Delta P}$

Market impact considerations

The realized price differential must account for market impact cost:

$\Delta P_{effective} = \Delta P_{observed} - \lambda \times V$

Where:

• $\lambda$ = Market impact coefficient
• $V$ = Trading volume

This adjustment leads to a modified profit expectation:

$E[P]_{adjusted} = P(S) \times V \times (\Delta P_{observed} - \lambda V) - C$

Temporal opportunity windows

The duration of arbitrage opportunities ($T$) affects strategy profitability through:

$E[P]_{total} = \sum_{i=1}^{n} E[P]_i \times I(T_i > \Delta t_{execution})$

Where:

• $I()$ = Indicator function
• $T_i$ = Duration of opportunity $i$
• $\Delta t_{execution}$ = Required execution time

Implementation considerations

Network topology optimization

The physical distance component of latency can be modeled as:

$t_{physical} = \frac{d}{c} + t_{processing}$

Where:

• $d$ = Physical distance
• $c$ = Speed of light in fiber
• $t_{processing}$ = Processing overhead

Risk management constraints

Risk-adjusted profit expectations must consider:

$E[P]_{risk} = E[P] - \sigma_P \times Z$

Where:

• $\sigma_P$ = Profit volatility
• $Z$ = Risk tolerance factor

Applications in modern markets

Cross-venue arbitrage

For cross-market surveillance and arbitrage between venues:

$\Delta P_{cross} = P_{venue1} - P_{venue2} - TC_{cross}$

Where:

• $TC_{cross}$ = Cross-venue transaction costs

Regulatory considerations

Modern high-frequency trading risk management must incorporate regulatory constraints into the arbitrage formula:

$E[P]_{compliant} = E[P] \times (1 - P_{violation})$

Where:

• $P_{violation}$ = Probability of regulatory violation

Practical implementation

The complete implementation requires:

1. Real-time latency measurement
2. Dynamic threshold adjustment
3. Risk limit incorporation
4. Regulatory compliance verification

Trading systems must balance:

• Execution speed
• Risk management
• Compliance requirements
• Infrastructure costs

Future developments

Emerging trends affecting latency arbitrage formulas include:

• Speed bumps and latency floors
• Artificial intelligence optimization
• Quantum computing applications
• Regulatory evolution

The continuous evolution of market structure requires regular formula refinement and adaptation to maintain effectiveness.