1. Introduction

With the rise of algorithmic trading, particularly high-frequency trading (HFT), financial markets have undergone significant transformation. HFT strategies rely on executing a large volume of trades within milliseconds, capitalizing on market inefficiencies, arbitrage opportunities, and latency advantages. Proponents of HFT argue that it enhances market efficiency by tightening bid-ask spreads and increasing overall liquidity. However, critics contend that HFT may contribute to excessive market volatility, flash crashes, and liquidity evaporation during periods of stress, thereby raising concerns about long-term market stability. Market liquidity, defined as the ease with which assets can be bought or sold without significantly affecting their prices, is a fundamental determinant of financial stability. Traditional market-making models suggest that liquidity providers stabilize prices by absorbing order flow imbalances. However, in an HFT-dominated environment, market participants often act as both liquidity providers and takers, often leading to rapid and sometimes unpredictable shifts in liquidity conditions. This dual role of HFT firms introduces complexity into market microstructure and challenges conventional liquidity models.

This paper develops a mathematical framework to investigate the impact of HFT on market liquidity. Using stochastic processes, Markov chains, and game theory, the author constructs theoretical models to analyze price impact, order book dynamics, and liquidity resilience. By exploring market conditions under different trading intensities, this study offers insights into the role of HFT in shaping modern financial markets.

2. Mathematical modeling of HFT and market liquidity

2.1. Liquidity supply model

Market liquidity, which refers to the ability to execute large transactions with minimal price impact, is crucially influenced by HFT activity. To quantify this, we define market liquidity \( {L_{(t)}} \) as a function of limit order book (LOB) dynamics and HFT participation. A simplified model for liquidity supply can be expressed as:

\( {L_{(t)}}={L_{0}}+{β_{1}}HF{T_{t}}-{β_{2}}{δ_{t}} \) (1)

Where \( {L_{0}} \) represents baseline market liquidity in the absence of HFT, \( HF{T_{t}} \) measures high-frequency trading activity at time t, \( {δ_{t}} \) denotes market volatility, \( {β_{1}} \) > 0 captures the liquidity-enhancing role of HFT under normal conditions, and \( {β_{2}} \) >0 reflects liquidity withdrawal due to volatility spikes.

Empirical studies suggest that HFT enhances liquidity during stable market conditions but may contribute to its depletion when volatility rises. This relationship is formalized by the elasticity measure:

\( {ε_{L}}=\frac{δL}{δHFT}×\frac{HFT}{L} \) (2)

If \( {ε_{L}} \) > 1, HFT is a dominant liquidity provider. However, when volatility increases beyond a critical threshold \( {δ_{C}} \) , the sign of \( {ε_{L}} \) may reverse, indicating liquidity withdrawal. Additionally, the limit order book depth function can be modeled as:

\( D(t)={D_{0}}+{γ_{1}}HF{T_{t}}-{γ_{2}}{δ_{t}} \) (3)

Where \( {D_{0}} \) represents passive liquidity from non-HFT traders, \( {γ_{1}} \) and \( {γ_{2}} \) determine the sensitivity of order book depth to HFT and volatility.

These formulations illustrate the dual effect of HFT—while it generally increases liquidity in stable markets, it may exacerbate liquidity dry-ups in turbulent periods.

2.2. Queue dynamics in high-frequency trading

Order execution in electronic markets follows a queuing mechanism, where orders compete for priority in the limit order book. HFT firms employ advanced strategies to optimize execution speed, thereby securing advantageous positions in the queue. We model the limit order execution process as an \( M∕M∕1 \) queue, where: Arrivals follow a Poisson process with rate \( {λ_{H}}FT \) , execution times follow an exponential distribution with mean \( 1∕μ \) . The expected queue waiting time for an HFT order is given by:

\( {w_{q}}=\frac{1}{μ-{λ_{H}}FT} \) .

If \( {λ_{H}}FT→μ \) , the queue length diverges, leading to order execution delays and increased market impact. This suggests that excessive HFT participation can increase congestion, negatively affecting execution quality for slower traders. Furthermore, incorporating priority dynamics, we express the probability of an HFT order executing before a non-HFT order as:

\( {P_{HFT}}=\frac{{λ_{H}}FT}{{λ_{H}}FT+{λ_{non-HFT}}} \)

This ratio underscores how HFT dominance can marginalize traditional investors, potentially leading to order anticipation effects and unfair trading advantages.

2.3. Volatility and market impact

HFT firms influence price dynamics by continuously submitting and canceling orders, affecting market impact and short-term volatility. A widely used metric for market impact is:

\( ΔP=η{Q^{α}} \)

where: \( ΔP \) represents the price change induced by an order of size Q, \( η \) is a proportionality factor capturing market depth, and \( α \) typically ranges between 0.5 and 1, indicating sublinear price impact for small trades. For high-frequency strategies, the aggregate market impact function can be approximated as:

\( R[ΔP|HFT]={θ_{1}}HF{T_{t}}-{θ_{2}}D(t) \)

where \( {θ_{1}} \) captures HFT’s contribution to price adjustments, \( {θ_{2}} \) reflects the mitigating effect of deep order books.

This suggests that when HFT activity is high but liquidity depth remains stable, market impact is minimized. However, in fragmented markets with limited depth, HFT can increase short-term volatility.

2.4. Empirical implications of the model

By combining liquidity supply, queue dynamics, and market impact models, we derive key empirical predictions. HFT enhances order book depth during normal conditions but may withdraw liquidity during market stress, leading to fluctuations in liquidity availability. Additionally, the prevalence of HFT strategies reduces execution times, improving market efficiency; however, it also increases order fragmentation, making trade execution more complex. While HFT dampens price fluctuations under normal conditions, it can amplify market stress during high-volatility periods, exacerbating sudden price swings and liquidity shortfalls.

3. Game-theoretic modeling of HFT competition

3.1. Strategic interactions between HFT firms

HFT firms engage in complex strategic interactions, leveraging advanced algorithms to optimize order execution. These interactions can be analyzed through game-theoretic models, particularly non-cooperative games, where each firm aims to maximize its profit while anticipating the actions of competitors. We model HFT competition as a Nash equilibrium problem in a continuous-time setting. Consider \( N \) HFT firms, each selecting a trading intensity \( {Q_{i}} \) to maximize its expected profit function:

\( {Π_{i}}({Q_{1}}{,Q_{2}},,{Q_{N}})=R[\int _{0}^{T}(π{(Q_{i}}){x_{i}} \) -c( \( {x_{i}})-\sum _{\dot{J}≠i}ρ({Q_{i}},{Q_{j}} \) ))]

where: \( ρ({Q_{i}}{Q_{j}}) \) models market-making competition, penalizing aggressive trading, c( \( {x_{i}}) \) represents inventory holding costs, \( π({Q_{i}}) \) denotes market impact, reflecting price changes induced by order flow.

Differentiating \( {Π_{i}} \) with respect to \( {Q_{i}} \) and solving the first-order condition, we obtain the Nash equilibrium trading intensities:

\( Q_{i}^{*}=argmax(λQ_{i}^{γ}{x_{i}}-c({x_{i}})-\sum _{j≠i}ρ({Q_{i}}{Q_{j}}) \)

From this equilibrium, we derive key insights: HFT firms optimize execution by submitting smaller but more frequent orders to minimize price impact, while increasing competition among HFT firms reduces profitability, driving the need for greater efficiency; at the same time, aggressive trading intensifies liquidity fragmentation, affecting order book stability. Additionally, strategic behaviors such as “sniping” (detecting and exploiting stale quotes) and “latency arbitrage” (capitalizing on microsecond price discrepancies) emerge in this competitive setting.

3.2. Market stability under HFT competition

Market stability in the presence of competing HFT firms can be analyzed using a mean-field game framework. The fundamental price process \( {P_{t}} \) evolves as:

\( d{P_{t}}=(μ-α{Q_{total}})dt+δ({Q_{total}})ⅆ{w_{t}} \) .

where \( {Q_{total}} \) = \( Σ_{ⅈ=1}^{N}{Q_{i}} \) represents aggregate HFT activity, and volatility follows the functional form:

\( δ({Q_{total}})={δ_{0}}+kQ_{total}^{δ} \)

This relationship suggests that: If \( δ \) > 1, HFT amplifies volatility, increasing flash crash risks; If \( δ \) < 1, HFT stabilizes prices by absorbing liquidity shocks. To ensure market stability, we impose the stability criterion:

\( R[δ{(Q_{total}}{)] \lt Q_{critical}} \)

where \( {Q_{critical}} \) denotes the threshold beyond which instability emerges. This implies that regulatory interventions, such as minimum resting times for orders and order-to-trade ratios, may be necessary to curb excessive HFT-driven volatility.

4. Empirical evidence on HFT’s impact on liquidity and volatility

4.1. HFT and market liquidity

Empirical studies provide mixed evidence regarding the impact of HFT on market liquidity. Some research suggests that HFT enhances liquidity provision, while others argue that it may lead to liquidity withdrawal during periods of market stress.

Several studies highlight the positive role of HFT in improving market liquidity: Brogaard et al. analyzed NASDAQ order book data and found that HFT market makers reduce bid-ask spreads by 20–30% on average, enhancing price efficiency [1]. Hasbrouck & Saar demonstrated that HFT reduces effective spreads by improving quote depth, leading to lower trading costs for institutional investors [2]. Foucault et al. used high-frequency data from European markets and found that HFT firms provide continuous liquidity, reducing the probability of order book imbalances. These studies support the hypothesis that HFT contributes positively to liquidity by narrowing spreads and increasing order book depth.

Despite the apparent benefits, critics argue that HFT firms engage in opportunistic liquidity withdrawal, particularly during market stress: Kirilenko et al. examined the 2010 Flash Crash and found that HFT firms withdrew liquidity just before large price swings, exacerbating market instability [3]. Menkveld documented that passive HFT liquidity evaporates in moments of extreme volatility, increasing order execution risks for slower traders [4]. Cartea & Penalva showed that HFT firms strategically cancel limit orders when they anticipate adverse price movements, leading to sudden liquidity shortfalls. These findings suggest that while HFT enhances steady-state liquidity, it may destabilize order books when volatility rises. To quantify this effect, we estimate the HFT-Liquidity Elasticity (HLE) using the regression model:

\( {L_{t}}={α_{0}}+{α_{1}}HF{T_{t}}+{α_{2}}{δ_{t}}+{ε_{t}} \)

Empirical results indicate that \( {α_{1}} \) > 0 in stable markets (HFT enhances liquidity), but \( {α_{1}} \) < 0 during stress periods, confirming that HFT withdraws liquidity when volatility exceeds a critical threshold.

4.2. HFT and market volatility

The impact of HFT on price volatility is another area of intense debate. Some studies argue that HFT reduces volatility by enhancing price efficiency, while others claim it amplifies short-term price swings. Some researchers suggest that HFT dampens volatility by quickly correcting pricing inefficiencies: Zhang & Riordan used high-frequency data and found that HFT reduces short-term volatility by dampening transitory price shocks, leading to smoother price movements [5]. Boehmer et al. found that HFT enhances price efficiency, reducing price deviations from fundamental values [6]. Jovanovic & Menkveld showed that HFT improves the price discovery process, leading to more stable intraday returns [7]. These studies indicate that HFT may act as a stabilizing force, particularly in liquid markets.

Conversely, other studies suggest that HFT exacerbates market volatility by engaging in aggressive trading strategies: Benos et al. found that aggressive HFT strategies amplify price swings, particularly in illiquid stocks, leading to increased market fragmentation [8]. Easley et al. noted that order anticipation strategies employed by HFT firms can trigger momentum ignition, causing prices to spike or drop sharply [9]. Biais et al. showed that HFT can create self-reinforcing feedback loops, where rapid order placement and cancellation lead to excessive short-term price fluctuations [10]. To quantify this effect, we estimate the HFT-Volatility Elasticity (HVE) using the regression model:

\( {δ_{t}}={β_{0}}+{β_{1}}HF{T_{t}}+{β_{2}}Liquidity+{ε_{t}} \)

Empirical results suggest that \( {β_{1}} \) > 0 during high-stress periods, confirming that HFT increases short-term volatility under certain conditions.

One of the most well-documented examples of HFT-induced volatility is the May 6, 2010, Flash Crash, during which the Dow Jones Industrial Average plunged nearly 1,000 points within minutes. Investigations revealed that HFT firms initially absorbed sell orders, but quickly withdrew liquidity, exacerbating the crash [11]. The event demonstrated how HFT can both provide and remove liquidity within extremely short time frames, leading to market instability.

These findings highlight the dual impact of HFT—while it enhances liquidity under normal conditions, it can destabilize markets during high-stress periods.

4.3. Summary of empirical findings

Empirical research on HFT’s impact on market liquidity and volatility presents contrasting results:

Table 1: Empirical findings on HFT’s impact on liquidity and volatility

These findings shown in Table 1 suggest that HFT’s effect is context-dependent—while it improves liquidity and price efficiency under normal conditions, it may contribute to instability and market fragility during crises.

5. Regulatory implications and future research directions

5.1. Regulatory challenges and policy proposals

The rise of HFT has led to increased regulatory concerns, prompting financial authorities worldwide to implement policies aimed at reducing systemic risks and ensuring fair market conditions. Regulatory efforts primarily focus on preventing market manipulation, reducing excessive volatility, and improving transparency in high-frequency trading.

Several challenges arise in regulating HFT, including market manipulation, liquidity fragmentation, and flash crashes. HFT strategies like spoofing (placing orders without intent to execute) and quote stuffing (flooding order books with fake liquidity) distort price discovery and undermine market efficiency [12]. Besides, the presence of dark pools and alternative trading venues makes it harder for regulators to track HFT’s true market impact, leading to concerns about information asymmetry [13]. And high-speed trading can exacerbate liquidity evaporation in extreme market conditions, leading to sudden price crashes, as seen in the 2010 Flash Crash [11].

To mitigate the risks associated with HFT while preserving its benefits, several policy measures have been proposed. One approach is the imposition of Financial Transaction Taxes (FTTs), which levy a small tax on each HFT trade to discourage excessive order submissions, thereby reducing market noise and improving order book stability, though critics argue that such taxes could lower overall market liquidity [14]. Another regulatory tool is the implementation of speed bumps, such as the IEX exchange’s 350-microsecond delay, which aims to prevent latency arbitrage and reduce unfair advantages for ultra-fast traders [15]. Additionally, mandating market maker obligations can ensure that HFT firms provide liquidity continuously rather than opportunistically, thereby reducing market fragility, particularly during periods of stress [6]. Regulators have also considered imposing order-to-trade ratios, which limit the number of orders an HFT firm can submit per executed trade, thereby preventing excessive quote cancellations and market manipulation [16]. Finally, circuit breakers, which automatically halt trading during extreme price fluctuations, serve as a safeguard against cascading failures caused by algorithmic trading errors [17]. These regulatory interventions collectively seek to strike a balance between curbing abusive HFT practices and maintaining its role in enhancing market liquidity.

5.2. Future research directions

While this paper provides a mathematical modeling framework for analyzing HFT’s liquidity impact, further research is needed in the following areas. AI and Machine learning in HFT: with the rise of artificial intelligence (AI) and deep learning, AI-driven HFT strategies are becoming increasingly sophisticated. Future research should explore. The impact of reinforcement learning algorithms on high-frequency trading strategies, the ways AI-driven HFT affects market microstructure and liquidity provision, and the potential risks of self-reinforcing AI trading loops, where multiple AI-driven HFT firms react to each other’s signals, leading to unintended volatility spikes.

Most existing studies focus on HFT’s impact within single asset classes (e.g., equities). However, HFT plays a crucial role in cross-asset trading, such as stock-option arbitrage: How HFT influences liquidity transmission between stock and option markets.Whether HFT improves or destabilizes liquidity across futures and cash markets. How high-frequency trading strategies affect foreign exchange and bond market volatility. Understanding HFT’s role in multi-asset market dynamics is critical for developing comprehensive regulatory frameworks.Cryptocurrency markets, characterized by high volatility and 24/7 trading, provide a unique environment for HFT.

Research should investigate: How HFT strategies adapt to decentralized exchanges (DEXs) with varying levels of liquidity, the impact of algorithmic trading bots on crypto price discovery and arbitrage efficiency and whether crypto HFT exacerbates flash crashes and liquidity crunches in digital asset markets. With the growing institutional adoption of cryptocurrencies, understanding HFT’s role in crypto market stability is an emerging research priority.

6. Conclusion

High-frequency trading (HFT) has fundamentally transformed modern financial markets, influencing liquidity provision, price discovery, and volatility dynamics. This paper provides a comprehensive analysis of HFT’s impact through theoretical modeling and empirical evidence. By integrating liquidity supply models, queue dynamics, and market impact frameworks, we examine the strategic behavior of HFT firms and their implications for market stability. Our findings suggest that while HFT enhances order book depth and improves execution efficiency under normal market conditions, it can also contribute to liquidity fragmentation and exacerbate price swings during periods of stress. Empirical research further confirms this dual effect, demonstrating that HFT reduces bid-ask spreads and enhances price efficiency in stable markets, yet may withdraw liquidity and amplify short-term volatility when uncertainty rises. From a regulatory perspective, policymakers face the challenge of balancing the benefits of HFT with its potential risks. Market manipulation concerns, such as spoofing and quote stuffing, necessitate stricter surveillance and enforcement mechanisms, while structural reforms like financial transaction taxes, order-to-trade ratios, and circuit breakers could help mitigate excessive volatility without hindering market efficiency. The introduction of speed bumps and market-making obligations may also play a role in ensuring fairer competition and reducing systemic risk.

Future research should focus on emerging developments in algorithmic trading, particularly the increasing role of artificial intelligence (AI) and machine learning in HFT decision-making. Additionally, the expansion of HFT into multi-asset markets and decentralized finance (DeFi) raises new questions about liquidity transmission, regulatory oversight, and market stability. Given the rapid evolution of trading technologies, ongoing research and regulatory adaptation will be crucial in shaping the future landscape of high-frequency trading and its role in financial markets.