1. Introduction

Algorithmic trading has become an integral part of modern financial markets, utilizing computer programs to automatically execute trading decisions, thereby improving market efficiency and liquidity [1-2]. Among the various algorithmic trading strategies, Volume Weighted Average Price (VWAP) strategies have gained significant attention due to their advantages in reducing transaction costs and evaluating execution quality [3-4].

VWAP is defined as the ratio of the value traded to the volume traded over a particular time horizon. It serves as both a trading benchmark and an execution strategy. As a benchmark, VWAP is widely used to assess the quality of trade execution. While as a strategy, it aims to minimize the variance between the execution price and the VWAP of the trading period [5-6].

The development and optimization of VWAP strategies have been the focus of extensive research in recent years. With the advancement of technology and the increasing complexity of financial markets, traditional VWAP strategies face new challenges and opportunities for improvement [7-8].

2. Literature Review

Algorithmic trading has rapidly evolved over the past few decades and has become an integral part of financial markets. This trading method, which uses computer programs to automatically execute trading decisions, has played a significant role in improving market efficiency and liquidity [9-10]. As technology has advanced, algorithmic trading strategies have become increasingly sophisticated, with VWAP strategies gaining particular attention due to their advantages in reducing transaction costs and evaluating execution quality [11-12].

The development of VWAP as a trading benchmark and execution strategy can be traced back to the late 1980s. Early research primarily focused on VWAP's application in assessing transaction costs and execution quality [13-14]. Over time, the research focus gradually shifted towards optimizing VWAP strategies to improve trading execution. Some scholars proposed dynamic VWAP methods, aiming to enhance traditional VWAP strategies by predicting market volume patterns [15-16].

In recent years, the application of machine learning and artificial intelligence techniques in VWAP strategy optimization has become a research hotspot. Researchers have explored various advanced prediction models and adaptive algorithms to improve the execution efficiency and adaptability of VWAP strategies [17-18]. However, despite significant progress, VWAP strategies still face numerous challenges in practical applications, such as changes in market microstructure, the impact of high-frequency trading, and performance under abnormal market conditions [19-20].

3. Understanding VWAP and Its Implementation

3.1. VWAP Algorithm

VWAP was introduced in 1988 as a method to evaluate the impact of substantial trades on the NYSE, providing a new perspective on price movements in relation to trading volume [21]. This approach offers a more comprehensive view of average pricing over time by taking into account the size of each trade. VWAP calculates the average trading price by weighting each transaction price according to its trading volume, thus providing a market-average price over a specified period. This measure helps assess and optimize the execution of large orders, aiming to minimize their impact on market prices and set a benchmark for trading performance.

VWAP is computed by dividing the cumulative trading value by the total trading volume over the specified period, as given by the following formula:

\( VWAP=\frac{\sum _{i=1}^{n}{v_{i}}{p_{i}}}{\sum _{i=1}^{n}{v_{i}}}=\sum _{i=1}^{n}{w_{i}}{p_{i}} \) (1)

In a given trading period, suppose there are n transactions, with each transaction having a price \( {p_{i}}(i=1,2,…,n) \) and a volume \( {v_{i}}(i=1,2,…,n) \) . Here, \( ∑_{i=1}^{n} {v_{i}}{p_{i}} \) represents the cumulative total transaction amount for the period,while \( {w_{i}}=\frac{{v_{i}}}{∑_{j=1}^{n} {v_{j}}} \) represents the proportion of the \( i \) -th transaction volume relative to the total transaction volume. The formula reflects the weighted average of all transaction prices according to their volumes.

VWAP has become a fundamental tool in financial markets, serving dual purposes as a performance metric for institutional investors and a key component in various algorithmic trading strategies [22]. It enables traders to assess their execution efficiency by comparing their achieved prices against the market VWAP [23].

3.2. VWAP trading strategy

In 2002, a trading strategy based on VWAP was developed, aiming to align executed trades as closely as possible with the market VWAP [24]. This approach involves forecasting volume distribution patterns and strategically dividing large orders into smaller, timed executions throughout the trading day [25].

The goal of the VWAP algorithm is to ensure that the executed VWAP closely aligns with the actual market VWAP. The core of this strategy lies in predicting the market's volume distribution, breaking large orders into smaller sub-orders, and executing these sub-orders throughout the trading day according to the predicted volume distribution ratio.

Let \( {{w_{i}}^{s}} \) represent the predicted trading volume distribution ratio for the \( i \) -th interval, and \( {{w_{i}}^{m}} \) represent the market's actual trading volume distribution ratio for the i-th interval. Based on the execution principle of the VWAP trading strategy, it is necessary to first calculate the VWAP price obtained from the trading volume prediction model and the actual market VWAP price over a specified period. The difference between the two can be calculated using the following formula:

\( min |({{w_{1}}^{s}}{p_{1}}+{{w_{2}}^{s}}{p_{2}}+⋯+{{w_{n}}^{s}}{p_{n}})-({{w_{1}}^{m}}{p_{1}}+{{w_{2}}^{m}}{p_{2}}+⋯+{{w_{n}}^{m}}{p_{n}})| \) (2)

In practice, the VWAP calculation involves dividing the total daily trading volume (V) into several time intervals and averaging the transaction prices and volumes for each interval. Suppose the total trading volume for security is V, and the volume ratio for each time interval is \( {w_{i}} \) . The VWAP calculation must consider these time interval volume distributions. If \( {w_{i}} \) represents the actual trading volume distribution ratio for each time interval in the market, and if investors can accurately predict these ratios, then orders distributed according to these predictions will produce a VWAP price close to the market's actual VWAP price [25].

The effectiveness of VWAP strategies heavily relies on accurate predictions of intraday volume distributions. By understanding these patterns, traders can enhance the timing of their order executions to minimize deviations from the market VWAP, potentially leading to improved trading outcomes and reduced costs [22].

However, the implementation of the VWAP strategies has proven to be challenging. Factors such as market structure complexities, fluctuations in liquidity, and information disparities have an impact on performance [3]. Ongoing research in this field focuses on refining volume prediction models, developing adaptive execution mechanisms, and integrating VWAP with other trading approaches to address these challenges [25].

4. VWAP Strategy Research and Improvements

Traditional VWAP trading strategies are widely used to reduce trading costs, but these strategies have two significant drawbacks.

Firstly, traditional VWAP strategies heavily rely on accurate predictions of intraday trading volume distribution. Inaccurate forecasts can result in unstable execution performance of the VWAP strategy. This reliance on predictions means that any deviation between forecasted and actual trading volumes can lead to significant discrepancies between the planned VWAP price and the actual execution price, thereby diminishing the effectiveness of the strategy [6].

Secondly, VWAP strategies are completely static execution strategies. During the trading period, they only follow the predefined strategy and cannot incorporate the latest market information. This static nature means that they are unable to adapt to sudden market changes or dynamic fluctuations, which may result in suboptimal execution outcomes [13]. Therefore, it is crucial to conduct research and make improvements to address these two major shortcomings of traditional VWAP strategies.

Researchers have categorized improvements to traditional static VWAP methods into two main approaches to address its primary shortcomings: 1) enhancing the accuracy of volume predictions, and 2) developing dynamic VWAP algorithms by incorporating historical price and volume data or real-time market data [7].

For algorithmically complex trading, particularly for execution algorithms such as the Volume-Weighted Average Price (VWAP) trading strategy, the accuracy of execution and prediction of intraday closing period trading volumes is crucial [10].

Before 2000, research primarily focused on how market activity or new information affects trading volume to explain specific intraday volume patterns. Biais et al. [14] first identified the V-shaped periodic characteristic of stock trading volumes in their paper. Following this, Dufour and Engle [14], along with Fol and Gouriroux, further corroborated the U-shaped (or V-shaped) intraday volume cycle characteristic in their research on trading volume [14].

Subsequently, scholars have increasingly focused on predicting intraday trading volume.

Lo et al. conducted an in-depth analysis of stock trading volume data from the New York Stock Exchange (NYSE) and the American Stock Exchange (AMEX) between 1962 and 1996 [14]. Through the application of Principal Component Analysis (PCA), they identified two primary components within the volume data. These components were crucial for capturing the cyclical and random characteristics of trading volume, which they termed the "common" and "specific" components. This discovery laid the groundwork for understanding how various factors influence overall trading volume behavior in financial markets.

Following this, Hasbrouck et al. used a similar PCA method in their study of the common factors affecting prices, order flows, and liquidity [22]. Their research focused on the trading volume data of 30 stocks within the Dow Jones Industrial Average (DJIA) and revealed the presence of a significant common factor in the volume data of these stocks.

Building on this foundation, Darolles et al. (2003) applied a theoretical framework to volume decomposition, drawing on the Capital Asset Pricing Model (CAPM), which separates returns into market and idiosyncratic components [7]. Their work has established a robust framework for comprehending the impact of various factors on trading volume.

Le Fol et al. extended the application of volume decomposition techniques to intraday high-frequency data, thereby enhancing the precision and granularity of volume analysis within shorter time intervals [7]. This extension provided deeper insights into the behavior of trading volume throughout a single trading day.

He et al. further advanced volume decomposition by identifying three independent factors: common factors, latent information factors, and specific factors [7]. They classified the first two as stable components of volume, while the specific factor was categorized as a random component, offering new perspectives on the nature of trading volume.

Bialkowski et al. addressed the issue of intraday trading volume prediction by using principal component analysis to decompose the volume into two components [7]. One component represents the average changes in volume due to market fluctuations, based on historical data for similar time intervals. The other component captures specific trading volume patterns unique to the market. Their study focused on modeling these specific volume patterns for 40 stocks in the CAC 40 index using 20-minute K-line data, employing ARMA and SETAR models for their analysis. Compared to traditional rolling average VWAP strategies, the SETAR model reduced the average absolute percentage error in volume prediction by 16.91%, and the VWAP strategy's tracking error was reduced by 7%.

Brownlees et al. decomposed intraday trading volume into three distinct components: the daily component, the intraday cyclical component, and the intraday non-cyclical component [9]. The study employed a multiplicative model, where intraday trading volume is expressed as the product of these three components. The data used in the analysis came from 15-minute K-line data of three popular U.S. stock index ETFs (SPY, DIA, and QQQQ). The researchers applied the Component Multiplication Error Model (C-MEM) and the General Method of Moments (GMM) to model and estimate the parameters of these components. Compared to the traditional moving average method, this enhanced VWAP trading strategy reduced the mean squared error of volume prediction by 12.7% and decreased the VWAP tracking error by 6.5%.

5. Dynamic VWAP Strategies

While traditional VWAP strategies heavily rely on accurate volume predictions and remain static throughout the trading period, dynamic VWAP strategies aim to address these limitations by incorporating real-time market data and adapting to changing market conditions [7].

Bialkowski et al. proposed one of the early dynamic VWAP strategies. Their approach employs principal component analysis and factor models to decompose trading volume, followed by ARMA and SETAR models for prediction [7]. This dynamic strategy demonstrated notable improvements over traditional VWAP methods.

Kakade et al. introduced a competitive algorithm for online trading, which can be applied to create dynamic VWAP strategies [8]. Their algorithm continuously updates trading decisions based on the most recent market data, allowing for more flexible execution in changing market conditions.

Almgren and Chriss developed a framework for optimal execution of portfolio transactions, which can be adapted to create dynamic VWAP strategies [10]. Their model balances the trade-off between market impact costs and the risk of unfavorable price movements, allowing for dynamic adjustment of trading speed based on market conditions.

More recently, machine learning techniques have been applied to develop adaptive VWAP strategies. For instance, Nevmyvaka et al. utilized reinforcement learning to optimize execution strategies, which can be applied to create dynamic VWAP algorithms that learn and adapt to market patterns in real-time [15].

Cont et al. proposed a model for limit order books that can be used to create more sophisticated dynamic VWAP strategies [25]. Their model captures the complex dynamics of order flow and market impact, allowing for more accurate predictions of price movements and optimal execution timing.

These dynamic approaches to VWAP trading offer the potential to significantly improve upon traditional static strategies by adapting to real-time market conditions and incorporating a wider range of relevant information [19]. By continuously adjusting the trading strategy based on current market dynamics, dynamic VWAP strategies can potentially achieve better execution prices and reduce overall trading costs [17].

Furthermore, the integration of machine learning and artificial intelligence techniques in dynamic VWAP strategies has opened up new possibilities for strategy optimization [18]. These advanced algorithms can process vast amounts of market data in real-time, identify complex patterns, and make rapid trading decisions. This capability allows for more nuanced and responsive trading strategies that can better navigate the complexities of modern financial markets [20].

However, it's important to note that dynamic VWAP strategies also come with their own challenges. The increased complexity of these strategies can render it more challenging for them to implement and monitor [11]. Additionally, the reliance on real-time data and complex algorithms introduces new sources of potential errors or system failures. Therefore, while dynamic VWAP strategies offer significant potential benefits, they also require careful implementation and robust risk management systems [12].

As financial markets continue to evolve and become increasingly complex and fast-paced, the development and refinement of dynamic VWAP strategies are likely to remain an active area of research and innovation in algorithmic trading [21].

6. Conclusion

This paper has provided a comprehensive review of the Volume-Weighted Average Price (VWAP) strategy, its evolution, and recent advancements in algorithmic trading. The historical development of VWAP has been traced from its introduction as a benchmark to its current status as a fundamental strategy in algorithmic trading. The traditional VWAP strategy, while widely used, has limitations primarily due to its reliance on accurate volume predictions and its static nature. Research efforts to address these limitations have focused on two main areas: improving the accuracy of volume predictions and developing dynamic VWAP strategies.

Advancements in volume prediction have leveraged various statistical and machine-learning techniques, significantly enhancing the accuracy of intraday trading volume forecasts. The development of dynamic VWAP strategies represents a significant leap forward, offering the potential for more robust and effective trading execution by incorporating real-time market data and adapting to changing market conditions.

As financial markets continue to evolve and become increasingly complex, the importance of sophisticated algorithmic trading strategies like VWAP is likely to grow. Future research directions may include the further integration of artificial intelligence and machine learning techniques, the development of multi-asset VWAP strategies, and the exploration of VWAP applications in emerging financial technologies. In conclusion, while the VWAP strategy has come a long way since its inception, ongoing research and technological advancements continue to refine and improve its effectiveness. The evolution of VWAP and similar strategies will play a crucial role in shaping the future of financial markets.