AUTHOR=Webb Abe 

TITLE=Applications of fractional stochastic volatility models to market microstructure theory and optimal execution strategies

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=10

YEAR=2024

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2024.1456746

DOI=10.3389/fams.2024.1456746

ISSN=2297-4687

ABSTRACT=<p>In this paper, we explore the applications of fractional stochastic volatility (FSV) models within the realm of market microstructure theory and optimal execution strategies. FSV models extend traditional stochastic volatility frameworks by incorporating fractional differentiation, allowing for more flexible and realistic representations of asset price dynamics over time. Our investigation begins with an introduction to FSV models, highlighting their ability to capture long-memory effects and volatility clustering observed in financial markets. These models provide a robust framework for understanding market microstructure dynamics, including order flow behavior, price impact functions, and liquidity provision mechanisms. Furthermore, we discuss recent advancements and empirical findings using FSV models, emphasizing their role in uncovering intraday volatility patterns and their implications for trading strategies under varying market conditions. By incorporating these nuanced volatility dynamics, FSV models contribute to the development of optimal execution algorithms that enhance transaction cost efficiency and market stability. The FSV model, when the Hurst exponent <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math></inline-formula> is set to 0.5, effectively reduces to a standard stochastic volatility model. THis nested relationship can be formally demonstrated by considering the FSV model’s general form and showing that for <inline-formula><mml:math id="M2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math></inline-formula> = 0.5, the fractional Brownian motion <inline-formula><mml:math id="M3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math></inline-formula><sub>H</sub>(<inline-formula><mml:math id="M4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math></inline-formula>) becomes a standard Brownian motion <inline-formula><mml:math id="M5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi><mml:mfenced open="(" close=")"><mml:mi>t</mml:mi></mml:mfenced></mml:math></inline-formula>, thereby aligning the FSV model with traditional models. Overall, our analysis underscores the significance of FSV models in advancing both theoretical insights and practical applications in modern finance, offering new avenues for research in high-frequency trading strategies and market efficiency.</p>