Computational Techniques in Financial Engineering

Liangfei Song, Hubert Li, David Weinberg

Abstract

Financial engineering represents a multidisciplinary field that employs mathematical techniques, computational methods, and financial theory to solve complex financial problems and create innovative financial products. This review paper provides an in-depth analysis of the recent advancements in financial engineering techniques, including risk management, derivative pricing, portfolio optimization, and algorithmic trading. By exploring key methodologies and their applications, this review aims to offer a comprehensive understanding of the current state of the field and highlight future directions for research.

1. Introduction

Financial engineering integrates principles from finance, mathematics, statistics, and computer science to address financial challenges and innovate in the financial markets ¹ . The rapid evolution of financial markets and the increasing complexity of financial instruments have necessitated the development of sophisticated financial engineering techniques ^2,3 . This review covers the major techniques and advancements in the field, focusing on their theoretical foundations, practical applications, and implications for financial practice.

2. Risk Management

Risk management is a critical aspect of financial engineering, aiming to identify, assess, and mitigate financial risks ⁴ . Several techniques have been developed to manage and quantify risk:

2.1. Value at Risk

One of the most widely utilized tools in risk management is Value at Risk (VaR) ⁵ , which provides a statistical estimate of the maximum loss a portfolio might experience over a specified period, given a certain confidence level. For instance, a VaR of $1 million at a 95% confidence level indicates that there is only a 5% chance that the portfolio will lose more than $1 million over the given time frame. Despite its widespread adoption, VaR has notable limitations. It primarily measures potential losses under normal market conditions and often fails to account for extreme, tail-end events—those that fall outside the usual statistical ranges but can have severe financial impacts ⁶ . This limitation means that VaR might not fully capture the risk of catastrophic events or market shocks, which can lead to an underestimation of potential losses ^7,8 . Consequently, while VaR remains a valuable tool for assessing risk, it is essential for financial engineers and risk managers to complement it with other risk assessment techniques and models to obtain a more comprehensive view of potential risks and their implications ⁹ .

2.2. Conditional Value at Risk (CVaR)

Conditional Value at Risk (CVaR), often referred to as Expected Shortfall, offers a more nuanced perspective on risk compared to Value at Risk (VaR) by concentrating on the tail end of the loss distribution ¹⁰ . While VaR provides a threshold below which losses are expected to fall with a certain probability, it does not account for the magnitude of losses that exceed this threshold. CVaR, on the other hand, takes this a step further by calculating the average loss in scenarios where losses surpass the VaR level. This means CVaR not only considers the probability of extreme losses but also provides insight into the severity of those losses ¹¹ . By focusing on the tail risk, CVaR addresses the shortcomings of VaR, especially in capturing the risk of extreme outcomes and providing a more comprehensive measure of potential losses. This makes CVaR a valuable tool for risk management, as it helps in understanding and mitigating the impact of rare but potentially catastrophic events.

2.3. Risk-Based Pricing Models

Risk-based pricing models are crucial for determining the appropriate pricing of financial instruments by incorporating various risk factors into their valuation ¹² . These models are particularly important for adjusting the price of securities, loans, or derivatives to reflect the underlying risk associated with them. One primary technique used in this context is credit spread modeling, which focuses on the difference between the yield on a risky bond and the yield on a risk-free benchmark bond ¹³ . This spread compensates investors for the additional risk they undertake. The size of the credit spread is influenced by factors such as the creditworthiness of the issuer, which is assessed using credit ratings or scores ¹⁴ . The models estimate credit spreads by evaluating default probabilities, recovery rates, and prevailing market conditions. Default probability estimation, a key component of credit spread modeling, involves quantifying the likelihood that a borrower or issuer will default on their obligations ^15,16 . This estimation can be achieved through various methods, including credit scoring models, which use historical data to predict default likelihood, or structural and reduced-form models that rely on financial statements and observed credit spreads.

In loans and mortgages, risk-based pricing adjusts interest rates based on the borrower's credit risk profile ^17-19 . This approach ensures that lenders are adequately compensated for lending to borrowers with varying levels of creditworthiness. Loan pricing models consider factors such as credit scores and loan-to-value ratios to determine appropriate interest rates, with higher-risk borrowers typically charged higher rates ²⁰ . Similarly, in mortgage pricing, additional elements like property value and economic conditions are factored in ²¹ . For derivatives, such as options and futures, risk-based pricing models take into account the underlying asset's risk factors, including volatility and time to expiration, to determine fair value ²² . Credit derivatives, such as credit default swaps, are priced by considering the credit risk of the reference entity, default probabilities, and recovery rates ²³ . Overall, risk-based pricing models are essential for ensuring that financial instruments are priced accurately in accordance with their associated risks, helping investors and financial institutions make informed decisions and manage risk effectively.

3. Derivative Pricing

Derivative pricing has been revolutionized by the development of sophisticated mathematical models and computational techniques:

3.1. Black-Scholes Model

The Black-Scholes model, introduced by Fischer Black and Myron Scholes in 1973, is foundational in the field of derivative pricing ²⁴ . It provides a closed-form solution (Figure 1a) for European call and put options, which are financial instruments giving the holder the right, but not the obligation, to buy or sell an asset at a specified price on a fixed date ^25,26 . The model assumes constant volatility, a constant risk-free interest rate, no dividends, and log-normally distributed asset prices. The Black-Scholes formula for a European call option is given by:

where

and N(•) denotes the cumulative distribution function of the standard normal distribution^27,28. Overall, the Black-Scholes model revolutionized financial markets by providing a systematic approach to option pricing, contributing to the growth of the options market and modern financial engineering.

Figure 1. (a) Solution to the Black-Scholes equation. (b) Example of Monte Carlo simulation to predict stock volatility using random sampling.

3.2. Monte Carlo Simulation
Monte Carlo simulation is a powerful computational technique employed in derivative pricing, especially for complex derivatives where traditional models may fall short^29-31. This method leverages random sampling and statistical analysis to estimate the price of financial instruments, particularly those with path-dependent characteristics such as Asian options or barrier options³². Unlike models that provide closed-form solutions, Monte Carlo simulation does not rely on simplifying assumptions about the nature of asset price movements or their distributions³³. Instead, it generates a large number of possible price paths for the underlying asset by simulating random fluctuations according to specified probabilistic rules^34,35. By averaging the results of these simulations, it calculates an estimate of the derivative's price.

This approach is particularly advantageous for derivatives whose payoff depends on the entire history of the underlying asset's price rather than just its final value³⁶. For instance, Asian options, which have payoffs based on the average price of the underlying asset over a specified period, can be challenging to value using closed-form solutions³⁷. Monte Carlo simulation offers a flexible and accurate method for pricing these and other complex derivatives by capturing a wide range of possible outcomes and their probabilities³¹. Additionally, the method can accommodate various models of asset price dynamics, including those with stochastic volatility or jumps, making it a versatile tool in modern financial engineering³⁸. An example of a Monte Carlo simulation to predict stock volatility is shown in Figure 1b. Overall, Monte Carlo simulation has become an essential technique for valuing complex financial products and managing risk in a dynamic market environment.

3.3. Finite Difference Methods
Finite difference methods are numerical techniques used to solve partial differential equations (PDEs) that arise in the pricing of derivatives³⁹. These methods approximate the solutions to PDEs by discretizing the underlying space and time variables. The Black-Scholes PDE, which governs the pricing of European options, is a key example and is given by:

where V represents the option price, S is the asset price, t is time, σ is volatility, and r is the risk-free rate⁴⁰. Finite difference methods, including explicit, implicit, and Crank-Nicolson schemes, are employed to solve this equation⁴¹. The explicit method, while straightforward, can become unstable if the time step is too large. Conversely, the implicit method offers better stability and is suitable for longer maturities, though it requires solving a system of linear equations. The Crank-Nicolson method strikes a balance between the explicit and implicit approaches, providing both stability and accuracy by averaging the results from both methods⁴². Despite their advantages, finite difference methods can be complex to implement and computationally intensive, particularly in high-dimensional cases. Nonetheless, their versatility in handling various boundary conditions and derivative types makes them a valuable tool in financial engineering⁴³.

4. Portfolio Optimization
Portfolio optimization techniques are central to modern finance, aiming to maximize returns while minimizing risk^44-47. Over the years, advancements in this field have led to more sophisticated methods that cater to varying investor needs and market conditions.

4.1. Mean-Variance Optimization
Mean-variance optimization, introduced by Harry Markowitz in 1952, is a cornerstone of modern portfolio theory⁴⁸. This technique involves constructing a portfolio of assets that maximizes expected returns for a given level of risk, or conversely, minimizes risk for a given level of expected return⁴⁹. Markowitz's framework uses the concept of variance as a measure of risk and applies quadratic programming to find the optimal portfolio⁵⁰. The original model assumes that investors are rational and markets are efficient, focusing on the trade-off between risk and return⁵¹. Since its inception, the mean-variance approach has been extended and refined to include various constraints such as budget constraints, limits on the proportion of assets, and specific risk constraints^52-54. For example, investors might impose constraints on the minimum or maximum investment in certain asset classes or industries, or they might consider factors like transaction costs and tax implications⁵⁵. Despite its foundational role, mean-variance optimization is often critiqued for its reliance on historical data and its sensitivity to changes in the input parameters, which can lead to unstable and unrealistic portfolio allocations⁵⁶.

4.2. Multi-Objective Optimization
Multi-objective optimization represents an evolution from the traditional mean-variance approach by incorporating multiple criteria into the portfolio construction process⁵⁷. Unlike single-objective optimization, which typically focuses solely on balancing risk and return, multi-objective optimization addresses a broader set of goals, such as maximizing returns, minimizing risk, and ensuring adequate liquidity^58-60. This approach is particularly useful in scenarios where investors have complex preferences and constraints⁶¹. For instance, an investor might prioritize not only high returns and low risk but also ethical investing criteria or liquidity needs⁶². Techniques such as the Pareto frontier are used to identify portfolios that provide the best possible trade-offs among competing objectives⁶³. Researchers like Michaud (1998) have developed models that allow for the inclusion of additional factors, such as transaction costs and constraints on asset weights, to better align with real-world investing scenarios⁶⁴. These models provide a more comprehensive view of the trade-offs involved and help investors make decisions that are more in line with their specific preferences and constraints.

4.3. Machine Learning Approaches
Machine learning approaches to portfolio optimization have emerged as a transformative force in the field of finance, offering innovative methods to improve asset allocation and risk management^65-70. These techniques leverage the power of computational algorithms to analyze large datasets, identify patterns, and make more accurate predictions about financial markets.

Deep learning, a subset of machine learning involving neural networks with multiple layers, has proven particularly valuable in portfolio optimization⁷¹. Deep learning models can capture complex, non-linear relationships in financial data that traditional models might overlook^72-74. For instance, convolutional neural networks (CNNs) and recurrent neural networks (RNNs) can analyze time-series data, such as historical prices and trading volumes, to identify underlying trends and anomalies⁷⁵. These models excel in extracting features from raw data and generating insights that can enhance predictive accuracy⁷⁶. By training on extensive datasets, deep learning algorithms can adapt to various market conditions and improve forecasts of asset returns, volatility, and correlations⁷⁷.

Reinforcement learning (RL) represents another significant advancement in portfolio optimization⁷⁸. RL algorithms are designed to learn optimal decision-making strategies through interactions with an environment^79-81. In the context of finance, this means developing investment strategies that maximize returns while managing risk⁸². RL models operate by exploring different portfolio configurations, receiving feedback in the form of rewards or penalties based on their performance, and gradually refining their strategies to improve outcomes⁸³. Techniques such as Q-learning and deep Q-networks (DQN) allow for the optimization of asset allocation by balancing exploration (trying new strategies) and exploitation (refining known strategies)⁸⁴. In this approach, the Q value is updated using the following algorithm,

where Q(s,a) is the expected reward for taking action a in state s, α is the learning rate, r is the reward received after taking action a, γ is the discount factor, and max⁡a′Q(s′,a′) represents the maximum expected reward for the next state s′ and possible actions a′⁸⁵. Therefore, Q-learning is particularly useful in adapting to dynamic and evolving market conditions, where traditional models may struggle to keep pace.

One of the key advantages of machine learning approaches is their ability to handle vast amounts of data and uncover hidden patterns that are not immediately apparent⁸⁶. For example, natural language processing (NLP) techniques can be used to analyze news articles, financial reports, and social media sentiments to gauge market sentiment and its potential impact on asset prices⁸⁷. By integrating these alternative data sources with traditional financial metrics, machine learning models can provide a more comprehensive view of the market and enhance decision-making.

However, the adoption of machine learning in portfolio optimization also presents several challenges. One major concern is the risk of overfitting, where a model becomes too closely aligned with historical data and performs poorly when faced with new or unseen market conditions⁸⁸. To mitigate this, techniques such as cross-validation and regularization are employed to ensure that models generalize well to out-of-sample data. Additionally, machine learning models can be complex and opaque, making it difficult for investors to interpret the rationale behind their recommendations⁸⁹. This "black-box" nature of some algorithms necessitates careful consideration and transparency in their application⁹⁰.

In recent years, researchers have focused on integrating machine learning with traditional portfolio optimization methods to harness the strengths of both approaches⁹¹. Hybrid models combine the predictive power of machine learning with the foundational principles of modern portfolio theory, aiming to create more robust and adaptable investment strategies⁹². For example, machine learning can be used to enhance the estimation of expected returns and risk parameters, which are then incorporated into traditional optimization frameworks like mean-variance optimization⁹³.

5. Algorithmic Trading
Algorithmic trading involves the use of algorithms to automate trading decisions and execution:

5.1. High-Frequency Trading (HFT)
High-Frequency Trading (HFT) involves the use of sophisticated algorithms to execute a large volume of trades at extremely high speeds, exploiting small, short-term market inefficiencies^94-96. HFT strategies rely on advanced technology and infrastructure to process and act on market data in fractions of a second. One prominent technique within HFT is market-making, where traders provide liquidity by simultaneously placing buy and sell orders, profiting from the bid-ask spread⁹⁷. Another technique is statistical arbitrage, which leverages mathematical models to identify and capitalize on pricing discrepancies across different markets or related securities⁹⁸. The ability to quickly analyze vast amounts of data and execute trades with minimal latency is crucial for HFT success, as these strategies often depend on capturing fleeting opportunities that exist for only a brief moment^99-100.

5.2. Algorithmic Execution Strategies
Algorithmic execution strategies focus on minimizing trading costs and mitigating market impact, which can occur when large trades are executed in the market¹⁰¹. Techniques such as Volume Weighted Average Price (VWAP) and Time Weighted Average Price (TWAP) are commonly employed in this domain¹⁰². VWAP aims to match the average price of a stock over a specific period with the volume-weighted average of its trades, helping to ensure that trades are executed at a price that is reflective of the market's average price¹⁰³. TWAP, on the other hand, breaks down a large order into smaller, evenly distributed trades over time, reducing the potential impact on the market price and minimizing the risk of moving the market against the trader^104-105. Both methods are designed to achieve optimal execution while minimizing the costs associated with large-scale trading activities.

5.3. Sentiment Analysis and NLP
Sentiment analysis and Natural Language Processing (NLP) techniques are employed to extract actionable insights from unstructured data sources such as news articles, social media posts, and financial reports¹⁰⁶. By analyzing the sentiment expressed in these texts, algorithms can gauge market sentiment and predict potential price movements¹⁰⁷. For example, NLP can be used to parse news headlines or social media conversations to assess the overall mood towards a particular stock or market sector¹⁰⁸. This information can then be integrated into trading strategies to anticipate market trends or react to emerging news more swiftly¹⁰⁹. The use of sentiment analysis helps traders and algorithms incorporate qualitative data into their decision-making processes, providing a more comprehensive view of market dynamics beyond traditional quantitative data¹¹⁰.

6. Conclusion and Future Directions
As financial engineering advances, the integration of cutting-edge computational power, expanded data accessibility, and refined analytical techniques drives ongoing evolution in the field¹¹¹. One of the pivotal areas of future research is the enhancement of risk management models. Traditional models often struggle to accurately predict and mitigate complex, high-dimensional risks in real-time market environments¹¹². Researchers are focusing on developing models that can better capture the dynamic nature of financial markets and incorporate a broader range of risk factors, including systemic and tail risks. Additionally, there is a growing emphasis on creating more sophisticated derivative pricing methods. As financial instruments become increasingly complex, there is a need for advanced models that can more precisely value these instruments and accommodate various market conditions, such as extreme volatility or liquidity crises¹¹³. Another significant research avenue involves the application of advanced machine learning techniques. The utilization of machine learning and artificial intelligence promises to revolutionize financial engineering by enhancing predictive accuracy, automating decision-making processes, and uncovering hidden patterns in vast datasets¹¹⁴. However, this also brings challenges related to model interpretability, overfitting, and the robustness of predictions. Thus, balancing these sophisticated techniques with traditional financial theories is a crucial task for future research. Moreover, as algorithmic trading becomes more prevalent, exploring its ethical implications becomes increasingly important¹¹⁵. The use of high-frequency trading algorithms, for example, raises concerns about market fairness, potential manipulation, and the impact on market stability. Researchers are tasked with addressing these ethical considerations by developing frameworks and regulations that ensure algorithmic trading contributes positively to market efficiency without compromising integrity.

In summary, the future of financial engineering promises significant advancements in risk management, derivative pricing, and machine learning applications, while also necessitating a careful examination of the ethical dimensions of algorithmic trading. These directions will shape the future landscape of financial markets, striving for a balance between innovation and responsible practice.

References

1. Lyuu, Yuh-Dauh. Financial engineering and computation: principles, mathematics, algorithms. Cambridge University Press, 2001.
2. Neftci, Salih N. Principles of financial engineering. Academic Press, 2008.
3. Zopounidis, Constantin, Michael Doumpos, and Panos M. Pardalos, eds. Handbook of financial engineering. Vol. 18. Springer Science & Business Media, 2010.
4. Van Gestel, Tony, and Bart Baesens. Credit Risk Management: Basic concepts: Financial risk components, Rating analysis, models, economic and regulatory capital. Oxford University Press, 2009.
5. Benninga, Simon, and Zvi Wiener. "Value-at-risk (VaR)." matrix 32 (1998): s33.
6. Dowd, Kevin, and David Blake. "After VaR: the theory, estimation, and insurance applications of quantile‐based risk measures." Journal of Risk and Insurance 73.2 (2006): 193-229.
7. Bee, Marco, and Fabrizio Miorelli. "Dynamic VaR models and the peaks over threshold method for market risk measurement: An empirical investigation during a financial crisis." Journal of Risk Model Validation 9 (2010): 1-40.
8. Lockwood, Erin. "Predicting the unpredictable: Value-at-risk, performativity, and the politics of financial uncertainty." Review of international political economy 22.4 (2015): 719-756.
9. Best, Philip. Implementing value at risk. John Wiley & Sons, 2000.
10. Rockafellar, R. Tyrrell, and Stanislav Uryasev. "Optimization of conditional value-at-risk." Journal of risk 2 (2000): 21-42.
11. Filippi, Carlo, Gianfranco Guastaroba, and Maria Grazia Speranza. "Conditional value‐at‐risk beyond finance: a survey." International Transactions in Operational Research 27.3 (2020): 1277-1319.
12. Mukatuni, Awelani Lynn. "Risk-based pricing of financial products." (2021).
13. Maltritz, Dominik, and Alexander Molchanov. "Analyzing determinants of bond yield spreads with Bayesian Model Averaging." Journal of Banking & Finance 37.12 (2013): 5275-5284.
14. Haque, Nadeem Ul, Donald Mathieson, and Nelson Mark. "Rating the raters of country creditworthiness." Finance & Development 34.1 (1997): 10-13.
15. Kealhofer, Stephen. "Quantifying credit risk I: default prediction." Financial Analysts Journal 59.1 (2003): 30-44.
16. Fernandes, João. "Corporate credit risk modeling: Quantitative rating system and probability of default estimation." Available at SSRN 722941 (2005).
17. Staten, Michael. "Risk-based pricing in consumer lending." JL Econ. & Pol'y 11 (2015): 33.
18. White, Alan M. "Risk‐based mortgage pricing: Present and future research." Housing Policy Debate 15.3 (2004): 503-531.
19. Spader, Jonathan S. "Beyond disparate impact: Risk-based pricing and disparity in consumer credit history scores." The Review of Black Political Economy 37.2 (2010): 61-78.
20. Avery, Robert B., et al. "Credit risk, credit scoring, and the performance of home mortgages." Fed. Res. Bull. 82 (1996): 621.
21. Kau, James B., and Donald C. Keenan. "An overview of the option-theoretic pricing of mortgages." Journal of housing research (1995): 217-244.
22. Schönbucher, Philipp J. Credit derivatives pricing models: models, pricing and implementation. John Wiley & Sons, 2003.
23. Mengle, David. "Credit derivatives: An overview." Economic Review-Federal Reserve Bank of Atlanta 92.4 (2007): 1.
24. Scholes, Myron S. "Derivatives in a dynamic environment." The American Economic Review 88.3 (1998): 350-370.
25. Mondello, Enzo. "Options: Basics and Valuation." Applied Fundamentals in Finance: Portfolio Management and Investments. Wiesbaden: Springer Fachmedien Wiesbaden, 2023. 479-535.
26. Zhang, Peter Guangping. Exotic options: a guide to second generation options. World scientific, 1997.
27. Geske, Robert, and Richard Roll. "On valuing American call options with the Black‐Scholes European formula." The Journal of Finance 39.2 (1984): 443-455.
28. Wu, Hsien-Chung. "Pricing European options based on the fuzzy pattern of Black–Scholes formula." Computers & Operations Research 31.7 (2004): 1069-1081.
29. Jäckel, Peter. Monte Carlo methods in finance. Vol. 5. John Wiley & Sons, 2002.
30. Zhang, Yunyu. "The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives." PloS one 15.2 (2020): e0229737.
31. Boyle, Phelim, Mark Broadie, and Paul Glasserman. "Monte Carlo methods for security pricing." Journal of economic dynamics and control 21.8-9 (1997): 1267-1321.
32. Gylfadottir, Gudbjort. Path-dependent option pricing: Efficient methods for Lévy models. University of Florida, 2010.
33. Avellaneda, Marco, et al. "Weighted Monte Carlo: a new technique for calibrating asset-pricing models." International Journal of Theoretical and Applied Finance 4.01 (2001): 91-119.
34. Chiarella, Carl, Roberto Dieci, and Xue-Zhong He. "Heterogeneity, market mechanisms, and asset price dynamics." Handbook of financial markets: Dynamics and evolution. North-Holland, 2009. 277-344.
35. Böhm, Volker, and Carl Chiarella. "Mean variance preferences, expectations formation, and the dynamics of random asset prices." Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics 15.1 (2005): 61-97.
36. Bingham, Nicholas H., and Rüdiger Kiesel. Risk-neutral valuation: Pricing and hedging of financial derivatives. Springer Science & Business Media, 2013.
37. Li, Weiping, and Su Chen. "Pricing and hedging of arithmetic Asian options via the Edgeworth series expansion approach." The Journal of Finance and Data Science 2.1 (2016): 1-25.
38. Carmona, René, and Michael Ludkovski. "Pricing asset scheduling flexibility using optimal switching." Applied Mathematical Finance 15.5-6 (2008): 405-447.
39. Duffy, Daniel J. Finite difference methods in financial engineering: a partial differential equation approach. John Wiley & Sons, 2013.
40. Kaur, Jaspreet, and Srinivasan Natesan. "A novel numerical scheme for time-fractional Black-Scholes PDE governing European options in mathematical finance." Numerical Algorithms 94.4 (2023): 1519-1549.
41. Chen, Wenting, Xiang Xu, and Song-ping Zhu. "Analytically pricing European-style options under the modified Black-Scholes equation with a spatial-fractional derivative." Quarterly of Applied Mathematics 72.3 (2014): 597-611.
42. Pooley, David M., Peter A. Forsyth, and Ken R. Vetzal. "Numerical convergence properties of option pricing PDEs with uncertain volatility." IMA Journal of Numerical Analysis 23.2 (2003): 241-267.
43. Malevergne, Yannick, and Didier Sornette. Extreme financial risks: From dependence to risk management. Springer Science & Business Media, 2006.
44. Korn, Ralf, and Elke Korn. Option pricing and portfolio optimization: modern methods of financial mathematics. Vol. 31. American Mathematical Soc., 2001.
45. Zhao, Ziping, et al. "Portfolio optimization." (2019).
46. Best, Michael J. Portfolio optimization. CRC Press, 2010.
47. Fabozzi, Frank J., et al. Robust portfolio optimization and management. John Wiley & Sons, 2007.
48. Mangram, Myles E. "A simplified perspective of the Markowitz portfolio theory." Global journal of business research 7.1 (2013): 59-70.
49. Bechis, Luca, Francesco Cerri, and Marco Vulpiani. "Machine learning portfolio optimization: hierarchical risk parity and modern portfolio theory." LUISS Guido Carli, Roma, Italy (2020).
50. Zhang, Yuanyuan, Xiang Li, and Sini Guo. "Portfolio selection problems with Markowitz’s mean–variance framework: a review of literature." Fuzzy Optimization and Decision Making 17 (2018): 125-158.
51. Briec, Walter, Kristiaan Kerstens, and Jean-Baptiste Lesourd. "Single-period Markowitz portfolio selection, performance gauging, and duality: a variation on the Luenberger shortage function." Journal of Optimization Theory and Applications 120 (2004): 1-27.
52. King, Alan J. "Asymmetric risk measures and tracking models for portfolio optimization under uncertainty." Annals of Operations Research 45 (1993): 165-177.
53. Mencarelli, Luca, and Claudia d'Ambrosio. "Complex portfolio selection via convex mixed‐integer quadratic programming: a survey." International Transactions in Operational Research 26.2 (2019): 389-414.
54. El Karoui, Noureddine. "High-dimensionality effects in the Markowitz problem and other quadratic programs with linear constraints: Risk underestimation." (2010): 3487-3566.
55. Das, Sanjiv, et al. "Portfolio optimization with mental accounts." Journal of financial and quantitative analysis 45.2 (2010): 311-334.
56. Kalayci, Can B., Okkes Ertenlice, and Mehmet Anil Akbay. "A comprehensive review of deterministic models and applications for mean-variance portfolio optimization." Expert Systems with Applications 125 (2019): 345-368.
57. Silva, Yuri Laio TV, Ana Beatriz Herthel, and Anand Subramanian. "A multi-objective evolutionary algorithm for a class of mean-variance portfolio selection problems." Expert Systems with Applications 133 (2019): 225-241.
58. Saborido, Rubén, et al. "Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection." Applied soft computing 39 (2016): 48-63.
59. Chiam, Swee Chiang, Kay Chen Tan, and A. Al Mamum. "Evolutionary multi-objective portfolio optimization in practical context." International Journal of Automation and Computing 5 (2008): 67-80.
60. Zheng, Yongjie, and Jiming Zheng. "A novel portfolio optimization model via combining multi-objective optimization and multi-attribute decision making." Applied Intelligence 52.5 (2022): 5684-5695.
61. Lwin, Khin, Rong Qu, and Graham Kendall. "A learning-guided multi-objective evolutionary algorithm for constrained portfolio optimization." Applied Soft Computing 24 (2014): 757-772.
62. Wang, Bo, et al. "A multi-objective portfolio selection model with fuzzy value-at-risk ratio." IEEE Transactions on Fuzzy Systems 26.6 (2018): 3673-3687.
63. Bai, Qiang, et al. "A hybrid Pareto frontier generation method for trade‐off analysis in transportation asset management." Computer‐Aided Civil and Infrastructure Engineering 30.3 (2015): 163-180.
64. Ang, Andrew. Asset management: A systematic approach to factor investing. Oxford University Press, 2014.
65. Perrin, Sarah, and Thierry Roncalli. "Machine learning optimization algorithms & portfolio allocation." Machine Learning for Asset Management: New Developments and Financial Applications (2020): 261-328.
66. Mashrur, Akib, et al. "Machine learning for financial risk management: a survey." Ieee Access 8 (2020): 203203-203223.
67. Ngo, Vu Minh, Huan Huu Nguyen, and Phuc Van Nguyen. "Does reinforcement learning outperform deep learning and traditional portfolio optimization models in frontier and developed financial markets?." Research in International Business and Finance 65 (2023): 101936.
68. Mwangi, Mary. "The Role of Machine Learning in Enhancing Risk Management Strategies in Financial Institutions." International Journal of Modern Risk Management 2.1 (2024): 44-53.
69. Irfan, Mohammad, et al., eds. Advanced machine learning algorithms for complex financial applications. IGI Global, 2023.
70. Nguyen, Duc Khuong, Georgios Sermpinis, and Charalampos Stasinakis. "Big data, artificial intelligence and machine learning: A transformative symbiosis in favour of financial technology." European Financial Management 29.2 (2023): 517-548.
71. Zhang, Zihao, Stefan Zohren, and Stephen Roberts. "Deep learning for portfolio optimization." arXiv preprint arXiv:2005.13665 (2020).
72. Jiang, Zhengyao, Dixing Xu, and Jinjun Liang. "A deep reinforcement learning framework for the financial portfolio management problem." arXiv preprint arXiv:1706.10059 (2017).
73. Ozbayoglu, Ahmet Murat, Mehmet Ugur Gudelek, and Omer Berat Sezer. "Deep learning for financial applications: A survey." Applied soft computing 93 (2020): 106384.
74. Wang, Wuyu, et al. "Portfolio formation with preselection using deep learning from long-term financial data." Expert Systems with Applications 143 (2020): 113042.
75. Dargan, Shaveta, et al. "A survey of deep learning and its applications: a new paradigm to machine learning." Archives of Computational Methods in Engineering 27 (2020): 1071-1092.
76. Obeidat, Samer, et al. "Adaptive portfolio asset allocation optimization with deep learning." International Journal on Advances in Intelligent Systems 11.1 (2018): 25-34.
77. Singh, Vinay, et al. "How are reinforcement learning and deep learning algorithms used for big data based decision making in financial industries–A review and research agenda." International Journal of Information Management Data Insights 2.2 (2022): 100094.
78. Hu, Yuh-Jong, and Shang-Jen Lin. "Deep reinforcement learning for optimizing finance portfolio management." 2019 amity international conference on artificial intelligence (AICAI). IEEE, 2019.
79. Aboussalah, Amine Mohamed, and Chi-Guhn Lee. "Continuous control with stacked deep dynamic recurrent reinforcement learning for portfolio optimization." Expert Systems with Applications 140 (2020): 112891.
80. Hambly, Ben, Renyuan Xu, and Huining Yang. "Recent advances in reinforcement learning in finance." Mathematical Finance 33.3 (2023): 437-503.
81. Yu, Pengqian, et al. "Model-based deep reinforcement learning for dynamic portfolio optimization." arXiv preprint arXiv:1901.08740 (2019).
82. Benhamou, Eric, et al. "Bridging the gap between Markowitz planning and deep reinforcement learning." arXiv preprint arXiv:2010.09108 (2020).
83. Sahu, Santosh Kumar, Anil Mokhade, and Neeraj Dhanraj Bokde. "An overview of machine learning, deep learning, and reinforcement learning-based techniques in quantitative finance: recent progress and challenges." Applied Sciences 13.3 (2023): 1956.
84. Talaat, Fatma M. "Effective deep Q-networks (EDQN) strategy for resource allocation based on optimized reinforcement learning algorithm." Multimedia Tools and Applications 81.28 (2022): 39945-39961.
85. Saeed, Sultan, et al. "Reinforcement Learning for Automated Financial Trading and Portfolio Management: Integrating Control Systems for Adaptive Risk-Aware Decision Making." (2024).
86. Khalil, Faisal, and Gordon Pipa. "Is deep-learning and natural language processing transcending the financial forecasting? Investigation through lens of news analytic process." Computational Economics 60.1 (2022): 147-171.
87. Xing, Frank Z., Erik Cambria, and Roy E. Welsch. "Natural language based financial forecasting: a survey." Artificial Intelligence Review 50.1 (2018): 49-73.
88. Gao, Ruizhuo, et al. "A review of natural language processing for financial technology." International Symposium on Artificial Intelligence and Robotics 2021. Vol. 11884. SPIE, 2021.
89. Deveikyte, Justina, et al. "A sentiment analysis approach to the prediction of market volatility." Frontiers in Artificial Intelligence 5 (2022): 836809.
90. Wan, Xingchen, et al. "Sentiment correlation in financial news networks and associated market movements." Scientific reports 11.1 (2021): 3062.
91. Ta, Van-Dai, Chuan-Ming Liu, and Direselign Addis Tadesse. "Portfolio optimization-based stock prediction using long-short term memory network in quantitative trading." Applied Sciences 10.2 (2020): 437.
92. Day, Min-Yuh, and Jian-Ting Lin. "Artificial intelligence for ETF market prediction and portfolio optimization." Proceedings of the 2019 IEEE/ACM international conference on advances in social networks analysis and mining. 2019.
93. Bengio, Yoshua, Andrea Lodi, and Antoine Prouvost. "Machine learning for combinatorial optimization: a methodological tour d’horizon." European Journal of Operational Research 290.2 (2021): 405-421.
94. Virgilio, Gianluca Piero Maria. "High-frequency trading: a literature review." Financial markets and portfolio management 33.2 (2019): 183-208.
95. Miller, Rena S., and Gary Shorter. High frequency trading: Overview of recent developments. Vol. 4. Washington, DC: Congressional Research Service, 2016.
96. Lhabitant, Francois-Serge, and Greg N. Gregoriou. "High-frequency trading: past, present, and future." Handbook of high frequency trading (2015): 155-166.
97. McGowan, Michael J. "The rise of computerized high frequency trading: use and controversy." Duke L. & Tech. Rev. 9 (2010): 1.
98. Serbera, Jean-Philippe, and Pascal Paumard. "The fall of high-frequency trading: A survey of competition and profits." Research in International Business and Finance 36 (2016): 271-287.
99. Linton, Oliver, and Soheil Mahmoodzadeh. "Implications of high-frequency trading for security markets." Annual Review of Economics 10.1 (2018): 237-259.
100. Golub, Anton. "Overview of high frequency trading." Manchester Business School, April 15 (2011).
101. Yadav, Yesha. "How algorithmic trading undermines efficiency in capital markets." Vand. L. Rev. 68 (2015): 1607.
102. Li, Handong, and Xunyu Ye. "A Dynamic, Volume‐Weighted Average Price Approach Based on the Fast Fourier Transform Algorithm." Asia‐Pacific Journal of Financial Studies 42.6 (2013): 969-991.
103. Pinto, Mário André Ferreira. "Design and implementation of an algorithmic trading system for the Sifox application." (2014).
104. Chakravarty, Satya R., and Palash Sarkar. "Algorithmic Trading Strategies." An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain. Emerald Publishing Limited, 2020. 79-89.
105. Li, Xiaodong, Pangjing Wu, and Rongrong Bo. "Improving Order Execution Cost Estimation by Semisupervised Learning." IEEE Access 7 (2019): 178342-178356.
106. Boubekeur, Amal. "Text Mining and Natural Language Processing: Investigating text mining and natural language processing techniques for extracting insights from unstructured text data." Australian Journal of Machine Learning Research & Applications 3.2 (2023): 158-170.
107. Ghavami, Peter. Big data analytics methods: analytics techniques in data mining, deep learning and natural language processing. Walter de Gruyter GmbH & Co KG, 2019.
108. Dash, Bibhu. Information Extraction from Unstructured Big Data: A Case Study of Deep Natural Language Processing in Fintech. University of the Cumberlands, 2022.
109. Suresh, M., et al. "Analyse Customer Behaviour and Sentiment Using Natural Language Processing (NLP) Techniques to Improve Customer Service and Personalize Banking Experiences." Educational Administration: Theory And Practice 30.5 (2024): 8802-8813.
110. Hansen, Kristian Bondo, and Christian Borch. "Alternative data and sentiment analysis: Prospecting non-standard data in machine learning-driven finance." Big Data & Society 9.1 (2022): 20539517211070701.
111. Ionescu, Sergiu-Alexandru, and Vlad Diaconita. "Transforming financial decision-making: the interplay of AI, cloud computing and advanced data management technologies." International Journal of Computers Communications & Control 18.6 (2023)
112. Theodorakopoulos, Leonidas, Alexandra Theodoropoulou, and Yannis Stamatiou. "A State-of-the-Art Review in Big Data Management Engineering: Real-Life Case Studies, Challenges, and Future Research Directions." Eng 5.3 (2024): 1266-1297.
113. Amihud, Yakov, Haim Mendelson, and Lasse Heje Pedersen. Market liquidity: asset pricing, risk, and crises. Cambridge University Press, 2012.
114. Schmitt, Marc. "Artificial intelligence in business analytics, capturing value with machine learning applications in financial services." (2020).
115. Lenglet, Marc. "Conflicting codes and codings: How algorithmic trading is reshaping financial regulation." Theory, Culture & Society 28.6 (2011): 44-66.