[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117320-en":3,"doc-seo-117320-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},117320,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Mathematics of Differential Machine Learning in Derivative Pricing and Hedging","This article develops a financial differential machine learning algorithm within a rigorous mathematical framework, focusing on how theoretical assumptions in pricing models shape algorithm construction. It addresses the challenge of obtaining unbiased estimates of differential labels from data, which matters as finance increasingly adopts data-driven valuation and hedging for derivatives. Using a Hilbert-space formulation and functional analysis tools, it links loss-function derivations to algorithmic implementations and supports the differential method’s optimality through comparative experimental results.","arXiv :2405 .01233v1 [ q-fin .MF] 2 May 2024  \nMathematics of Differential Machine Learning in Derivative Pricing and  \nHedging  \nRunning Title: Diff. ML in Derivative Pricing  \nPedro Duarte Gomes  \nDepartment of Mathematics, University of Copenhagen  \nAbstract  \nThis article introduces the groundbreaking concept of the financial differential machine learning algorithm through a rigorous mathematical framework. Diverging from existing literature on financial machine learning, the work highlights the profound implications of theoretical assumptions within financial models on the construction of machine learning algorithms.  \nThis endeavour is particularly timely as the finance landscape witnesses a surge in interest towards data-driven models for the valuation and hedging of derivative products. Notably, the predictive capabilities of neural networks have garnered substantial attention in both academic research and practical financial applications.  \nThe approach offers a unified theoretical foundation that facilitates comprehensive comparisons, both at a theoretical level and in experimental outcomes. Importantly, this theoretical grounding lends substantial weight to the experimental results, affirming the differential machine learning method’s optimality within the prevailing context.  \nBy anchoring the insights in rigorous mathematics, the article bridges the gap between abstract financial concepts and practical algorithmic implementations.  \n1 Keywords  \nDifferential Machine Learning, Risk Neutral valuation, Derivative Pricing, Hilbert Spaces Orthogonal Projection, Generalized Function Theory  \nContents  \n1 Keywords 2  \n2 Introduction 4  \n3 Set up 5  \n4 From Classical Results into Differential Machine Learning 6  \n4.1 Risk Neutral Valuation Approach ........................ 6  \n4.2 Differential Machine learning: building the loss function ............ 7  \n5 Example: Digital Options 10  \n6 Choice of Basis 11  \n6.1 Limitations of the Fixed-basis .......................... 12  \n6.2 Parametric Basis: Neural Networks ....................... 13  \n6.2.1 Depth ................................... 14  \n6.2.2 Width ................................... 14  \n7 Simulation-European Call Option 15  \n7.1 Black-Scholes ................................... 15  \n7.2 Hedging Experiment ............................... 16  \n7.3 Least Squares Monte Carlo Algorithm ...................... 16  \n7.3.1 Monomial Basis .............................. 16  \n7.3.2 Neural Network Basis ........................... 16  \n7.4 Differential Machine Learning Algorithm .................... 17  \n7.4.1 Neural Network basis ........................... 18  \n8 Numerical Results 19  \n9 Conclusion 21  \n10 Conflict of Interests Statement 21  \n2 Introduction  \nWithin the dynamic landscape of financial modelling, the quest for reliable pricing and hedging mechanisms persists as a pivotal challenge. This article aims to introduce an encompassing theory of pricing valuation uniquely rooted in the domain of machine learning. A primary focus lies in overcoming a prominent hurdle encountered in implementing the differential machine learning algorithm, specifically addressing the critical need for unbiased estimation of differential labels from data sources, as highlighted in studies by Huge (2020) and Broadie (1996) . This breakthrough holds considerable importance for contemporary practitioners across diverse institutional settings, offering tangible solutions and charting a course toward refined methodologies. Furthermore, this endeavour not only caters to the immediate requirements of practitioners but also furnishes invaluable insights that can shape forthcoming research endeavours in this domain.  \nThe article sets off from the premise that the pricing and hedging functions can bethought of as elements of a Hilbert space, in a similar way as Pelsser and Schweizer, 2016 . A natural extension of these elements across time, originally attained in the current article, is accomplish","cbCaikQBBFRRXu9w","https://ap.wps.com/l/cbCaikQBBFRRXu9w","pdf",532798,1,23,"English","en",105,"# Keywords\n# Introduction\n# Set up\n# From Classical Results into Differential Machine Learning\n## Risk Neutral Valuation Approach\n## Differential Machine learning: building the loss function\n# Example: Digital Options\n# Choice of Basis\n## Limitations of the Fixed-basis\n## Parametric Basis: Neural Networks\n### Depth\n### Width\n# Simulation-European Call Option\n## Black-Scholes\n## Hedging Experiment\n## Least Squares Monte Carlo Algorithm\n### Monomial Basis\n### Neural Network Basis\n## Differential Machine Learning Algorithm\n### Neural Network basis\n# Numerical Results\n# Conclusion\n# Conflict of Interests Statement","[{\"question\":\"What core problem does the article address in differential machine learning for derivatives?\",\"answer\":\"It targets the need for unbiased estimation of differential labels from data sources, which is a key hurdle when implementing the differential machine learning algorithm.\"},{\"question\":\"How is the pricing and hedging problem formulated in the paper?\",\"answer\":\"The paper treats pricing and hedging functions as elements of a Hilbert space and extends them across time using the Hahn–Banach extension theorem.\"},{\"question\":\"What is the role of basis choice in the proposed framework?\",\"answer\":\"Modeling in Hilbert spaces reduces the task to choosing a loss function and an appropriate basis function, then comparing major basis classes and their limitations and 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core problem does the article address in differential machine learning for derivatives?","Question",{"text":74,"@type":75},"It targets the need for unbiased estimation of differential labels from data sources, which is a key hurdle when implementing the differential machine learning algorithm.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How is the pricing and hedging problem formulated in the paper?",{"text":79,"@type":75},"The paper treats pricing and hedging functions as elements of a Hilbert space and extends them across time using the Hahn–Banach extension theorem.",{"name":81,"@type":72,"acceptedAnswer":82},"What is the role of basis choice in the proposed framework?",{"text":83,"@type":75},"Modeling in Hilbert spaces reduces the task to choosing a loss function and an appropriate basis function, then comparing major basis classes and their limitations and 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