[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120879-en":3,"doc-seo-120879-105":30,"detail-sidebar-cat-0-en-105":91},{"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":27,"seo_description":14,"update_tm":28,"read_time":29},120879,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","An excursion into Differential Machine Learning and Applications to Finance - Bachelor’s Degree Thesis","Differential Machine Learning connects deep learning with the pricing of financial derivatives by introducing neural network structures for computing derivative prices and sensitivities. This bachelor thesis provides a mathematical understanding of the core concepts from a 2020 paper, then extends the twin network framework to include the remaining Greeks, generates predictions, and evaluates them against closed-form Black-Scholes results. The work also demonstrates applications in valuing European call options and estimating their Greeks.","Degree in Mathematics  \nTitle: An excursion into Differential Machine Learning and Applications to Finance.  \nAuthor: Amàlia Simón i Ribas  \nAdvisor: Argimiro Arratia Quesada  \nDepartment: Ciències de la Computació  \nAcademic year: 2022-2023  \n2  \nUniversitat Polit`ecnica de Catalunya Facultat de Matem`atiques i Estad´ıstica  \nBachelor’s Degree Thesis  \nAn Excursion into Differential Machine Learning and Applications to Finance  \nAm`alia Sim´on i Ribas  \nSupervised by  \nArgimiro Arratia Quesada  \nDepartment of Computer Science & BGSMath & IMTech  \nAcknowledgements  \nI would like to thank my thesis supervisor, Argimiro Arratia, for his dedication and hours invested in this project.  \nI would also like to express my gratitude to my family and friends, for always encouraging me and giving me unconditional support.  \nAbstract  \nThe paper ”Differential Machine Learning”, published in 2020 by Antoine Savine and Brian Huge opens a new door to the calculation of financial derivatives through a design of a new Neural Network, the core of Deep Learning, creating a link between these two fields. This thesis aims to give a mathematical understanding of all the concepts introduced in the paper, complementing it and extending it. The other part of this thesis works around the possibility of extending the twin network and its calculations to include the rest of the Greeks, implementing its predictions and comparing its results to the original closed formulas obtained by the Black-Scholes model. In conclusion, this thesis contributes to the understanding of the twin network and showcases some applications of Deep Learning in Finance, specifically in the calculation of European call options and its respective Greeks.  \nKeywords: Neural Networks (NN), Twin network, Black-Scholes, Option Valuation, Financial Derivative, Greeks, Differential Machine Learning  \nContents  \nIntroduction 1  \n1 Option Pricing and Black-Scholes-Merton Model 3  \n1.1 Option pricing basics ........................... 3  \n1.2 Black-Scholes-Merton model ....................... 4  \n1.3 Black-Scholes-Merton Formula ...................... 6  \n1.4 Greeks ................................... 7  \n1.5 Monte Carlo simulation .......................... 9  \n1.6 Longstaff-Schwartz Least Square MC method ............. 11  \n1.7 Heston Stochastic Volatility Model ................... 12  \n2 Neural Networks 13  \n2.1 Feedforward Neural Networks ...................... 13  \n2.1.1 Architecture of a Neural Network ................ 13  \n2.1.2 Forward Pass ........................... 14  \n2.1.3 Gradient Descent ......................... 15  \n2.1.4 Backpropagation ......................... 15  \n2.1.5 Algorithm of a NN ........................ 18  \n3 Differential Machine Learning 19  \n3.1 Introduction to Differential Machine Learning ............. 19  \n3.2 Automatic Adjoint Differentiation .................... 21  \n3.2.1 Description ............................ 21  \n3.2.2 Implementation .......................... 21  \n3.3 The Model: Twin Network ........................ 23  \n3.4 Benefits of Diff ML ............................ 26  \n4 Implementation of Differential Machine Learning 27  \n4.1 Motivation ................................. 27  \n4.2 Description of the Original Code ..................... 27  \n4.3 Black & Scholes Approach ........................ 30  \n4.4 Heston Model Approach ......................... 33  \n4.5 Modified Code for the Black & Scholes Approach ........... 34  \nConclusion 37  \nBibliography 38","cbCaiuFNYzF2YLNC","https://ap.wps.com/l/cbCaiuFNYzF2YLNC","pdf",777763,1,50,"English","en",105,"# Introduction\n## Option Pricing and Black-Scholes-Merton Model\n## Neural Networks\n## Differential Machine Learning\n## Implementation of Differential Machine Learning\n# Conclusion","[{\"question\":\"What is the main goal of this thesis?\",\"answer\":\"To provide a mathematical understanding of Differential Machine Learning from the referenced paper and to extend the twin network for pricing and Greek calculations in finance.\"},{\"question\":\"How does the thesis validate neural network predictions?\",\"answer\":\"It compares the twin network outputs for Greeks with the original closed-form formulas obtained from the Black-Scholes model.\"},{\"question\":\"Which financial products and sensitivities are analyzed?\",\"answer\":\"The thesis focuses on European call options and computes their corresponding Greeks using deep learning methods.\"}]","An excursion into Differential Machine Learning and Applications to Finance - Bachelor’s Degree Thesis | PDF",1785732465,126,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"an-excursion-into-differential-machine-learning-and-applications-to-finance-bachelors-degree-thesis","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/an-excursion-into-differential-machine-learning-and-applications-to-finance-bachelors-degree-thesis/120879/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main goal of this thesis?","Question",{"text":75,"@type":76},"To provide a mathematical understanding of Differential Machine Learning from the referenced paper and to extend the twin network for pricing and Greek calculations in finance.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the thesis validate neural network predictions?",{"text":80,"@type":76},"It compares the twin network outputs for Greeks with the original closed-form formulas obtained from the Black-Scholes model.",{"name":82,"@type":73,"acceptedAnswer":83},"Which financial products and sensitivities are analyzed?",{"text":84,"@type":76},"The thesis focuses on European call options and computes their corresponding Greeks using deep learning methods.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":21,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]