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Mathematics of Differential Machine Learning in Derivative Pricing and Hedging

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  • Pedro Duarte Gomes

Abstract

This 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. This 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. The 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. By anchoring the insights in rigorous mathematics, the article bridges the gap between abstract financial concepts and practical algorithmic implementations.

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  • Pedro Duarte Gomes, 2024. "Mathematics of Differential Machine Learning in Derivative Pricing and Hedging," Papers 2405.01233, arXiv.org.
    • Handle: RePEc:arx:papers:2405.01233
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    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Magnus Grønnegaard Frandsen & Tobias Cramer Pedersen & Rolf Poulsen, 2022. "Delta force: option pricing with differential machine learning," Digital Finance, Springer, vol. 4(1), pages 1-15, March.
    3. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    5. Giorgio Gnecco, 2012. "A Comparison between Fixed-Basis and Variable-Basis Schemes for Function Approximation and Functional Optimization," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-17, January.
    6. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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