Deep Learning of Transition Probability Densities for Stochastic Asset Models with Applications in Option Pricing

Haozhe Su (), M. V. Tretyakov () and David P. Newton ()
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Haozhe Su: Nottingham Business School, Nottingham Trent University, Nottingham NG1 4FQ, United Kingdom
M. V. Tretyakov: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
David P. Newton: School of Management, University of Bath, Bath BA2 7AY, United Kingdom

Management Science, 2025, vol. 71, issue 4, 2922-2952

Abstract: Transition probability density functions (TPDFs) are fundamental to computational finance, including option pricing and hedging. Advancing recent work in deep learning, we develop novel neural TPDF generators through solving backward Kolmogorov equations in parametric space for cumulative probability functions. The generators are ultra-fast, very accurate and can be trained for any asset model described by stochastic differential equations. These are “single solve,” so they do not require retraining when parameters of the stochastic model are changed (e.g., recalibration of volatility). Once trained, the neural TDPF generators can be transferred to less powerful computers where they can be used for e.g. option pricing at speeds as fast as if the TPDF were known in a closed form. We illustrate the computational efficiency of the proposed neural approximations of TPDFs by inserting them into numerical option pricing methods. We demonstrate a wide range of applications including the Black-Scholes-Merton model, the standard Heston model, the SABR model, and jump-diffusion models. These numerical experiments confirm the ultra-fast speed and high accuracy of the developed neural TPDF generators.

Keywords: deep learning; transition probability density; parametric PDEs; neural networks; option pricing (search for similar items in EconPapers)
Date: 2025
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