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.

翻译：转移概率密度函数是计算金融学（包括期权定价与对冲）的基础。通过深化深度学习领域的最新研究成果，我们开发了新型神经转移概率密度生成器，通过求解参数空间中的后向柯尔莫哥洛夫方程来生成累积概率函数。这些生成器具有超高速、高精度的特点，可适用于任何由随机微分方程描述的资产模型。该方法属于"单次求解"模式：当随机模型参数（如波动率重校准）发生变化时无需重新训练。训练完成后，神经转移概率密度生成器可迁移至性能较低的计算机，在期权定价等场景下实现与闭式解相当的计算速度。通过将所提出的神经近似转移概率密度函数嵌入数值期权定价方法，我们验证了其计算效率。数值实验涵盖布莱克-斯科尔斯-默顿模型、标准赫斯顿模型、SABR模型以及跳扩散模型等广泛场景，证实了所开发神经转移概率密度生成器兼具超高速与高精度的特性。