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The deep parametric PDE method and applications to option pricing

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  • Glau, Kathrin
  • Wunderlich, Linus

Abstract

We propose, formalise and analyse the deep parametric PDE method to solve high-dimensional parametric partial differential equations with a focus on financial applications. A single neural network approximates the solution of a whole family of PDEs after being trained without the need of sample solutions. As a practical application, we compute option prices and Greeks in the multivariate Black–Scholes model as there is an urgent need for highly efficient methods. After a single training phase, the prices and sensitivities for different times, states and model parameters are available in milliseconds. Exploiting the PDE framework and incorporating a-priori knowledge of no-arbitrage bounds improves the performance significantly. We evaluate the accuracy in the price, the Greeks and the implied volatility with examples of up to 25 dimensions. A comparison with alternative machine learning methods confirms the effectiveness of the new approach and reveals advantages of the underlying PDE formulation.

Suggested Citation

  • Glau, Kathrin & Wunderlich, Linus, 2022. "The deep parametric PDE method and applications to option pricing," Applied Mathematics and Computation, Elsevier, vol. 432(C).
    • Handle: RePEc:eee:apmaco:v:432:y:2022:i:c:s0096300322004295
    DOI: 10.1016/j.amc.2022.127355
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    Cited by:

    1. Christian Bayer & Chiheb Ben Hammouda & Antonis Papapantoleon & Michael Samet & Ra'ul Tempone, 2024. "Quasi-Monte Carlo for Efficient Fourier Pricing of Multi-Asset Options," Papers 2403.02832, arXiv.org.
    2. Guo, Jingjun & Kang, Weiyi & Wang, Yubing, 2024. "Multi-perspective option price forecasting combining parametric and non-parametric pricing models with a new dynamic ensemble framework," Technological Forecasting and Social Change, Elsevier, vol. 204(C).
    3. Dupret, Jean-Loup & Hainaut, Donatien, 2024. "Deep learning for high-dimensional continuous-time stochastic optimal control without explicit solution," LIDAM Discussion Papers ISBA 2024016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Lee, Hangsuck & Ha, Hongjun & Kong, Byungdoo & Lee, Minha, 2024. "Valuing three-asset barrier options and autocallable products via exit probabilities of Brownian bridge," The North American Journal of Economics and Finance, Elsevier, vol. 73(C).
    5. Donatien Hainaut & Alex Casas, 2024. "Option pricing in the Heston model with physics inspired neural networks," Annals of Finance, Springer, vol. 20(3), pages 353-376, September.
    6. Hainaut, Donatien & Casas, Alex, 2024. "Option pricing in the Heston model with Physics inspired neural networks," LIDAM Discussion Papers ISBA 2024002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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