Paper: arXiv 2311.07211

Abstract

In this work, we present a novel machine learning approach for pricing high-dimensional American options based on the modified Gaussian process regression (GPR). We incorporate deep kernel learning and sparse variational Gaussian processes to address the challenges traditionally associated with GPR. These challenges include its diminished reliability in high-dimensional scenarios and the excessive computational costs associated with processing extensive numbers of simulated paths Our findings indicate that the proposed method surpasses the performance of the least squares Monte Carlo method in high-dimensional scenarios, particularly when the underlying assets are modeled by Merton’s jump diffusion model. Moreover, our approach does not exhibit a significant increase in computational time as the number of dimensions grows. Consequently, this method emerges as a potential tool for alleviating the challenges posed by the curse of dimensionality.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper involves advanced mathematical concepts like Gaussian Processes, deep kernel learning, and variational inference, with detailed derivations in the theory sections. It also presents empirical experiments comparing performance against Least Squares Monte Carlo under jump diffusion models, discussing computational time and curse of dimensionality, indicating a backtest-ready approach.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Pricing High-dim American Options"] --> B["Data & Model Setup<br>Simulated Paths, Merton Jump Diffusion"]
  B --> C["Core Methodology<br>Modified Gaussian Process Regression"]
  C --> D["Key Innovations<br>Deep Kernel Learning & Sparse Variational GPs"]
  D --> E["Computational Process<br>Addressing Curse of Dimensionality"]
  E --> F["Key Findings<br>Outperforms LS Monte Carlo"]
  F --> G["Outcome<br>Scalable, Computationally Efficient"]

Cited References (29)

  1. Valuation of the early-exercise price for options using simulations and nonparametric regression (1996)
  2. Regression methods for pricing complex American-style options (2001)
  3. Valuing American options by simulation: a simple least-squares approach (2001)
  4. Kernel-based Monte Carlo simulation for American option pricing (2009)
  5. Pricing high-dimensional American options by kernel ridge regression (2020)
  6. American option pricing with machine learning: An extension of the Longstaff-Schwartz method (2021)
  7. Pricing of High-Dimensional American Options by Neural Networks (2010)
  8. Neural network regression for Bermudan option pricing (2021)
and 21 more
  1. Pricing Bermudan options using regression trees/random forests (2023)
  2. Machine learning for quantitative finance: fast derivative pricing, hedging and fitting (2018)
  3. Gaussian process regression for derivative portfolio modeling and application to credit valuation adjustment computations (2020)
  4. KrigHedge: Gaussian process surrogates for delta hedging (2021)
  5. Kriging metamodels and experimental design for Bermudan option pricing (2018)
  6. Variance reduction applied to machine learning for pricing Bermudan/American options in high dimension (2019) — link
  7. Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models (2020)
  8. DGM: A deep learning algorithm for solving partial differential equations (2018)
  9. Deep kernel learning (2016)
  10. Stochastic variational deep kernel learning (2016)
  11. Variational learning of inducing variables in sparse Gaussian processes (2009)
  12. Gaussian Processes for Big Data (2013)
  13. Gaussian Processes for Machine Learning (2005) — link
  14. Universal Kernels. (2006)
  15. Gaussian processes and kernel methods: A review on connections and equivalences (2018) — link
  16. On the Surprising Behavior of Distance Metrics in High Dimensional Space (2001)
  17. Gpytorch: Blackbox matrix-matrix gaussian process inference with gpu acceleration (2018)
  18. Deep optimal stopping (2019)
  19. PyTorch: An Imperative Style, High-Performance Deep Learning Library (2019)
  20. Scikit-learn: Machine learning in Python (2011)
  21. Scalable variational Gaussian process classification (2015)

0 of 29 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.