Research & Resources
Papers
Machine Learning for Risk Calculations, Chebyshev Tensors & Deep Neural Nets
Dynamic Sensitivities and Initial Margin via Chebyshev Tensors
Published in Risk Magazine, Cutting Edge Section, March 2021
Denting the FRTB IMA computational challenge via Orthogonal Chebyshev Sliding Technique
Published in Wilmott Magazine, January 2021
Tensoring Volatility Calibration
Published in Wilmott Magazine, July 2024
In this piece of research, we built Chebyshev Tensors — either directly or with the help of the Tensor Extension Algorithms — to tackle the computational bottleneck associated with the calibration of the rough Bergomi volatility model. Results are encouraging as the accuracy of model calibration via Chebyshev Tensors is similar to that when using Deep Neural Nets, but with building efforts that range between 5 and 100 times more efficient in the experiments run. Our tests indicate that when using Chebyshev Tensors, the calibration of the rough Bergomi volatility model is around 40,000 times more efficient than if calibrated via “brute-force” (using the pricing function).
The FRTB-IMA computational challenge for Equity Autocallables
The Orthogonal Chebyshev Sliding Technique was introduced and applied to a portfolio of swaps and swaptions within the context of the FRTB-IMA capital calculation. The computational cost associated to the computation of the ES values – an essential component of the capital calculation under FRTB-IMA – was reduced by more than 90% while passing PLA tests.
This paper extends the use of the Orthogonal Chebyshev Sliding Technique to portfolios of equity autocallables defined over a range of spot underlyings. Results are very positive as computational reductions are of about 95% with passing PLA metrics.
Since equity autocallables are a commonly traded exotic trade type, with significant FRTB-IMA computational costs, the extension presented in this paper constitutes an important step forward in tackling the computational challenges associated to an efficient FRTB-IMA implementation.
By other authors
Chebyshev Interpolation for Parametric Option Pricing
By Maximilian Gaß, Kathrin Glau, Mirco Mahlstedt, Maximilian Mair
Inspired by a series of remarkable papers in recent years that use Deep Neural Nets to substantially speed up the calibration of pricing models, we investigate the use of Chebyshev Tensors instead of Deep Neural Nets. Given that Chebyshev Tensors can be, under certain circumstances, more efficient than Deep Neural Nets at exploring the input space of the function to be approximated — due to their exponential convergence — the problem of calibration of pricing models seems, a priori, a good case where Chebyshev Tensors can excel.
In this piece of research, we built Chebyshev Tensors — either directly or with the help of the Tensor Extension Algorithms — to tackle the computational bottleneck associated with the calibration of the rough Bergomi volatility model. Results are encouraging as the accuracy of model calibration via Chebyshev Tensors is similar to that when using Deep Neural Nets, but with building efforts that range between 5 and 100 times more efficient in the experiments run. Our tests indicate that when using Chebyshev Tensors, the calibration of the rough Bergomi volatility model is around 40,000 times more efficient than if calibrated via “brute-force” (using the pricing function).
Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing
Kathrin Glau, Daniel Kressner, Francesco Statti
Inspired by a series of remarkable papers in recent years that use Deep Neural Nets to substantially speed up the calibration of pricing models, we investigate the use of Chebyshev Tensors instead of Deep Neural Nets. Given that Chebyshev Tensors can be, under certain circumstances, more efficient than Deep Neural Nets at exploring the input space of the function to be approximated — due to their exponential convergence — the problem of calibration of pricing models seems, a priori, a good case where Chebyshev Tensors can excel.
In this piece of research, we built Chebyshev Tensors — either directly or with the help of the Tensor Extension Algorithms — to tackle the computational bottleneck associated with the calibration of the rough Bergomi volatility model. Results are encouraging as the accuracy of model calibration via Chebyshev Tensors is similar to that when using Deep Neural Nets, but with building efforts that range between 5 and 100 times more efficient in the experiments run. Our tests indicate that when using Chebyshev Tensors, the calibration of the rough Bergomi volatility model is around 40,000 times more efficient than if calibrated via “brute-force” (using the pricing function).
The Chebyshev method for the implied volatility
Kathrin Glau, Paul Herold, Dilip B. Madan, Christian Pötz
Inspired by a series of remarkable papers in recent years that use Deep Neural Nets to substantially speed up the calibration of pricing models, we investigate the use of Chebyshev Tensors instead of Deep Neural Nets. Given that Chebyshev Tensors can be, under certain circumstances, more efficient than Deep Neural Nets at exploring the input space of the function to be approximated — due to their exponential convergence — the problem of calibration of pricing models seems, a priori, a good case where Chebyshev Tensors can excel.
In this piece of research, we built Chebyshev Tensors — either directly or with the help of the Tensor Extension Algorithms — to tackle the computational bottleneck associated with the calibration of the rough Bergomi volatility model. Results are encouraging as the accuracy of model calibration via Chebyshev Tensors is similar to that when using Deep Neural Nets, but with building efforts that range between 5 and 100 times more efficient in the experiments run. Our tests indicate that when using Chebyshev Tensors, the calibration of the rough Bergomi volatility model is around 40,000 times more efficient than if calibrated via “brute-force” (using the pricing function).
Deep Learning Volatility
Blanka Horvath, Aitor Muguruza, Mehdi Tomas
Inspired by a series of remarkable papers in recent years that use Deep Neural Nets to substantially speed up the calibration of pricing models, we investigate the use of Chebyshev Tensors instead of Deep Neural Nets. Given that Chebyshev Tensors can be, under certain circumstances, more efficient than Deep Neural Nets at exploring the input space of the function to be approximated — due to their exponential convergence — the problem of calibration of pricing models seems, a priori, a good case where Chebyshev Tensors can excel.
In this piece of research, we built Chebyshev Tensors — either directly or with the help of the Tensor Extension Algorithms — to tackle the computational bottleneck associated with the calibration of the rough Bergomi volatility model. Results are encouraging as the accuracy of model calibration via Chebyshev Tensors is similar to that when using Deep Neural Nets, but with building efforts that range between 5 and 100 times more efficient in the experiments run. Our tests indicate that when using Chebyshev Tensors, the calibration of the rough Bergomi volatility model is around 40,000 times more efficient than if calibrated via “brute-force” (using the pricing function).
Differential Machine Learning
Brian Huge, Antoine Savine
Inspired by a series of remarkable papers in recent years that use Deep Neural Nets to substantially speed up the calibration of pricing models, we investigate the use of Chebyshev Tensors instead of Deep Neural Nets. Given that Chebyshev Tensors can be, under certain circumstances, more efficient than Deep Neural Nets at exploring the input space of the function to be approximated — due to their exponential convergence — the problem of calibration of pricing models seems, a priori, a good case where Chebyshev Tensors can excel.
In this piece of research, we built Chebyshev Tensors — either directly or with the help of the Tensor Extension Algorithms — to tackle the computational bottleneck associated with the calibration of the rough Bergomi volatility model. Results are encouraging as the accuracy of model calibration via Chebyshev Tensors is similar to that when using Deep Neural Nets, but with building efforts that range between 5 and 100 times more efficient in the experiments run. Our tests indicate that when using Chebyshev Tensors, the calibration of the rough Bergomi volatility model is around 40,000 times more efficient than if calibrated via “brute-force” (using the pricing function).
An Extension of Chebfun to two Dimensions
Alex Townsend, Lloyd N. Trefethen
Inspired by a series of remarkable papers in recent years that use Deep Neural Nets to substantially speed up the calibration of pricing models, we investigate the use of Chebyshev Tensors instead of Deep Neural Nets. Given that Chebyshev Tensors can be, under certain circumstances, more efficient than Deep Neural Nets at exploring the input space of the function to be approximated — due to their exponential convergence — the problem of calibration of pricing models seems, a priori, a good case where Chebyshev Tensors can excel.
In this piece of research, we built Chebyshev Tensors — either directly or with the help of the Tensor Extension Algorithms — to tackle the computational bottleneck associated with the calibration of the rough Bergomi volatility model. Results are encouraging as the accuracy of model calibration via Chebyshev Tensors is similar to that when using Deep Neural Nets, but with building efforts that range between 5 and 100 times more efficient in the experiments run. Our tests indicate that when using Chebyshev Tensors, the calibration of the rough Bergomi volatility model is around 40,000 times more efficient than if calibrated via “brute-force” (using the pricing function).
Book
Machine Learning for Risk Calculations
In this book, I. Ruiz and M. Zeron share the line of research they have taken for several years on the topic of how to optimally use Machine Learning models to accelerate and decrease the compute burden of pricing and risk calculations.
Part I – Fundamental Approximation Methods
Chapter 1. Machine Learning
Chapter 2. Deep Neural Networks
Chapter 3. Chebyshev Tensors
Part II – The toolkit, plugging in approximation methods
Chapter 4. Introduction, why a toolkit is needed
Chapter 5. Composition techniques
Chapter 6. Tensors in TT format and tensor extension algorithms
Chapter 7. Sliding technique
Chapter 8. The Jacobian projection technique
Part III – Hybrid solutions, approximations methods and the toolkit
Chapter 9. Introduction to hybrid solutions
Chapter 10. The toolkit and Deep Neural Nets
Chapter 11. The toolkit and Chebyshev Tensors
Chapter 12. Hybrid Deep Neural Nets and Chebyshev Tensors frameworks
Part IV – Applications
Chapter 13. The aim
Chapter 14. When to use Deep Neural Networks and when to use Chebyshev Tensors
Chapter 15. Counterparty credit risk
Chapter 16. Market risk
Chapter 17. Dynamic sensitivities
Chapter 18. Pricing model calibration
Chapter 19. Approximation of the implied volatility function
Chapter 20. Optimisation problems
Chapter 21. Pricing cloning
Chapter 22. XVA sensitivities
Chapter 23. Sensitivities of exotic derivatives
Chapter 24. Software libraries relevant to the book
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