- The paper extends signature-based approaches to solve optimal stopping problems in rough volatility frameworks for American options.
- It demonstrates that deep-signatures yield tight pricing bounds, achieving as low as a 1% duality gap in the rough Bergomi model.
- The research outlines computational trade-offs and motivates future work to improve efficiency in non-Markovian financial modeling.
An Analysis of Pricing American Options under Rough Volatility Using Deep-Signatures and Signature-Kernels
The paper under review presents a comprehensive paper on the application of signature-based methods, specifically employing deep-signatures and signature-kernels, for the pricing of American options in non-Markovian frameworks characterized by rough volatility. The primary focus lies in extending primal and dual solutions to optimal stopping problems, particularly under the challenging conditions presented by rough volatility models such as the rough Heston and rough Bergomi models.
Summary
Signatures of paths, which have their origins in rough path theory, serve as a powerful tool to approximate solutions for stochastic control problems that deviate from Markovian assumptions. The paper discusses how signature-based methods can be leveraged to address the non-Markovian nature of rough volatility processes, which are commonly used in financial modelling but pose significant computational challenges.
The proposed methodology consists of utilizing deep-signature and signature-kernel frameworks to extend traditional approaches to pricing American options, which amount to solving optimal stopping problems. The novelty of this approach is rooted in the ability of signature methods to represent non-Markovian processes efficiently and facilitate numerical approximations that capture the rough characteristics of volatility.
Key Results
- The investigation encompasses both the primal and dual formulations of the optimal stopping problem, which yield lower and upper bounds for option prices. Three signature-based methods were explored: linear functionals of truncated signatures, the application of deep neural networks to truncated signatures, and linear combinations of signature kernels.
- The numerical results indicate that the proposed methods can achieve tight bounds on option prices. In particular, the rough Bergomi model with a Hurst index of 0.07 demonstrated a notably small duality gap of about 1%, indicating the high accuracy of the approach.
- Although no single method consistently outperforms in all scenarios, the deep-signature approach shows competitive results, especially in calculating tighter upper bounds in the dual formulation, where it appears to capture the non-linearity associated with the Doob martingale integrand effectively.
- The paper highlights computational challenges and trade-offs among different methods. The deep-signature approach, while computationally expensive, showed promising improvements in capturing complex dynamics, especially in the rougher volatility regime.
Practical and Theoretical Implications
Practically, this research extends the toolkit available for practitioners in financial mathematics aiming to model and price derivatives in markets characterized by rough volatility. The application of signature methods offers robust alternatives to traditional methods constrained by Markovian assumptions. Theoretically, the paper enhances the understanding of how path signatures can be effectively utilized within kernel regression frameworks to approximate functional outcomes in non-Markovian scenarios. This paves the way for future exploration into more generalized applications of signature methods beyond financial derivatives.
Future Developments
The paper opens numerous avenues for future research, including the exploration of more efficient algorithms to handle the computational burdens observed, particularly with respect to deep neural networks. The development of techniques to further reduce the dimensionality of signatures while maintaining expressiveness will be crucial to improving computational efficiency. Additionally, extending the framework to incorporate other forms of non-Markovian or even path-dependent financial models could illustrate the broader applicability and robustness of signature-based methods in advanced financial engineering contexts.
This paper represents a significant contribution to both the theoretical and practical aspects of option pricing under conditions of rough volatility, proposing advanced mathematical tools to tackle the inherent complexity of non-Markovian financial models.