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At the Intersection of Deep Sequential Model Framework and State-space Model Framework: Study on Option Pricing

Ziyang Ding, Sayan Mukherjee

TL;DR

This work introduces the Unscented Reservoir Smoother (URS), a framework that unifies reservoir computing (specifically Echo State Networks) with state-space modeling via the Unscented Kalman Filter to address nonlinear, noisy time series while providing uncertainty quantification. URS leverages ESP-compatible ESN dynamics for rich temporal structure and uses the unscented transform to handle nonlinearity, with forward filtering and RTS smoothing to produce robust latent-state estimates and smoothed posteriors. Inference is performed offline via Generalized EM (with joint unscented transform or Taylor linearization) and online via a joint UKF that updates both reservoir weights and latent states. Empirical results on option-pricing tasks demonstrate competitive forecasting accuracy, particularly for longer horizons, and improved uncertainty calibration compared to traditional stochastic-volatility models and LSTM baselines, highlighting URS as a promising bridge between deep sequential models and probabilistic state-space methods.

Abstract

Inference and forecast problems of the nonlinear dynamical system have arisen in a variety of contexts. Reservoir computing and deep sequential models, on the one hand, have demonstrated efficient, robust, and superior performance in modeling simple and chaotic dynamical systems. However, their innate deterministic feature has partially detracted their robustness to noisy system, and their inability to offer uncertainty measurement has also been an insufficiency of the framework. On the other hand, the traditional state-space model framework is robust to noise. It also carries measured uncertainty, forming a just-right complement to the reservoir computing and deep sequential model framework. We propose the unscented reservoir smoother, a model that unifies both deep sequential and state-space models to achieve both frameworks' superiorities. Evaluated in the option pricing setting on top of noisy datasets, URS strikes highly competitive forecasting accuracy, especially those of longer-term, and uncertainty measurement. Further extensions and implications on URS are also discussed to generalize a full integration of both frameworks.

At the Intersection of Deep Sequential Model Framework and State-space Model Framework: Study on Option Pricing

TL;DR

This work introduces the Unscented Reservoir Smoother (URS), a framework that unifies reservoir computing (specifically Echo State Networks) with state-space modeling via the Unscented Kalman Filter to address nonlinear, noisy time series while providing uncertainty quantification. URS leverages ESP-compatible ESN dynamics for rich temporal structure and uses the unscented transform to handle nonlinearity, with forward filtering and RTS smoothing to produce robust latent-state estimates and smoothed posteriors. Inference is performed offline via Generalized EM (with joint unscented transform or Taylor linearization) and online via a joint UKF that updates both reservoir weights and latent states. Empirical results on option-pricing tasks demonstrate competitive forecasting accuracy, particularly for longer horizons, and improved uncertainty calibration compared to traditional stochastic-volatility models and LSTM baselines, highlighting URS as a promising bridge between deep sequential models and probabilistic state-space methods.

Abstract

Inference and forecast problems of the nonlinear dynamical system have arisen in a variety of contexts. Reservoir computing and deep sequential models, on the one hand, have demonstrated efficient, robust, and superior performance in modeling simple and chaotic dynamical systems. However, their innate deterministic feature has partially detracted their robustness to noisy system, and their inability to offer uncertainty measurement has also been an insufficiency of the framework. On the other hand, the traditional state-space model framework is robust to noise. It also carries measured uncertainty, forming a just-right complement to the reservoir computing and deep sequential model framework. We propose the unscented reservoir smoother, a model that unifies both deep sequential and state-space models to achieve both frameworks' superiorities. Evaluated in the option pricing setting on top of noisy datasets, URS strikes highly competitive forecasting accuracy, especially those of longer-term, and uncertainty measurement. Further extensions and implications on URS are also discussed to generalize a full integration of both frameworks.

Paper Structure

This paper contains 33 sections, 1 theorem, 65 equations, 5 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Case Background
  4. Option
  5. Option Pricing: Black-Scholes Model
  6. Model Formulation
  7. Echo State Neural Network
  8. Echo State Property
  9. Modification to the model
  10. Unscented Kalman Filter
  11. Unscented Transform
  12. Forward Filtering
  13. Backward Smoothing
  14. Model Parameter Inference
  15. Offline Parameter Inference: Generalized EM
  16. ...and 18 more sections

Key Result

Proposition 4.1

Jaeger2010 Consider an echo state neural network with $\rho(\mathbf{G}) < 1$. Suppose the network's sigmoidal function is implemented as logistic function Then, $\forall t \in \mathbb{N}$, $\forall \bm{u}_{t_0:+\infty}$, $\forall \boldsymbol{\theta}_t \neq \boldsymbol{\theta}'_t \in \mathcal{H}$ compact, we have

Figures (5)

  • Figure :
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  • Figure :

Theorems & Definitions (3)

  • Definition 4.1
  • Definition 4.2
  • Proposition 4.1