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Filling in Missing FX Implied Volatilities with Uncertainties: Improving VAE-Based Volatility Imputation

Achintya Gopal

TL;DR

This work shows that simple modifications to the architecture of the VAE lead to significant imputation performance improvements, and modify the VAE imputation algorithm in order to better handle the uncertainty in data, as well as to obtain accurate uncertainty estimates around imputed values.

Abstract

Missing data is a common problem in finance and often requires methods to fill in the gaps, or in other words, imputation. In this work, we focused on the imputation of missing implied volatilities for FX options. Prior work has used variational autoencoders (VAEs), a neural network-based approach, to solve this problem; however, using stronger classical baselines such as Heston with jumps can significantly outperform their results. We show that simple modifications to the architecture of the VAE lead to significant imputation performance improvements (e.g., in low missingness regimes, nearly cutting the error by half), removing the necessity of using $β$-VAEs. Further, we modify the VAE imputation algorithm in order to better handle the uncertainty in data, as well as to obtain accurate uncertainty estimates around imputed values.

Filling in Missing FX Implied Volatilities with Uncertainties: Improving VAE-Based Volatility Imputation

TL;DR

This work shows that simple modifications to the architecture of the VAE lead to significant imputation performance improvements, and modify the VAE imputation algorithm in order to better handle the uncertainty in data, as well as to obtain accurate uncertainty estimates around imputed values.

Abstract

Missing data is a common problem in finance and often requires methods to fill in the gaps, or in other words, imputation. In this work, we focused on the imputation of missing implied volatilities for FX options. Prior work has used variational autoencoders (VAEs), a neural network-based approach, to solve this problem; however, using stronger classical baselines such as Heston with jumps can significantly outperform their results. We show that simple modifications to the architecture of the VAE lead to significant imputation performance improvements (e.g., in low missingness regimes, nearly cutting the error by half), removing the necessity of using -VAEs. Further, we modify the VAE imputation algorithm in order to better handle the uncertainty in data, as well as to obtain accurate uncertainty estimates around imputed values.

Paper Structure

This paper contains 43 sections, 1 theorem, 32 equations, 28 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background: VAE
  4. Deep Latent Variable Model (DLVM)
  5. Variational Inference (VI)
  6. Reparameterization Trick
  7. Handling Missing Data
  8. Reviewing the Definition of VAE
  9. Why We Do Not Use Beta-VAE
  10. Improving Training Methodology of VAEs
  11. Heteroscedasticity
  12. Importance Weighted Autoencoder (IWAE)
  13. Residual Networks
  14. Intuition of Efficacy of Residual Networks
  15. Distributional Imputation
  16. ...and 28 more sections

Key Result

Theorem 7.1

A set of call prices $\{p_{k, t}\}$ are consistent with no arbitrage (assuming the forward price is equal for all $t$) if and only if

Figures (28)

  • Figure 1: A diagrammatic representation of a vanilla VAE. The two terms on the right side of the diagram are the two terms of the loss function. Interestingly, the KL term is independent of $\theta$ and hence can be calculated before running the decoder (generator) network.
  • Figure 2: Mixture of two Gaussians.
  • Figure 3: Comparison of generations by different VAEs.
  • Figure 4: Mixture of two Gaussians.
  • Figure 5: $\sigma$-VAE
  • ...and 23 more figures

Theorems & Definitions (1)

  • Theorem 7.1