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Functional quantization of rough volatility and applications to volatility derivatives

Ofelia Bonesini, Giorgia Callegaro, Antoine Jacquier

Abstract

We develop a product functional quantization of rough volatility. Since the quantizers can be computed offline, this new technique, built on the insightful works by Luschgy and Pages, becomes a strong competitor in the new arena of numerical tools for rough volatility. We concentrate our numerical analysis to pricing VIX Futures in the rough Bergomi model and compare our results to other recently suggested benchmarks.

Functional quantization of rough volatility and applications to volatility derivatives

Abstract

We develop a product functional quantization of rough volatility. Since the quantizers can be computed offline, this new technique, built on the insightful works by Luschgy and Pages, becomes a strong competitor in the new arena of numerical tools for rough volatility. We concentrate our numerical analysis to pricing VIX Futures in the rough Bergomi model and compare our results to other recently suggested benchmarks.

Paper Structure

This paper contains 32 sections, 13 theorems, 122 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Gaussian Volterra processes on $\mathbb R_+$
  3. Series expansion
  4. The Riemann - Liouville case
  5. Functional quantization and error estimation
  6. Stationarity
  7. Application to VIX derivatives in rough Bergomi
  8. The generalized Bergomi model
  9. VIX Futures in the generalized Bergomi
  10. Properties of $Z^{T}$
  11. Series expansion
  12. The truncated RL case
  13. VIX Derivatives Pricing
  14. Quantization error of VIX Derivatives
  15. Numerical results for the RL case
  16. ...and 17 more sections

Key Result

Lemma 2.4

For any $s,t \in [0,1]$, the covariance function of $Y$ is given by

Figures (8)

  • Figure 1: Product functional quantizations of the RL process with N-quantizers, for $H\in \{0.1, 0.25, 0.5\}$, for $N=10$ and $N=100$.
  • Figure 2: Product functional quantization of $Z^{H,T}$ via N-quantizers, with $H=0.1$, $T=0.7$, for $N\in\{10, 100\}$.
  • Figure 3: VIX Futures prices (left) and relative error (right) computed with quantization and with Monte-Carlo as a function of the maturity $T$, for different numbers of trajectories, for each forward variance curve scenario.
  • Figure 4: Log-log (natural logarithm) plot of the empirical absolute error with the theoretically predicted one for Scenario 1, with $T \in \{ 1, 12\}$ months.
  • Figure 5: Prices of ATM Call Options on the VIX via quantization.
  • ...and 3 more figures

Theorems & Definitions (36)

  • Remark 2.2
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • ...and 26 more