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Rough differential equations for volatility

Ofelia Bonesini, Emilio Ferrucci, Ioannis Gasteratos, Antoine Jacquier

Abstract

We introduce a canonical way of performing the joint lift of a Brownian motion $W$ and a low-regularity adapted stochastic rough path $\mathbf{X}$, extending [Diehl, Oberhauser and Riedel (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations]. Applying this construction to the case where $\mathbf{X}$ is the canonical lift of a one-dimensional fractional Brownian motion (possibly correlated with $W$) completes the partial rough path of [Fukasawa and Takano (2024). A partial rough path space for rough volatility]. We use this to model rough volatility with the versatile toolkit of rough differential equations (RDEs), namely by taking the price and volatility processes to be the solution to a single RDE. We argue that our framework is already interesting when $W$ and $X$ are independent, as correlation between the price and volatility can be introduced in the dynamics. The lead-lag scheme of [Flint, Hambly, and Lyons (2016). Discretely sampled signals and the rough Hoff process] is extended to our fractional setting as an approximation theory for the rough path in the correlated case. Continuity of the solution map transforms this into a numerical scheme for RDEs. We numerically test this framework and use it to calibrate a simple new rough volatility model to market data.

Rough differential equations for volatility

Abstract

We introduce a canonical way of performing the joint lift of a Brownian motion and a low-regularity adapted stochastic rough path , extending [Diehl, Oberhauser and Riedel (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations]. Applying this construction to the case where is the canonical lift of a one-dimensional fractional Brownian motion (possibly correlated with ) completes the partial rough path of [Fukasawa and Takano (2024). A partial rough path space for rough volatility]. We use this to model rough volatility with the versatile toolkit of rough differential equations (RDEs), namely by taking the price and volatility processes to be the solution to a single RDE. We argue that our framework is already interesting when and are independent, as correlation between the price and volatility can be introduced in the dynamics. The lead-lag scheme of [Flint, Hambly, and Lyons (2016). Discretely sampled signals and the rough Hoff process] is extended to our fractional setting as an approximation theory for the rough path in the correlated case. Continuity of the solution map transforms this into a numerical scheme for RDEs. We numerically test this framework and use it to calibrate a simple new rough volatility model to market data.
Paper Structure (17 sections, 14 theorems, 147 equations, 4 figures, 1 table)

This paper contains 17 sections, 14 theorems, 147 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. General framework, applications overview and frequently used notations.
  3. The Itô lift of an adapted rough path
  4. A rough path for rough volatility
  5. Applications to volatility modelling
  6. Lead-lag approximations
  7. Delayed piecewise linear approximation
  8. Case a): $T_1, T_2$ are partition points
  9. Case b): $T_1, T_2$ are on the same partition interval
  10. Case c): $T_1, T_2$ are on different partition intervals
  11. Hybrid lead-lag approximations
  12. Lagged mollifier approximations
  13. Simulation and calibration
  14. Numerical tests
  15. Pricing in the quadratic RDE Heston model
  16. ...and 2 more sections

Key Result

lemma 1

The set shuffle-generates $\mathrm{Sh} \lceil \mathbb{R}^{d+e} \rceil$ and it does so freely modulo the shuffle relations in $\mathbb{R}^d$ and $\mathbb{R}^e$. Namely, calling $\mathcal{B} \coloneqq \mathrm{span}(\mathfrak{B})$ and given any algebra $A$ and linear map $\phi \colon \mathcal{B} \to A$ restric

Figures (4)

  • Figure 1: Convergence along mesh refinement
  • Figure 2: No lead-lag $\implies$ explosion
  • Figure 3: Correct drift $\iff$ equality with exponential martingale
  • Figure 4: Call option prices and errors (differences divided by $S_0$) in the RDE quadratic rough Heston model for maturity $T = 0.548$ years.

Theorems & Definitions (47)

  • definition 1: Rough path
  • definition 2: Adapted $H$-integrable rough path
  • lemma 1
  • proof
  • definition 3: Itô lift
  • theorem 1
  • proof
  • remark 1: Non-geometric joint lifts
  • proposition 1
  • proof
  • ...and 37 more