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Microstructural Foundation of Rough Log-Normal Volatility Models

Paul P. Hager, Ulrich Horst, Thomas Wagenhofer, Wei Xu

Abstract

We establish a microstructural foundation of the rough Bergomi model. Specifically, we consider a sequence of order driven financial market models where orders to buy or sell an asset arrive according to a Poisson process and have a long lasting impact on volatility. Using a recently established C-tightness result for càdlàg processes we establish the weak convergence of the price-volatility process to a log-normal rough volatility model. Our weak convergence result is accompanied by weak error rates that employ a recently established Clark-Ocone formula for Poisson processes and turn our microstructure model into viable alternative to classical simulation schemes. The weak error rates strongly hinge on Poisson arrival dynamics and are novel to the rough microstructure literature.

Microstructural Foundation of Rough Log-Normal Volatility Models

Abstract

We establish a microstructural foundation of the rough Bergomi model. Specifically, we consider a sequence of order driven financial market models where orders to buy or sell an asset arrive according to a Poisson process and have a long lasting impact on volatility. Using a recently established C-tightness result for càdlàg processes we establish the weak convergence of the price-volatility process to a log-normal rough volatility model. Our weak convergence result is accompanied by weak error rates that employ a recently established Clark-Ocone formula for Poisson processes and turn our microstructure model into viable alternative to classical simulation schemes. The weak error rates strongly hinge on Poisson arrival dynamics and are novel to the rough microstructure literature.
Paper Structure (32 sections, 29 theorems, 226 equations, 4 figures)

This paper contains 32 sections, 29 theorems, 226 equations, 4 figures.

Table of Contents

  1. Introduction and Problem Formulation
  2. Microstructural Foundation of Rough Bergomi Model
  3. Weak Convergence Rates
  4. Assumptions and Main Results
  5. The Benchmark Model
  6. Main Results
  7. Numerical Simulations
  8. Proof of Weak Convergence
  9. A-priori Estimates
  10. $C$-tightness of $\{\widetilde{V}^{(n),0}\}$
  11. The term $A^{(n)}_{\epsilon}$.
  12. The term $B^{(n)}_{\epsilon}$.
  13. $C$-tightness of $\{ \widetilde{V}^{(n),1} \}$
  14. Convergence for $\widetilde{V}^{(n),1,1}$.
  15. Convergence for $\widetilde{V}^{(n),1,2}$.
  16. ...and 17 more sections

Key Result

Theorem 2.7

The rescaled logarithmic price-volatility process $\{(\widetilde{P}^{(n)},\widetilde{V}^{(n)})\}_{n \in \mathbb{N}}$ converges weakly to the process $(\widetilde{P}^*,\widetilde{V}^*)$ in $\mathbb{D}([0,T];\mathbb{R}^2)$ as $n\to\infty$ where with $\sigma_p$ and $\sigma_v$ from Assumption Ass:Compensator. Here $W$ is a standard Brownian motion and $B^H$ is a fractional Brownian motion with Hurst

Figures (4)

  • Figure 1: Plots of negatively-correlated log-price and Log-volatility of a sample path with Gaussian jumps for the volatility, kernel $\phi_n(t)=(1/n+t)^{H-1/2}$ and scaling parameter $n={10}$.
  • Figure 2: Plots of negatively-correlated log-price and Log-volatility of a sample path with Bernoulli jumps for the volatility, kernel $\phi_n(t)=(1/n+t)^{H-1/2}$ and scaling parameter $n={20}$.
  • Figure 3: Depiction of Skorokhod neighborhoods for two prelimit sample price processes with Gaussian price- and Bernoulli volatility-jumps for $n={200}$, respectively $n=1000$.
  • Figure 4: Weak error for the degree-four Hermite polynomial in the Poisson prelimit price model with the optimized kernel sequence $\{\widehat{\phi}_n\}_{n\in\mathbb{N}}$, shown for $H=0.15$ (left) and $H=0.30$ (right). The blue curve displays the Monte Carlo estimate of the absolute weak error relative to a benchmark computed with the Euler scheme on a time grid of size $5000$, while the dashed black line represents the proven theoretical convergence rate with a fitted constant prefactor. The shaded regions indicate approximate $68\%$, $95\%$, and $99.7\%$ Monte Carlo uncertainty bands around this theoretical prediction. Both panels are shown on log-log scales.

Theorems & Definitions (65)

  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Remark 2.9
  • Lemma 3.1: Kunita_2004, Theorem 2.11
  • Lemma 3.2
  • proof
  • ...and 55 more