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Strong Solutions and Quantization-Based Numerical Schemes for a Class of Non-Markovian Volatility Models

Martino Grasselli, Gilles Pagès

TL;DR

The paper addresses non-Markovian volatility models in finance where volatility depends on the past via memory kernels. It introduces a functional quantization framework built on an extended Lamperti transform that converts the diffusion with memory into a Brownian motion plus a drift, and solves the resulting ODEs for quantizer codewords. It establishes existence and uniqueness of strong solutions for the motivating SDEs using a path-dependent SDE theorem, ensuring correctness of the numerical scheme. It then applies the method to two models (GuyonVolMostlyPathDep2022 and platrendek18) with numerical illustrations showing accurate replication of implied volatility smiles and zero-coupon bond pricing, while highlighting the importance of Stratonovich correction terms. The approach yields a computationally efficient alternative to Monte Carlo with clear error structure and potential for further acceleration.

Abstract

We investigate a class of non-Markovian processes that hold particular relevance in the realm of mathematical finance. This family encompasses path-dependent volatility models, including those pioneered by [Platen and Rendek, 2018] and, more recently, by [Guyon and Lekeufack, 2023]. Our study unfolds in two principal phases. In the first phase, we introduce a functional quantization scheme based on an extended version of the Lamperti transformation that we propose to handle the presence of a memory term incorporated into the diffusion coefficient. In the second phase, we study the problem of existence and uniqueness of a strong solution for the SDEs related to the examples that motivate our study, in order to provide a theoretical basis to correctly apply the proposed numerical schemes.

Strong Solutions and Quantization-Based Numerical Schemes for a Class of Non-Markovian Volatility Models

TL;DR

The paper addresses non-Markovian volatility models in finance where volatility depends on the past via memory kernels. It introduces a functional quantization framework built on an extended Lamperti transform that converts the diffusion with memory into a Brownian motion plus a drift, and solves the resulting ODEs for quantizer codewords. It establishes existence and uniqueness of strong solutions for the motivating SDEs using a path-dependent SDE theorem, ensuring correctness of the numerical scheme. It then applies the method to two models (GuyonVolMostlyPathDep2022 and platrendek18) with numerical illustrations showing accurate replication of implied volatility smiles and zero-coupon bond pricing, while highlighting the importance of Stratonovich correction terms. The approach yields a computationally efficient alternative to Monte Carlo with clear error structure and potential for further acceleration.

Abstract

We investigate a class of non-Markovian processes that hold particular relevance in the realm of mathematical finance. This family encompasses path-dependent volatility models, including those pioneered by [Platen and Rendek, 2018] and, more recently, by [Guyon and Lekeufack, 2023]. Our study unfolds in two principal phases. In the first phase, we introduce a functional quantization scheme based on an extended version of the Lamperti transformation that we propose to handle the presence of a memory term incorporated into the diffusion coefficient. In the second phase, we study the problem of existence and uniqueness of a strong solution for the SDEs related to the examples that motivate our study, in order to provide a theoretical basis to correctly apply the proposed numerical schemes.

Paper Structure

This paper contains 23 sections, 1 theorem, 133 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivating examples
  3. Path-Dependent Volatility Models
  4. The Model of platrendek18
  5. Functional Quantization via Lamperti Transform
  6. Step 1: Functional Product Quantization of the Brownian Motion
  7. Step 2: Reduction to a Brownian Motion Plus a Drift
  8. Step 3: Reduction to a System of Ordinary Differential Equations
  9. Application to the Model of GuyonVolMostlyPathDep2022
  10. Existence and Uniqueness of a Strong Solution
  11. Parametric Restrictions for the Positivity of the Volatility Process
  12. Lamperti Transform
  13. Numerical Illustration for the Model of GuyonVolMostlyPathDep2022
  14. Application to the Model of platrendek18
  15. Existence and Uniqueness for the Volatility Process
  16. ...and 8 more sections

Key Result

Proposition 3.1

$(a)$Functional codewords. Assume that $h,g_1,g_2,b, a$ are Borel measurable functions satisfying, for all $(t,y,\tilde{y}^{g_1},\tilde{y}^{g_2},\tilde{y}^{h})\in [0,T]\times \mathbb{R}^4$, the linear growth conditions Assume that $a$ is continuously differentiable with respect to both arguments and $g_2\!\in C^1([0,T]\times\mathbb R)$. Then, the functional codewords of the quantizer of a weak so

Figures (11)

  • Figure 1: Visualization of 100 sample paths of the process $S$ (top panels) and the process $y$ (bottom panels) for the 2-factor specification of the model of GuyonVolMostlyPathDep2022. The left panels correspond to trajectories obtained via functional quantization, while the right panels display trajectories generated from a Monte Carlo simulation with $10^5$ paths. The model parameters are as follows: $\beta_0=0.08;\ \beta_1=-0.08;\ \beta_2=0.5;\ \lambda_1=62;\ \lambda_2=40;\ R_{1,0}=-0.044;\ R_{2,0}=0.007;\ S_0=100;\ r=0;\ T=1$ with $500$ steps for the time discretization and $\sigma_0=y_0 = 0.1254$.
  • Figure 2: Impact of variations in $\lambda_1$ and $\lambda_2$ on the mean and standard deviation of the (time-averaged) codeword $y_t$.
  • Figure 3: Impact of variations in $\beta_1$ and $\beta_2$ on the mean and standard deviation of the (time-averaged) codeword $y_t$.
  • Figure 4: Comparison between implied volatility smiles and skews across a range of maturities, from one month up to twelve months, generated using the model parameters via product functional quantization ("QF smiles" on the top left of the panel), with those obtained from Monte Carlo simulation ("Monte Carlo Smiles", on the top right of the panel) and those obtained when the corrective term is omitted in the functional quantization of the asset price process $S$, while keeping the correctly quantized codewords for the volatility process $y$ ("FQ Smiles -- No Stratonovich correction on S" on the bottom of the panel.
  • Figure 5: One hundred sample paths of the processes $y$ and $\hat{S}$ obtained via functional quantization (left panels) and those generated by Monte Carlo simulation with $10^5$ paths. The model parameters are $\alpha=\beta= \sigma =1, \xi=0.05, \eta=0.002, \lambda=0.2$. $S_0=100;\ r=0;\ T=1$ with $500$ steps for the time discretization.
  • ...and 6 more figures

Theorems & Definitions (3)

  • Definition 3.1
  • Proposition 3.1: Quantization of $Y$
  • proof