Rough PDEs for local stochastic volatility models

TL;DR

This work develops a robust pricing framework for general non-Markovian local stochastic volatility models by conditioning on the volatility-driving Brownian motion and interpreting the integrated volatility as a rough path. This yields time-inhomogeneous Markov dynamics for the X process and a Feynman-Kac type RPDE that governs conditional option prices, enabling unconditional prices to be obtained by averaging RPDE solutions over volatility samples. The methodology extends to multivariate LSV models and encompasses classical stochastic volatility as well as rough volatility settings, with initial numerical schemes for RPDEs and partial Monte Carlo variance reduction demonstrated on European options. The approach offers a principled, PDE-based pricing tool for non-Markovian models, facilitating efficient computation of prices and Greeks in rough/heterogeneous volatility environments and providing a bridge between rough-path theory and practical financial pricing.

Abstract

In this work, we introduce a novel pricing methodology in general, possibly non-Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time-inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so-called rough partial differential equations (RPDEs), through a Feynman-Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non-Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.

Figures (3)

• Figure 1: Strong relative errors for fixed time step-size $\Delta t = 1/30$ and increasing number of space-steps, for both finite-difference scheme. In (a): Bachelier SV-model Example \ref{['LSVexnum']}, and in (b): SABR SV-model Example \ref{['LSVex1num']} with $\beta = 0.6$.
• Figure 2: The Greeks Delta (left) and Gamma (right) for European put options in the SABR model, using full Monte-Carlo simulation vs. partial Monte-Carlo via RPDE solutions. We choose $\Delta t = 1/120$, $\Delta x = (b-a)/90$, $\rho = -0.4$, $\beta = 0.6$ and $h=0.05$.
• Figure 3: European put option in SABR local stochastic volatility model: Monte-Carlo values of $u^{\mathbf{I}^{(m)}}(0,x)$ vs. finite-difference solutions to the RPDE with first-order ($\times$) and second-order ($\boldsymbol{\cdot}$) schemes, along several sample paths of $(I,[I])$. The black line corresponds to the mean along all $M=10'000$ samples. We choose $\Delta t = 1/120$, $\Delta x = (b-a)/90$, $\rho = -0.4$ and $\beta = 0.6$.

Theorems & Definitions (40)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Proposition 2.5
• Remark 2.6
• Definition 2.7
• Lemma 2.8
• Definition 2.9
• Example 2.10