[Paper Review] Pathwise asymptotics for Volterra type stochastic volatility models

M. Cellupica, Barbara Pacchiarotti|arXiv (Cornell University)|Feb 15, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance22 references3 citations

TL;DR

This paper establishes a pathwise large deviation principle for log-price processes in Volterra-type stochastic volatility models driven by Gaussian Volterra processes. Using Chaganty's theorem on joint and marginal distributions, it derives a sample path large deviation principle with speed $\varepsilon_n^{-2}$, generalizing prior results to infinite-dimensional settings and allowing for correlated stochastic volatility and drift terms.

ABSTRACT

We study stochastic volatility models in which the volatility process is a positive continuous function of a continuous Volterra stochastic process. We state some pathwise large deviation principles for the scaled log-price.

Motivation & Objective

• To extend large deviation principles to infinite-dimensional Volterra-type stochastic volatility models with general Gaussian Volterra processes.
• To establish a small-noise pathwise large deviation principle for the log-price process under minimal regularity assumptions on the drift and volatility functions.
• To generalize prior results by Forde, Zhang, and Gulisashvili to models with correlated stochastic volatility and non-zero drift.
• To provide a novel analytical framework based on Chaganty's theorem, avoiding reliance on Freidlin-Wentzell theory.
• To demonstrate applicability to fractional models, including fractional Ornstein-Uhlenbeck processes, via covariance structure analysis.

Proposed method

• Formalize the asset price dynamics using a stochastic differential equation with drift $\mu(\hat{B}_t)$ and volatility $\sigma(\hat{B}_t)$, where $\hat{B}_t$ is a Volterra Gaussian process.
• Introduce a small-noise scaling $\varepsilon_n$ to the volatility and the Volterra process, leading to the scaled log-price $Z_t^n = \log S_t^n$.
• Apply Chaganty's theorem on large deviations for joint and marginal distributions to derive the rate function for the scaled log-price process.
• Establish exponential tightness of the approximating family $\hat{B}^n$ to ensure convergence of the large deviation principle.
• Use a two-step approximation: first, define a sequence $Z^{n,m}$ with piecewise-constant approximations of $\hat{B}^n$, then prove exponential equivalence to $Z^n$.
• Derive the explicit good rate function $I_Z(x)$ involving the $H^1_0[0,T]$ norm and a quadratic variation term that depends on the derivative of the path and the volatility structure.

Experimental results

Research questions

• RQ1 Can a pathwise large deviation principle be established for log-price processes in Volterra-type stochastic volatility models with general Gaussian Volterra processes?
• RQ2 How does the inclusion of a non-zero drift term and correlated stochastic volatility affect the large deviation rate function?
• RQ3 Can Chaganty's theorem be used as a viable alternative to Freidlin-Wentzell theory for deriving large deviations in infinite-dimensional settings?
• RQ4 What conditions on the kernel $K(t,s)$ and the functions $\mu$, $\sigma$ ensure the validity of the large deviation principle?
• RQ5 How do fractional models, such as the fractional Ornstein-Uhlenbeck process, fit into this framework and what is their asymptotic behavior?

Key findings

• A pathwise large deviation principle is established for the scaled log-price process $Z_t^n$ with speed $\varepsilon_n^{-2}$, valid for continuous $\mu$ and positive continuous $\sigma$.
• The good rate function is given by $I_Z(x) = \inf_{f \in H_0^1[0,T]} \left[ \frac{1}{2}\|f\|_{H_0^1}^2 + \frac{1}{2}\int_0^T \left( \frac{\dot{x}(t) - \mu(\hat{f}(t)) - \rho \dot{\Psi}(f,\hat{f})(t)}{\bar{\rho} \sigma(\hat{f}(t))} \right)^2 dt \right] $ for $x \in H_0^1[0,T]$, and $+\infty$ otherwise.
• The result generalizes prior work by Forde and Zhang and Gulisashvili to models with correlated volatility and drift, under weaker regularity assumptions.
• Exponential tightness of the Volterra process $\hat{B}^n$ ensures the validity of the large deviation principle even in the infinite-dimensional setting.
• For the fractional Ornstein-Uhlenbeck process, the large deviation speed is $\varepsilon_n^{-2H}$, and the rate function retains the same structure with $\hat{f}(t) = \int_0^t K_H(t,s) \dot{f}(s) ds$.
• The method avoids Freidlin-Wentzell theory and instead relies on Chaganty's theorem, enabling a more direct derivation of the rate function for joint and marginal distributions.