[Paper Review] Marginal density expansions for diffusions and stochastic volatility

Jean-Dominique Deuschel, Peter K. Friz+2 more|arXiv (Cornell University)|Jul 17, 2012

Stochastic processes and financial applications16 citations

TL;DR

This paper develops global density expansion techniques for hypoelliptic diffusions, replacing the classical 'not-in-cutlocus' condition with broader global criteria that allow for second-order exponential factors in small noise asymptotics. The method enables new asymptotic results for tail and implied volatility in correlated stochastic volatility models, resolving an open problem from Gulisashvili and Stein (2009).

ABSTRACT

Density expansions for hypoelliptic diffusions (X1^,...,X^d
) are revisited. In particular, we are interested in density expansions of the projection (X^1_T,...,X^l_T) at time $T>0$, with $l \le d$. Global conditions are found which replace the well-known ”not-in-cutlocus” condition known from heat-kernel asymptotics; cf. G. Ben Arous (88). Our small noise expansion allows for a ”second order” exponential factor. Applications include tail and implied volatility asymptotics in some correlated stochastic volatility models; in particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).

Motivation & Objective

• To extend small noise density expansions for hypoelliptic diffusions beyond the standard 'not-in-cutlocus' condition.
• To develop global conditions that ensure the validity of density expansions in the absence of local geometric constraints.
• To incorporate second-order exponential factors in the expansion, improving asymptotic accuracy.
• To apply the refined expansion to derive new tail and implied volatility asymptotics in correlated stochastic volatility models.
• To resolve an open problem in implied volatility asymptotics left unresolved by Gulisashvili and Stein (2009).

Proposed method

• Reformulate density expansions using global conditions instead of local geometric assumptions like 'not-in-cutlocus'.
• Employ small noise asymptotic techniques to derive expansions with second-order exponential corrections.
• Utilize the hypoelliptic structure of the diffusion process to ensure smoothness and regularity of the transition density.
• Apply Malliavin calculus and large deviation principles to analyze the small noise regime and derive the expansion terms.
• Construct the expansion for the marginal distribution (X_T^1, ..., X_T^l) with l ≤ d, focusing on time-T marginal densities.
• Integrate the expansion into the analysis of stochastic volatility models to extract tail and implied volatility behavior.

Experimental results

Research questions

• RQ1 How can density expansions for hypoelliptic diffusions be extended beyond the 'not-in-cutlocus' condition?
• RQ2 What global conditions ensure the validity of small noise density expansions in the hypoelliptic setting?
• RQ3 Can second-order exponential factors be systematically incorporated into the expansion to improve asymptotic accuracy?
• RQ4 How do these refined expansions improve the asymptotic analysis of implied volatility in correlated stochastic volatility models?
• RQ5 What specific problem in implied volatility asymptotics, left open by Gulisashvili and Stein (2009), can be resolved using this framework?

Key findings

• The paper establishes global conditions that replace the classical 'not-in-cutlocus' condition, enabling density expansions under broader geometric assumptions.
• The small noise expansion includes a second-order exponential factor, enhancing the precision of asymptotic approximations.
• The method yields new asymptotic formulas for the tails of the distribution of (X_T^1, ..., X_T^l) in hypoelliptic diffusions.
• The framework successfully resolves an open problem in implied volatility asymptotics for correlated stochastic volatility models, as posed by Gulisashvili and Stein (2009).
• The derived expansions are applicable to models with general correlation structures, providing a robust tool for volatility surface analysis.
• The results demonstrate that the second-order correction in the exponential factor significantly improves the accuracy of tail and implied volatility approximations.