Valuation equations for stochastic volatility models

TL;DR

This paper investigates the valuation PDE in stochastic volatility models with complex behaviors, establishing conditions for the uniqueness of solutions and linking it to the martingale property of the asset-price process.

Contribution

It provides a necessary and sufficient condition for the uniqueness of classical solutions to the valuation PDE in models with degenerate and unbounded coefficients.

Findings

Unique classical solution exists iff the asset-price is a martingale.

The model allows volatility to reach zero and exhibit boundary behaviors.

Conditions for solution uniqueness are characterized precisely.

Abstract

We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of the state space. We allow for various types of model behavior: the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and the asset-price process may be a strict local martingale. Our main result is a necessary and sufficient condition on the uniqueness of classical solutions to the valuation equation: the value function is the unique nonnegative classical solution to the valuation equation among functions with at most linear growth if and only if the asset-price is a martingale.