Singular risk-neutral valuation equations

ABSTRACT. Many risk-neutral pricing problems proposed in the nance literature do not admit closed-form expressions and have to be dealt with by solving the corresponding Partial Integro-Dierential Equation. Often, these PIDE's have singular diusion matrices and coefficients that are not Lipschitz continuous up to the boundary. In addition, in general, boundary conditions are not specied. In this paper we prove existence and uniqueness of (continuous) viscosity solutions for linear PIDE's with all the above features, under a Lyapunov type condition. Our results apply to European and Asian option pricing, in jump-diusion stochastic volatility and path dependent volatility models. We verify our Lyapunov type condition in several examples, among which the Arithmetic Asian option in the Heston model.