A Spectral Method Approach to Quadratic Normal Volatility Diffusions

Abstract

This paper is concerned with the quadratic volatility family of driftless stochastic differential equations (SDEs), also known in the literature as Quadratic Normal Volatility models (QNV), which have found applications primarily in mathematical finance, but can also model dynamics of stochastic processes in other fields such as mathematical biology and physics. These SDE models are characterized by a quadratic volatility term and can be reduced to one of four distinct possibilities depending on the roots of the quadratic volatility term and the position of initial value. We find explicit weak solutions for each case by a combination of Itô calculus and Fourier analysis, which can be described as a ’spectral method’. Furthermore, for all cases we also express the weak solutions as fairly simple functions of Brownian motion, which allows for efficient one-step Monte Carlo evaluation of functionals of the solutions Xt of the form E(f(Xt)) and also more general functionals of the solution. The method used to compute the solutions may also be of interest itself more generally in other fields where SDEs play a fundamental role.

Keywords

• stochastic processes
• stochastic differential equations
• Itô calculus
• martingales with continuous parameter
• quadratic volatility