Polar Coordinates for the 3/2 Stochastic Volatility Model

ABSTRACT The 3/2 stochastic volatility model is a continuous positive process s with a correlated infinitesimal variance process . The exact definition is provided in the Introduction immediately below. By inspecting the geometry associated with this model, we discover an explicit smooth map from to the punctured plane for which the process satisfies an SDE of a simpler form, with independent Brownian motions and the identity matrix as diffusion coefficient. Moreover, is recoverable from the path by a map that depends only on the distance of from the origin and the winding angle around the origin of . We call the process together with its map to the polar coordinate system for the 3/2 model. We demonstrate the utility of the polar coordinate system by using it to write this model's asymptotic smile for all strikes at t = 0 . We also state a general theorem on obstructions to the existence of a map that trivializes the infinitesimal covariance matrix of a stochastic volatility model.