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 $\nu$. The exact definition is provided in the Introduction immediately below. By inspecting the geometry associated with this model, we discover an explicit smooth map $\psi$ from ${\left(R^{+}\right)}^2$ to the punctured plane $R^2-(0,0)$ for which the process $(u,v)=\psi (\nu ,s)$ satisfies an SDE of a simpler form, with independent Brownian motions and the identity matrix as diffusion coefficient. Moreover, $({\nu }_t,s_t)$ is recoverable from the path ${(u,v)}_{[0,t]}$ by a map that depends only on the distance of $(u_t,v_t)$ from the origin and the winding angle around the origin of ${(u,v)}_{[0,t]}$. We call the process $(u,v)$ together with its map to $(\nu ,s)$ 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.