Polar Coordinates for the 3/2 Stochastic Volatility Model

The 3/2 stochastic volatility model is a continuous positive process s with a correlated infinitesimal variance process nu$\nu $. The exact definition is provided in the Introduction immediately below. By inspecting the geometry associated with this model, we discover an explicit smooth map psi$ \psi $ from (R+)2$({\mathbb{R}}<^>+)<^>2 $ to the punctured plane R2-(0,0)${\mathbb{R}}<^>2-(0,0)$ for which the process (u,v)=psi(nu,s)$(u,v)=\psi(\nu,s)$ satisfies an SDE of a simpler form, with independent Brownian motions and the identity matrix as diffusion coefficient. Moreover, (nu t,st)$(\nu_t,s_t)$ is recoverable from the path (u,v)[0,t]$(u,v)_{[0,t]}$ by a map that depends only on the distance of (ut,vt)$(u_t,v_t)$ from the origin and the winding angle around the origin of (u,v)[0,t]$(u,v)_{[0,t]}$. We call the process (u,v)$(u,v)$ together with its map to (nu,s)$(\nu,s)$ 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.