MARKOVIAN STOCHASTIC VOLATILITY WITH STOCHASTIC CORRELATION — JOINT CALIBRATION AND CONSISTENCY OF SPX/VIX SHORT-MATURITY SMILES

Abstract

In this paper, we show how to calibrate a general Markovian stochastic volatility model with stochastic correlation to the VIX implied volatility smile and the overall level, slope and curvature of the SPX smile in the [Formula: see text] limit. Explicit formulae are obtained for the asymptotic VIX smile for Heston and SABR-type models with mean reversion, and the Lewis CEV-p-model. We also discuss how the Bass martingale can be used to give an exact fit to a single VIX smile for [Formula: see text]. In the second half of this paper, we derive a more involved integral equation for the correlation function [Formula: see text] to be perfectly consistent with the short-maturity SPX and VIX smiles at all strikes (or all strikes in an interval) as [Formula: see text], and discuss consistency conditions between the wings of the two asymptotic smiles and how to avoid [Formula: see text] for the calibrated [Formula: see text] in practice.