Dispersion-Constrained Martingale Schrödinger Bridges : Joint Entropic Calibration of Stochastic Volatility Models to S&P 500 and VIX Smiles

We extend the discrete-time construction of [Guyon, J.: The Joint S&P 500/VIX Smile Calibration Puzzle Solved, Risk, April 2020] and explain how to build a continuous-time stochastic volatility (SV) model which jointly and exactly calibrates S&P 500 (SPX) smiles, VIX futures, and VIX smiles at discrete dates, via minimum entropy. The Schrödinger problem approach of [Guyon 2020] is now formulated in continuous time: among all the (arbitrage-free) jointly calibrating semimartingale models, we build the one that minimizes the relative entropy with respect to a reference SV model. This naturally leads to "Dispersion-constrained martingale Schrödinger bridges'', a new type of Schrödinger bridges which are constrained by the martingality of the SPX and by the prices of VIX derivatives. The calibrated model is identical to the reference model, but for an extra drift in the SV which, by duality, is written explicitly. The dual formulation consists of solving Hamilton-Jacobi-Bellman PDEs and maximizing a concave functional of those solutions over SPX and VIX payoffs. We show that the gradient of this dual functional simply reads as the difference between model and market prices, and therefore can be easily computed using classical finite difference methods such as the alternating-direction implicit (ADI) scheme. The dual maximization can then be performed by classical least squares techniques. Interestingly, the extra drift in SV is path-dependent in both the SPX and the instantaneous volatility. Numerical results on model-generated data illustrate the accuracy of the method. This article is dedicated to the memory of Marco Avellaneda, a pioneer in minimum entropy calibration