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

We extend the discrete-time construction of [J. Guyon, 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 Schr\" odinger problem approach of [J. Guyon, Risk, April (2020)] is now formulated in continuous time: among all the (arbitrage-free) jointly calibrated semimartingale models, we aim to build the one that minimizes the relative entropy with respect to a reference SV model. This naturally leads to dispersion-constrained martingale Schr\" odinger bridges, a new type of Schr\" odinger bridges which is constrained by the martingality of the SPX and by the prices of VIX derivatives. A dual problem is derived heuristically and gives rise to a candidate model, which is then jointly calibrated provided a number of conditions are met. The dual problem consists of solving Hamilton––Jacobi––Bellman PDEs and maximizing a concave functional of those solutions over SPX and VIX payoffs. The candidate model is identical to the reference model, but for an extra drift in the SV which explicitly depends on the solution to the dual HJB PDE. Focusing on the practical case where we calibrate to a finite number of vanilla options, we show that the gradient of the dual functional simply reads as the difference between model and market prices and is therefore easily computed using classical finite difference methods, such as the alternating-direction implicit scheme. The dual maximization is then performed by classical least squares techniques and is shown to yield a jointly calibrated model. 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.