Calibration of local‐stochastic volatility models by optimal transport

Abstract

In this paper, we study a semi-martingale optimal transport problem and its application to the calibration of local-stochastic volatility (LSV) models. Rather than considering the classical constraints on marginal distributions at initial and final time, we optimize our cost function given the prices of a finite number of European options. We formulate the problem as a convex optimization problem, for which we provide a PDE formulation along with its dual counterpart. Then we solve numerically the dual problem, which involves a fully nonlinear Hamilton-Jacobi-Bellman equation. The method is tested by calibrating a Heston-like LSV model with simulated data and foreign exchange market data.

Martingale Benamou–Brenier: A probabilistic perspective

“…Starting with [23] this observation underpinned new interplay between Skorokhod embeddings and robust finance, e.g., [12,15], and subsequently led to the introduction of martingale optimal transport in [7,18] and the ensuing rapid and rich growth of this field. More recently, optimal transport techniques have also been used as means for non-parametric calibration: OT is used as a means to project one's favourite model onto the set of calibrated martingales, i.e., martingales which satisfy a set of given distributional constraints, see [21,22]. In general, this OT-calibration problem is solved via its dual, numerically optimizing over solutions to a nonlinear PDE, which can be challenging.…”

Optimal Control of Diffusion Processes with Terminal Constraint in Law

“…There has recently been a surge of interest for this kind of problems. For instance [22] and [21] use similar formulations to study respectively the problem of calibration of local-stochastic volatility models and the problem of portfolio allocation with prescribed terminal wealth distribution. Probability constraints of the form P rhpX T q ď 0s ď 1 ´ǫ also fall into our analysis since they can be written as functions of the law LpX T q of X T .…”

“…We refer to the works of Blaquière [7] and more recently, Mikami [29] and Mikami and Thieullen [30] where similar approaches are developed in connection with the so-called Schrödinger problem. This approach has been followed recently by Guo, Loeper and Wang [22] and Guo, Loper, Langrené and Ning [21] for problems with various expectation constraints. In both paper, the authors show that their original problem is in duality with a problem of optimal control of sub-solutions of an HJB equation.…”