A guaranteed deterministic approach to margining on exchange-traded derivatives market: Numerical experiment

The article discusses a modern approach to risk management of the central counterparty,primarily the issue of the sufficiency of its financial resources, including the provision of clearingmembers, the capital of the central counterparty and the mutual liability fund. The main subject is the margining system, responsible for an adequate level of collateral for clearing members, that plays critical role in risk management, being the vanguard in protecting against losses associated with default by clearing members and the most sensitive to market risk part of the central counterparty’s skin of the game. A system of margining a portfolio of options and futures in the derivatives market is described, with default management based on the methodology proposed by a number of inventors, registered in 2004. For this system, a mathematical model of margining (i.e. determining the required level of the collateral) is built, based on the ideology of a guaranteed deterministic approach to superhedging: Bellman–Isaacs equations are derived from the economic meaning of the problem. A form of these equations, convenient for calculations, is obtained. Lipschitz constants for the solutions of Bellman–Isaacs equations are estimated. A computational framework for efficient numerical solution of these equations is created. Numerical experiments are carried out on some model examples to demonstrate the efficiency of the system. These experiments also show practical implications of marginsubadditivity — a crucial property of the mathematical model.

Unified analysis of discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations

We provide a unified analysis of a posteriori and a priori error bounds for a broad class of discontinuous Galerkin and $C^0$-IP finite element approximations of fully nonlinear second-order elliptic Hamilton--Jacobi--Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions in $H^2$ of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for the a priori error analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda--Talenti inequality to piecewise polynomial vector fields.

Probabilistic interpretation of a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations

By introducing a stochastic differential game whose dynamics and multi-dimensional cost functionals form a multi-dimensional coupled forward-backward stochastic differential equation with jumps, we give a probabilistic interpretation to a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations. For this, we generalize the definition of the lower value function initially defined only for deterministic times $t$ and states $x$ to stopping times $\tau$ and random variables $\eta\in L^2(\Omega,\mathcal {F}_\tau,P; \mathbb{R})$. The generalization plays a key role in the proof of a strong dynamic programming principle. This strong dynamic programming principle allows us to show that the lower value function is a viscosity solution of our system of multi-dimensional coupled Hamilton-Jacobi-Bellman-Isaacs equations. The uniqueness is obtained for a particular but important case.