Toward a coherent Monte Carlo simulation of CVA

Abstract

This paper is devoted to the simulation of the Credit Valuation Adjustment (CVA) using a pure Monte Carlo technique with Malliavin Calculus (MCM). The procedure presented is based on a general theoretical framework that includes a large number of models as well as various contracts, and allows both the computation of CVA and its sensitivity with respect to the different assets. Moreover, we provide the expression of the backward conditional density of assets vector that can be simulated off-line in order to reduce the variance of the CVA estimator. Regarding computational aspects, both complexity and accuracy are studied for MCM and regression methods and compared to the square Monte Carlo benchmark.

Resolving small random symmetric linear systems on graphics processing units

“…Book [7] is one of the first references that presents the industry practices in computing CVA. Among research papers, maybe the most devoted to computing CVA are [1], [14] and [21].…”

“…where R is the recovery made by the counterparty when it defaults, E denotes the expectation operator, P t is the process of the value exposure to the counterparty, τ is the random default time of the counterparty, T is the protection time horizon and the positive part function is denoted by + . As already explained in [1], one of the most important challenges of the CVA comes from the fact that the exposure P t is generally the price of a basket of different contracts that are written with our counterparty. If these contracts can be priced by closed expressions, the CVA can be calculated thanks to a onestage simulation using either the discretization of a partial differential equation or Monte Carlo method as in [10].…”

“…However, when the underlying contracts must be simulated, it is natural from expression (1) to perform a two-stage simulation: The inner stage to compute P t and the outer stage for the CVA. This two-stage simulation leads to the square Monte Carlo simulation used in [1] as a benchmark algorithm.…”

“…When American contracts are involved, the CVA can be simulated thanks to a nested Monte Carlo (NMC) that performs a Dynamic Programming Algorithm (DPA) on the inner trajectories. This procedure constitutes a straight extension of the square Monte Carlo presented in [1] as a benchmark method for CVA on European contracts. Basically, the main ingredient of this extension relies on the resolution of a large number of small random symmetric linear systems.…”

Xva Principles, Nested Monte Carlo Strategies, and Gpu Optimizations

“…Although extremely fast and accurate whenever applicable, this approach is restricted to models written as piecewise time homogeneous Markov chains (for applicability of the matrix exponentiation formula) with at most three factors (unless advanced techniques are used for circumventing the curse of dimensionality), which may not be an option in all banks. Malliavin calculus can be used for extending such an approach to more standard (space continuous) models (see Abbas-Turki, Bouselmi, and Mikou (2014)), but the curse of dimensionality issue remains.…”