Computing Credit Valuation Adjustment solving coupled PIDEs in the Bates model

TL;DR

This paper introduces an efficient numerical method for computing Credit Valuation Adjustment (CVA) under the Bates model, which accounts for stochastic volatility and jumps, by solving coupled PIDEs with a hybrid tree-finite difference approach.

Contribution

It develops a novel hybrid numerical method replacing Monte Carlo steps with finite differences to solve coupled PIDEs for CVA in the Bates model, improving efficiency.

Findings

The proposed method is effective for European options.

The method is reliable for American options.

Numerical tests demonstrate improved accuracy and efficiency.

Abstract

Credit value adjustment (CVA) is the charge applied by financial institutions to the counterparty to cover the risk of losses on a counterpart default event. In this paper we estimate such a premium under the Bates stochastic model (Bates [4]), which considers an underlying affected by both stochastic volatility and random jumps. We propose an efficient method which improves the finite-difference Monte Carlo (FDMC) approach introduced by de Graaf et al. [11]. In particular, the method we propose consists in replacing the Monte Carlo step of the FDMC approach with a finite difference step and the whole method relies on the efficient solution of two coupled partial integro-differential equations (PIDE) which is done by employing the Hybrid Tree-Finite Difference method developed by Briani et al. [6, 7, 8]. Moreover, the direct application of the hybrid techniques in the original FDMC approach is also considered for comparison purposes. Several numerical tests prove the effectiveness and the reliability of the proposed approach when both European and American options are considered.