CVA under Bates Model with Stochastic Default Intensity

Abstract

Counterparty credit risk has received increasing attention and become a topical issue since 2007 credit crisis, particularly for its impact on the valuation of the OTC derivatives. Credit Value Adjustment (CVA) has become an import field and it is required in Basel III. This paper studies CVA for European options under Bates model with stochastic default intensity. We develop a Monte Carlo and finite difference method framework for assessing exposure profiles and impact of counterparty credit risk in pricing. The exposures are computed by solving a partial integro-differential Equation (PIDE) using implicit-explicit (IMEX) time discretization schemes. CVA in presence of wrong way risk (WWR) is embedded in the correlation between risk factor and default intensity. Meanwhile, the jump-at-default feature of the models offers an effective means to assess WWR. Our results show that both jump and WWR play an important role in evaluating CVA and exposures. The impact is significant and it is crucial for risk management purpose.

Computing credit valuation adjustment solving coupled PIDEs in the Bates model

“…In this section we propose the results of some numerical experiments, which aim to compare the goodness of the proposed methods. In particular, we compare the standard FDMC method employed by Feng [12], the HTFD-HTMC approach and the C-HTFD approach. We employ the numerical methods according to 4 configurations (A, B, C, D), each of them with an increasing number of steps, determined in order to achieve approximately these run times Table 1.…”

“…This method is based on the direct application of the hybrid techniques to the Finite Difference Monte Carlo (FDMC) method, which was first developed by de Graaf et al [11] for the Heston model and then adapted for the Bates model by Feng [12]. First of all, an estimation of the value function V (t, S, V ) is computed through a grid of values by solving equation (2.5) by employing the Alternating Direction Implicit (ADI) method and specifically the scheme proposed by Haentjens and In't Hout [16].…”

“…Such an approach combines the finite-difference method and the Monte Carlo method to solve a partial differential equation (PDE) and to estimate the mean exposure respectively. Feng [12] adapted the FDMC approach to deal with the case of an underlying evolving according to the Bates model: in this particular case, the PDE to be solved is replaced by a partial integral differential equation (PIDE), which implies an additional computational effort. Other recently introduced alternative approaches consist in employing the fast Fourier transform (Borovykh et al [5]) or marked branching diffusions (Henry-Labordère [17]).…”

Computing Credit Valuation Adjustment solving coupled PIDEs in the Bates model

“…In this section we propose the results of some numerical experiments, which aim to compare the goodness of the proposed methods. In particular, we compare the standard FDMC method employed by Feng [12], the HTFD-HTMC approach and the C-HTFD approach. We employ the numerical methods according to 4 configurations (A, B, C, D), each of them with an increasing number of steps, determined in order to achieve approximately these run times 1 : (A) 0.25 s, (B) 1 s, (C) 4 s, (D) 16 s. The employed mesh and configuration parameters are reported in Table 1.…”