Computing credit valuation adjustment solving coupled PIDEs in the Bates model

Abstract

Credit Value Adjustment is the charge applied by financial institutions to the counter-party 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 in The Review of Financial Studies 9(1): 69–107, 1996), 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. (Journal of Computational Finance 21, 2017) 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 which is done by employing the Hybrid Tree-Finite Difference method developed by Briani et al. (arXiv:1603.07225 2016;IMA Journal of Management Mathematics 28(4): 467–500, 2017;The Journal of Computational Finance 21(3): 1–45, 2017). 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. Subject classification numbers as needed.

A proof of PIDE (8)

This appendix provides a derivation of the PIDE (8) followed by the the CVA price. First of all, we present the proof for the simple case of an underlying evolving according to the Black–Scholes model. Then we consider the Bates model.

1.1 Black–Scholes model

Let S denote the underlying, which is assumed to evolve according to

$$\begin{aligned} \frac{dS_{t}}{S_{t}}=(r-\eta )dt+\sigma \,dW_{t}, \end{aligned}$$

where $\eta$ denotes the continuous dividend rate, $S_{0}$ is a positive value and $W_{t}$ is a Brownian motion. We assume that counter-party credit risk is diversifiable across a large number of counter-parties. In the case that this assumption is not justified, then the risk-neutral value of the contract can be adjusted using an actuarial premium principle (Gaillardetz and Lakhmiri 2011). Moreover, the fraction of the original counter-parties of the contract who have defaulted before time t is given by

$$\begin{aligned} PD\left( t\right) =1-\exp \left( -\int _{0}^{t}\delta \left( s\right) ds\right) . \end{aligned}$$

Let us consider a financial product which pays $\left( 1-R\right) E\left( t\right)$ if the counter-party defaults at time t and 0 otherwise. Then the value of such a product at time 0 - the discounted value of future cash-flows - is equal to the CVA. Therefore, we can consider the CVA as a derivative itself and we denote its financial value at time t with ${\mathcal {C}}\left( t,S_{t},V_{t}\right)$, that is

$$\begin{aligned} {\mathcal {C}}\left( t,S_{t},V_{t}\right) ={\mathbb {E}}\left[ \int _{t}^{T}D\left( 0,s\right) \left( 1-R\right) E\left( s\right) dPD_{s}|{\mathcal {F}}_{t}\right] , \end{aligned}$$

having $CVA={\mathcal {C}}\left( 0,S_{0},V_{0}\right)$.

Suppose that the writer of a CVA derivative forms a self-financing portfolio portfolio $\varPi$ which, in addition to being short to the CVA, is long x units of the index S, i.e.

$$\begin{aligned} \varPi =-{\mathcal {C}}\left( t,S,V\right) +xS. \end{aligned}$$

Then, by Itô’s lemma,

$$\begin{aligned} d\varPi= & {} -\left[ \frac{\partial {\mathcal {C}}}{\partial t}+\frac{\sigma ^{2}S^{2}}{2}\frac{\partial ^{2}{\mathcal {C}}}{\partial S^{2}}+\mu S\frac{\partial {\mathcal {C}}}{\partial S}\right] dt-\sigma S\frac{\partial {\mathcal {C}}}{\partial S}dW+x\mu Sdt+x\sigma SdW \nonumber \\&\quad -\frac{dPD\left( t\right) }{dt}\left( 1-R\right) E\left( t\right) dt \end{aligned}$$

(17)

where the last term reflects the cash-flows from the fraction of the original counterparties who default between t and $t+dt$. Setting $x=\frac{\partial {\mathcal {C}}}{\partial t}$ in equation (17) gives

$$\begin{aligned} d\varPi =-\left[ \frac{\partial {\mathcal {C}}}{\partial t}+\frac{\sigma ^{2}S^{2}}{2}\frac{\partial ^{2}{\mathcal {C}}}{\partial S^{2}}\right] dt-\frac{dPD\left( t\right) }{dt}\left( 1-R\right) E\left( t\right) dt. \end{aligned}$$

Since the portfolio is now (locally) risk-less, it must earn the risk-free interest rate r. Setting $d\varPi =r\varPi dt$ results in

$$\begin{aligned} -\left[ \frac{\partial {\mathcal {C}}}{\partial t}+\frac{\sigma ^{2}S^{2}}{2}\frac{\partial ^{2}{\mathcal {C}}}{\partial S^{2}}\right] dt-\frac{dPD\left( t\right) }{dt}\left( 1-R\right) E\left( t\right) dt=r\left[ -{\mathcal {C}}+\frac{\partial {\mathcal {C}}}{\partial t}S\right] dt. \end{aligned}$$

Therefore, we get

$$\begin{aligned} \frac{\partial {\mathcal {C}}}{\partial t}+\frac{\sigma ^{2}S^{2}}{2}\frac{\partial ^{2}{\mathcal {C}}}{\partial S^{2}}+rS\frac{\partial {\mathcal {C}}}{\partial S}-r{\mathcal {C}}+\left( 1-R\right) E\left( t\right) \frac{dPD\left( t\right) }{dt}=0. \end{aligned}$$

1.2 Bates model

The proof of PIDE (8) in the case of the Bates model may be done by employing the same approach followed in the previous paragraphs for the Black–Scholes model. Alternatively, we can obtain the PIDE (8) by a straightforward application of the Feynman-Kac formula for Lévy processes (Rong 1997; Glau 2016), which can be applied since the CVA is defined as the expected value of a time integral of function depending on a Lévy process.