Efficient solution of structural default models with correlated jumps and mutual obligations

Abstract

The structural default model of Lipton and Sepp [Credit value adjustment for credit default swaps via the structural default model, J. Credit Risk 5(2) (2009), pp. 123–146] is generalized for a set of banks with mutual interbank liabilities whose assets are driven by correlated Lévy processes with idiosyncratic and common components. The multi-dimensional problem is made tractable via a novel computational method, which generalizes the one-dimensional fractional partial differential equation method of Itkin [Efficient solution of backward jump-diffusion PIDEs with splitting and matrix exponentials, J. Comput. Financ. (2014), forthcoming. Available at http://arxiv.org/abs/1304.3159] to the two- and three-dimensional cases. This method is unconditionally stable and of the second order of approximation in space and time; in addition, for many popular Lévy models it has linear complexity in each dimension. Marginal and joint survival probabilities for two and three banks with mutual liabilities are computed. The effects of mutual liabilities are discussed, and numerical examples are given to illustrate these effects.

Keywords:

• structural default model
• joint survival probability
• marginal survival probability
• mutual liabilities
• Lévy processes
• PIDE
• splitting
• matrix exponential

2010 AMS Subject Classifications:

• 91G40
• 91G60
• 91G80
• 47G30
• 35R09
• 65L12

Acknowledgments

We thank Peter Carr, Darrel Duffie, Peter Forsyth, Igor Halperin and Rajeev Virmani for useful comments. We assume full responsibility for any remaining errors. The views represented herein are the authors' own views and do not necessarily represent the views of BAML or its affiliates and are not a product of BAML Research.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. It should be emphasized that Vasicek model considers a single period setting, whereas Lévy models have to be analysed in continuous time. In addition, Lévy models use infinitely divisible distributions, rather than standard Gaussian random variables.

2. That is, $O(N_1\times N_2)$ in the 2D case and $O(N_1\times N_2\times N_3)$ in the 3D case.

3. Below expression assumes that the bank assets are allowed to be below its liabilities up to some value determined by the recovery rate. In this case there is no default if such a breach is observed at some time before the maturity T. In this setup the default boundary has a kink at $t=T$. Also in a sense of [5] we assume the default barrier to be continuous.

4. In order to better fit the market data, we can replace ${\sigma }_i$ with the local volatility function ${\sigma }_i(t,A_{i,t})$.

5. In particular, this is the case for the Merton, Kou, exponential, CGMY and Meixner models.

6. As we use splitting on financial processes, pure jump models are naturally covered by the same method. In the latter case there is no diffusion at the first and third step of the method, so one has to solve a pure convection equation. This could be achieved by applying various methods known in the fluid mechanics literature [40].