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Published in:
Basics of the Finite Difference Method Read chapter Read first chapter

2017 | OriginalPaper | Chapter

8. Multidimensional Structural Default Models and Correlated Jumps

Author : Andrey Itkin

Published in: Pricing Derivatives Under Lévy Models

Publisher: Springer New York

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Abstract

In this chapter, we extend the MPsDO to the multidimensional case. To make our description more transparent, we use a concrete example, first considered in Itkin and Lipton (Int. J. Comput. Math. 92(12):2380–2405, 2015). In that paper, the structural default model of Lipton and Sepp (J. Credit Risk 5(2):123–146, 2009) is generalized to a set of banks with mutual interbank liabilities whose assets are driven by correlated Lévy processes with idiosyncratic and common components. Below we show how efficient FD schemes can be constructed using the MPsDO under this model in two- and three-dimensional cases. Also, the effects of mutual liabilities are discussed, and numerical examples are given to illustrate these effects.

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previous chapter High-Order Splitting Methods for Forward PDEs and PIDEs
next chapter LSV Models with Stochastic Interest Rates and Correlated Jumps
Footnotes
1
It should be emphasized that the Vasicek model considers a single-period setting, whereas Lévy models have to be analyzed in continuous time. In addition, Lévy models use infinitely divisible distributions, rather than standard Gaussian random variables.
 
2
The expression given below 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.
 
3
In order to better fit the market data, we can replace σ i with the local volatility function σ i (t, A i, t ).
 
4
Since 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 steps of the method, so one has to solve a pure convection equation. This can be achieved by applying various methods known in the fluid mechanics literature; see, e.g., [41].
 
5
By definition of A 2 B , the matrix M 2 is a lower triangular matrix with three nonzero diagonals. The main and the first lower diagonals are positive, and the second lower diagonal is negative. However, the former two dominate the latter.
 
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Metadata
Title
Multidimensional Structural Default Models and Correlated Jumps
Author
Andrey Itkin
Copyright Year
2017
Publisher
Springer New York
Book
Pricing Derivatives Under Lévy Models

Print ISBN: 978-1-4939-6790-2

Electronic ISBN: 978-1-4939-6792-6

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-1-4939-6792-6_8

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