# Explicit solution to an inverse first-passage time problem for L\'{e}vy processes. Application to counterparty credit risk

## Author Info

• M. H. A Davis
• M. R. Pistorius
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## Abstract

For a given Markov process $X$ and survival function $\ovl H$ on $\mbb R_+$, the {\em inverse first-passage time problem} (IFPT) is to find a barrier function $b:\mbb R_+\to[-\infty,+\infty]$ such that the survival function of the first-passage time $\tau_b=\inf\{t\ge0: X(t) \leq b(t)\}$ is given by $\ovl H$. In this paper we consider a version of the IFPT problem where the barrier is {\em fixed at zero} and the problem is to find an entrance law $\mu$ and a time-change $I$ such that for the time-changed process $X\circ I$ the IFPT problem is solved by a constant barrier at the level zero. For any L\'{e}vy process $X$ satisfying a Cram\'{e}r assumption, we identify explicitly the solution of this problem, which is given in terms of a quasi-invariant distribution of the process $X$ killed at the epoch of first entrance into the negative half-axis. For a given multi-variate survival function $\ovl H$ of generalised frailty type we construct subsequently an explicit solution to the corresponding IFPT with the barrier level fixed at zero. We apply these results to the valuation of financial contracts that are subject to counterparty credit risk.

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File URL: http://arxiv.org/pdf/1306.2719

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1306.2719.

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Handle: RePEc:arx:papers:1306.2719

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## References

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1. Christophette Blanchet-Scalliet & Fr\'ed\'eric Patras, 2008. "Counterparty risk valuation for CDS," Papers 0807.0309, arXiv.org.
2. Merton, Robert C., 1973. "On the pricing of corporate debt: the risk structure of interest rates," Working papers 684-73., Massachusetts Institute of Technology (MIT), Sloan School of Management.
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