Killed Brownian motion with a prescribed lifetime distribution and models of default

Abstract

The inverse first passage time problem asks whether, for a Brownian motion B and a nonnegative random variable ζ, there exists a time-varying barrier b such that P{Bs > b(s), 0 ≤ s ≤ t} = P{ζ > t}. We study a "smoothed" version of this problem and ask whether therewhere λ is a killing rate parameter, and ψ : R → [0, 1] is a nonincreasing function. We prove that if ψ is suitably smooth, the function t → P{ζ > t} is twice continuously differentiable, and the condition 0 < − d log P{ζ>t} dt < λ holds for the hazard rate of ζ, then there exists a unique continuously differentiable function b solving the smoothed problem. We show how this result leads to flexible models of default for which it is possible to compute expected values of contingent claims.

The Inverse First Passage Time Problem for killed Brownian motion

“…In [EEH14] the IFPT for the random time τ defined in (1.5) was analyzed thoroughly. We note that τ is an approximation of the more natural choice of stopping time τ .…”

“…Remark 1.4. It was key in the arguments from [EEH14] to assume that ψ was a smooth enough approximation of the indicator 1 (−∞,0] . Theorem 4.1 from [EEH14] shows that there exists a solution to the IFPT problem for τ .…”

“…It was key in the arguments from [EEH14] to assume that ψ was a smooth enough approximation of the indicator 1 (−∞,0] . Theorem 4.1 from [EEH14] shows that there exists a solution to the IFPT problem for τ . However, it does not yield the uniqueness of the barrier function b nor any regularity properties.…”

Explicit solution of an inverse first-passage time problem for Lévy processes and counterparty credit risk

“…Avellaneda and Zhu [3] derive a free boundary problem for the density of a diffusion killed upon first hitting the boundary, where the free boundary is the solution to the IFPT, and Cheng et al [12] established the existence and uniqueness of a solution to this free-boundary problem. A related "smoothed" version of the IFPT problem is considered in Ettinger et al [15]: for any prescribed life-time it is shown that there exists a unique continuously differentiable boundary for which a standard Brownian motion killed at a rate that is a given function of this boundary has the prescribed life-time. 3 In this paper, we consider a related inverse problem where the barrier is fixed to be equal to zero, and the problem is to identify in a given family a stochastic process whose first-passage time below the level zero has the given probability distribution.…”

“…The quantification of this type of risk, named counterparty risk, requires the joint modelling of asset values and the risk of default of the company in question (see Cesari et al [10] for background on counterparty risk). Various aspects of the modelling of counterparty risk in default barrier models have been investigated, for instance, in [7,9,15,27,28,31]; in these papers, the model and market quotes are matched by calibration of the model parameters. Next, we present an explicit example of the valuation of a call option under counterparty risk in a default-barrier model that is by construction consistent with a given risk-neutral probability of default, using the solution to the RIFPT problem given in Corollary 2.7.…”