The inverse first passage time problem asks whether, for a Brownian motion $B$ and a nonnegative random variable $\zeta$, there exists a time-varying barrier $b$ such that $\mathbb{P}\{B_s>b(s),0\leq s\leq t\}=\mathbb{P}\{\zeta>t\}$. We study a "smoothed" version of this problem and ask whether there is a "barrier" $b$ such that $ \mathbb{E}[\exp(-\lambda\int_0^t\psi(B_s-b(s))\,ds)]=\mathbb{P}\{\zeta >t\}$, where $\lambda$ is a killing rate parameter, and $\psi:\mathbb{R}\to[0,1]$ is a nonincreasing function. We prove that if $\psi$ is suitably smooth, the function $t\mapsto \mathbb{P}\{\zeta>t\}$ is twice continuously differentiable, and the condition $0<-\frac{d\log\mathbb{P}\{\zeta>t\}}{dt}<\lambda$ holds for the hazard rate of $\zeta$, 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.
1. Introduction. Investors are exposed to credit risk, or counterparty risk, due to the possibility that one or more counterparties in a financial agreement will default, that is, not honor their obligations to make certain payments. Counterparty risk has to be taken into account when pricing a transaction or portfolio, and it is necessary to model the occurrence of default jointly with the behavior of asset values.
The default time is sometimes modeled as the first passage time of a credit index process below a barrier. Black and Cox [2] were among the first to use this approach. They define the time of default as the first time the ratio of the value of a firm and the value of its debt falls below a constant level, and they model debt as a zero-coupon bond and the value of the firm as a geometric Brownian motion. In this case, the default time has the distribution of the first-passage time of a Brownian motion (with constant drift) below a certain barrier.
Hull and White [6] model the default time as the first time a Brownian motion hits a given time-dependent barrier. They show that this model gives the correct market credit default swap and bond prices if the time-dependent barrier is chosen so that the first passage time of the Brownian motion has a certain distribution derived from those prices. Given a distribution for the default time, it is usually impossible to find a closed-form expression for the corresponding time-dependent barrier, and numerical methods have to be used.
We adopt a perspective similar to that of Hull and White [6]. Namely, we model the default time as
where the diffusion Y is some credit index process, U is an independent mean one exponentially distributed random variable, 0 ≤ ψ ≤ 1 is a suitably smooth, nonincreasing function with lim x→-∞ ψ(x) = 1 and lim x→+∞ ψ(x) = 0, and λ > 0 is a rate parameter. Then
The random time τ is a “smoothed-out” version of the stopping time of Hull and White; instead of killing Y as soon at it crosses some sharp, timedependent boundary, we kill Y at rate λψ(y -b(t)) if it is in state y ∈ R at time t ≥ 0. That is, lim ∆t↓0 P{τ ∈ (t, t + ∆t) | (Y s ) 0≤s≤t , τ > t}/∆t = λψ(Y t -b(t)).
When the credit index value Y t is large, corresponding to a time t when the counterparty is in sound financial health, the killing rate λψ(Y t -b(t)) is close to 0 and default in an ensuing short period of time is unlikely, whereas the killing rate is close to its maximum possible value, λ, when Y t is low and default is more probable. Note that if we consider a family of [0, 1]-valued, nonincreasing functions ψ that converges to the indicator function of the set {x ∈ R : x < 0} and λ tends to ∞, then the corresponding stopping time τ converges to the Hull and White stopping time inf{t > 0 : Y t < b(t)}.
The hazard rate of the random time τ is .
On the other hand, suppose that ζ is a nonnegative random variable with survival function t → G(t) := P{ζ > t}. Writing g for the derivative of G, the corresponding hazard rate is
As a result, a necessary condition for a function b to exist such that the corresponding random time τ has the same distribution as ζ is that 0 < -g(t) < λG(t), t ≥ 0. (1.4) We show in Theorem 2.1 that if Y is a Brownian motion with a given suitable random initial condition, assumption (1.4) holds, and the survival function G is twice continuously differentiable, then there is a unique differentiable function b such that the stopping time τ has the same distribution as ζ. In particular, we establish that the function b can be determined by solving a system consisting of a parabolic linear PDE with coefficients depending on b and a nonlinear ODE for b with coefficients depending on the solution of the PDE. Note from (1.2) that changing the function b on a set with Lebesgue measure zero does not affect the distribution of τ , and so we have to be careful when we talk about the uniqueness of b. This minor annoyance does not appear if we restrict to continuous b.
In Theorem 4.1 we give an analogue of the existence part of the above result when ψ is the indicator of the set {x ∈ R : x < 0}.
Having proven the existence and uniqueness of a barrier b, we consider the pricing of certain contingent claims in Section 5. For simplicity, we take the asset price (X t ) t≥0 to be a geometric Brownian motion
where W is a standard Brownian motion. We take the credit index (Y t ) t≥0 to be given by
where B is another standard Brownian motion, and take the default time to be given by (1.1), where the exponential random variable U is independent of the asset price X and the credit index Y . We assume that the Brownian motions W and B are correlated; that is, that their covariation is [B, W ] t = ρt for some constant ρ ∈ [-1, 1]. We consider claims with a payoff of the form F (X T )1{τ > T } for some fixed maturity T . We show how it is possible to compute conditional expected values such as
In Section 6 we report the results of some experiments where we
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