Survival Analysis in Affine Setting

We study the survival probability of an homogeneous group of economic agents by adopting the reduced-form approach and assuming an affine evolution of the default intensities. We consider both the cases of continuous and discrete-times observations, and propose a modified likelihood function. We discuss the estimation of the parameters of interest and the forecast of default probabilities. 1 Setting We consider an homogeneous set of economic agents with the aim of studying their default probability. We observe them from a common starting point (say t = 0) and until their respective default or censuring times, either in a continuous way or in a discrete-time setting. We define the censure of an agent as the exit from the sample for any reason independent of the occurrence of default, which is bounded by a fixed future date (say t = T ), time horizon of our observations. For each firm i = 1, . , N , denote τi the default time, ξi the censure time, and Ti = τi ∧ ξi the first-exit time. Moreover, let fi (resp. Fi, Si, hi) be the density function (resp. cdf, survival function, hazard function) associated to τi: Fi(t) = P(τi ≤ t) = ∫ t 0 fi(u)du, Si(t) = 1− Fi(t), hi(t) = fi(t) Si(t) . (1) ∗University of Perugia Dep. Economy Finance and Statistics, Via A.Pascoli 20 06123 Perugia, and Vienna University of Technology Dep. Financial and Actuarial Mathematics, Wiedner Hauptstrasse 8/105-1 A-1040 Vienna. email: beatrice.acciaio@stat.unipg. it, acciaio@fam.tuwien. ac.at. Financial support from the European Science Foundation (ESF) “Advanced Mathematical Methods for Finance” (AMaMeF) under the exchange grant 1192 is gratefully acknowledged. †University of Perugia Dep. Economy Finance and Statistics, Via A.Pascoli 20 06123 Perugia. email: elena.stanghellini@stat.unipg. it 1 In our analysis we assume each default time τi to admit an intensity process λi = {λi(t), t ≥ 0}, meaning that, conditionally on λi, τi is the first jump of an inhomogeneous Poisson process with intensity λi. This leads to the interpretation of λi as the conditional expected default rate (also called hazard-rate in survival analysis) of agent i: given the current value λi(t), for small ∆t > 0 and t < τi, the product λi(t)∆t approximates the probability that agent i defaults in the time interval (t, t + ∆t]: P(τi ≤ t + ∆t|λi(t), τi > t) ≈ λi(t)∆t. This doubly-stochastic property implies that, for any time t, the probability that agent i survives to a given future time s is P(τi > s|Ft) = 1{τi>t}E[e R s t |Ft], (2) where Ft is the σ-algebra that contains all the information available at time t, so that in particular we have Si(t) = E[e− R t 0 i]. (3) Intensity-based models go back to Cox and Isham [5], see also Artzner and Delbaen [1], Duffie and Singleton [10] and Jarrow and Turnbull [12] for applications in this context, as well as Lando [13, Chapter 5] for a survey. Here, as in a large financial literature, we consider affine processes both for their computational tractability and flexibility in capturing certain properties exhibited by many financial time series. Simple examples of use of this type of processes are the Gaussian Ornstein-Uhlenbeck model, applied to interest rates by Vasicek [15], and the Feller diffusion model, applied to interest-rate modeling by Cox, Ingersoll, and Ross [6]. The affine framework provides a powerful class of intensity models that admit closed-form solutions for default probabilities up to the solution of an ordinary differential equation (the reader is referred to Duffie & Kan (1996)). In this setting we can calculate the value in (3) as E[e− R t 0 i] = ei, (4) where the coefficients α(t) and β(t) are solutions of some Riccati ODEs. From this fact we deduce the following representation for the functions in (1): Si(t) = e i, (5) Fi(t) = 1− Si(t) = 1− ei, (6) fi(t) = dFi(t) dt = −(α′(t) + β(t)λi(0))ei, (7) hi(t) = fi(t) Si(t) = −(α′(t) + β(t)λi(0)), (8)