From the decompositions of a stopping time to risk premium decompositions

ESAIM: PROCEEDINGS AND SURVEYS, June 2017, Vol. 56, p. 1-21 S. Crépey, M. Jeanblanc and A. Nikeghbali Editors FROM THE DECOMPOSITIONS OF A STOPPING TIME TO RISK PREMIUM DECOMPOSITIONS ∗ Delia Coculescu 1 Abstract. The occurrence of some events can impact asset prices and produce losses. The amplitude of these losses are partly determined by the degree of predictability of those events by the market investors, as risk premiums build up in an asset price as a compensation of the anticipated losses. The aim of this paper is to propose a general framework where these phenomena can be properly defined and quantified. Our focus are the default events and the defaultable assets, but the framework could apply to any event whose occurrence impacts some asset prices. We provide the general construction of a default time under the so called (H) hypothesis, which reveals a useful way in which default models can be built, using both market factors and idiosyncratic factors. All the relevant characteristics of a default time (i.e. the Azéma supermartingale and its Doob-Meyer decomposition) are explicitly computed given the information about these factors. We then define the default event risk premiums and the default adjusted probability measure. These concepts are useful for pricing defaultable claims in a framework that includes possible economic shocks, such as jumps of the recovery process or of some default-free assets at the default time. These formulas are not classic and we point out that the knowledge of the default compensator (or the intensity process when the default time is totally inaccessible) is not a sufficient quantity for finding explicit prices; the Azéma supermartingale and its Doob-Meyer decomposition are needed. The progressive enlargement of a filtration framework is the right tool for pricing defaultable claims in non standard frameworks where non defaultable assets or recovery processes may react at the default event. 1. Introduction Negative financial events such as defaults can sometimes be predicted by investors or, on the opposite, they can occur in an abrupt way and produce losses. In this paper, the properties of a stopping time which models a default event are analyzed in relation with the losses that it produces to debt-holders when it occurs, using standard properties of the jump times of martingales. In the ”general theory of processes”, one classifies stopping times as predictable, accessible and totally inaccessible stopping times (see Definition 2.1 below). Traditionally, in the default risk literature structural models have produced default times that are predictable stopping times (for instance the first hitting time of a fixed level by a diffusion), whereas in the reduced form approach defaults are modeled as totally inaccessible stopping times (first jump of a Cox process). The difference between the two classes of stopping times can be ∗ I would like to thank Monique Jeanblanc and an anonymous referee for their comments and suggestions which helped to improve the present text. 1 Department of Banking and Finance University of Zürich, Plattenstrasse 32 Zürich 8032, Switzerland. c EDP Sciences, SMAI 2017 Article published online by EDP Sciences and available at http://www.esaim-proc.org or https://doi.org/10.1051/proc/201756001 2 ESAIM: PROCEEDINGS AND SURVEYS eliminated by a change of the underlying filtration: a predictable stopping time can become totally inaccessible in a smaller filtration, as illustrated by several so-called incomplete information models including [16], [29], [23]. See also the paper [22] where this information-based connection between the structural and reduced-form approaches is explained. Using no arbitrage arguments, we show (Section 2) that defaults that are thought to produce losses for a financial asset do not have a predictable part and it is hence natural to model them as totally inaccessible stopping times in the market filtration. It has become standard to construct reduced-form default models in two steps (as originally proposed in [17], [24]): one begins with a filtration where the default time is not observable, and then obtain the market filtration after progressively enlarging the original filtration so that the default time becomes a stopping time. In Section 3 we present this construction and some fundamental results that we shall subsequently use in the paper. Our leading assumption in this paper will be that in the enlarged filtration the martingales from the original filtration remain martingales, a property known in the literature as the (H) hypothesis. In Section 4 we study some general properties of a stopping time τ (i.e., the default time) under this two-step construction. In particular, we emphasize that there exist decomposing sequences involving some stopping times (T i ), i ≥ 1 of the initial filtration F and a random time avoiding the F stopping times T 0 (Proposition 4.2). We use the properties of these times (i.e., their compensators) in order to characterize the Azéma supermartingale of the default time and give its Doob-Meyer decomposition (Proposition 4.10). This represents a generalization of the classical models, where the Azéma supermartingale is supposed to be continuous. The particular case when τ is totally inaccessible is also analyzed and we give a useful economic interpretation to this decomposition. In Section 5, we apply our results (in particular the decompositions studied in Section 4) in order to obtain pricing formulas for defaultable claims. The usual reduced-form framework is extended in order to include possible economic shocks, and in particular jumps of the recovery process at the default time. Indeed, there has been increasing support in the empirical literature that both the probability of default and the loss given default are correlated and driven by macroeconomic variables (see [2], [6]). A perfect illustration of this phenomenon is the rapid decline in property prices the 2008 US mortgage credit crisis, where defaults coincided with a wave of asset liquidation. Our aim is to extend the usual reduced-form setup in order to include possible jumps in the recoveries at the default time and to give an expression for the risk premiums attached to such jumps. Building on the method developed in [10], we develop pricing formulas which are not classic (precisely because of the possible jumps of some default-free assets at the default time): the knowledge of the default compensator or the intensity process is not anymore a sufficient quantity for finding explicit prices, but we need indeed the Azéma supermartingale and its Doob-Meyer decomposition. We also propose a definition for the default event risk premium, which measures the compensation investors should require for the losses that occur at the default time; this turns out to be strictly positive when the default time is totally inaccessible, but null for a predictable default time. We provide (...truncated)