Affine Modelling of Credit Risk, Pricing of Credit Events and Contagion Alain Monfort 1 Fulvio Pegoraro 1,2 Jean-Paul Renne 3 Guillaume Roussellet 4 1 CREST 2 Banque de France and ECB 3 HEC Lausanne 4 McGill University BCB Sao Paulo Conference August 2017 Views presented here are not necessarily those of the Banque de France or of the ECB.
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices Introduction � Credit risk models: pricing of cross-sections of credit-sensitive instruments (bonds and/or CDS, different entities, different maturities). � Term-structure models are tractable if � exp( u ′ w t + h ) � is known in closed form ( A ) E t ( w t = state vector). � Credit-risk model = model of the joint dynamics of w t = [ w ∗ ′ , d ′ t ] ′ : t w ∗ : real-valued common ( y t ) and entity-specific ( x t ) factors ( w ∗ t = [ y ′ t , x ′ t ] ′ ), t : binary default indicators ( d t = [ d 1 , t , . . . , d E , t ] ′ ). d t ′ , d ′ � Not obvious to build general models satisfying (A) with w t = [ w ∗ t ] ′ . t � Standard credit risk models: (over)simplifying assumptions [next slide]. 2 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices Standard credit-risk framework: Assumptions Assumption S.1 (No systemic entities, or “ no-jump condition ”) { d t } does not Granger-causes { w ∗ t } . Assumption S.2 (No contagion) No contagion between entities (notation: w t = { w t , w t − 1 , . . . } ): p ( d i , t | w ∗ p ( d i , t | w ∗ for i � = j . t , d j , t ) = t ) Assumption S.3 (Defaults are not "priced") [SDF] The default events (or credit events) are not priced, in the sense that: M t , t +1 ( w ∗ M t , t +1 ( w t +1 ) = t +1 ) . � �� � � �� � SDF between dates t and t + 1 does not depend on d t 3 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices Standard credit-risk framework: Pricing formula P ( d e , t = 1 | d e , t − 1 = 0 , w ∗ t ) = 1 − exp( − λ e , t ) ( ≈ λ e , t if small) ���� � �� � def. intensity def. probability � If λ e , t , r t and log( M t − 1 , t ) are affine in w ∗ t then date- t price of a zero-coupon bond (ZCB) issued by e : − � h − � h i =1 r t + i − 1 + λ e , t + i ) = E t ( e i =0 u ′ i w ∗ B e ( t , h ) = E Q t + i ) t ( e ( say ) . (1) � Closed-form formula if { w ∗ t } follows an affine process, i.e. if, for all u : E � � = exp � t + b ( u ) � exp( u ′ w ∗ t +1 ) | w ∗ a ( u ) ′ w ∗ . t ⇒ In the standard affine credit-risk framework, tractability is reached under (i) the no-jump condition, (ii) the absence of contagion (between entities), (iii) the absence of default-event pricing. 4 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper � We propose a general affine credit-risk pricing model jointly allowing for: i ) Systemic entities. We break down the no-jump condition. [Collin-Dufresne, Goldstein and Hugonnier (2004)] ii ) Contagion effects between entities. Economic/financial linkages. [Ait-Sahalia, Laeven and Pelizzon (2014)] iii ) Pricing of credit events. Credit spread puzzle. [Gourieroux, Monfort and Renne (2014)] iv ) Flexible specifications of stochastic recovery rates (RR). [Altman and Sironi (2005)] � Properties: Our state vector –including credit-event variables– is affine ⇒ explicit pricing formulas (under flexible RR assumptions). 5 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t Small entity Large DEFAULTS 𝜀 �,��� > 0 entity Common factor (macro): 𝑧 ��� 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t persistence Small Small entity entity Large DEFAULTS 𝜀 �,��� > 0 entity Common factor Common factor (macro): 𝑧 ��� (macro): 𝑧 � persistence 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t persistence Small Small entity entity Large DEFAULTS Large 𝜀 �,��� > 0 entity entity Common factor Common factor (macro): 𝑧 ��� (macro): 𝑧 � persistence 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t persistence Small Small Pr (𝜀 �,� > 0) ↑↓? entity entity Large DEFAULTS Large 𝜀 �,��� > 0 entity entity Common factor Common factor (macro): 𝑧 ��� (macro): 𝑧 � persistence 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t persistence Small Small Pr (𝜀 �,� > 0) ↑↓? entity entity Large DEFAULTS Large 𝜀 �,��� > 0 entity entity Common factor Common factor (macro): 𝑧 ��� (macro): 𝑧 � persistence Impact on SDF and prices: 𝑁 ���,� ↑ 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t persistence Small Small Pr (𝜀 �,� > 0) ↑↓? entity entity Large DEFAULTS Large 𝜀 �,��� > 0 entity entity Common factor Common factor (macro): 𝑧 ��� (macro): 𝑧 � persistence Impact on SDF and prices: 𝑁 ���,� ↑ 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t persistence Small Small entity entity Large DEFAULTS Large 𝜀 �,��� > 0 entity entity Common factor Common factor (macro): 𝑧 ��� (macro): 𝑧 � persistence Impact on SDF and prices: 𝑁 ���,� ↑ 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t persistence Small Small Pr (𝜀 �,� > 0) ↑ entity entity Large DEFAULTS Large 𝜀 �,��� > 0 entity entity Common factor Common factor (macro): 𝑧 ��� (macro): 𝑧 � persistence Impact on SDF and prices: 𝑁 ���,� ↑ 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t persistence Small Small Pr (𝜀 �,� > 0) ↑ entity entity Large Large DEFAULTS Large entity 𝜀 �,��� > 0 entity entity Common factor Common factor (macro): 𝑧 ��� (macro): 𝑧 � persistence Impact on SDF and prices: 𝑁 ���,� ↑ 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Causality scheme Time t-1 Time t persistence Small Small Pr (𝜀 �,� > 0) ↑ entity entity Large Large DEFAULTS Large entity 𝜀 �,��� > 0 entity entity Default pricing Common factor Common factor (macro): 𝑧 ��� (macro): 𝑧 � persistence Impact on SDF and prices: 𝑁 ���,� ↑ 6 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices This paper: Applications 1) Pricing of sovereign CDS in an Epstein-Zin endowment economy. • Consumption affected by sovereign defaults implies that sovereign default events (i.e. disasters) appear in the SDF. � �� � cannot be captured by standard credit-risk models ⇒ Measurement of sovereign credit-event premiums. 2) Pricing of quanto CDS. • Sovereign defaults affect exchange rate. � �� � cannot be captured by standard credit-risk models ⇒ Assessment of market-expected depreciations-at-default. 3) Ability of our model to replicate banks’ CDS spread variations observed in the aftermath of the Lehman bankruptcy. • Usefulness of contagion effects. 7 / 24
Introduction The General Affine Credit Risk Model General Valuation of Defaultable Securities Illustrations Appendices Default Time Modeling Assumption H.1 The default date of any entity e ∈ { 1 , . . . , E } is: � � τ ( e ) = inf t > 0 : δ ( e ) > 0 t where δ ( e ) is called credit-event variable . t 8 / 24
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