Credit Derivatives Modelling: Background Intensity

1. Credit Derivatives Modelling: Background Intensity Sebastien Hitier The views expressed here are the authors own, and may not represent those of BNP Paribas group or any of its affiliates.

2. Contents • Some context for credit quantitative modelling • Introducing Background Intensity • Five useful applications of background intensity modelling

3. Some context for credit quantitative modelling • Overview of the mathematical tools used, specificities of the problem • Methodological differences between actuarial and derivatives pricing • Historical perspective in credit quantitative modelling

4. Overview of mathematical tools used • One Period Modelling • Default occurrence is a {0, 1} valued random variable. • Recovery upon default is a [0, 1] valued variable, but little is know about its mean • Credit Portfolio loss is defined by : • For credit portfolio, the goal is to calculate the density of LT • Given the marginals are bernouli variables, joint density of loss is determined by the default copula. • Multi Period Modelling • Default event is modelled as the first jump of a poisson process with intensity l • As default risk varies over time, intensity can be made stochastic, which leads to a survival probability formula that looks like a rates discount factor: • Using stochastic intensity allows to leverage all of the rates modelling implementations

5. Specificities of credit • Empirical studies of credit events • Default events are rare occurrences in the investment grade world, little statistics • Large number of correlated variables • Possible regime changes • Less market depth and liquidity • Need a parsimonious model for credit correlation • Complexity of multi period modelling • Positivity on intensity is a must have, compared to interest rate models, the asset values have jumps. • Just as rates model are made difficult by the discounting being stochastic, credit models are difficult because of the future payoff is conditional on (stochastic) survival • Difficulties raised by multi period models for multiple credit • Theoretical difficulties: What is a correct filtration setup for multiple credit? • Multi period modelling is relatively well known for single name, challenge is multiple names

6. Differences between actuarial and derivatives pricing • Actuarial pricing • based on historical default realisation • parameters are estimated, and realism is more important than parsimony • The goal is to compute an “expectation”. Single period modeling is not an issue. • Derivatives pricing • Derivative pricing entail replication by simpler instruments • Calculating price as a “risk neutral” expectation implies • Multi period modelling is relatively well known for single name, challenge is multiple names

7. Historical perspective in credit quantitative modelling • Initial modeling: • Intensity models (used wherever we need a random arrival time) • Structural models introduced in the 70s (Merton model) when stochastic calculus was introduced in finance • Tools from actuarial finance: copula (used to model joint accidents) • Progress prompted by market developments: • CDS replaced asset swaps as the credit derivative of choice in 98, allow to isolate bond basis • Market develops as new derivatives become available: e.g., credit index • Crisis (Enron, Worldcom) push new regulations (observability of derivatives) which was the first reason to push an index tranche market and base correlation 15 August 2014 7

8. Introducing background intensity • Motivation for the concept of background intensity • The default realisation marker, background filtration and intensity • Revisiting the H hypothesis, and Kusuoka’s “remark” • Generalised credit dynamics and risk neutral dynamics

9. Motivation for the concept of background intensity • Default intensity can always be defined (in distribution sense) w.r. to a filtration: • General modeling: default time is a stopping time. • Default compensator is defined by: • The default intensity is 0 after default. Therefore, it is typically not independent from default occurrence. • Background Intensity • The concept of background intensity obtained by projection onto a background filtration is already well known. But the number of possible subfiltrations is bewildering. What good properties do we seek? • Reduced form models have a natural choice of background filtration, this background intensity is independent of the default realisation variable. • Other models (structural) are often used for multi-name modeling, do they have a background intensity (defined in distribution sense), and what can be said of their systemic risk? • Our goal is to clarify what are the useful hypotheses for background intensity in the context of single name an multi-name pricing.

10. The default realisation marker, background filtration and intensity • Approach can be summarized as follow: • Orthogonalise into • The default marker is defined by: • The useful assumption concerning background intensity is:

11. Revisiting the H hypothesis, and Kusuoka’s “remark” • How does HH1 compare with usual assumptions and H hypothesis? • Decreasing background survival with t • Any martingale is a martingale ( ) • Kusuoka’s remark (as seen by french bankers) • Extremly relevant due to the fundamental difference between finance and measure theory. Finance is about replication after measure change. • Measure change introduces drifts for brownians, for poisson process means change of intensity (see Cont) • H hypothesis not necessary hold under a non Ft adapted measure change • P intensity is the natural default intensity, Q intensity is the market price of instantaneous protection. What is the condition on dQ/dP for the H hypothesis to hold under risk neutral pricing?

12. Generalised credit dynamics and risk neutral dynamics • Definition of hazard rates: • Survival Probability Diffusion and Martingality: • Risk Neutral Intensity • Market price of risk: and • Hedging a T maturity derivative requires T credit delta and instant protection:

13. Generalised credit dynamics and risk neutral dynamics • Under what condition can we talk of a Risk Neutral Intensity? • Valuing a T zero coupon: change from P to QT changes intensity • Change from Qb to QT does not change intensity unless IR term structure jumps on default • Radon-Nikodym derivative needs to be in the background filtration for the H hypothesis to hold under QT when it holds under P • What is the subfiltration of choice? • General dynamics are valid under any subfiltration of Gt • If the subfiltration does not verify HH1, the hazard rates are warped by the projection. We want to study the market dynamics, so we use a subfiltration that verifies HH1

14. Five useful applications of background intensity modelling • General credit dynamics formula, Introducing conditionally independent default • Diversification effect: results on forward loss distribution • Stronger results for spot loss: conditional independence • Introducing the canonical copula for portfolio loss • Properties of the portfolio loss copula

15. General credit dynamics formula, Introducing conditionally independent default • General credit dynamics formula • Introducing conditionally independent default HH2 • Assumption required for arbitrarily large sequence of exchangeable names: Other names hazard rates do not jump on default • When this assumption is not verified, projection on the largest filtration independent to default markers yields pseudo-intensities, not market intensities • Our concern beyond martingality and intensity is to get dynamics that describe moves of market implied probability.

16. Diversification effect: defining systemic information • Background information is a good step towards the definition of systemic information • If there are c arbitrarily large sequences of exchangeable names, and each name hazard rate is driven by a brownian motion, we can define the systemic filtration Fc • Hazard rate still have each an idiosyncratic evolution, but it is expected to diversify away.

17. Diversification effect: results on forward loss distribution • Main result:

18. Stronger results for spot loss: conditional independence • Loss convergence

19. Market info on forward loss distribution The previous result describe the dynamics of the forward loss density. The tranche market is a loss option market What information do we get on loss density from the tranche market? For instance, Dupire formula allows to extract local volatility information from option prices given: Here we do not have a term structure of options on the same underlying, but options on the spot loss Lt where t varies In the case of the coterminal option, we could get information on systemic volatility, in the case of spot loss options, we get only information on systemic intensity. 15 August 2014 19

20. Conclusion on diversification We defined systemic information and showed diversification effects that apply to the forward loss E(LT Gt) We showed that the right variable conditional on which we obtain diversification is the projection onto Fc, which is a martingale We see that the right variable for the spot loss diversification is the same projection indexed by t, which is a predictable process drifting by the systemic intensity. 15 August 2014 20

21. Introducing the canonical copula for portfolio loss • Conditional independence is much stronger than diversification • It allows to price finite discrete portfolio by conditioning on the copula factor.

22. Properties of the portfolio loss copula

23. Conclusion on Canonical Copula The De Finetti theorem already granted us similar results for the distribution of the sum of excheangable bernouli variables (default indicators) What we explicited is the link between survival dynamics and the one factor copula variable obtainable with De Finetti theorem We also obtained results concerning the copula for non exchangeable variables The large pool and exchangeable assumption helped us to Rule out contagion Explicit what happens when no single name is supposed to have a systemic role in the economy But ultimately, we can relax the exchangeability assumption and still obtain results for discrete, heterogeneous portfolios. 15 August 2014 23