1. Intensity Based Models Advanced Methods of Risk Management Umberto Cherubini

2. Learning Objectives In this lecture you will learn 1.The concept of hazard rate and of intensity 2.Model the dynamics of intensity (Poisson, double stochastic models and Cox models) 3.Extending the model to include positive recovery rates 4.Models incorporating loss-given-default in market data. 5.Calibrating intensities from market data.

3. Reduced form models In the reduced form models, default probability and recovery rate (LGD) are modelled based on statistical assumptions instead of an economic model of the firm. The modelling techniques that are used are very close to those applied in insurance mathematics, specifying the frequency of occurrence of the event (default probability) and the severity of loss in case the event takes place (loss given default) These models use then the concept of intensity and for this reason are also called intensity based

4. Credit spread and survival analysis Denote, in a structural model, Q the probability of survival of the obligor after the maturity of the obligation, (the default probability is then DP = 1 – Q) and LGD the loss given default figure. Then, the credit spread is given by Credit Spread = – ln[1 – (1 – Q )LGD]/(T – t) Assume now the most extreme case in which all the exposure is lost (LGD = 1). We have Credit Spread = – ln[Q]/(T – t) Models from survival analysis (actuarial science) can help design the credit spread.

5. Hazard rates Consider the conditional default probability

6. Assume the default event to be drawn from a Poisson distribution (remember that it describes the probability of a countable set of events in a period of time). The Poisson distribution is characterized by a single parameter, called intensity. The probability that no event takes place before time T (in our case meaning survival probability beyond that)is given by the formula Prob(  > T) = exp (– (T - t)) Poisson model

7. Applying the survival probability function Q = Prob(  > T) = exp ( – (T - t)) to the credit spread formula (again under the assumption LGD = 1) Credit Spread = – ln[Q]/(T – t) we get Credit Spread = Constant intensity model

8. Intensity vs structural Intensity denotes the probability of an event in an infinitesimal interval of time. The expected time before occurrence of the event is 1/. Differently from structural models, the default event comes as a “surprise”. Technically, it is said that default is an inaccessible time. The intensity corresponds to the concept of instantaneous forward rate in interest rate models.

9. If the intensity parameter is not fixed, but changes stochastically with time, the model is called Cox model (or double stochastic models) For every maturity we can consider an average intensity (t,T) and the credit spread curve will be Credit Spread(t,T) = (t,T) Notice that the relationship between, that is the instantaneous intensity, and the average intensity is the same as that between instantaneous spot rate and yield to maturity in term structure models Double stochastic models

10. Survival probability The survival probability beyond time T is recovered simply using the zero coupon bond formula

11. Assume the dynamics of default intensity is described by a diffusive process like d (t) = k( – (t))dt +   dz(t) where setting  = 0, 0.5 deliver standard affine term structure models for the credit spread Debt(t,T) = v(t,T)exp(A(T-t) - B(T -t) (t)) with A and B the affine functions in Vasicek (  = 0) or Cox Ingersoll Ross (  = 0.5) models Affine models

12. If we assume positive recovery rate (and so LGD < 1) and independence between interest rate in default intensity we can easily extend the analysis. Denote  the recovery rate and compute Debt(t,T;  )=v(t,T)[Prob(  > T)+  Prob(   T)] Debt(t,T;  )=  v(t,T) +(1-  ) Prob(  >T)v(t,T) Debt(t,T; 0)= Prob(  >T)v(t,T), from which... Debt(t,T;  )=  v(t,T) +(1-  ) D(t,T; 0) The price is obtained as a portfolio of the risk free asset and a defaultable exposure with recovery rate zero. Positive recovery rate

15. The spread of a BBB 10 exposure over the risk-free yield curve is 45 basis points. Assuming zero recovery rate we get Prob(  >T) = exp (–.0045 10) = 0.955997 and the probability of default is 1 - 0.955997 = 4.4003% Assuming a 50% recovery rate we have Prob(  >T) = [exp (–.0045 10) -  ]/(1-  ) = 0.911995 and default probability is 1 - 0.911995 = 8.8005% Default probabilities

16. Simulating default times F(T 1 ) = P(  > T 1 ) = exp(– T 1 ) = u which is uniformly distributed Generate: u = rnd() Compute T 1 = F -1 (u) = – ln(u)/ For double stochatic models: first simulate the trajectory of (t)

17. Stochastic interest rates Assume discrete time model, time span , stochastic interest rate and loss given default defined in terms of market value. R risky rate, r riskless rate