Diagnóstico RELANSA S.A.C:

We consider a probability space (Ω,F, P) and a random time to default τ defined on this space. As usual, we denote F(t) = P(τ ≤t) and S(t) := 1−F(t) as the corresponding survival probability function. Again we define the jump-to-default indicator process (Xt) associated withτ byXt=I_{τ≤t} fort≥0.

We assume that the only observable quantity is the random timeτ or, equivalently, the associated jump indicator process (Xt). The appropriate filtration is therefore

given by (Ht) with

Ht=σ({Xu :u≤t}).

By definition,τ is an (Ht)-stopping time, as{τ ≤t}={Xt= 1} ∈ Ht for all t≥0;

moreover (Ht) is the smallest filtration with this property.

The following Lemma, stated without proof, lays out a useful way of expressing expectations conditional on the filtration (Ht).

Lemma 5.7.1. Letτ be a random time with jump indicator processXt=I_{τ≤t} and

Chapter 5 §5.7 Reduced-Form Credit Risk Models 76 we have E(I_{τ >t}V|Ht ) =I_{τ >t} E(V;τ > t) P(τ > t) .

Proposition 5.7.1. Letτ be a random time with absolutely continuous distribution functionF(t) and hazard-rate function λ(t). Then M˜t:=Xt−

∫t∧τ

0 λ(s)ds, t≥0,

is an (Ht)-martingale, where H is a Ht-measurable random variable.

The proofs of Lemma 5.7.1 and Proposition 5.7.1 can be found in [12].

Additional information is typically generated by background processes, often mod- elled as diffusions or continuous-time Markov chains, representing, for instance, eco- nomic activity in a country or in an industry sector, risk-free interest rates or rating transitions between the non-default states. Formally, we represent this additional information by some filtration (Ft) on (Ω,F, P).

This leads us to introduce a new filtration (Gt) by

Gt=Ft∨ Ht, t≥0,

meaning thatGt is the smallest σ-algebra that contains Ft and Ht. Obviously τ is

an (Ht) stopping time and hence also a (Gt)-stopping time. In the context of credit

risk models the filtration (Gt) contains information about the background processes

and the occurrence or non-occurrence of default up to time t, and thus typically corresponds to the information available to investors.

The following Lemma is stated without proof in [12].

Lemma 5.7.2.For everyGt-measurable random variableV there is someFt-measurable

random variable V˜ such thatV I_{{τ >t}}= ˜V I_{{τ >t}}.

In economic terms this Lemma tells us that before a default occurs all informa- tion is generated by the background filtration (Ft). Now we turn to conditional

expectations with respect to Gt.

Lemma 5.7.3. For every integrable random variable V we have

E(I_{{τ >t}}V|Gt ) =I_{{τ >t}}E ( I_{τ >t}V|Ft ) P(τ > t|Ft) .

Note that Lemma 5.7.3 allows us to replace certain conditional expectations with respect to Gt by conditional expectations with respect to background information

Ft.

Proof. E(I_{τ >t}V|Gt

)

isGt-measurable and zero on {τ ≤t}. By Lemma 5.7.2

there is therefore anFt-measurable random variable ˜Z such that E

(

I_{{τ >t}}V|Gt

)

=

I_{{τ >t}}Z˜. Since Ft⊂ Gt, taking conditional expectations with respect toFt yields

E(I_{τ >t}V|Ft

)

Chapter 5 §5.7 Reduced-Form Credit Risk Models 77

Hence ˜Z =E(I_{τ >t}V|Ft

)

/P(τ > t|Ft), which proves the lemma. 2

We can extend the reduced-form models by replacing the usually deterministic haz- ard rates with stochastically-modelled hazard rates. These are referred to as models with doubly stochastic random times - also calledconditional Poisson orCox ran- dom times in the literature. However, before we formally define doubly stochastic random times, let us first introduce the cumulative hazard rate.

Definition 5.7.1.Cumulative hazard function. The functionΛ (t) :=−ln (S(t))

is called the cumulative hazard function of the random time τ. If F is absolutely continuous with density f, then the corresponding hazard rate function is λ(t) :=

f(t)/(1−F(t)) =f(t)/S(t).

Definition 5.7.2. Doubly stochastic random times A random timeτ is called doubly stochastic with respect to the background filtration (Ft) if it admits the (Ft)-

conditional hazard-rate process (λt), if Λ(t) is strictly increasing, and if, for all

t >0,

P(τ ≤t|F_∞) =P(τ ≤t|Ft).

The above conditioning means that, given the past values of the information process, the future values do not contain any extra information for predicting the probability that the default time τ occurs before some future time t.

We have seen in proposition 5.7.1 that the jump indicator process (Xt) can be turned

into an (Ht)-martingale if we subtract the process

∫t∧τ

0 λ(s)ds. Here we generalize

this result to doubly stochastic random times.

Proposition 5.7.2. Letτ be a doubly stochastic random time with(Ft)-conditional

hazard-rate process(λt). Then M˜t:=Xt−

∫t∧τ

0 λsds is a (Gt)-martingale.

Given the set-up in this section, a non-negative (Gt)-adapted process (λt) is called

a (Gt)-martingale intensity process of the random time τ if ˜Mt:=Xt−

∫t∧τ

0 λsdsis

a (Gt)-martingale.

In reduced-form credit risk models, (λt) is usually called thedefault intensity of the

default timeτ. This martingale intensity is uniquely defined on {t < τ}.

We can also model doubly stochastic random times through a factor model with hazard rate λt = h(Ψt) (under a risk-neutral measure Q). Here (Ψ) is some d-

dimensional process representing economic factors, which is adapted to the back- ground filtration (Ft);h is a function from Rd toR+.

The risk-neutral default probability of a corporation can be estimated from credit- spread data for bonds issued by that corporation. Market quotes for CDS spreads can also be used to infer risk-neutral default probabilities.

Chapter 5 §5.7 Reduced-Form Credit Risk Models 78

Martingale Modelling

In a complete market, the only thing that matters for the pricing of derivative securities is theQ-dynamics of the traded underlying assets. When building a model for pricing derivatives it is therefore a natural shortcut to model the objects of interest - such as interest rates, default times and the price processes of traded bonds - directly under some exogenously specified measure Q. This approach is calledmartingale modelling.

Denoting by B(t) > 0 the default-free savings account and by Gt the information

available to investors at timet, we have the following formula for the price at time

t≤T of a security whose value atT is given by theFt-measurable random variable

H≥0: Ht=B(t)EQ ( B(T)−1H|Gt ) .

Martingale modelling ensures that the resulting model is arbitrage free. However, as pointed out in [12], it has two drawbacks. First, historical information is, to a large extent, useless in estimating model parameters, as these may change in the transition from real-world measure to equivalent martingale measure. Second, realistic models for pricing credit derivatives are typically incomplete, so that one cannot eliminate all risk by dynamic hedging. In those situations one is interested in the distribution of the remaining risk under the actual risk measureP, so martingale modelling alone is not sufficient. In summary, the martingale-modelling approach is most suitable in situations where the market for underlying securities is reasonably liquid.