This paper presents the study of reduced-form approach and hybrid model for the valuation of credit risk. Credit risk arises whenever a borrower is expecting to use future cash flows to pay a current debt. It is closely tied to the potential return of investment, the most notable being that the yields on bonds correlate strongly to their perceived credit risk. Credit risk embedded in a financial transaction, is the risk that at least one of the parties involved in the transaction will suffer a financial loss due to decline in creditworthiness of the counter-party to the transaction or perhaps of some third party. Reduced-form approach is known as intensity-based approach. This is purely probabilistic in nature and technically speaking it has a lot in common with the reliability theory. Here the value of firm is not modeled but specifically the default risk is related either by a deterministic default intensity function or more general by stochastic intensity. Hybrid model combines the structural and intensity-based approaches. While avoiding their difficulties, it picks the best features of both approaches, the economic and intuitive appeal of the structural approach and the tractability and empirical fit of the intensity-based approach.
As stock markets have become more sophisticated, so have their products. The simple buy or sell trades of the early markets have been replaced by more complex financial options and derivatives. These contracts can give investors various opportunities to tailor their deals to their investment needs.
The main emphasis in the intensity-based approach is put on the modelling of the random time of default, as well as evaluating condition expectations under a risk-neutral probability of functionals of the default time and corresponding cash follows. Typically, the random default time is defined as the jump time of some one-jump process.
In recent years, we see a spectacular growth in trading, especially in derivative instruments. There is also an increasing complexity of products in the financial markets with the growing complexity and trading size of financial markets; mathematical models have come to play an increasingly important role in financial decision making, especially in the context of pricing and hedging of derivative instruments. Models have become indispensable tools in the development of new financial products and the management of their risks.
Credit risk is defined as the changes in the credit quality of a borrower. This is called the spread risk. If a borrower has a lower quality ranking we expect that he will be less able to pay off his running-up debt. Therefore credit risk is characterized by two risks: default risk and spread risk. The importance of valuation and hedging models in derivatives markets cannot be over-emphasized. The financial risk can therefore be categorized into four (4) types namely: Market risk, Liquidity risk, Operational risk and Credit risk.
The first category of credit risk models are the ones based on the original framework developed by Merton [
Reduced-form models somewhat differ from each other by the manner in which the recovery rate is parameterized. For example, Jarrow and Turnbull [
For mathematical background, valuation of credit risk, some numerical method for options valuation and stochastic analysis based on the Ito integral, see [
In this approach, the value of the firm’s assets and its capital structure are not model at all, and the credit events are specified in terms of some exogenously specified jump process (as a rule, the recovery rates at default are also given exogenously). We can distinguish between the reduced-form models that are only concerned with the modelling of default time, and that are henceforth referred to as the intensity-based models, and the reduced form models with migrations between credit rating classes called the credit migration models.
The main emphasis in the intensity-based approach is put on the modelling of the random time of default, as well as evaluating condition expectations under a risk-neutral probability of functionals of the default time and corresponding cash follows. Typically, the random default time is defined as the jump time of some one-jump process. As well shall see, a pivotal role in evaluating respective conditional expectations is played by the default intensity process.
Modelling of the intensity process which is also known as the hazard rate process, is the starting point in the intensity approach.
Before going deeper in the analysis of the reduced-form approach, we shall first examine a related technical
question. Suppose we want to evaluate a conditional expectation
a probability space
In financial applications, it is quite natural and convenient to model the filtration G as G = FVH, where h is the filtration that carries full information about default events (that is, events such as
Using the intensity of τ with respect to F.
We study the case where the reference filtration F is trivial, so that it does not carry any information whatsoever. Consequently, we have that G = h. Arguably, this is the simplest possible used in practical financial applications, as it leads to relatively easy calibration of the model.
We start by recalling the notion of a hazard function of a random time. Let τ be a finite, non-negative random time.
Let τ be a finite, non-negative, variable on a probability space
The right continuous cumulative distribution function F of τ satisfies
We also assume that
We introduce the right-continuous jump process
We shall assume throughout that all random variables and processes that are used in what follows satisfy suitable integrability conditions. We begin with the following simple and important result.
Lemma 1
For any
For any
that is,
The hazard function is introduced through the following definition.
Definition 1: The increasing right-continuous function
is called the hazard function of a random time τ.
If the distribution function F is an absolutely continuous function, i.e., if we have
for some function
where we set
The function
Definition 2: The dividend process D of a defaultable contingent claim
D is a process of finite variation and
Note that if default occurs at some date t, the promised dividend
Remark: In principle, the promised payoff X could be incorporated into the promised dividends process C. However, this would inconvenient, since in practice the recovery rules concerning the promised dividend C as the promised claim X are different, in general. For instance, in the case of a defaultable coupon bond, it is frequently postulated that in case of default the future coupons are lost, but a strictly positive fraction of the face value is usually received by the bondholder.
Corollary 2: For any
Corollary 3: Let Y be
If, in addition, the random time τ admits the hazard rate function γ then we have
In particular, for any
and
Lemma 4: The process L, given by the formula
is an h-martingale.
The h-adapted process of finite variation L given by last formula is an h-martingale (for Γ continuous or a discontinuous function).
We examine further important examples of martingales associated with the hazard function, with the assumption that the hazard function Γ of a random time τ is continuous. Also we assume that the cumulative distribution function F is absolutely continuous function, so that the random time τ admits the intensity function γ, our goal is to establish a martingale characterization of γ.
More specifically, we shall check directly that the process
follows and h-martingale. To this end,
On the other hand, if we denote
Let us set
This shows that the process
Lemma 5: Assume that F (and this also the Hazard function Γ) is continuous function. Then the process
is h-martingale.
In view of the martingale in Lemma 5, the following definition is natural.
Definition 3: A function
Remarks: Since the bounded, increasing process H is constant after time τ its compensation is constant after τ as well. This explains why the function
It happens that the martingale hazard function can be found explicitly. In fact, we have the following.
Proposition 6: The unique martingale hazard function of τ with respect to the filtration h is the right-conti- nuous increasing function
Observe that the martingale hazard function
We conclude that the martingale hazard function
In general, we have
where
In order to value a defaultable claim, we need, of course, to specify the unit in which we would like to express all prices. Formally, this is done through a choice of discount factor (a numeraire). For the sake of simplicity, we shall take the savings account
as the numraire, where r is the short term interest rate process.
We also postulate that some probability measure
In accordance with our assumption that the reference filtration is trivial, we also assume that:
・ the default time τ admits the
・ the short-term interest rate
In view of the latter assumption, the price at time t of a unit default-free zero-coupon bond of maturity T equals
In the market practice, the interest rate (more precisely, the yield curve) can be derived from the market price of the zero-coupon bond. In a similar way the hazard rate can be deduced from the prices of the corporate zero- coupon bonds, or from the market values of other actively traded credit derivatives.
In view of our earlier notation for defaultable claims adopted, for the corporate unit discount bond we have
Consider first a corporate zero-coupon bond with unit face value, the maturity date T, and zero recovery at default (that is,
The price
where
According to this convention, we have
Using Corollary 3, we check that the pre-default value
where
Assume now that
Let us denote by
or equivalently,
In the case of full recovery, that is, for
Remarks. Similar representations can be derived also in the case when the reference filtration F is not trivial, and under the assumption that market risk and credit risk are independent that is:
・ the default time admits the F-intensity process γ,
・ the interest rate process r is independent of the filtration F.
In the previous section, it was assumed that the reference filtration F carries no information. However, for practical purposes it is important to study the situation where the reference filtration is not trivial. This section presents some results to this effect.
We assume that a martingale measure Q is given, and examine the valuation of defaultable contingent claims under this probability measure. Note that the defaultable market is incomplete if there are no defaultable assets traded on the market that are sensitive to the same default risk as the defaultable contingent claim we wish to price. Thus, the martingale measure may not be unique.
Let
We start by extending some definitions and results to the present framework. We denote
Definition 4: The F-hazard process Γ of a random time τ is defined through the equality
Notice that the existence of Γ implies that τ is not an F-stopping time. If the event
If the hazard process is absolutely continuous, so that
The valuation of the terminal payoff
The question is how to compute
Lemma 7: For any
If, in addition, Y is
Assume that Y is
The latter property can be extended to stochastic process: for any G-predictable process X there exists an F- predictable process
is valid for every
The following extension of Corollary 3 appears to be useful in the valuation of the recovery payoff
Lemma 8: Assume that the hazard process Γ is a continuous, increasing process, and let Z be a bonded, F-predictable process. Then for any
To value the promised dividends (that are paid prior to τ, it is convenient to make use of the following result.
Lemma 9: Assume that the hazard process Γ is continuous. Let C be a bounded, F-predictable process of finite variation. Then for event
We assume that τ is given on a filtered probability spaces
The probability
The ex-dividend price
Definition 5: For any date
we always set
Proposition 10: The value process of a defaultable claim
If the hazard process Γ is an increasing, continuous process, then
Corollary 11: Assume that the F-hazard process Γ is a continuous, increasing process. Then the value process of a defaultable contingent claim
Consider a defaultable zero-coupon bond with the par (face) value L and maturity date T. First, we re-examine the following recovery schemes: the fractional recovery of par value and the fractional recovery of Treasury value. Subsequently, we shall deal with the fractional recovery of pre-default value, but in this section using the stochastic intensity instead of the deterministic intensity used earlier. We assume that τ has the E-intensity γ.
Under this scheme, a fixed fraction of the face value of the bond is paid to the bondholders at the time of default. Formally, we deal here with a defaultable claim
If τ admits the F-intensity γ, the pre-default value of the bond equals
Remarks. The above setup is a special case of the fractional recovery of par value scheme with a general F- predictable recovery process
Here, in the case of default, the fixed fraction of the face value is paid to bondholders at maturity date T. A corporate zero-coupon bond is now represented by a defaultable claim
The price of this claim oat time
or equivalently,
The pre-default value
Again, the last formula is special case of the general situation where
Assume that
where
A challenging practical problem is the calibration of statistical properties of both the recovery process δ and the intensity process γ. The empirical evidence strongly suggests that the amount recovered at default is best modelled by the recovery of par value scheme. However, we conclude that recovery concept that specifies the amount recovered as fraction of appropriately discounted par value, that is, the fractional recovery of treasury value, has broader empirical support.
This is basically combination of ideas from both the structural and intensity-based approaches, this is by postulating that the hazard rate of default (intensity) event is directly linked to the current value of the firm’s assets (or the firm’s equity). Reduced-form models with this specific feature are referred to as hybrid model. In this setup, the default time is still a totally inaccessible stopping time, but the likelihood of default may grow rapidly when the total value of the firm’s assets approaches some barrier. Madan and Unal [
They postulate that the hazard rate of default equals
Under the martingale measure
We take a function
notice that the stochastic intensity
The futures price
In particular, the futures price
More explicitly,
for some function
By virtue of Equation (4.1) and the Feynman-Kao theorem, the function r satisfies, under mild technical assumptions, the following pricing partial differential equation
subject to the terminal condition
where the parameter v satisfies
For a fixed value of the parameter v, the function
with the initial conditions
serve to produce estimates of parameters of the hazard rate process, based on the observed market yields on defaultable bonds.
We have in our disposal two models for the valuation of credit risk named the reduced-form model and the hybrid model. It is worth noting that the cornerstone of credit risk and its modelling is based on the information one can perceive. This information can be complete (structural approach), partial (incomplete information model which is called hybrid model) or not available (reduced-form model). This perceived information defined the methodology that one can apply to model credit risk. Everything lies on whether information is available or not. And that is the very fundamental economic notion of credit risk. We conclude this paper by commenting on the advantages and disadvantages of the reduced-form model and the hybrid model for the valuation of credit risk.
・ The level of the credit risk is reflected in a single quantity: the risk-neutral default intensity.
・ The random time of default is an unpredictable stopping time, and thus the default event comes as an almost total surprise.
・ The valuation of defaultable claims is rather straightforward. It resembles the valuation of default-free contingent claims in term structure models, through well understood techniques.
・ Credit spreads are much easier to quantify and manipulate than in structural models of credit risk. Consequently, the credit spreads are more realistic and risk premia are easier to handle.
・ The intensity of the random time of default plays the role of a models input.
・ Valuation result for corporate bonds and credit derivatives are relatively simple, even in the case of basket credit derivatives.
・ In practice, the intensity of default can be inferred from observed prices of bonds (the calibrated or implied default intensity).
・ Value of the firm is not explicitly modelled.
・ Typically, current data regarding the level of the firm’s assets and the firm’s leverage are not taken into account.
・ Specific features related to safety covenants and debt’s seniority are not easy to handle.
・ All (important) issues related to the capital structure of a firm are beyond the scope of this approach.
・ Most practical approaches to portfolio’s credit risk are linked to the value-of-the-firm approach.
・ This is basically combination of ideas from both the structural and intensity based approaches. While avoiding their difficulties, it picks the best features of both approaches: the economic and intuitive appeal of the structural approach and the tractability and empirical fit of the intensity-based approach.
・ Hybrid model is of great importance in credit risk valuation because of the existence of a bankruptcy process.
・ Dependent defaults are easy to handle through correlation of processes corresponding to different names.
・ A stringent assumption that the total value of the firm’s assets can be easily observed. In practice, continuous-time observations of the value processes are not available. Thus the structural model with incomplete accounting data can be dealt with using the intensity-based methodology.
・ Most practical approaches to portfolio’s credit risk are linked to the value-of-the-firm approach.