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Assets

Efficient pricing of default risk: Different approaches for a single goal Damiano Brigo Head of Credit Models, Banca IMI

Massimo Morini University of Milan Bicocca

Abstract With the rapid development of the credit derivatives market, efficient pricing of default has become an extremely important issue for the credit risk management of banks and other investors. We consider here some of the opportunities and problems that the development of this market poses to quantitative research in academia and industry. We describe different modeling choices pointing out the practical pros and cons of the different frameworks. For all different frameworks, we present innovative solutions allowing both computational efficiency and high consistency with the increasingly liquid credit reference market, the market of credit default swaps.

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Efficient pricing of default risk: Different approaches for a single goal

The last few years have been described by many as a period of

outstanding increased from U.S.$170 billion in 1997 to almost

downturn for the financial markets, or at least of increased

U.S.$1,400 billion in 2001. At the same time, the range of

worry and lack of confidence. Crises have struck investors in

products is growing, in particular in portfolio credit derivatives

different markets, starting from the collapse of the LTCM

and options on existing credit derivatives. Besides represent-

hedge fund in 1998, to the burst of the Internet equity bubble

ing an important contribution to the stability of the financial

in 2000, and to the debt crisis of both sovereigns (Russia

market as a whole, the increasing liquidity of this market is

1998, Argentina 2001) and major industrial worldwide players

opening up unmissable opportunities for credit risk manage-

(Enron 2001, WorldCom 2002). These events have pushed

ment of banks and other investors.

many investors out of different sectors of the financial market. In the following section, we will explain the structure of the However, a closer look at the situation reveals that the context

most common credit derivatives. Then we consider the rele-

is twofold. While a proportion of investors were actually pushed

vance of the development of a liquid market for credit risk for

out by the recent crises, those who remained started to devel-

reliable relative value pricing through quantitative modeling.

op instruments to strengthen and defend themselves for the

We describe the general features of the major modeling

future. This process includes the development of a substantial-

frameworks, and present innovative solutions for exploiting

ly new market, the credit derivatives market, which has been

the opportunities and reducing the problems of the different

growing dramatically even when most other financial markets

approaches.

have been stagnating. Credit derivatives are mostly over-thecounter derivative securities whose final payout depends on

Credit default swaps (CDS)

the default of a reference entity. Default has several meanings,

Since the market is experiencing a fast evolution, it is not yet

including the risk that one counterparty will not honor some of

possible to give a precise or definitive taxonomy of credit

its obligations. While the types of structures that are being

derivatives. However, an important feature of these instru-

developed are quite complex and varied the common purpose

ments, pointed out for instance by Bielecki and Rutkowski

of these products remains the same, to allow market partici-

(2001), is the precise extent of credit risk that a product

pants to single out, transfer, and trade separately credit and

allows to be transferred, namely the intrinsic definition of

default risk, namely the part of the risk in a contract which is

credit risk. We have securities that allow the entire risk asso-

related to the credit reliability of an obligor.

ciated with a transaction vulnerable to credit risk to be transferred, such as ‘total rate of return swaps’ and ‘equity return

In parallel, the development of this market has been fueled by

swaps.’ We also have products that transfer the risk related to

the increased attention of regulatory agencies to exposures in

changes in the value of an agreement due to movements of

OTC derivatives by many of the world’s major financial institu-

the credit quality (financial reliability) of one or more oblig-

tions. Regulations, such as Basel I and II, made the advantages

ors, such as credit spreads swaps and options. Finally, there

of an efficient credit risk market even more apparent, in par-

are products precisely focused on separating and transfer-

ticular to major banks. The market for credit derivatives was

ring only the risk directly involved with a default event. This

created in parallel in Europe and in the U.S. during the 1990’s,

last category appears to be most attractive to market partic-

and it has recently experienced the highest growth rate of all

ipants. Credit default swaps (CDS) and similar products, for

derivatives markets. The vitality of this market is revealed

example, represented over two thirds of the entire credit

both in terms of quantitative expansion and qualitative devel-

derivatives business in 2002.

opment and financial innovation. According to a Risk magazine survey in 2002, the notional value of credit derivatives

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A credit default swap is a contract between two parties, called

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Efficient pricing of default risk: Different approaches for a single goal

the protection buyer and the protection seller, designed to

ance contracts and could not be traded separately from the

transfer the financial loss that the protection buyer would

reference obligation. This system did not favor efficiency, con-

suffer if a particular default event happened to a third party,

sistency, and competition in evaluating credit risk, and strong-

called the reference entity. Usually, the reference entity is a

ly limited the possibilities for credit risk management. Instead,

debtor of the protection buyer. The protection buyer agrees

the creation of an increasingly liquid market allowing investors

to pay a periodic amount R (less frequently an upfront fee) to

to trade credit risk separately as any other tradable asset was

the seller in exchange for a single protection payment, made

a real major financial innovation, since it led to a strong

by the seller only in case the pre-specified feared default

increase in attention, competition, and precision in evaluating

event happens. The CDS are quoted on the market though a

credit risk.

fair R, often called CDS spread, making the current price of the contract equal to zero, like in interest rate swaps.

From a quant’s perspective, it allowed for the extension of the advanced techniques for pricing and management that had

In spite of all possible variations, the basic structure of the

been developed for different markets, such as equity, interest-

product is always the simple one described. This structure

rate, and FX derivatives, to credit risk.

allows for a protection similar to an insurance contract, but a CDS is an autonomous security traded on a financial market

The increasing liquidity of the CDS market, in particular, has

like any other derivative. The relevance of this feature in a sit-

played an important role in shaping credit modeling, making it

uation of increasing liquidity will be further pointed out in the

more similar to those typical in other markets. The CDS, aimed

next section, with particular attention to its consequences on

at transferring only the risk directly involved with a default

quantitative valuation and risk management.

event, are the most interesting category also from a quant’s point of view. In fact, their evaluation poses in a clean way the

CDS and other credit derivatives from a quant’s perspective

problem of modeling and pricing the financial consequences

Traditional models for pricing and hedging contingent claims

quotations embed fundamental information on the market

written on equity or on the term structure of interest rates

assessment of the risk neutral default probabilities for the ref-

do not include a specific consideration for credit risk. The

erence obligor.

of the central specific event, the default. Consequently, CDS

increased attention of market participants and regulators to credit risk, and the growing amount of complex structured

Since, as is well known, pricing of contingent claims is cor-

products for which default risk could not be overlooked, called

rectly performed by expectation under risk neutral probabili-

for increased attention to credit risk also from quants in aca-

ties, this is actually what we are interested in for valuation of

demia and industry. In particular, because of the contempo-

default-dependent payoffs. Obviously this information is not

rary development of the credit derivatives market, giving the

transparent and requires setting a precise modeling frame-

possibility to transfer default risk through an OTC contract,

work for being assessed quantitatively (although we will see

the focus moved to the development of specific models for the

that default probabilities in CDS appear quite robust to the

valuation of these new contracts. In fact the agreements used

choice of the model). The procedure for determining the

for transferring credit risk needed to be correctly priced for

parameters of a model in such a way that it gives answers con-

making the transfer effective.

sistent with known reference market prices is called calibration. When the model has been calibrated to the basic prod-

Previous financial agreements which were used to protect

ucts, it can be used for pricing consistently more advanced

investors from credit risk had features often similar to insur-

default-risky payoffs.

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Efficient pricing of default risk: Different approaches for a single goal

However, before CDS reached a critical liquidity, the market

Structural models

price of credit risk expressed by them was not a reliable rep-

The first important contribution to quantitative credit model-

resentation of the market expectations or risk premia.

ing dates back to the seminal paper of Black and Scholes

Therefore modeling concentrated on explaining default risk

(1973), which introduced the principles of modern option pric-

based on information ‘external’ to its specific market. With the

ing. In this paper the valuation of company liabilities was indi-

increase in liquidity and diffusion of the CDS, the precise

cated as a possible application of the pricing technique intro-

information on the price of credit risk they provide, deter-

duced (later known as the Black and Scholes formula), and the

mined by a reliable level of demand and supply, becomes a

underlying idea was a structural explanation of default.

fundamental benchmark for fair valuations. This moved the

Although the distinction is at times rather subtle, financial

attention of quants to building models that are able to regis-

research labels as structural those models containing a styl-

ter efficiently, through calibration, the information provided

ized description of the economic causes for a market value or

by the reference CDS market. Such models are then reliable

event; they are opposed to those models, at times called

enough to be used for required applications.

reduced form models, based on a quite general mathematical framework to be specified consistently with historical or cross-

This last development set a context for relative value pricing

section data. In the Black and Scholes (1973) and then Merton

similar to that of traditional derivatives markets, but many

(1974) default model, a company defaults if, at maturity T of its

technical modeling problems posed by valuation of credit risk

debts, the value of the company is lower than the reimburse-

are peculiar. Most of them had been paid little attention to in

ment to be made. Based on this interpretation and by assum-

the past, since possible applications were less necessary and

ing a geometric Brownian Motion (GBM, also called lognormal

relevant. The possible structural causes of credit risk, or on

diffusion) dynamics for the value of the company, all the math-

the other hand the sudden consequences of a default event,

ematics required for computing default probabilities is that

are both examples of specific features making this risk inher-

typical of plain vanilla equity option pricing.

ently different from that involved in modeling the underlying asset of more traditional derivatives. Another distinctive fea-

Black and Cox (1976) also remain in this framework for explain-

ture, not treated in this work, is the fundamental incongruence

ing default based on balance sheet notions, but increase real-

between the nature of default interdependency and the con-

ism by introducing the so-called first passage time models.

cept of interdependency comfortably used for interest rate

Here the value of a company is again a GBM, and default still

and equity markets. Therefore, although many techniques and

happens if the value of this process is lower than the level of

models previously used have turned out to be useful for this

its debts, but now this event is not limited to an unrealistic sin-

new application, the extension of the quantitative apparatus

gle maturity time. Here the debt level is a barrier Dt, possibly

to this new market has called for a massive work on model

time varying, and the value of the company can fall below Dt

innovation and development of new techniques.

at any time, causing default. For first time passage models the mathematics required was already in use for equity barrier

The alternative models for credit risk pricing do not differ

options. The model was tractable (easy to make computations

simply due to technical details. The main modeling frame-

with) in case the underlying was assumed to have constant

works stem from different views on fundamental issues, such

parameters, in this case the explicit probability distribution for

as the choice of the variables to model or the correct descrip-

the first time that a GBM hit a barrier is known.

tion of a default event. We briefly review some of these frameworks in the next three sections.

One may be unconvinced by the hypothesis on the firm value underlying these approaches, but the ultimate reason that led

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to the development of alternative approaches was a practical

credit spreads, but not for their volatility. For this one has to

one. Structural models appeared empirically unable to

allow for continuous stochastic variation of intensity. In this

describe real market expectation of defaults. The gradual

case we move to Cox Processes. In this setting, conditional on

description of default implied by these structural approaches

the information on the path followed by intensity, default hap-

was not consistent with market expectation of default as a

pens at first jump time of an inhomogeneous Poisson process

sudden event, not fully predictable by observing book values

with this time-varying intensity. Thus survival probability is

of a company. In particular structural models tend to imply

P(τ > t) = E[P(τ > t | { λ(s): 0 ≤ s ≤ t})] = E[e–∫λ(s)ds]

unrealistically low default probability in the short-term. This approach can exploit many results and well-known modLater we will describe some possibilities to overcome these

els typical of stochastic modeling of the short interest rate

practical problems of structural models without leaving the

(instantaneous spot rate), already in use for interest rate

structural representation of default. In the next section,

derivatives. Notice in fact that the expressions for survival

instead, we see a radically different framework.

probabilities are analogous to the functions expressing bond prices in terms of the dynamics of the short rate. Advanced

Intensity models

intensity models [Duffie and Singleton (1999)] are more flexi-

The direct way to take into account the sudden nature of the

ble than structural models. However, depending on the dynam-

default event in the real word is modeling directly default as

ics chosen for the intensity, here too there can be differences

an unpredictable, exogenous event. This leads to reduced form

in terms of tractability and calibration power. We will see later

models, also called intensity models, where default happens at

a model designed with specific attention to both aspects.

the time of the first jump of a stochastic (jump) Poisson process with intensity λ. This means that the time of default τ

Market models

is exponentially distributed with a probability of survival for t

Intensity models, although developed for increasing consis-

years given by P(τ > t) = e–λt, with expected time to default 1/λ.

tency with the market, are based on modeling theoretical,

In such models there is no attempt made to model the eco-

non-observable quantities and lead to option formulas quite

nomic causes for default. Default is simply a random variable

different from those traders are used to. Traders on the inter-

with a distribution parameter λ to be determined consistently

est rates market are used to price options with the Black for-

with market evidence.

mula, developed originally for commodity futures. This formula is based on modeling the underlying as a lognormal

Also in this framework many extensions have been put for-

process under the pricing measure. Without this structure,

ward to increase realism and flexibility. A model with constant

even the typical concept of implied volatility cannot be used

default intensity is neither realistic nor flexible enough to

consistently. To recover the features of classic option formu-

embed market information about default probabilities on dif-

las in credit, the first step is modeling directly the underlying

ferent interims. As a solution, the intensity can be supposed to

rates in credit derivatives payoffs. So default is not to be

vary deterministically in time, also continuously (given some

explicitly modeled, neither as a predictable nor as a structur-

technical conditions). In this case, the default happens at the

al event.

first jump of an inhomogeneous Poisson process with intensity λ(t), leading to a survival probability P(τ > t) = e–∫λ(s)ds. The

In interest rate modeling, recent developments in probability

shape of this survival probability hints at the fundamental fact

made it possible to give a rigorous probabilistic foundation to

that the intensity can be interpreted as an instantaneous cred-

the aforementioned market standard of pricing most liquid

it spread. This model can account for the term structure of

options via heuristically derived Black-like formulas. This led

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Efficient pricing of default risk: Different approaches for a single goal

to the so-called market models, such as the Libor and the

enough flexibility to be calibrated to a term structure of CDS

Swap Market Models.

rates, and thus to the embedded default probability.

Generalizing the definition, in a market model we model

Is it possible to enrich the calibration power of structural mod-

directly market observables underlying common options, and

els while keeping tractability, namely consistent with the

they are assumed to follow the so-called standard market

mathematics of barrier options? Thanks to recent advances in

model, namely a lognormal diffusion under the reference pric-

this last field, the answer is now affirmative. Lo et al (2003)

ing measure. This allows us to price options with Black-like for-

show that, assuming a specific shape for the barrier, one can

mulas. Hull and White (2003) outline that a market model

have explicit formulas also when the underlying is modeled as

approach, which makes it possible to detect the implied volatil-

a GBM with time-varying parameters. For credit risk, this

ities in option quotations, can be useful also in a credit setting,

means that we can have a tractable structural model even if

in order to make option quotations more standard, and there-

allowing for time-varying volatility of the underlying. Such a

fore transparent and understandable. This would help enhance

model and its calibration to CDS are described in Brigo and

liquidity in some growing markets for credit options, such as

Tarenghi (2004).

options on CDS. But there are some relevant issues to be considered, ranging from detecting the correct specification of

Assuming that the volatility of the firm value is piecewise con-

the state variables helping to price real world derivatives, to

stant (a step function), the model can be easily calibrated to

technical issues in the definition of the reference probability

CDS quotes using both volatility and the debt barrier. In Brigo

measure.

and Tarenghi (2004) diagnostic tests are performed to verify that this is not only a formal consistency due to the many

A tractable structural model consistent with market implied default probabilities

parameters, but corresponds to an increased capability to pick

Typically credit models were specified according to credit

bilities are computed and compared with those implied by a

spreads or balance sheets information. However credit

standard intensity model (the framework designed for this

spreads appear to be largely determined by causes different

exact purpose). Surprisingly enough, the probabilities found

from default probabilities, for instance liquidity. On the other

are nearly the same. This evidence confirms two hypotheses.

hand, some recent major defaults (Enron, Parmalat) suggest

Firstly, that CDS information on default probabilities is robust.

that default often happens in conjunction with particularly

If the level of fitting power is kept equivalent, it is not highly

unreliable balance-sheets information, so that book values can

dependant on modeling assumptions. Secondly, that the

be even more misleading for assessing default probabilities.

tractable structural model of Brigo and Tarenghi (2004) also

out relevant implied market structures. Implied default proba-

matches, besides the CDS quotes, the whole structure of Therefore, it seems that nowadays, even when using a struc-

default probabilities as determined by intensity models.

tural model, it may be appropriate to refer to the credit deriv-

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atives market to extract default probabilities, and in particular

A recent development that is currently under investigation

to CDS. Indeed, the CDS market is becoming an increasingly

concerns the possibility to calibrate the CDS quotes via an

reliable and liquid source of information for market default

uncertain debt barrier, using the barrier scenarios and proba-

probabilities, and is rapidly updated when new information on

bilities as calibrating parameters and keeping the value of the

the real financial situation of a debtor is disclosed, sometimes

firm volatility as an exogenous input from the equity market.

even anticipating it. However, there are technical problems in

Among other issues, the work of Brigo and Tarenghi (2004)

implementing this approach. Structural models have not

suggests that practical problems of standard structural mod-

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Efficient pricing of default risk: Different approaches for a single goal

els in pricing are not an unavoidable consequence of the

an intensity) has the form dyt = k(µ – yt)dt + v(ytdWt)1/2, where

hypothesis made, but depend a lot on the data used for

the vector β = (k, µ, v, y0) has positive components. All param-

expressing these assumptions in quantitative terms, and on

eters have a clear and intuitive role in determining the dynam-

the fitting capability of the structural model to these data.

ics of the process.

Brigo and Tarenghi (2004) also show applications of the

The condition 2kµ>v2 ensures the process remains positive,

model. Firstly, they analyze a concrete default case, the

and Brigo and Alfonsi (2005) present a scheme to enforce this

Parmalat case, showing how the model can efficiently embed

property also when the model is discretized for simulation.

increasing proximity to default as information on the wors-

This process features a non-central chi-square distribution

ened credit quality of a company is disclosed in time. Secondly,

and is highly tractable, featuring closed-form formulas for

they consider the pricing of a claim embedding counterparty

pricing.

risk, and show how in this setting default correlation can be accounted for in a rather natural way, exploiting the proximity

CIR has often been used in interest rate derivatives pricing,

between the underlying being modeled and the equity market.

and also for the pricing of credit derivatives. With the development of the interest rate derivatives, it soon appeared

A positive intensity model with increased calibration power

strongly limited by the low number of parameters. In fact

We mentioned earlier that not all possible specifications of the

interest rates.

these models cannot fit exactly the initial term structure of

intensity dynamics are equivalent for applications. We can recall at least three different orders of problems. Firstly, the

As we noticed also in the last section, an increased calibration

intensity has to be strictly positive (thus forbidding Gaussian

power is now desirable in the credit market too. When a refer-

models) for default probabilities to be meaningful. Secondly,

ence market for a source of risk becomes reliable enough,

not all candidate intensity processes have the same calibration

using models fully consistent with that market price becomes

power. If the range of products considered increases, even in a

a necessary requisite for giving a fair price to more advanced

reasonable range, basic common processes can become inad-

products. Furthermore, increased liquidity reduces the risk of

equate. This was already revealed in interest rate modeling.

perfectly fitting unreliable quotations. Using a technique

Thirdly, computational times and complexity are extremely

developed on the interest rate market, one can extend the CIR

important in pricing. Models allowing for closed form formulas

dynamics above to fit a full term structure of CDS. This is done

(not requiring simulation) are usually preferred, in particular

in Brigo and Alfonsi (2005), where references are given, and

when a model has to be calibrated both to credit and interest

in the resulting ‘CIR++’ model they set the intensity to λt = yt

rate products.

+ ψ(t,β), where y has the dynamics seen before and ψ is a deterministic function. If ψ is determined consistently with the

In Brigo and Alfonsi (2005) a model is proposed that aims to

default probabilities extracted from the CDS market (say via a

give a more complete answer to these requirements than in

deterministic-intensity model), the model is exactly calibrated

preceding models. To ensure positive intensity, the family of

to CDS. And we are left with parameters β for calibrating dif-

stochastic processes considered are the square-root diffu-

ferent products, although this feature will probably show its

sions, introduced by Feller (1951) for application in the field of

usefulness when options in the credit market become more

genetics. In finance this process is usually called the CIR

liquid. More importantly, the model extended this way inherits

process, after the application to short interest rate modeling

the tractability of CIR.

of Cox, Ingersoll and Ross (1985). The process y (for instance

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Efficient pricing of default risk: Different approaches for a single goal

When building the general model, Brigo and Alfonsi (2005)

ferent variations of the payoff in the market are analyzed in

assume extended CIR dynamics in parallel to the default inten-

detail, and in particular one CDS payoff structure is singled

sity and the short interest rate, with the instantaneous corre-

out, realistic enough for application to the market but simple

lation between the two Brownian drivers set to ρ. This model

enough to allow for the construction of a market model. This

has the potential to be fully consistent with interest rates and

CDS structure is shown to lead to a CDS option equivalent to

CDS default probabilities, and parameters are left to include

a simplified callable defaultable floating rate note. Since this is

information on both credit and interest rate option markets. In

another relevant payoff in credit options, this latter CDS struc-

addition, when ρ is set to zero, the calibration of the full model

ture is taken as the central one for building the market model

is automatic due to the tractability of the extended CIR cou-

(although the derivation is outlined also for alternative speci-

pled with the interesting feature of separability. The latter

fications).

means that the credit derivatives desk of a bank can ask for interest rate parameters from the interest rates desk and add

Setting the price of the CDS to zero and deriving the spread

them in a model to be calibrated to CDS, keeping full consis-

R one finds the correct underlying of the option. Denote by

tency in valuation procedures.

Ta+1, …,Tb the payment dates of the CDS, with time intervals αi. Denote by Q the risk neutral probability and by D(t,Ti) the sto-

When ρ ≠ 0 this is no longer true, but Brigo and Alfonsi (2005)

chastic discount factor. The correct definition of CDS rate

show empirically that the implications of ρ on CDS prices is

turns out:

bounded by a small fraction of the market bid-ask spread, and is thus negligible. So one can keep the efficient zero-correla-

Ra,b (t) = [Σbi=a+1 E[D(t,Ti)1{Ti-1 t | Ft)

tion calibration and then set ρ to the desired value in pricing.

P (t,Ti)],

Approximated formulas are introduced for different payoffs, including CDS options, namely options to enter a CDS at a

with P (t,Ti) = [E[D(t,Ti)1{τ >Ti} |Ft]]/[ Q(τ > t | Ft)]

future time. And with deterministic interest rates and stochastic intensity an exact analytical pricing formula is derived.

Here τ is default time, while E[·| Ft)] represents expectation

These formulas have a particular use for detecting the capa-

conditional on all information (up to t) except the default time.

bility of this model to interpret and replicate the smile phe-

1A represents the indicator function for set A. The protection

nomenon. But for this we first have to see how the smile can

payment is set to 1. The corresponding CDS option, to enter

be detected on the CDS option market.

a CDS with fixed rate K at time Ta, has discounted payoff 1{τ >Ta} D(t,Ta) [Σbi=a+1 αiP(Ta,Ti)](Ra,b (ta) – K)+ and taking risk

A market model for real world credit payoffs with non-vanishing numeraire

neutral expectation one finds the price. The technical tools

We already described how a market model approach might be

Rutkowski (2001).

are omitted here and can be found for example in Bielecki and

useful for the development of the credit options market, and which issues should receive particular attention. Accordingly,

The technical tool to develop market models is the change of

rather than starting from an abstract definition, Brigo (2004)

numeraire theory. To put it in a nutshell, this allows us to com-

focuses on a specific payoff, the options on CDS, since they

pute a price, expressed by a risk neutral expectation, as the

are written on the most liquid single-name credit derivative,

expectation of a related quantity under a different probability

and their market is likely to expand in the future. In order to

measure (equivalent to the risk neutral one).

detect the most convenient state variable to be modeled, Brigo (2004) starts from the real world CDS payoff. The dif-

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The purpose of market models is to detect a suitable probabil-

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Efficient pricing of default risk: Different approaches for a single goal

ity measure under which both the underlying is a martingale

results, and the application of a model approximation to the

and the price expression reduces to a Black formula. In our

derivation of a pricing formula for Constant Maturity CDS, a

case, this implies that all but (Ra,b (ta) – K)+ should get out of

payoff receiving increasing attention in the market, are given

the expectation.

in Brigo (2004).

In Brigo (2004) a probability measure is considered, individua martingale. In addition, if Ra,b(t) is assumed to follow a log-

Understanding the smile and the parameters from a comparison of intensity and market models

normal process under this measure, through the change of

Another advantage of having obtained a rigorous market

numeraire the price of the option reduces, before default, to

model is that we can detect the implications of the use of a dif-

the simple formula

ferent model (for instance the CIR++ intensity model) on

ated by its so-called associated numeraire, such that Ra,b(t) is

options implied volatility. This means first of all understanding Σbi=a+1 αiP(t,Ti)(Ra,b(t)Φ(d1) – KΦ(d2)), where Φ(·) is the cumu-

the effects of the model parameters (i.e. β in CIR++) on implied

late Normal probability and d1 and d2 are as in the Black for-

volatility. Secondly, it means understanding the pattern of the

mula, with reference to the underlying Ra,b(t) at valuation

smile effect intrinsic in the model dynamics. As is well known,

time t.

the smile effect can be interpreted as a deviation from the lognormality assumption for underlying market observables

It is interesting to notice that, in this derivation of Brigo

which, when market quotations are made through a lognormal

(2004), the chosen numeraire cannot drop to zero, in the

model, is revealed by a non-flat shape of the graph of implied

spirit of Jamshidian (2002), thus ensuring equivalence of the

volatilities plotted against strike prices. From the perspective of a trader, a clear understanding of these implications is

pricing measures.

of fundamental importance, Rk,b(0)

K

Price

Volatility

often influencing the choice

Option 1

61

60

32.5

62.16%

of a model at least as much

Option 2

43.4

43

24.5

63.71%

as the technical advantages

Figure 1

of the model itself. For the CIR++ stochastic intensity

Parameter K↑ µ↑ ν↑ y0↑ K↑ ρ = 0 K↑ ρ = –1 K↑ ρ = 1 ρ↑

Implied volatility ↓ ↑ ↑ ↑ ↑/flat ↑/↓/ flat ↑/flat ↓

As long as the market has not yet reached a critical level of liq-

and interest rate model, this

uidity, this model would be hard to calibrate to reliable implied

kind of analysis is done in

volatilities for pricing different products. However, it plays a

Brigo and Cousot (2003).

very important role. It makes it possible to consistently trans-

Among other issues, they

late the prices of different options into implied volatilities,

obtain the following format, as presented in Figure 2, where K

helping to understand different implications of market options

is the option strike. Smile patterns implied by explicit intensity

quotations. Let us look at the quotations in Figure 1. It is hard

models such as CIR++ might be considered unsatisfactory

to compare them based on the Price column, provided by the

when this phenomenon is clearly established on the CDS

market. If we move to the Volatility column, provided by the

option market. In this case one can obtain more flexible pat-

model just presented, the information provides a much more

terns modeling directly the market observable R under the rel-

effective understanding and comparison. The specification of

evant measure, similarly to what we have described above, but

a general model, including the dynamics of R under a range of

replacing lognormality with a tractable dynamics allowing for

probability measures, is currently under investigation. Initial

smile (displaced diffusion, CEV, mixture dynamics, SABR).

Figure 2

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Pagina 160

Efficient pricing of default risk: Different approaches for a single goal

Conclusion

References

In this work we have outlined some general features of

• Bielecki T., and M. Rutkowski, 2001, “Credit risk: Modeling, valuation and hedging,” Springer Verlag • Black F., and M. Scholes, 1973, “The pricing of options and corporate liabilities,” Journal of Political Economy, 81, 637-654 • Black, F., and J. C. Cox, 1976, “Valuing corporate securities: Some effects of bond indenture provisions,” Journal of Finance, 31, 351-367 • Brigo, D., 2004, “Constant maturity credit default swap pricing with market models,” available at ssrn.com • Brigo, D., 2005a, “Market models for CDS options and callable floaters,” Risk Magazine, January 2005. Extended version available at damianobrigo.it • Brigo, D., and A. Alfonsi, 2005, “Credit default swaps calibration and derivatives pricing with the SSRD stochastic intensity model,” Available at damianobrigo.it. Finance and Stochastics, Vol. X (1). • Brigo, D., and L. Cousot, L., 2003, “A comparison between the SSRD Model and a market model for CDS options pricing,” Bachelier 2004 Conference • Brigo, D., and M. Tarenghi, 2004, “Credit default swap calibration and equity swap valuation under counterparty risk with a tractable structural model,” Paper presented at the 2004 FEA conference at MIT • Duffie D., K. Singleton, 1999, “Modeling term structures of defaultable bonds,” Review of Financial Studies, 12, 687-720 • Hull, J., and A. White, 2003, “The valuation of credit default swap options,” Rothman school of management working paper • Lo C. F., H. C. Lee, and C. H. Hui, 2003, “A simple approach for pricing barrier options with time-dependent parameters,” Quantitative Finance, 3 • Merton, R., 1974, “On the pricing of corporate debt: The risk structure of interest rates,” Journal of Finance, 29, 449-470

quantitative modeling for relative value pricing in the field of credit derivatives. In doing so, we had to address different approaches to modeling, pointing out pros and cons of different frameworks. Then, although avoiding most technical details, we presented three different models in different frameworks, all designed to overcome possible limitations and inadequacies typical of earlier solutions. A final remark on some of the results summarized here is in order. It may seem that some of the solutions presented here are actually ahead of the market, in that they would require a further development in liquidity of certain markets for being fully exploited. However, it has sometimes happened that financial markets have developed only when sufficiently sound technical tools for dealing with such developments had been provided by research. This research typically required a lot of analysis and attempts before reaching a level which, associated to different external factors, allowed the market to take some steps forward in terms of efficiency and stability.

160 - The

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