This book is a must-read for anyone wishing to understand credit risk from mathematical and intuitive perspectives. It is a point of reference for all credit risk modelling practitioners.
Collections Sociology []. Search DSpace. This Collection. The ability to quickly set up a model which allows one to experiment with different assumptions calls for a good collection of workhorses.
I have included a collection of tools here which I view as indispensable workhorses. Mastering these tech- niques will provide a nice toolbox. A book is of course a different game.
Some monographs use the opportunity to show in detail all the stuff that editors would not allow for reasons of space to be published. These can be extremely valuable in teaching the reader all the details of proofs, thereby making sure that the subtleties of proof techniques are mastered. This monograph does almost the opposite: it takes the liberty of not proving very much and worrying mainly about model structure.
In short, this book is my way of introducing the area to my intended audience. There are several other books in the area—such as Ammann , Arvanitis and Gregory , Bielecki and Rutkowski , Bluhm et al. I hope of course that my readers will agree. The original papers on which the book are based are listed in the bibliography.
I have attempted to relegate as many references as possible to the notes since the long quotes of who did what easily break the rhythm. Stochastic calculus involving jump processes, including state price densities for processes with jumps, is not assumed to be familiar.
It is my experience from teaching that there are many advanced students who are comfortable with Black—Scholes-type modeling but are much less comfortable with the mathematics of jump processes and their use in credit risk modeling. For this reader I have tried to include some stochastic calculus for jump processes as well as a small amount of general semimartingale theory, which I think is useful for studying the area further.
For years I have been bothered by the fact that there are some extremely general results available for semimartingales which could be useful to people working with models, but whenever a concrete model is at work, it is extremely hard to see whether it is covered by the general theory. The powerful results are simply not that accessible. I have included a few rather wimpy results, compared with what can be done, but I hope they require much less time to grasp.
I also hope that they help the reader identify some questions addressed by the general theory. I am also hoping that the book gives a useful survey to risk managers and regulators who need to know which methods are in use but who are not as deeply involved in implementation of the models.
There are many sections which require less technical background and which should be self-contained. Large parts of the section on rating estimation, and on dependent defaults, make no use of stochastic calculus. I have tried to boil down the technical sections to the key concepts and results.
Often the reader will have to consult additional sources to appreciate the details. The book tries in many cases to give an overview of the essentials. There are many people to whom I owe thanks for helping me learn about credit risk. In the work which became my thesis I received a lot of encouragement from my thesis advisor, Bob Jarrow, who knew that credit risk would become an important area and kept saying so.
Since then, several co-authors and co-workers in addition to Bob have helped me understand the topic, including useful technical tools, better. In the process of writing this book, I have received exceptional assistance from Jens Christensen.
Both the teaching and the Heurigen visits have been a source of inspiration. There are many other colleagues and friends who have contributed to my under- standing of the area over the years, by helping me understand what the important problems are and teaching me some of the useful techniques.
This continues to be a great source of inspiration. I gratefully acknowledge support from The Danish Social Science Research Foundation, which provided a much needed period of reduced teaching. Richard Baggaley at Princeton University Press has been extremely supportive and remarkably patient throughout the process.
I owe more to my wife Lise and my children Frederik and Christine than I can express. At some point, my son Frederik asked me if I was writing the book because I wanted to or because I had to. I fumbled my reply and I am still not sure what the precise answer should have been. This book is for him. The development of option-pricing techniques and the application to the study of corporate liabilities is where the modeling of credit risk has its foundations.
Obtaining such a link is a key problem of credit risk modeling. We make models describing the distribution of the default events and we try to deduce prices from these models. To answer this we must understand the full description of the variables governing default and we must understand risk premiums.
All of this is possible, at least theoretically, in the option-pricing framework. Chapter 2 starts by introducing the Merton model and discusses its implications for the risk structure of interest rates—an object which is not to be mistaken for a term structure of interest rates in the sense of the word known from modeling government bonds.
We present an immediate application of the Merton model to bonds with different seniority. There are several natural ways of generalizing this, and to begin with we focus on extensions which allow for closed-form solutions. One direction is to work with different asset dynamics, and we present both a case with stochastic interest rates and one with jumps in asset value. A second direction is to introduce a default boundary which exists at all time points, representing some sort of safety covenant or perhaps liquidity shortfall.
The Black—Cox model is the classic model in this respect. As we will see, its derivation has been greatly facilitated by the development of option-pricing techniques. Moreover, for a clever choice of default boundary, the model can be generalized to a case with stochastic interest rates. An Overview Having the closed-form expressions in place, we look at a numerical scheme which works for any hitting time of a continuous boundary provided that we know the transition densities of the asset-value process.
With a sense of what can be done with closed-form models, we take a look at some more practical issues. Coupon payments really distinguish corporate bond pricing from ordinary option pricing in the sense that the asset-sale assumptions play a critical role. We illustrate this by looking at the term-structure implications of different asset-sale assumptions. Another practical limitation of the models mentioned above is that they are all static, in the sense that no new debt issues are allowed.
A simple model is presented which takes a stationary leverage target as given and the consequences are felt at the long end of the term structure. This anticipates the models of Chapter 3, in which the choice of leverage is endogenized. One of the most practical uses of the option-based machinery is to derive implied asset values and implied asset volatilities from equity market data given knowledge of the debt structure.
They all take the asset-value process and its division between different claims as given, and the challenge is to price the different claims given the setup. Chapter 3 looks at the standard approach to obtaining an optimal capital structure within an option-based model. This involves looking at a trade-off between having a tax shield advantage from issuing debt and having the disadvantage of bankruptcy costs, which are more likely to be incurred as debt is increased.
It does have some conceptual problems and these are also dealt with in this chapter. This so-called strategic debt service is more easily explained in a binomial setting and this is how we conclude this chapter. First, classical discriminant analysis is reviewed.
While this model had great computational advantages before statistical computing became so powerful, it does not seem to be a natural statis- tical model for default prediction. Both logistic regression and hazard regressions have a more natural structure. They give parameters with natural interpretations and handle issues of censoring that we meet in practical data analysis all the time. Hazard regressions also provide natural nonparametric tools which are useful for exploring the data and for selecting parametric models.
And very importantly, they give an extremely natural connection to pricing models. While on the topic of default probability estimation it is natural to discuss some techniques for analyzing rating transitions, using the so-called generator of a Markov chain, which are useful in practical risk management.
Thinking about rating migra- tion in continuous time offers conceptual and in some respects computational im- provements over the discrete-time story. For example, we obtain better estimates of probabilities of rare events. We also discuss the role of the Markov assumption when estimating tran- sition matrices from generator matrices.
The natural link to pricing models brought by the continuous-time survival analy- sis techniques is explained in Chapter 5, which introduces the intensity setting in what is the most natural way to look at it, namely as a Cox process or doubly stochastic Poisson process.
Particularly elegant is the notion of recovery of market value, which we spend some time considering. For the intensity model framework to be completely satisfactory, we should under- stand the link between estimated default intensities and credit spreads. Is there a way in which, at least in theory, estimated default intensities can be used for pricing? An Overview an important part of this chapter is the description of what the sources of excess expected return are in an intensity model.
An important moral of this chapter is that even if intensity models look like ordinary term-structure models, the structure of risk premiums is richer. How do default intensities arise? However, this is not a very tractable approach from the point of view of either estimation or pricing credit derivatives. If we do not simply want to assume that intensities exist, can we still justify their existence? It turns out that we can by introducing incomplete information. Chapter 6 is about rating-based pricing models.
This is a natural place to look at those, as we have the Markov formalism in place. Typi- cal examples are the step-up clauses of bond issues used, for example, to a large extent in the telecommunication sector in Europe. When modeling the spreads for a given rating, it is desirable to model the joint evolution of the different term structures, recognizing that members of each category will have a probability of migrating to a different class.
In this chapter we will see how such a joint modeling can be done. We also present a model with stochastically varying spreads for different rating classes, which will become useful later in the chapter on interest-rate swaps.
The problem with implementing these models in practice are not trivial. The goal of this chapter is to get to the point at which the literature currently stands: counterparty credit risk on the swap contract is not a key factor in explaining interest-rate swap spreads.
But before we can get there, we review the foundations for reaching that point. While there may very well be institutional features such as differences in tax treatments which permit such advantages to exist, we focus in Chapter 7 on the fact that the comparative-advantage story can be set up as a sort of puzzle even in an arbitrage- free model. This puzzle is completely resolved. But the interest in understanding the role of two-sided default risk in swaps remains.
We look at this with a strong focus on the intensity-based models. With netting agreements in place, the effect is negligible. The difference between the swap rate and the corporate AA curve is highlighted in this case.
The difference is further illustrated by showing that theoretically there is no problem in having the AAA rate be above the swap rate—at least for long maturities. The result that counterparty risk is not an important factor in determining credit risk also means that swap curves do not contain much information on the credit qual- ity of its counterparties. Hence swaps between risky counterparties do not really help us with additional information for building term structures for corporate debt.
To get such important information we need to look at default swaps and asset swaps. In idealized settings we explain in Chapter 8 the interpretation of both the asset- swap spread and the default swap spread. An Overview Pricing of CDOs and analysis of portfolios of loans and credit-risky securities lead to the question of modeling dependence of defaults, which is the topic of the whole of Chapter 9. The curse is that techniques which offer elegant computation of default losses assume a lot of homogeneity among issuers.
Factor structures can mitigate but not solve this problem. We discuss correlation of rating movements derived from asset-value correlations and look at correlation in intensity models. Recently, a lot of attention has been given to the notion of copulas, which are really just a way of generating multivariate distributions with a set of given marginals.
After this the rest is technical appendices. A small appendix reviews arbitrage-free pricing in a discrete-time setting and hints at how a discrete-time implementation of an intensity model can be carried out. Two appendices collect material on Brownian motion and Markov chains that is convenient to have readily accessible. Finally, they look at some abstract results about special semi- martingales which I have found very useful.
The main goal is to explain the struc- ture of risk premiums in a structure general enough to include all models included in this book. Another part involves getting a better grip on the quadratic variation processes. Finally, there is an appendix containing a workhorse for term-structure model- ing. This approach dates back to Black and Scholes and Merton and it remains the key reference point for the theory of defaultable bond pricing. Some of this progress will be used below to build a basic arsenal of models.
However, the main focus is not to give a complete catalogue of the option-pricing models and explore their implications for pricing corporate bonds. Rather, the goal is to consider some special problems and questions which arise when using the machinery to price corporate debt. This is closely related to specifying what triggers default in models where default is assumed to be a possibility at all times.
While ordinary barrier options have barriers which are stipulated in the contract, the barrier at which a company defaults is typically a modeling problem when looking at corporate bonds. Typically, however, the future capital-structure changes are subsumed as part of the dynamics of the stock.
Corporate Liabilities as Contingent Claims corporate bonds, we will see models that take future capital-structure changes more explicitly into account. In standard option pricing, where we observe the value of the underlying asset, implied volatility is determined by inverting an option-pricing formula.
Here, we have to jointly estimate the underlying asset value and the asset volatility from the price of a derivative security with the asset value as underlying security. We will explain how this can be done in a Merton setting using maximum-likelihood estimation.
This chapter sets up the basic Merton model and looks at price and yield impli- cations for corporate bonds in this framework. We then generalize asset dynamics including those of default-free bonds while retaining the zero-coupon bond struc- ture. Next, we look at the introduction of default barriers which can represent safety covenants or indicate decisions to change leverage in response to future movements in asset value.
We also increase the realism by considering coupon payments. Finally, we look at estimation of asset value in a Merton model and discuss an application of the framework to default prediction.
Here, W is a standard Brownian motion under the probability measure P. Let the starting value of assets equal V0. The price of this asset is clearly the price of a liquidly traded security. At maturity of the bond, equity holders pay the face value of the debt precisely when the asset value is higher than the face value of the bond.
If assets are worth less than D, equity owners do not want to pay D, and since they have limited liability they do not have to either.
Appendix D contains further references. Corporate Liabilities as Contingent Claims The question is then how the debt and equity are valued prior to the maturity date T.
Note that no other parties receive any payments from V. In particular, there are no bankruptcy costs going to third parties in the case where equity owners do not pay their debt and there are no corporate taxes or tax advantages to issuing debt.
Hence, the choice of D by assumption does not change VT , so in essence the Modigliani—Miller irrelevance of capital structure is hard-coded into the model. We will sometimes suppress some of the parameters in C if it is obvious from the context what they are.
Applying the Black—Scholes formula to price these options, we obtain the Merton model for risky debt. Some consequences of the option representation are that the bond price Bt has the following characteristics. This is clear given the fact that the face value of debt remains unchanged. It is also seen from the fact that the put option decreases as V goes up. Again not too surprising. Increasing the face value will produce a larger state-by-state payoff.
It is also seen from the fact that the call option decreases in value, which implies that equity is less valuable. This is most easily seen by looking at equity. The call option increases, and hence debt must decrease since the sum of the two remains unchanged. The higher discounting of the riskless bond is the dominating effect here. It will, however, shift wealth from bond holders to shareholders, since both the long call option held by the equity owners and the short put option held by the bond holders will increase in value.
This possibility of wealth transfer is an important reason for covenants in bonds: bond holders need to exercise some control over the investment decisions. In this chapter we always look at the continuously compounded yield of bonds. The face value of debt is Note that a more accurate term is really promised yield, since this yield is only realized when there is no default and the bond is held to maturity. Hence the promised yield should not be confused with expected return of the bond.
To see this, note that in a risk-neutral world where all assets must have an expected return of r, the promised yield on a defaultable bond is still larger than r. In this book, the difference between the yield of a defaultable bond and a corresponding treasury bond will always be referred to as the credit spread or yield spread, i. We reserve the term risk premium for the case where the taking of risk is rewarded so that the expected return of the bond is larger than r.
The risk structure of interest rates is obtained by viewing s T as a function of T. In Figures 2. The risk structure cannot be used as a term structure of interest rates for one issuer, however. We will return to this discussion in greater detail later. For now, consider the risk structure as a way of looking, as a function of time to maturity, at the yield that a particular issuer has to promise on a debt issue if the issue is the only debt issue and the debt is issued as zero-coupon bonds.
Yields, and hence yield spreads, have comparative statics, which follow easily from those known from option prices, with one very important exception: the depen- dence on time to maturity is not monotone for the typical cases, as revealed in Fig- ures 2.
The maximum point of the spread curve can be at very short matu- rities and at very large maturities, so we can obtain both monotonically decreasing and monotonically increasing risk structures within the range of maturities typically observed. Corporate Liabilities as Contingent Claims Note also that while yields on corporate bonds increase when the riskless interest rate increases, the yield spreads actually decrease.
We therefore now consider the behavior of the risk structure in the short end, i. The result we show is that when the value of assets is larger than the face value of debt, the yield spreads go to zero as time to maturity goes to 0 in the Merton model, i.
It is important to note that this is a consequence of the fast rate at which the probability of ending below D goes to 0. Hence, merely noting that the default probability itself goes to 0 is not enough.
The result is easy to check for a Brownian motion and hence also easy to believe for diffusions, which locally look like a Brownian motion. We now show why this fact implies 0 spreads in the short end. A bond with a current price, say, of 80 whose face value is will have an enormous annualized yield if it only has say a week to maturity. Equity makes no promises, but it is worth remembering that the equity is, of course, far riskier than debt. This is to be compared with the much riskier return distribution of the stock shown in Figure 2.
To see this, note Table 2. Senior debt can be priced as if it were the only debt issue and equity can be priced by viewing the entire debt as one class, so the most important change is really the valuation of junior debt.
Corporate Liabilities as Contingent Claims 0. A discretized distribution of corporate bond returns over 1 year in a model with very high leverage.
The asset value is and the face value is The asset volatility is assumed to be 0. A discretized distribution of corporate stock returns over 1 year with the same parameter values as in Figure 2. Payoffs to senior and junior debt and equity at maturity when the face values of senior and junior debt are DS and DJ , respectively. Option representations of senior and junior debt. C V , D is the payoff at expiration of a call-option with value of underlying equal to V and strike price D.
First of all, interest rates on treasury bonds are stochastic, and secondly, there is evidence that they are correlated with credit spreads see, for example, Duffee When we use a standard Vasicek model for the riskless rate, the pricing problem in a Merton model with zero-coupon debt is a now standard application of the numeraire-change technique. This technique will appear again later, so we describe the structure of the argument in some detail.
The effect of interest-rate volatility in a Merton model with stochastic interest rates. The asset volatility is 0. We are then ready to analyze credit spreads in this model as a function of the parameters. We focus on two aspects: the effect of stochastic interest rates when there is no correlation; and the effect of correlation for given levels of volatility.
As seen in Figure 2. Letting the volatility be 0 brings us back to the standard Merton model, whereas a volatility of 0. Increasing volatility to 0. Correlation, as studied in Figure 2. Note that higher correlation produces higher spreads.
An intuitive explanation is that when asset value falls, interest rates have a tendency to fall as well, thereby decreasing the drift of assets, which strengthens the drift towards bankruptcy. The effect of correlation between interest rates and asset value in a Merton model with stochastic interest rates.
We will then use the pricing relationship to discuss the implications for the spreads in the short end and we will show how one compares the effect of volatility induced by jumps with that induced by diffusion volatility.
This section can be skipped without loss of continuity. The Merton Model with Jumps in Asset Value 21 and let this be the dynamics of the cumulative return for the underlying asset-value process. Recall that we can get the Black—Scholes partial differential equation PDE by performing the following steps in the classical setup. Identify the drift term and the martingale part.
We now perform the equivalent of these steps in our simple jump-diffusion case. Such equations can only be solved explicitly in very special cases. One could try to solve the integro-differential equation for contingent-claims prices. To understand some of the important changes that are caused by introducing jumps in a Merton model, we focus on two aspects: the effect on credit spreads in the short end, and the role of the source of volatility, i.
The effect of changing the mean jump size and the intensity in a Merton model with jumps in asset value. This makes recovery at an immediate default lower and hence increases the spread in the short end. From b to c the intensity is doubled, and we notice the exact doubling of spreads in the short end, since expected recovery is held constant but the default probability over the next short time interval is doubled.
The effect of the source of volatility in a Merton model with jumps in asset value. In both cases, three different current asset values are considered. The difference between the spread curves in the case of low leverage is very small.
Effects are also limited in the long end in all cases. We recognize the second term in the expression as the expected fractional loss given default. D An immediate consequence is that doubling the overall jump intensity should double the instantaneous spread. Another consequence is, as is intuitively obvious, that lowering the mean jump size should typically lead to higher spreads. Both facts are illustrated in Figure 2. This is done in Figure 2. As is evident from that graph, the main effect is in the short end of the risk structure of interest rates.
While it is tempting to think of quadratic variation as realized volatility, it is impor- tant to understand the difference between the volatility arising from the diffusion and the volatility arising from the jump part.
We cannot obtain the jump-induced volatility, even theoretically, as our observations get closer and closer in time. So, while the jump-diffusion model is excellent for illustration and simulating the effects of jumps, the problems in estimating the model make it less attractive in practical risk management.
The pricing of the coupon bond needs to look at all coupon payments in a single model and in this context our assumptions on asset sales become critical. To understand the problem and see how to implement a pricing algorithm, consider a coupon bond with two coupons D1 and D2 which have to be paid at dates t1 and t2. In this simple model with no information asymmetries, it does not matter which option they choose. Corporate Liabilities as Contingent Claims matter which option we consider.
The option to issue new debt is not considered here, where we assume that the debt structure is static. So, think of equity owners as deciding an instant before t1 whether to pay the coupon at date t1 out of their own pockets. Applying this line of reasoning leads to the following recursion when pricing coupon debt assuming no asset sales.
Given coupons D1 ,. This will give us prices of debt and equity using an assumption of no asset sales. What if asset sales are allowed? In this case we still work recursively backwards but we need to adjust both the default boundary and the asset value. Default Barriers: the Black—Cox Setup 29 paying out of their own pockets, but in fact, it is optimal for equity owners to sell assets instead of covering the payment themselves. To write down how to price the securities is a little more cumbersome even if the imple- mentation is not too hard.
We leave the details to the reader. If the assets are large enough, we subtract the coupon payment in the asset value. In the model with asset sales, we need to distinguish between the sequence of up-and-down moves, since we subtract an amount from the asset value at coupon dates that is not a constant fraction of asset value. The assumptions we make on asset sales are critical for our valuation and for term-structure implications.
We return to this in a later section. Note that we have only considered one debt issue. When there are several debt issues we of course need to keep track of the recovery assigned to the different issues at liquidation dates.
The idea is to let defaults occur prior to the maturity of the bond. In mathematical terms, default will happen when the level of the asset value hits a lower boundary, modelled as a deterministic function of time. In the original approach of Black and Cox, the boundary represents the point at which bond safety covenants cause a default.
Corporate Liabilities as Contingent Claims due to liquidity constraints where we approximate frequent small coupon payments by a continuous stream of payments.
First-passage times for diffusions have been heavily studied. If one is looking for closed-form solutions, it is hard to go much beyond Brownian motion hitting a linear boundary although there are a few extensions, as mentioned in the bibliographical notes. This mathematical fact almost dictates the type of boundary for asset value that we are interested in, namely boundaries that bring us back into the familiar case after we take logarithms.
To simplify notation, let the current date be 0 so that the maturity date T is also time to maturity. We assume that the starting value V is above C1. We will value the contribution from these two parts separately. From Proposition Now we will use this expression to get the price B m of the bond payout at maturity of a bond in the Black—Cox model. Corporate Liabilities as Contingent Claims The trick is to rewrite the default event and use a different underlying price process.
For an argument on how to proceed, see Appendix B. This in turn saves us the computational burden of computing B b to get the bond value, since we can just determine this value as the difference between asset value and equity. While the existence of a default barrier increases the probability of default in a Black—Cox setting compared with that in a Merton setting, note that the bond holders actually take over the remaining assets when the boundary is hit and this in fact leads to higher bond prices and lower spreads.
That is, if we consider a Merton model and a Black—Cox model where the face value of debt is the same in the two models, then the Black—Cox version will have lower spreads, as illustrated in Figure 2.
The fact that credit spreads decrease is of course consistent with the boundary representing a safety covenant and in a diffusion model the boundary makes it impossible to recover less than the value at the boundary.
But if instead we make bankruptcy costly to the debt holders, the presence of a barrier will be capable of increasing credit spreads compared with a Merton model.
Using an exogenous recovery in a structural model leads to a sort of hybrid form of model in which we focus on modeling asset value but do not distribute all of the asset value to the different parts of the capital structure. Since the analytical derivation is not given in Black and Cox, I have been unable to locate the source of the discrepancy.
A comparison between Merton and Black—Cox. Note that despite a higher default probability in the Black—Cox model, the spreads are smaller in the comparable case with no dividend payments on the underlying asset. The smaller spread is due to the favorable recovery from the viewpoint of the bond holders.
In fact, when the maturity is long, the default triggering boundary becomes larger than the face value for long time horizons and the payoff to bond holders is therefore larger than face value in some states.
This implies negative spreads. The easiest case to handle is one with asset sales allowed, since payments on coupons then continue until maturity as long as the asset value does not hit zero. The boundary conditions then depend on which assumptions we make on asset sales.
To get analytic solutions for this general case, we assume that the debt is perpetual, so that only changes in the asset value and not the passing of time itself affect the value of the debt. Before look- ing at boundary conditions, an explicit solution to the homogeneous part and then the full ODE must be found, and this is done in Merton and in greater gen- erality in Black and Cox , who employ a substitution to obtain a differential equation whose solution can be expressed using special functions.
We will not dwell on the explicit form of the solution here but instead focus on the boundary conditions employed, since these reveal the assumptions made on asset sales.
The simpler boundary conditions arise when asset sales are permitted. With no asset sales, we need to decide what is necessary to determine an optimal point for the equity owners to stop paying coupons. The approach taken here closely resembles the approach we will see in the next section.
Plugging the solution to this problem into the boundary condition then gives the solution for the valuation of debt. The situation here is analogous to the situation considered in the Leland model of the next chapter after the coupon level of the debt issue has been chosen.
Corporate Liabilities as Contingent Claims 2. In this section we consider what can be done with stochastic interest rates when there is a default barrier. This solution is considered since it seems to be a natural way of modeling asset pro- cesses for insurance or reinsurance companies, who invest their reserves in products following an index say but who occasionally suffer large insurance losses.
Assume that the payoff at the default boundary is a fraction f1 of the asset value at that point, i. This means that we can treat all payoffs as occurring at date T and this brings the numeraire-change technique back on track. Since the riskless rate is a Gaussian diffusion, p 0, T has a closed-form solution as in the Vasicek model or the Hull—White extension of the Vasicek model.
Hence we need only compute the expectation of B T under the forward measure. Under QT , Zt is a martingale and it has a deterministic volatility.
All we need to do then is to translate our results for barrier options into the case with 0 drift and a deterministic volatility and this is done simply by changing the timescale. Corporate Liabilities as Contingent Claims The particular functional form of the boundary is what ensures the analytical solution in this model. We will return to this when we consider stationary leverage ratios.
Kim et al. Stochastic Interest Rates and Jumps with Barriers 39 technique is not helpful. Two of the important conclusions to draw from this model are the following. First, the default triggering boundary has a large effect on yield spreads. But this is mainly due to the fractional recovery, since as we have seen for the Black—Cox model, a boundary triggering default may well serve to lower spread because it transfers assets to debt holders in states where equity would still have positive value in a model with no boundary.
In Zhou b a numerical scheme is proposed for dealing with this situation. The model is in fact used to price corporate claims in a model with endogenous default, which we will meet in Chapter 3. Here we merely quote part of the results. This suggests a very natural application to insurance companies suffering occasional large claims from policy holders. While the method works for cases other than than those of exponentially distributed downwards jumps, we write it out here only for that case.
Without loss of generality, assume that X starts at x and that the default boundary is 0. The expressions for the transforms are given in Hilberink and Rogers , but inversion is not a trivial numerical problem. We will not discuss the numerical issues here.
Results Citations. Topics from this paper. Financial risk modeling. Citation Type. Has PDF. Publication Type. More Filters. The credit literature is replete with topical issues on bank credit risk management in the aftermath of the — financial meltdown. Scholarly articles have focused on the problem—albeit from … Expand. This doctoral thesis is devoted to estimation and examination of default probabilities PDs within credit risk management and comprises three various studies.
In the first study, we introduce a … Expand. Essays on Risk and Fair Pricing. This thesis is concerned with the valuation of contracts on financial markets, specifically the estimation of adjustments to the risk-free price of a derivative due to the inclusion of different … Expand. This thesis studies the modelling of credit risk in static credit portfolios, where the emphasis lies on the modelling of the dependence between defaults of the obligors.
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