portfolio losses and the term structure of loss transition rates- a new methodology for the pricing...

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Working PaperSeries

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National Centre of Competence in ResearchFinancial Valuation and Risk Management

Working Paper No. 264

Portfolio Losses and the Term Structure of LossTransition Rates: A New Methodology for the Pricing

of Portfolio Credit Derivates

Philipp J. Schnbucher

First version: February 2005Current version: September 2005

This research has been carried out within the NCCR FINRISK project onCredit Risk

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PORTFOLIO LOSSES AND THE TERM STRUCTURE OF LOSS

TRANSITION RATES: A NEW METHODOLOGY FOR THE PRICING OF

PORTFOLIO CREDIT DERIVATIVES

PHILIPP J. SCHONBUCHER

Department of Mathematics, ETH Zurich

February 2005this version: September 2005

Abstract. In this paper, we present a model for the joint stochastic evolution of the cu-mulative loss process of a credit portfolio and of its probability distribution. At any giventime, the loss distribution of the portfolio is represented using forward transition rates, i.e.the transition rates of a hypothetical time-inhomogeneous Markov chain which reproduces thedesired transition probability distribution. This approach allows a straightforward calibrationof the model (e.g. to a full initial term- and strike structure of synthetic CDOs includingthe correlation smile) and it is shown that (except for regularity restrictions) every arbitrage-free loss distribution admits such a representation with forward transition rates. To capturethe stochasticevolution of the loss distribution, the transition rates are then equipped withstochastic dynamics of their own, and martingale / drift restrictions on these dynamics arederived which ensure absence of arbitrage in the model. Furthermore, we analyze the dy-namics of spreads and STCDO-prices that are implied by the model and show that the inputparameters can be viewed as spread move parameters and correlation move parameters. Wealso show how every dynamic model for correlated individual defaults can be cast into thisframework.

1. Introduction

The markets for portfolio credit derivatives have become more standardized. Two importantreference indices of CDS portfolios have been created (iTraxx for Europe and CDX for theUSA) and there are now liquid markets for index-CDS and for standardized synthetic single-tranche CDOs (STCDOs) on these index portfolios. Simultaneously, more exotic portfolio credit

JEL Classification. G 13.Key words and phrases. Default Correlation, Stochastic Correlation, CDO Pricing, HJM-Models.Authors Address: ETH Zurich, D-MATH, Ramistr. 101, CH-8092 Zurich, Switzerland.

[email protected], www.schonbucher.deThe author would like to thank the participants at the 2nd Inaugural Fixed-Income Conference in Prague, and atthe JPMorgan Workshop on Credit Risk Modelling, New York, for their feedback. All errors are my own. Furthercomments and suggestions are welcome.Financial support by the National Centre of Competence in Research Financial Valuation and Risk Management(NCCR FINRISK), Project 5: Credit Riskis gratefully acknowledged. The NCCR FINRISK is a research programsupported by the Swiss National Science Foundation.

1


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2 PHILIPP J. SCHONBUCHER

Attachment 3Y 5Y 7Y 10Y

Low High Bid Offer BC Bid Offer BC Bid Offer BC Bid Offer BC

0 3 6.0 7.5 14.12 29.50 30.25 12.05 47.1 48 9.51 58.25 59.25 9.303 6 18 28 24.40 96 100 23.13 193 200 20.96 505 520 13.346 9 6 13 31.25 33 36 31.43 52 57 30.84 100 106 23.909 12 13 15 38.91 29 34 38.87 48 55 32.61

12 22 7.50 8.75 57.49 12 15 59.97 22 25 55.8522 100 2.25 4.00 5.25 7.25 8.25 10.75

Index 22 38 47 58

Table 1. Market quotes for tranched loss protection of different maturities onEuropean iTraxx Series 4, on Sept. 26th, 2005. Lower and upper attachmentpoints are in % of notional, base correlation (BC) is given in %. Prices for the0-3 tranche are % of notional upfront plus 500bp running, all other prices are

bp p.a.. Source (including BC calculations): JPMorgans Bloomberg page.

derivatives have arisen, in particular so-called bespoke STCDOs, CDOs of CDOs (or CDO2s),forward starting STCDOs, STCDOs with embedded options, and outright options on STCDOsor on indices.

The fact that prices on STCDOs on standardized portfolios are quoted very frequently and withrelatively narrow bid-ask spreads has uncovered several shortcomings of the existing pricing mod-els for CDOs. In particular, it turned out that the standard one-factor Gauss copula model (seeLi (2000)) was unable to simultaneously price all traded STCDOs with the same correlationparameter: Different STCDOs require different correlation parameters giving rise to the corre-

lation smile. A typical examples are the quotes and base correlation parameters in table 1.To account for this effect, several modifications of the one-factor Gauss copula model have beenproposed, see e.g. Andersen and Sidenius (2004), Hull and White (2003).

One reason for the introduction of CDS indices and the corresponding index tranches was thecreation ofhedge instrumentsfor the management of the risk of the more exotic portfolio creditderivatives. But when it comes to the dynamic hedging of transactions, the Gauss copula and itsextensions have a problem because these models are essentially static. As shown by Schonbucherand Schubert (2001), copula models can be equipped with consistent intensity dynamics but theresulting dynamics are not necessarily realistic (in particular if a Gaussian copula or one of itsmodifications is used), and the necessary analytics can become very involved very quickly. Fur-thermore, it turned out that Gauss copula models and their modifications had highly unrealistic

forwardprices for STCDO tranches which is problematic if these models are to be used to priceoptions or forward contracts on CDOs. In this respect, alternative approaches may be moresuccessful, e.g. the frailty approach (Schonbucher (2003)), fully multivariate intensity models(Duffie et al. (2000)), or the intensity-Gamma model by Joshi and Stacey (2005).

As opposed to the approaches described above which start from a specification of the individualobligors default processes, the model presented in this paper takes a top-down approach whichfocuses on thecumulativeloss process of thewholeportfolio. To motivate this approach, considerthe situation in equity markets: If we have to price an option on the S&P 500 index, it is natural


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TERM STRUCTURE OF LOSS TRANSITION RATES 3

to model the S&P 500 index directly instead of modelling the 500 individual share price processesof its constituents. Although some information on the individual components of the index is lost

in the transition to the aggregate top-down index model, an aggregate index model is oftenbetter able to capture other features such as a volatility smile or the empirically observed indexdynamics.

The situation in credit index markets is very similar, with liquid markets for the indices (iTraxxand CDX) and a set of traded derivatives on the index loss distributions. If there are liquidmarkets for STCDOs of all strikes1, we can even infer an implied distribution for the cumulativeloss process of the portfolio at all time-horizons at which we have a full strike structure ofSTCDOs. Thus, in order to provide a framework which allows to capture the price dynamicsof STCDOs we directly model the stochastic evolution of the distribution of the cumulative lossprocess of the underlying reference portfolio, and we defer the link back to the individual obligorsdefaults to a later stage.

The idea of a top-down approach for credit portfolio losses is not new. First, any model whichassumes homogeneity of obligors like the popular large-pool approximations (e.g. Vasicek(1987), Lucas et al. (2001) or Gordy (2003)) follows the same philosophy, and also more recentpapers, e.g. Giesecke and Goldberg (2005), Frey and Backhaus (2004) are inspired by similarideas. The distinguishing feature of the model presented here is that we model the full forwarddistribution of the loss process, i.e. we provide a framework which we can model the forwardlooking loss probabilities simultaneously for al lpossible time-horizons and loss levels. All othertop-down models that we are aware of do not attempt to model forward loss distributions butmodel the spot loss process. Unfortunately, a spot-modelling approach quickly leads intolarge complications when one tries to calibrate the model to a correlation smile.

The forward-modelling approach of this paper is inspired by the famous Heath et al. (1992)(HJM) approach in interest-rate models where a full forward-looking term structure of interest-rates is modelled. While Heath et al. (1992) model a single term structure of zero couponbond prices, here we shall model the evolution of a loss process, and we have to find a suitableparametric representation for a full term- and strike structure of prices of STCDOs on this lossprocess. Essentially, we first have to invent the forward rates for portfolio losses, before wecan make them stochastic.

We propose to to represent the loss distribution using forward transition ratesof a hypotheticaltime-inhomogeneous Markov chain which is constructed in such a way that its distribution co-incides with the given loss distribution. It is shown that (up to very weak regularity conditions)the set of loss distributions that can be reached based upon such forward transition-rates coin-

cides with the set ofallarbitrage-free loss distributions, i.e. we can truly say thatwithout lossof generalitywe represent the loss distribution with a set of forward transition rates. Further-more, this representation withforwardtransition rates allows a straightforward calibration of themodel to a full initial term- and strike structure of synthetic CDOs on the underlying portfolio(including the correlation smile), and (under the additional assumption of one-step transitions)this representation is even unique. This representation is introduced and analyzed in section 2.

1In this paper we talk of strikes for STCDOs to emphasize their similarity to options. The technical termin the market is attachment point and exhaustion point.


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In the next step (section 3), we allow the forward transition rates (and thus the loss distribution)to change stochastically over time. In this situation, it is not clear a priori whether the transition

probabilities that are derived from the current transition rates really are the true transitionprobabilities of the loss process, i.e. whether the transition rate dynamics and the loss processdynamics are consistent. We precisely define consistency and derive and characterize necessaryand sufficient conditions of consistency for the case with one-step transitions in section 3, andfor the multi-step case in section 4.

Section 3.3 discusses possible specifications of the model dynamics. In particular, we show howchanges in index-spread, correlation level or correlation smile can be represented with a simplespecification of the transition rate volatilities. We also show how the risk of a spread-explosionon an individual name (e.g. a Ford/GM episode like in May 2005) can be captured in the model.

In subsection 3.4 we allow the forward transition rates to jump at defaults (i.e. increments of

the loss process) by extending the loss process to a marked-point process, where the marker may(amongst other things) represent the identity of the defaulted obligor. If the model is equippedwith this feature, we show in section 3.5 that this modelling framework is now equivalent toa multivariate individual-intensity based model: By standard thinning of the loss process, wecan derive the individual default intensities (i.e. go from portfolio loss to individual loss), andclearly, any multivariate model for individual intensities implies a loss distribution which (by therepresentation results of section 2) can be represented using forward transition-rate dynamicsand these dynamics must then satisfy the necessary conditions for consistency that were givenearlier. Thus, the results in this paper can be viewed as a general modelling framework forportfolio credit risk, it becomes a concrete model once the initial term- and strike structure oftransition rates, and the transition rate volatilities are specified.

2. Representation of Loss Distributions using Markov Transition Rates

2.1. Obligors and Loss Process. The model is set up on a filtered probability space (, (Ft)(t0),Q),whereQ is a spot martingale measure, the filtration (Ft)(t0) satisfies the usual conditions, andis rich enough to support all processes which are introduced in the following. All processes are(Ft)(t0)-adapted and cadlag. There is a large but finite time horizonT

, and all expectationsand probabilities are with respect to Q (unless explicitly stated otherwise).Furthermore, we assume that default-free interest-rates r and defaults are independent under Q.

This assumption will be lifted in a companion paper.

We now introduce the two key quantities of the model: The loss process and the loss distribution:

1. Assumption (Loss Process and Loss Distribution).

(i) There arei = 1, . . . , I obligors with default timesi and default indicator processesDi(t) =1{it}. All obligors are assumed to have identical default losses which we normalize toone.


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(ii) The defaultloss process is defined as

L(t) :=

Ii=1

Di(t). (2.1)

(iii) TheQ-distribution ofL(T)at timet Tis described withp(t, T) = (p0(t, T), . . . , pI(t, T)),

where

pn(t, T) := P [L(T) = n | Ft] for all n= 0, . . . , I , t T (2.2)

are the regular conditional probabilities defining the distribution of L(T). p(t, ) is calledthe loss distribution at time t.

(iv) If there exists a liquid, arbitrage-free market for derivatives on the loss processL(T) (e.g.STCDOs) which uniquely determines at any time t the loss distribution, this distributionis called the implied loss distribution at timet.

(v) UnderQ, the default-free interest-rates are independent from the defaultsDi.

The existence of an impliedloss distribution (iv above) is not essential, as the loss distributionunder Q is well-defined in any case. Nevertheless, we believe that it describes an importantspecial case. For example, if a full strike- and maturity structure of STCDOs existed, sucha distribution would be uniquely determined in the same way as a full strike- and maturitystructure of European option prices determines the implied distribution of the underlying asset.For credit indices, such a full strike and maturity structure of STCDOs does not exist, yet themarket is moving in that direction. At the time of writing, there is a liquid market for STCDOson the standard CDS index portfolios (iTraxx and CDX) with 3, 5 and 10Y maturities and avariety of strikes. These prices restrict the set of possible loss distributions, although they donot uniquely determine it.

The assumption of identical losses given default (i above) is commonly made when modellingderivatives on CDS indices. Later, we will show how this assumption be relaxed. Assumption (v)of independence of interest-rates from the loss process was made merely to simplify the expositionof the model at this stage, it too will be lifted later on.

2.2. Properties of the Loss Distribution. Being an integer-valued, piecewise constant, boundednon-decreasing process,L(t) is a submartingale with all regularity properties that we may desire.In particular, the regular conditional probabilities defining the loss distribution will exist. Werequire some very mild regularity properties from the loss distribution:

2. Assumption (Regular Loss Distribution).Lett T, n I. The implied distributionp(t, T) = (p0(t, T), . . . , pI(t, T))

satisfies:

(i) pn(t, T) is continuous inT for alln I andT t.(ii) pn(t, T) is at least once continuously differentiable inT, almost everywhere.

(iii) Ifpn(t, T0)> 0 for someT0, thenpn(t, T)> 0 for allT > T0.

Continuity and differentiability (property (i) and (ii)) are very convenient to ensure the the exis-tence of transition rates, it is hard to imagine situations where it may not be satisfied (althoughit is theoretically possible). In combination with the initial condition pn(t, t) = 1{L(t)=n} it alsoensures that L(t) is continuous in probability, i.e. limh0 P [L(t + h) = L(t)| Ft] = 1 for all


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t. Point (iii) is also a technical condition which makes much sense economically: Essentially, itsays that in every state there is a positive probability that there will be no more defaults.

In some situations it can be convenient to assume in addition that the probability of simultaneousdefaults is zero, i.e. Lhas only increments of one. (For example, this makes the correspondencebetween transition rates and loss distribution unique.) Furthermore, simultaneous defaults alsohave probability zero in most bottom-up portfolio credit risk models. Nevertheless, all results inthe following hold for the general situation.

From the path structure ofL we can already derive some properties ofp(t, T):

3. Lemma. For all0 n I andt T we have:

(i) pn(t, T) 0 andIm=0pm(t, T) = 1.

(ii) pn(t, t) = 1{L(t)=n}.(iii) pm(t, T) = 0 for allm < L(t).(iv) Monotonicity: The following function is non-increasing inT:

P [L(T) n| Ft] =n

m=0

pm(t, T).

Proof. Properties (i) to (iii) are immediate, so is property (iv) which follows from the monotonic-ity ofL.

Clearly, properties (i) to (iv) are necessary conditions for the initial loss distribution to be

arbitrage-free. Proposition 4 will show that these properties are also sufficient.

2.3. Time-Inhomogeneous Markov Chains. As we plan to represent the loss distributionusing Markov-chain transition rates, we now give a brief reminder of the main results that wewill need. Let us consider a time-inhomogeneous Markov chain L(t) with state space{0, 1, . . . , I }and generator matrix A(t) = (aij(t))0i,jI, i.e. fori =j , aij(t) is the transition rate from state

L(t) = i to L(t+dt) = j at time t. It is well-known that any matrixA(t) which satisfies theconditions

anm(t) 0,I

k=0, k=n

ank(t) = ann(t) =:an(t) (2.3)

for all 0 m, n I, t 0 can be used to define a finite-state time-inhomogeneous Markovchain. and that the transition probability matrix2 P(t, T) of this Markov chain can be found byintegrating the Kolmogorov ordinary differential equations:

d

dTP(t, T) = P(t, T)A(T), with initial condition: P(t, t) = Id. (2.4)

2The transition probability matrix is defined as a regular version of Pij(t, T) := Ph

L(T) =j L(t) =i

ifor

0 i, j I. Note that in our case the matrix indices start with zero in order to correspond to the respectivevalues ofL.


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IfL(t) is non-decreasing, then its transition matrix and its generator matrix must be upper-

triangular with a zero last row (the state L(t) = I is absorbing):

A(t) =

a0(t) a01(t) a02(t) . . . a0I(t)0 a1(t) a12(t) . . . a1I(t)...

......

......

0 0 0 aI1(t) aI1,I(t)0 0 0 0 0

(2.5)

Integrating the Kolmogorov equations (2.7) we see that the transition probability matrix isconnected to the entries of the generator matrix in the following way:

Pnm(t, T) =

0 for m < n,

exp{T

t an(s)ds} for m= n,

Pmm(t, T)T

tm1

k=n

Pnk(t,s)

Pmm(t,s) akm(t, s) ds for m > n.

(2.6)

Proof. (of (2.6)) The correctness of the solutions is easily checked by substitution back into theorigional differential equation.

Form > n the Kolmogorov equations (2.7) reduce (for m > n, t T) to

d

dTPnm(t, T) = Pnm(t, T)am(T) +

m1k=n

Pnk(t, T)akm(T), Pnm(t, t) = 0, (2.7)

or equivalently

ddT

Pnm(t, T) =

m1k=n

Pnk(t, T)akm(T) Pnm(t, T)

Nl=m+1

aml(T), Pnm(t, t) = 0, (2.8)

with solution

Pnm(t, T) = Pmm(t, T)

Tt

m1k=n

Pnk(t, s)

Pmm(t, s)akm(t, s) ds. (2.9)

Thus, there are closed-form solution for the transition probability matrix in terms of the transi-tion rates of the generator matrixA. In a practical implementation of the model, these integralscan be integrated numerically starting from the diagonal and going to the right (increasing m).

2.4. Representation of the Loss Distribution with a Forward Transition Rates. Tofind a representation of the loss distribution (2.2) as solution (2.6) of the Kolmogorov equationswith suitable transition rates anm(T), we need to show that such transition rates exist (underassumption 2). This is ensured by the following proposition:

4. Proposition (Representation of the Loss Distribution). Let t 0 and let p(t, T), T tbe a loss distribution satisfying assumptions 2 and having the properties stated in lemma 3(i)-

(iii). Then there exists a time-inhomogeneous Markov chainL(T), T t on {0, . . . , I } withgenerator matrixA(t, T) such that the the solutionP(t, T) to the Kolmogorov forward equations(2.4)reproduces the loss distributionp(t, T), i.e.

pn(t, T) = PL(t),n(t, T) 0 n I, T t. (2.10)


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Proof. See appendix.

From now on, justified by proposition 4, we will use the generator matrix A to represent thecurrent state of the conditional transition probabilities of the loss process L, or any arbitrage-freeconditional loss distribution. This is merely a matter of representation and considering only lossdistributions which can be represented with a generator matrix does not restrict the space ofpossible loss distributions (except for the technical regularity assumptions in assumption 2).

5. Definition (Forward Transition Rates). Letp(t, T) be the loss distribution to a loss processL(t) satisfying assumptions 1 and 2. We call the generator matrixA(t, T) of proposition 4 theforward transition rates ofL at timet.

6. Corollary. For every non-decreasing loss process, the associated loss distribution will satisfylemma 3. Thus, the space of loss distributions that satisfy the regularity assumption 2 and havethe properties of lemma 3 coincides with the space of loss distributions that can be represented

with a time-dependent generator matrix.

Loosely speaking3, this means that every arbitrage-free loss distribution can be represented witha matrix of forward transition rates. It is worth remarking that alternative approaches, inparticular the popular base correlation curves are not able to represent all arbitrage-free lossdistributions, nor are all base correlation curves arbitrage-free (see e.g. OKane and Livesey(2004)).

Note furthermore that in this proposition, L(t) i s a general non-decreasing process on thestate space {0, . . . , I } which can have much more complex dynamics than the a simple time-inhomogeneous Markov chain which is used to represent its transition probabilities. For exam-ple, L(t) could be driven by a multivariate firms value model or a multivariate intensity-based

model.

2.5. One-Step Representation of the Loss Distribution. The following proposition shows,that under very weak additional assumptions, we may even restrict the matrix of forward-transition rates to a matrix with only one-step transitions. This has the additional advantage thatunder the assumptions of proposition 7, the representation with one-step forward transition-ratesisunique.

7. Proposition (One-Step Representation of the Loss Distribution). Let t 0 and letp(t, T),T t be a loss distribution which satisfies the following properties:

(i) p(t, T) is at least once continuously differentiable inT,

(ii) p(t, T) satisfies the properties (i)-(iv) of lemma 3,(iii) pn(t, T)> 0 for allT > t andn L(t),(iv) for all0 n I, let

an(t, T) := 1

pn(t, T)

nk=0

Tpk(t, T). (2.11)

We assume that limTt an(t, T)< .

3Loosely speaking, because we did not specify the admissible trading strategies.


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which reproduces this loss distribution. Furthermore,

P

kn T

= P

n

nl=1

Ekl + T

= T

0P [n T e] f

n,k

(e)de,

where fn,k(e) is the density ofn

l=0 Ekl . If we now let the distributions of the E

kl converge

uniformly to a point mass at Ekl = 0 as k , then Pkn T

P [n T ] at all T

where P [n T ] is continuous, which implies weak convergence. A possible choice would be

Ejk normally distributed N(0, 1/k) truncated at mn.

Thus, it is only a very slight deviation from full generality to assume the existence of a set offorward transition-rates with only one-step transitions. Nevertheless it may be possible that amodel with multi-step transitions better captures an economic reality, e.g. when random recoveryrates are to be modelled (see section 4 for more details).

3. Stochastic Transition Rates: One-Step Transitions

In this section we treat the case of loss processes with only one-step transitions. As shown inproposition 7, up to very weak conditions this case encompasses all possible arbitrage-free lossdistributions.

3.1. Consistency.

9. Assumption (One-Step Transitions). The matrix of transition ratesA(t, T) follows a sto-chastic process int with the following properties: For allt T T, we have:

(i) A(t, T) is bi-diagonal, i.e. for all 0 n < I, we have anm(t, T) = 0 for m {n, n+ 1}.Furthermore, an(t, T) := ann(t, T) =an,n+1(t, T) 0 andaI(t, T) = 0.

(ii) The solutions (3.1) to the Kolmogorov equations are well-defined for all0 n, m I

Pnm(t, T) =

0 for m < n,

exp{T

t an(t, s)ds} for m= n,T

t Pn,m1(t, s) am1(t, s)e

RT

s am(t,u)du ds for m > n.

(3.1)

(iii) There exists an integrable random variable Y() such that A(t, T;) Y() for all t T T a.s.

In the first part of the assumption we assume that only one-step transitions are necessary torepresent the loss distribution, essentially ruling out simultaneous defaults of multiple obligorsat exactly the same time. This restriction has the advantage of making the correspondence fromloss distributions to transition rates unique (see e.g. the proof to proposition 4) and of makingthe martingale restrictions that we will derive later on unique, too. Point (ii) is a very weakassumption on the realizations of possible forward transition rates, essentially we only rule outrealizations that are not integrable with respect to T. If (ii) holds, the Kolmogorov equationsalso hold

TPnm(t, T) = am(t, T)Pnm(t, T) + am1(t, T)Pn,m1(t, T) (3.2)


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for 0 n m I (setting a1(t, T) = 0 = P0,1(t, T)). Furthermore, we will always havePnm(t, T) [0, 1]. Point (iii) of assumption 9 is also purely technical to allow the interchange of

differentiation, integration and expectation of the transition probabilities.

Given that we now have a stochastic matrix A(t, T), we must find criteria which allow us tocheck whether the dynamics ofA are consistent with the dynamics ofL.

10. Definition(Consistency). Let A(t, T) and P(t, T) be stochastic processes according to as-sumption 9. LetL(t)be a given loss process, whose loss distribution at timet is given byp(t, T).A(t, T)is said to beconsistent withL if for every givenT tand every0 n Ithe transitionprobabilitiesPL(t),n(t, T) satisfy

PL(t),n(t, T) =pn(t, T) =P [L(T) =n | Ft] . (3.3)

The following is an immediate consequence of this definition:

11. Proposition. LetA(t, T)andP(t, T)be stochastic processes according to assumption 9. LetL(t) be a given loss process. A(t, T) is consistent to L(t) if and only if

PL(t),n(t, T)

(viewed as a process int) is a martingale for all0 n I, T T.

Proof. Let PL(t),n(t, T) be a martingale. By the final condition of (3.1),PL(T),n(T, T) = 1{L(T)=n}.

Thus, PL(t),n(t, T) = E

1{L(T)=n} Ft , and A(t, T) and L(t) are consistent.

The converse direction only if follows because every conditional expectation of an integrablerandom variable is a uniformly integrable martingale.

Under consistency, the predictable compensator ofL can be characterized using a consistent setof forward transition rates A(t, T):

12. Proposition. Let assumption 9 hold, and A(t, T) be consistent to L(t). Then the pointprocessL(t) must have an arrival intensityL(t) which is a.e. given by

L(t) =aL(t)

(t, t). (3.4)

Proof. The proof is a direct application of Avens theorem (see Aven (1985)):W.l.o.g. we assumeL(t)< I. (L(t) = Iis trivial.) Let{k}k0 be a sequence decreasing to zero.Define

Zk(t) = 1

kE [L(t +k) L(t)| Ft]


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We have to show that limk Zk(t) =aL(t)(t, t) a.s. This follows from

Zk(t) = 1k

In=1

(n L(t))pn(t, t +k) = 1k

In=L(t)+1

(n L(t)) PL(t),n(t, t +k)

=I

n=L(t)+1

(n L(t)) 1

k

PL(t),n(t, t) +k

TPL(t),n(t, t) + o(

2k)

=I

n=L(t)+1

(n L(t))

an(t, t)PL(t),n(t, t) + an1(t, t)PL(t),n1(t, t)

+o(k)

k

=aL(t)(t, t)PL(t),L(t)(t, t) + O(k) = aL(t)(t, t) + o(1),

where used a Taylor approximation, substituted the Kolmogorov equations, and used thatPL(t),n(t, t) = 0 for n > L(t). Thus, limk Zk(t) = aL(t)(t, t), and this convergence is bounded

by the random variable Y introduced in assumption 9. Thus, all conditions of Avens theoremare satisfied and the claim follows.

Conversely, proposition 12 implies that it is a necessary condition for existence of a consistentset of dynamic forward transition rates that the cumulative loss process L(t) admits an intensity.This means that default times in this setup are totally inaccessible and that this framework isa generalized version of the intensity-based framework. In particular, most firms value modelscannot be cast into this framework, although (by proposition 4) it is still possible to representthe loss distribution of such models at almost all times with a set of forward transition rates.The problem is that these transition rates will explode as a default is approached.

The following result characterizes more clearly when pn(t, T) is a martingale.13. Proposition. LetA(t, T) satisfy assumption 9, and letL(t) be a loss process. The processPL(t),n(t, T) is a Q-martingale for all n N, t T if and only if an(t, T)PL(t),n(t, T) is aQ-martingale for alln N, t T.

Proof. IfPL(t),n(t, T) is a martingale for all T, then also 1

h(PL(t),n(t, T +h) PL(t),n(t, T)) is

a martingale for all h > 0. The mean value theorem in conjunction with the boundedness ofA(t, T) and the Kolmogorov equations ensures that the difference quotients remain bounded aswe let h 0. So, ifpn(t, T) is a martingale, then

TPL(t),n(t, T) = an(t, T)PL(t),n(t, T) + an1(t, T)PL(t),n1(t, T) (3.5)

is also a martingale.Forn L(t), PL(t),n(t, T)an(t, T) = 0 is trivially a martingale.

At n= L(t), we have T

PL(t),n(t, T) = an(t, T)PL(t),n(t, T), which means that an(t, T)PL(t),n(t, T)must be a martingale. The induction step (n 1) n for n > L(t) follows directly from (3.5)using the fact that the l.h.s. of (3.5) must be a martingale and that an1(t, T)PL(t),n1(t, T)has already been shown to be a martingale.Conversely, ifan(t, T)PL(t),n(t, T) are martingales for all n, then by the Kolmogorov equationsT

PL(t),n(t, T) are also martingales, then PL(t),n(t, S) = PL(t),n(t, t) +S

tT

PL(t),n(t, T)dT isalso a martingale.


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Note that we did not assume consistency with L(t) in proposition 13, this is a statement onA(t, T) and P(t, T). There is a related representation of the criterion given in proposition 13

which is of independent interest as it identifies the survival measures that are best suited in thiscontext:

14. Corollary (Martingale Condition PT

n ). LetA be consistent to L and define then-survival

measurePT

n with the Radon-Nikodym density

dPT

n

dQ

t

=LTn (t) := EQ

1{L(T)=n}

Ft pn(0, T)

= PL(t),n(t, T)

PL(t),n(0, T). (3.6)

an(t, T)PL(t),n(t, T) is aQ-martingale if and only ifan(t, T) is aPT

n -martingale.

Proof. Change the measure in the martingale condition.

3.2. A Diffusion-Based Specification. We now need to get more concrete regarding the dy-namics ofA in order to make the model operational.

15. Assumption (Dynamics of A). There are only one-step transitions (assumption 9 holds)and for alln < I, the dynamics ofan(t, T) are given by

dan(t, T) = n(t, T)dt +n(t, T)dW, (3.7)

whereW is ad-dimensionalQ-Brownian motion. n(t, T) andn(t, T) are predictable stochas-tic processes taking values in R and Rd respectively which are sufficiently regular to allow theapplication of Fubinis theorem4 in the following.

Choosing diffusion-based dynamics for A(t, T) essentially rules out that the transition rates arealso driven by the jumps in L, i.e. we never have jumps in the transition rates at defaults. Webelieve that this is not very restrictive, as the transition rates are already conditioned on L(t),anyway, so the intensity of the next default will still change upon default events. This can alsobe seen from the fact that L(t) = aL(t)(t, t) does depend on the past number of defaults viathe index ofa. On the other hand, the identityof the defaulted obligor usually also affects thetransition rates for future defaults. If, for example, there is one obligor with very high defaultrisk defaulted, the intensity of the next default may be lower than if a low-risk obligor defaultedbecause in the latter case the very risky obligor is still in the portfolio. In the general version ofthe model we will therefore allowA to jump at times of defaults.

To characterize the dynamics of the transition probabilities Pnm(t, T), we writedPnm(t, T) =unm(t, T)dt + vnm(t, T)dW, (3.8)

and dPL(t),m(t, T) =um(t, T)dt + vm(t, T)dW+m(t, T)dL(t), (3.9)

whereum(t, T) =uL(t),m(t, T) and vm(t, T) =vL(t),m(t, T). As the dynamics ofPL(t),n(t, T)are already uniquely determined by the dynamics of L and A, this is notation and not anassumption.

4 See e.g. Protter (1990), ch. IV.4 for a discussion of such conditions.


15/31


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(ii) the intensity ofL(t) is given by equation (3.4)

L(t) =aL(t)(t, t). (3.4)

Proof. only if:AssumeA and L are consistent. Condition (3.4) follows from proposition 12. We must have thatPL(t)m(t, T) and am(t, T)PL(t)m(t, T) are martingales. But

d

am(t, T)PL(t)m(t, T)

= am(t, T)dPL(t)m(t, T) + PL(t)m(t, T)dam(t, T) + d

am(t, T), PL(t)m(t, T)

,

and thus

E

d

am(t, T)PL(t)m(t, T) Ft = PL(t)m(t, T)m(t, T)dt +m(t, T), vL(t)m(t, T)dt.

PL(t)m(t, T) is a martingale with zero expected increments, and the covariation of a and Pequals the predictable variation from the diffusion terms because a does not jump. But, if

am(t, T)PL(t)m(t, T)

is to be a martingale, then it must also have zero expected increments,hence (3.15) follows.

if:We need to show that PL(t),m(t, T) are martingales for all 0 m I. Form = L(t), the claimhas already been shown in Schonbucher (1998) which anchors the induction.

Thus, letm > L(t) and assume that we have already shown that PL(t),m(t, T) are martingales forall 0 m < m. Then, by the same argument leading to proposition 13,am(t, T)PL(t),m(t, T)will also be martingales. This means in particular that the drift of am1(t, T)PL(t),m1(t, T)must compensate its jumps, i.e.

P aL(t),m1(t, T) = am1(t, T)m1(t, T)L(t).

Ifm 1 =L(t), substituting this and (3.15) into (3.11) yields the drift ofPL(t)m(t, T) as

um(t, T) = aL(t)(t, t)PL(t)+1,m(t, T)

Tt

eRT

s am(t,u)du

am1(t, u)m1(t, u)L(t)

du

= L(t)PL(t)+1,m(t, T) +L(t)

Tt

eRT

s am(t,u)du am1(t, u)PL(t),m1(t, u)du

= L(t)

PL(t)+1,m(t, T) PL(t),m(t, T)

= L(t)m(t, T).

Ifm 1> L(t), the drift ofPL(t)m(t, T) is

um(t, T) = T

t

eRT

s am(t,u)du am1(t, u)m1(t, u)L(t)du

= L(t)

Tt

eRT

s am(t,u)du am1(t, u)

PL(t)+1,m1(t, u) PL(t),m1(t, u)

du

= L(t)

PL(t)+1,m(t, u) PL(t),m(t, u)

= L(t)m(t, T).

Thus, the drift ofPL(t),m(t, T) will in all cases exactly compensate its jumps, PL(t),m(t, T) haszero expected increments which makes it a local martingale. But PL(t),m(t, T) is bounded aboveand below, so it is also a full martingale. Repeating the induction yields that PL(t),m(t, T) is amartingale for all m, hence A is consistent with L.


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3.3. Implementation and Parameter Specification. Proposition 17 is the key result of thissection, it has a similar role to the drift restrictions derived by Heath et al. (1992) for interest-rate

models.

Using this result, a loss model can be specified by directly prescribing the volatilities of theforward transition rates. From there, the drifts of the forward transition rates under Q followfrom the drift condition 3.15, and the intensity of the loss process from (3.4). If this recipe isfollowed, absence of arbitrage is ensured. Furthermore, given a suitable choice of initial forwardtransition rates, the model will be automatically calibrated to a given term- and strike structureof STCDOs. It is now straightforward to simulate the future evolution of the model with Monte-Carlo simulation.

We would like to emphasize again, that the whole simulation will start off from a full initial term-and strike-structure of the forward transition rates. Thus, the model is automatically calibrated

to STCDOs and all other portfolio credit derivative prices that were used to determine the initialcondition. Furthermore, at any timet in the simulation of the future evolution of the forwardloss transition rates, the loss distribution at that particular point in time is easily recovered bysimply integrating the Kolmogorov equations with the then current transition rates an(t

, T).This means, that option payoffs that depend on the loss distribution (e.g. payoffs of optionson STCDOs) can be directly evaluated at maturity of the option, without needing any furthersimulation into the future.

E.g. to price a European option with maturity T1 on a STCDO with maturity T2, we wouldproceed as follows:

1. Draw a unit exponential random variableE

2. Simulate a path of{A(t, T)}0t(1) under the assumption L(t) = 0, where (1) is reached assoon as

(1)0

a0(s, s)ds= E.3. Set L(t) = 1 of the forward transition rates and the loss process L(t). (Note that we only

need to simulate t until T1, and that we only need to consider T T2.)4. AtT1, use A(T1, T) to calculateP(T1, T) for T [T1, T2]5. Calculate the conditional transition probabilities P(0, T1) using the conditional transition

matrixA(t).6. For each possible leveln = 0, . . . , I ofL(T1), calculate the price of the underlying STCDO

using then-th row ofP(T1, T). Calculate the option payofffrom this, weight it with the con-ditional probability P0,n(0, T1), multiply with an appropriate discount factor and aggregate.

7. Repeat.

Of course, hedge ratios and other risk parameters can also be determined by a very similarstrategy.

Another interesting issue is the specification of volatilities and factor loadings for the forwardtransition rates. Here, several alternatives arise which (besides being of practical interest) furtherdemonstrate the flexibility of the model and the variety of market moves that can be capturedin this framework.


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There are two dimensions in which we must specify factor loadings: The time dimension 0 T T, and the loss dimension 0 n I. The effects and implications of different factor

specifications in the time-dimension have already been studied extensively in the context ofmodelling the term structure of risk-free interest-rates, therefore we now concentrate on thenew dimension in the loss direction. In the following, we show possible factor loadings for onecomponent of the driving Brownian motion, of course several such factors should be combinedto achieve a realistic overall model of the dynamics of the forward transition rates.

No transition rate volatility:

n(t, T) = 0 for all 0 n I 1.

In this case, the model reduces to a simple time-dependent Markov chain for the loss process.There is no spread volatility, yet there can still be default dependence, but that dependencearises from the transitions of the loss process and the corresponding changes in the transition

rates alone.

Factor Parallel Shift:

n(t, T)> 0 for all 0 n I 1.

With this specification, a positive increment in the driving factor will increase all transition rates,while a negative increment in the driving factor will decrease the transition rates. Thus, thisfactor captures moves in the overall default risk in the portfolio as they would be reflected bymoves in the underlying portfolio-CDS (i.e. moves in the level of iTraxx or CDX). Thus, theparallel shift factor can be directly calibrated to the highly liquid index-CDS market and itsvolatility.

Factor Tilt around EL or Correlation Move:

n(t, T)< 0 for n EL(T)

n(t, T)> 0 for n > EL(T),

whereE L(T) =E [L(T)| Ft] is the expected loss for time T. A positive increment in a factorwith this loading would decease transition rates for states that are below the expected loss, andincrease them for states that are above the expected loss. With a suitabe parametrization wecan ensure that the expected loss remains unchanged, then this factor will drive the standarddeviation of the loss distribution, with positive increments increasing the standard deviation,and negative increments decreasing the standard deviation. But (for given expected losses) lossdistributions with high standard deviation correspond to loss distributions arising from portfolioswith high default correlation, while low standard deviation is roughly equivalent to low defaultcorrelation. As this factor drives market-implied forward transition rates, this means that this

factor will essentially drive changes in the level of the implied correlations of the underlyingSTCDOs.

Factor Individual Spread Blow-Out (GM/Ford Factor) :

n(t, T) 0 for n = L(t)

n(t, T) = 0 for n > L(t).

This factor only drives the rate of the nextdefault, while keeping the rest of the transition ratesessentially unaffected. A positive increment in this factor would capture scenarios in which an


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individual obligor has a sudden, very large increase in his spread, but the rest of the portfolioremains unaffected (essentially an idiosyncratic default/downgrade). Such an episode caused

considerable turmoil in the synthetic credit derivatives market in June 2005 when General Motorsand Ford Motor Company suffered downgrades. It is interesting that such a very idiosyncraticevent can be captured in this model, even though the default risk of individual obligors was neverdirectly modelled.

3.4. Jumps at Defaults. As mentioned in the discussion of assumption 15, it may be desirableto equip the events of defaults (i.e. the increments of L) with an additional random variable(a marker) which reveals additional relevant information such as the identity of the obligorthat defaulted. This can be achieved in the framework of marked point processes. We modifyassumption 15 as follows:

18. Assumption (Dynamics ofA with Marker).

(i) There are only one-step transitions (assumption 9 holds). Furthermore, at the times of theincrements in the loss processL, a random marker is drawn from the mark space(E) RdE .The timesT1, T2, . . .of increments inLand the corresponding markersE1, E2, . . .define amarked point process with compensator measure

(d, dt) = K(d)L(t)dt, (3.16)

whereK(d) is a predictable probability measure onEand its Borel sets.(ii) For alln < I, the dynamics ofan(t, T) are driven by ad-dimensionalQ-Brownian motion

Wand the marked point process defined above:

dan(t, T) =n(t, T)dt +n(t, T)dW

+an(t, T; EL(t)+1)dL

E

an(t, T; )K(d)L(t)dt, (3.17)

where an(t, T; ) is a predictable function and n(t, T) and n(t, T) are predictable sto-chastic processes taking values in R and Rd respectively which are sufficiently regular toallow the application of Fubinis theorem in the following.

In (3.17), we already compensated the jumps in a this is no loss of generality as we can alwaysabsorb the compensator in the drift term. Given thata is driven by the marker, we also need to

derive the new dynamics ofPnm(t, T).19. Proposition. Let assumption 18 hold. Then the dynamics ofPnm(t, T) are given by

dPnm(t, T) =unm(t, T)dt + vnm(t, T)dW

+ T

t

eRT

s am(y)dycnm(t, s)ds

L(t)dt

+Pnm(t, T; EL(t)+1)dL

E

Pnm(t, T; )K(d)L(t)dt, (3.18)


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whereunm(t, T) andnm(t, T) are given as in proposition 16, and

cnm(t, s) =E

e

RT

s a

m(t,u;)du

1

P a

n,m1(t, s; )K(d)

Pnm(t, T; ) =

Tt

eRT

s am(t,u)+

a

m(t,u;)du

Pn,m1(t, s)am1(t, s) +

P an,m1(t, s; )

ds Pnm(t, T)

P anm(t, T; ) =am(t, T)Pnm(t, T; ) + Pnm(t, T)

am(t, T; ) +

Pnm(t, T; )

am(t, T; ).

The dynamics ofPL(t),m are given by

dPL(t)m(t, T) =um(t, T)dt + vm(t, T)dW

+ T

t

eRT

s am(y)dycn+1,m(t, s)ds

L(t)dt

+m(t, T)dL(t)

+Pn+1,m(t, T; EL(t)+1)dL

E

Pn+1,m(t, T; )K(d)L(t)dt. (3.19)

All predictable functions and processes are understood to be evaluated at their left limits.

Proof. See appendix A.3.

As before, the dynamics ofPL(t),m(t, T) imply necessary and sufficient conditions for consistency:

20. Proposition (Drift Restrictions). Let assumption 18 hold. The dynamics (3.17)are consis-tent with the loss processL(t) if and only if the following conditions are satisfied:

(i) The diff

usion parameters ofA satisfy for all0

m

I, t

T

T

PL(t),m(t, T)m(t, T) = m(t, T)vnm(t, T) cL(t)m(t, T)aL(t)(t, t), (3.20)

(ii) the intensity ofL(t) is given by equation (3.4)

L(t) =aL(t)(t, t). (3.4)

Proof. Assume that (3.20) holds. Compared to (3.15), we now have an additional term in thedrift condition (3.20) which exactly cancels the additional drift term in the second line of (3.19).Thus, we are left with the sum of two martingales: The integrals of line 1 and 3 of (3.19) (whichwere shown to be martingales in proposition 17), and the integral of the last line of (3.19) whichis a compensated jump process and hence a martingale.

Conversely, assume that A is consistent with L. Then PL(t),m(t, T) is a martingale. Hence,it must have expected increments of zero, which yields (3.20). (3.4) follows from proposition12.

Equipped with proposition 20 we can again build arbitrage-free models for the evolution of thetransition rates and the loss distribution. The starting point is now not only a specification ofthe transition rate volatilitiesn(t, T), but also a specification of

an(t, T; e), i.e. the size of their

jumps conditional on the realization of the marker.


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3.5. Comparison to Individual-Intensity Based Models. We define the canonical markerspace E = {1, . . . , I }, and at loss times with dL() = 1, we let the marker E be the index

of the obligor who defaulted at time . Furthermore we assume that there are no simultaneousdefaults. With this setup, we can switch between a forward-transition-rate based aggregate viewand an individual-intensity based view, using standard results on thinning and aggregation ofpoint processes:

Given the aggregate default intensity L(t) and the compensator measure of the marked pointprocess, the individual default intensities can be found by thinning: For all i S(t), we have

i(t) =

E

1{=i}K(d)L(t). (3.21)

Conversely, given the individual intensities, the intensity of the next default equals

L(t) =

iS(t)i(t), (3.22)

whereS(t) :={1 i I |i> t}is the set of survivors at time t, so the aggregate portfolio lossintensity is well-defined. Furthermore, given the individual default intensitiesi, there exists acorresponding loss distribution (defined as in equation (2.2)), and typically this loss distributionwill also satisfy 2 which means that there exits a set of associated forward transition rates. Thenthe dynamics of these forward transition rates must in turn satisfy proposition 20, because thetransition rates are consistent with the loss process by construction.

In particular, the modelling framework presented in this paper is no less general than a modellingapproach based upon individual default intensities. Despite this equivalence, we believe theforward transition-rate modelling approach has a number of significant advantages:

Calibration: The model can be fitted effortlessly and exactly to most common creditderivatives and portfolio credit derivatives prices.

Specification: The modelling quantities (transition rate volatilities) can be interpretedeasily in terms of shifts in index-CDS-spreads or STCDO implied correlations.

The framework of this paper is more general than most intensity-based specifications (wedo not need a Cox-process / hypothesis H-assumption), for example the modelling ofspecial effects like default contagion is straightforward, here.

4. Stochastic Transition Rates: Multi-Step Transitions

The main purpose of allowing multi-step transitions in this modelling framework is not to capturesimultaneous defaults of several obligors at the same time (which we believe to be an event thatcan be safely ignored in many cases), but rather to capture the effects of random or varyinglosses in default.

If, for example, the loss of the next default takes values of 2.5, 5, 7.5 and 10 with equal proba-bilities, then this situation can be captured by specifying a loss unit of 2.5 and equal transition


22/31


intensities for losses of 1,2,3 and 4 unit-steps. Thus, by a suitable choice of loss unit and the corre-sponding relative sizes of the transition intensities, a discrete distribution for the next increment

in L(t) can be modelled.

Alternatively, stochastic recovery rates can also be captured with the specification of a mappingC : L C(L) from the number of losses L to the dollar-amount of losses C(L). Identical andfixed lossesc given default correspond to a linear map C(L) = cL, but nonlinearC, e.g. a convexC, imply that losses in default tend to be larger in recessions (i.e. when there are many losses).

4.1. Multi-Step Transitions. Except for the new form of the solutions to the Kolmogorovequations (4.1), the generalization of assumption 9 to the multi-step case is straightforward:

21. Assumption (Multi-Step Transitions). The matrix of transition rates A(t, T) follows astochastic process in t with the following properties: For allt T T, we have:

(i) A(t, T) is upper-triangular, i.e. for all 0 n I, we haveanm(t, T) = 0 form < n and

anm(t, T) 0 for m > n. Furthermore, an(t, T) := ann(t, T) =I

k=n+1 an,k(t, T) andaIn (t, T) = 0 for all0 n I.

(ii) The solutions (3.1) to the Kolmogorov equations are well-defined for all0 n, m I

Pnm(t, T) =

0 for m < n,

exp{T

t an(t, s)ds} for m= n,m1

k=n

Tt

Pnk(t, s) akm(t, s)e

RT

s am(t,u)du ds for m > n.

(4.1)

(iii) There exists an integrable random variable Y() such that A(t, T;) Y() for all t T T a.s.

In this situation we need an extension of our original concept of consistency:

22. Definition(Multi-Step Consistency). LetA(t, T)andP(t, T)be stochastic processes accord-ing to assumption 21. LetL(t) be a given loss process, whose loss distribution at timet is givenbyp(t, T).A(t, T) is said to be consistent with L if for every given T t and every 0 n k I thefollowing process is a martingale

PL(t),n(t, T)ank(t, T). (4.2)

Multi-step consistency implies one-step consistency (by the same argument leading to proposition13), but it is a stronger property than one-step consistency, the converse implication need nothold.

Most of the results of the previous section carry over to the multi-step case with only minormodifications. Proposition 12 becomes proposition 23 which is proved analogously to 12:

23. Proposition. Let assumption 21 hold, and A(t, T) be consistent to L(t). Then the pointprocessL(t) must have an intensityL(t) which is a.e. given by

L(t) = aL(t)(t, t) (4.3)

and the event of a transition fromL(t) to k, wherek > L(t), has an arrival intensity of

Lk (t) = aL(t),k(t, t). (4.4)


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Extending assumption 15 the dynamics of the multi-step transitions are:

24. Assumption (Dynamics ofA). For all0 n,m < I, the dynamics ofanm(t, T) are

danm(t, T) = nm(t, T)dt +nm(t, T)dW, (4.5)

whereW is ad-dimensionalQ-Brownian motion. nm(t, T) andnm(t, T) are predictable sto-chastic processes taking values in R and Rd respectively which are sufficiently regular to allowthe application of Fubinis theorem in the following.

. . . and we need to modify the notation for the dynamics of the transition probabilities slightlyin order to allow Lto take several different values:

dPnm(t, T) = unm(t, T)dt + vnm(t, T)dW, (4.6)

dPL(t),m(t, T) = um(t, T)dt + vm(t, T)dW+

km(t, T; k)1{L(t)=kL(t)}. (4.7)

Again, the results regarding the dynamics of the transition probabilities in the multi-step caseand the consistency conditions on the drifts of the transition rates will be very similar to theresults for the one-step case.

25. Proposition (Multi-Step Transition Probability Dynamics). Let assumption 24 hold. Theparameters of the dynamics (4.6) and (4.7) of the transition probabilities are given by

m(t, T; k) =

0 form < L(t)

PL(t),m(t, T) form= L(t)

Pkm(t, T) PL(t),m(t, T) form > L(t)

(4.8)

unm(t, T) =

0 form < n

an(t, t) T

t n(t, u)du +

12

T

t n(t, u)du

2form= n

anm(t, t)Pmm(t, T) +T

t e

RT

s am(t,u)du

Pnm(t, u)m(t, u)

m(t, u)vnm(t, u) +m1

k=n+1 P ankm(t, u)

du form > n.

(4.9)

vnm(t, T) =

0 form < n

T

t n(t, s)ds form= nT

t e

RT

s am(t,u)du

m1k=n+1

P ankm(t, s) Pnm(t, s)m(t, s)

ds, form < n

(4.10)

whereP ankm(t, T) andP ankm(t, T)are the drift and diffusion coefficients ofPnk(t, T)akm(t, T) for

m > n

P ankm(t, T) = akm(t, T)unk(t, T) + Pnk(t, T)km(t, T) +km(t, T)vnk(t, T) (4.11)

P ankm(t, T) = Pnk(t, T)km(t, T) + akm(t, T)vnk(t, T). (4.12)

Proof. From equation (2.6) we know that for m > n

Pnm(t, T) =m1k=n

Tt

eRT

s am(t,u)duPnk(t, s)akm(t, s) ds. (4.13)


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The terms in the sum are of the same form as the expressions that were analyzed in the caseof one-step transitions, with the only difference that k replaces m 1 and akm(t, s) replaces

am1(t, s). Thus, we can directly transfer the calculations of appendix A.2 and sum them up toreach the claim.

This leads us directly to the generalization of proposition 17:

26. Proposition (Multi-Step Drift Restrictions). Let assumption 24 hold. The dynamics (4.5)are consistent with the loss processL(t) if and only if the following conditions are satisfied:

(i) The diffusion parameters ofA satisfy for all0 m k I, t T T

PL(t),m(t, T)mk(t, T) = mk(t, T)vL(t),m(t, T) (4.14)

(ii) the intensities of the individual transitions ofL(t) are given by equation (4.4)

Lk (t) = aL(t),k(t, t). (4.4)

The proof is again analogous to the proof of proposition 17.

Appendix A. Proofs

A.1. Proof of proposition 4.

Proof. W.l.o.g. we set t = 0 and L(0) = 0. We also suppress the first time argument in p andP. The proof is constructive, building the transition rates starting from n = 0. To reach anupper-triangular matrix we directly setaij = 0 for i > j . Then, we start from

Kk=0

Tpk(T) =

0kK

K+1lN

pk(T)akl(T), (A.1)

If equation (A.1) holds for all 0 K N, then this is equivalent to the Kolmogorov equationsbeing satisfied for all 0 K N. The intuitive interpretation of (A.1) is, that the amount bywhich the combined probability mass of states 0, . . . , K decreases (the derivative on the l.h.s.)equals the probability-weighted rates at which transitions occur from these states to states beyondK(the sum on the r.h.s.).

LetT 0 be given, and letn 0 be such thatpn(T)> 0. We set M:= min{m > n|pm(T)> 0}to the index of the next probability which is not zero. If such an M does not exist, we havepm(T) = 0 for all n < m Nwhich is a special case of the following construction (simply omitthe construction ofanM). Typically except for T = 0 there will be no intermediate stateswith zero probabilities, in this case M=n + 1.

Assume that we have already constructed the transition rates amk(T) for all 0 m < n, and all0 k N, where


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the firstn Kolmogorov equations are satisfied (for 0 m < n), and amk(T) = 0 for all 0 m < n < k N,

i.e. there are no transitions beyond state n from states below n.

We first set the transition rates from all intermediate states m = n+ 1, . . . , M 1 (the stateswith zero probabilities pm(T) = 0)

amk(T) = 0 for all 0 k N. (A.2)

The choice of the value of zero is arbitrary, any nonnegative number would have been admissible.Next, we set the transition rates from state n to these zero-states:

anm(T) =T

pm(T)

pn(T) . (A.3)

As pm(T) = 0, the T-derivative of pm(T) must be non-negative otherwise we would havepm(T+h)< 0 for some small h >0. pm(T) is strictly positive by assumption, so anm(T) 0is well-defined. Under this specification, the Kolmogorov equations are satisfied for statesm=n + 1, . . . , M 1.

Next, we setank(T) = 0 for allk > M. Thus, there are no transitions beyond stateM from anystate below M.

Using (A.1) for K= n yields:

n

k=0

Tpk(T) =

N

k=n+1pn(T)ank(T)

=pn(T)anM(T) +M1

k=n+1

pn(T)ank(T)

=pn(T)anM(T) +M1

k=n+1

Tpk(T)

(using (A.3)), which yields

anM(T) = 1

pn(T)

M1

k=0

Tpk(T) 0.

The inequality stems from the monotonicity (lemma 3 (iv)). Note that the transition rateanM(T)is the only transition rate to state Mthat can be non-negative.

We now have specified all transition rates away from state n. Thus, the holding rate of staten

is ann(T) = M

m=n+1 anm(T). By assumption, equation (A.1) is satisfied for all K < n. Byconstruction, it is now also satisfied for K=n. Thus, the first n + 1 Kolmogorov equations (upto index n) hold. Furthermore, the Kolmogorov equations for the statesn < m < Malso holdby construction. Thus, the induction step is complete.


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Finally, we need to anchor the induction at n = 0. But, inspection of the construction aboveshows that it also works for n= 0: We only needed the fact that there are no transitions from

states below n beyond n directly to states above n. But this property is trivially satisfied atn= 0. And positivity p0(T)> 0 also holds for all T.

Thus, for everyT 0 we can start at n = 0 and iteratively construct all transition ratesanm(T)such that the Kolmogorov equations are satisfied.

Given a generator matrix, it is a direct consequence of standard existence theorems for Markovchains that there always exists a loss process L(t) to support this generator matrix.

This construction avoids multi-step transitions wherever possible. This may introduce very largetransition rates if the state probabilities approach zero, in particular if the true model has multi-step transitions. This can be corrected by allowing multi-step transitions and re-distributing theprobability mass of transitions. If for example k < l andakl is very large (i.e. ifaklpk Tpl),then there will be a m > l such that alm > 0 is large. In this case, we can decrease aklpk bya certain amount , and simultaneously decrease almpl by while increasing akmpk by ; theKolmogorov equations will remain unaffected by this procedure and the two large transition rateswill have been decreased.

A.2. Proof of Proposition 16: Dynamics ofPnm(t, T).

Proof. The jump sizesn(t, T) follow directly from the changes in the first index ofPL(t,m)(t, T).

Let 0 n m I(the case n > m is trivial), and let 0 t T T. To simplify notation,we set (for s [t, T])

M(t, T) := Pn,m1(t, T)am1(t, T) (A.4)

dM(t, T) =:M(t, T)dt +M(t, T)dW. (A.5)

X(t,s,T) := exp{

Ts

am(t, u)du}. (A.6)

Pnm(t, T) =

Tt

M(t, s)X(t,s,T)ds forn < m

X(t,t,T) forn = m.(A.7)

The dynamics ofM andX are

dM(t, T) =

un,m1(t, T) + m1(t, T) +m1(t, T)vn,m1(t, T)

dt

+

Pn,m1(t, T)m1(t, T) + am1(t, T)vn,m1(t, T)

dW (A.8)

dX(t,s,T)

X(t,s,T) =

Ts

m(t, u)du +1

2

Ts

m(t, u)du2

dt T

s

m(t, u)du

dW. (A.9)

TheM-dynamics are a simple application of the Ito-product rule, while dynamics of expressionslike (A.6) are well-known from the analysis of Heath et al. (1992), where also the case m = n


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has been solved: The dynamics ofPnn(t, T) = X(t,t,T) are

dPnn(t, T)Pnn(t, T)

=

an(t, t) T

t

n(t, u)du + 12

Tt

n(t, u)du

2

dt T

t

n(t, u)du

dW.

(A.10)

We now take m > nand start from

Pnm(t, T) =

Tt

Pn,m1(t, s) am1(t, s)e

RT

s am(u)du ds. (A.11)

Substituting into (A.7) and using Itos product rule and the stochastic Fubini theorem yields

dPnm(t, T) = M(t, t)X(t,t,T)dt

+ T

t

M(t, s)X(t,s,T) T

s

m(t, u)du +1

2

T

s

m(t, u)du2

ds+

Tt

X(t,s,T)M(t, s)ds

Tt

X(t,s,T)M(t, s) T

s

m(t, u)du

ds

dt

+

Tt

X(t,s,T)M(t, s) M(t, s)X(t,s,T) T

s

m(t, u)du

ds

dW.

In the following, we will use frequently the following rule for interchange of order of integration

T

t

T

s

f(s, u)du ds= T

t

u

t

f(s, u)ds du, (A.12)

and the fact that X(t,s,T) =X(t,s,u)X(t,u,T) for all u [s, T].

We consider the parts of the dynamics ofPnm(t, T) in turn:

First, M(t, t) = 0 ifm > n + 1, and ifm = n + 1 thenM(t, t) =am1(t, t) in which caseX(t,t,T) =Pm,m(t, T).

Next, the part with m:

Tt

Ts

M(t, s)X(t,s,T)m(t, u)du ds=

Tt

ut

M(t, s)X(t,s,T)m(t, u)ds du

= T

t

ut

M(t, s)X(t,s,T)ds m(t, u)du

using X(t,s,T) =X(t,s,u)X(t,u,T) for all u [s, T] yields

=

Tt

ut

M(t, s)X(t,s,u)ds X(t,u,T)m(t, u)du

=

Tt

Pnm(t, u)X(t,u,T)m(t, u)du. (A.13)


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In the same way we reach for the volatility term:

Tt

Ts

M(t, s)X(t,s,T)m(t, u)du ds= T

tPnm(t, u)X(t,u,T)m(t, u)du.

For the part of the drift term with 12

Ts m(t, u)du

2, we use that

1

2

Ts

m(t, u)du2

=

Ts

y

1

2

ys

m(t, u)du2

dy

=

Ts

m(t, u) T

u

m(t, y)dy

du. (A.14)

This yields

Tt M(t, s)X(t,s,T)

1

2 T

s m(t, u)du2

ds= T

t

Ts M(t, s)X(t,s,T)m(t, u)

Tu m(t, y)dy

du ds

=

Tt

Pnm(t, u)X(t,u,T)m(t, u) T

u

m(t, y)dy

du

=

Tt

yt

Pnm(t, u)X(t,u,T)m(t, u)m(t, y)du dy

=

Tt

m(t, y)X(t,y,T) y

t

Pnm(t, u)X(t,u,y)m(t, u)du

dy.

The part with M(t, s) is left unchanged. The part with M(t, s) is again transformed using (A.12):

Tt

Ts

X(t,s,T)M(t, s)m(t, u)du ds= T

t

ut

X(t,s,T)M(t, s)m(t, u)ds du

=

Tt

X(t,u,T)m(t, u) u

t

X(t,s,u)M(t, s)ds

du.

Collecting terms, we reach

dPnm(t, T) = am1(t, t)Pm,m(t, T)1{m=n+1} dt

+

Tt

X(t,u,T)Pnm(t, u)m(t, u)du

+ T

t X(t,u,T)m(t, u) u

t X(t,s,u)Pnm(t, s)m(t, s)ds

du

+

Tt

X(t,u,T)M(t, u)du

Tt

X(t,u,T)m(t, u) u

t

X(t,s,u)M(t, s)ds

du

dt

+

Tt

X(t,s,T)M(t, s) Pnm(t, s)m(t, s)

ds

dW,


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or, a bit more simplified

dPnm(t, T) = am1(t, t)Pm,m(t, T)1{m=n+1} dt

+

Tt

X(t,u,T)Pnm(t, u)m(t, u)du +

Tt

X(t,u,T)M(t, u)du

Tt

X(t,u,T)m(t, u) u

t

X(t,s,u)M(t, s) Pnm(t, s)m(t, s)

ds

du

dt

+

Tt


ds

dW,

and finally,

dPnm(t, T) = am1(t, t)Pm,m(t, T)1{m=n+1}dt

+ T

t

X(t,u,T)

Pnm(t, u)m(t, u) + M(t, u) m(t, u)vnm(t, u)

du

dt

+ vnm(t, T)dW,

where

vnm(t, T) =

Tt


ds.

Collecting the appropriate terms yields the statement of the proposition.

A.3. Proof of Proposition 19.

Proof. We use the same notation as in appendix A.2. We write

dM(t, T) =M(t, T)dt +M(t, T)dW+M (L L)

dX(t, T) =X (t, T)dt +X (t, T)dW+X (L L),

where L(de, dt) is the jump measure of the marked point process, L(de, dt) its compensatormeasure, and L =

E(e)L(de, dt) denotes integration of the predictable function over

the mark space Ewith respect to the measure L (similarly for L).

Then Itos product rule for semimartingales states that

d(M X) =MX dt +MX dW+MX (L L),

where

MX =MX+ X M+

MX + (XY ) L +MX (L L),

MX =MX+X M

MX =MX+X M+

XM.

From this, it follows directly that P anm(t, T; ) of proposition 19 is the predictable jump functionofPnm(t, T)am(t, T), in particular

P an,m1(t, T; ) =M(t, T; ).


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REFERENCES 29

Furthermore, it is easy to see that

X (t,s,T) =X(t,s,T)e RTs am(t,u;)du 1.A direct substitution of these two jump measures combined with the It o product rule into (A.7)now yields the jump function Pnm(t, T; ) and the dynamics ofPnm(t, T) as they are stated inthe proposition. (Note that Pnm(t, T; ) can be found more easily by directly evaluating Pnmafter the jump minus Pnm before the jump.)

For the dynamics ofPL(t),m(t, T) we must recognize that there are two simultaneous jumps inPL(t),m(t, T): the jump in the index, and the jump in the transition rates. If is the time of ajump, we can thus decompose the jump in PL(t),m(t, T) as follows:

PL(),m(, T)PL(),m(, T) =

PL(),m(, T)PL(),m(, T)

+

PL(),m(, T)PL(),m(, T)

.

This decomposition shows that we can view the jumps in sequence: First, the change in theindex, and then the jumps in the transition rates am at the new levelL() = L() + 1. Hence,we have to use cL(t)+1,m and not cL(t),m in equation (3.19).

References

Leif Andersen and Jakob Sidenius. Extensions to the Gaussian copula: random recovery andrandom factor loadings. Journal of Credit Risk, 1(1):2970, 2004.

T. Aven. A theorem for determining the compensator of a counting process. ScandinavianJournal of Statistics, 12(1):6972, 1985.

Darrel Duffie, Jun Pan, and Kenneth Singleton. Transform analysis and asset pricing for affine

jump diffusions. Econometrica, 68(6):13431376, 2000.R. Frey and J. Backhaus. Portfolio credit risk models with interacting default intensities : a

Markovian approach. Working paper, Department of Mathematics, Universitt Leipzig, 2004.Kay Giesecke and Lisa Goldberg. A top down approach to multi-name credit. Working paper,

Cornell University, September 2005.Michael B. Gordy. A risk-factor model foundation for ratings-based bank capital rules. Journal

of Financial Intermediation, 12(3):199232, July 2003.David Heath, Robert Jarrow, and Andrew Morton. Bond pricing and the term structure of

interest rates: A new methodology for contingent claims valuation. Econometrica, 60:77105,1992.

John Hull and Alan White. Valuation of a CDO and an n-thto-default CDS without MonteCarlo simulation. Working paper, University of Toronto, 2003.

Mark S. Joshi and Alan Stacey. Intensity Gamma: a new approach to pricing portfolio creditderivatives. Working paper, Royal Bank of Scotland, Quantitative Research Centre, May 2005.

David. X. Li. On default correlation: A Copula function approach. working paper 99-07, RiskMetrics Group, April 2000.

Andre Lucas, Pieter Klaassen, Peter Spreij, and Stefan Staetmans. An analytic approach tocredit risk of large corporate bond and loan portfolios. Journal of Banking and Finance, 25(9):16351664, 2001.

Dominic OKane and Matthew Livesey. Base correlation explained. QCR Quarterly Q3/4,Lehman Brothers Fixed Income Quantitative Research, November 2004.

Philip Protter. Stochastic Integration and Differential Equations. Springer, 1990.


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30 REFERENCES

Philipp J. Schonbucher. Term structure modelling of defaultable bonds. The Review of Deriva-tives Studies, Special Issue: Credit Risk, 2(2/3):161192, 1998.

Philipp J. Schonbucher. Information-driven default contagion. Working paper, D-MATH, ETHZrich, December 2003.

Philipp J. Schonbucher and Dirk Schubert. Copula-dependent default risk in intensity models.Working paper, Department of Statistics, Bonn University, 2001.

Oldrich Vasicek. Probability of loss on loan portfolio. Working paper, KMV Corporation, 1987.

P. Schonbucher:

Mathematics Department, ETH Zurich, ETH Zentrum HG-F 42.1, Ramistr. 101, CH 8092 Zurich, Switzerland,Tel: +41-1-63226409

E-mail address: , P. Schonbucher [email protected], www.schonbucher.de

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