Credit Risk Models with Filtered Market Information R udiger Frey - - PowerPoint PPT Presentation

Credit Risk Models with Filtered Market Information

R¨ udiger Frey Universit¨ at Leipzig Bressanone, July 2007 ruediger.frey@math.uni-leipzig.de www.math.uni-leipzig.de/~frey joint with Abdel Gabih and Thorsten Schmidt

1

2

• 1. Introduction

Market for portfolio credit derivatives increases and becomes more and more liquid ⇒ suitable models for pricing and hedging these products are needed. Criteria for a good model (idealistic wish-list)

• Realistic dynamics of credit spread allowing for spread risk and

contagion.

• Realistic dependence structure of default times in order to capture
• bserved properties of credit derivative prices (in particular the

correlation-skew on CDO markets)

• Tractability.

computation of prices/hedge ratios and calibration with reasonable computational effort.

3

Existing model classes

• Models with conditionally independent defaults and observable

factors such as [Duffie and Garleanu, 2001], or [Graziano and Rogers, 2006 + Tractability, both theoretically and - for simple parameterizations

• also numerically;

+ Inherently dynamic models; – no default contagion; – Realistic levels of default correlation and credit spread dynamics require complicated models;

4

• (Factor)

copula models such as [Laurent and Gregory, 2005] [Sch¨

• nbucher, 2003] or [Hull and White, 2004] (market standard.)

+ Easy to calibrate to defaultable term structure (CDS-spreads). + Sophisticated parameterizations can explain CDO prices. – Inherently static models; in particular hedging is based on ad-hoc methods.

• Models with interacting intensities.

Default contagion and is explicitly modelled, often using Markov chains. Examples include [Jarrow and Yu, 2001], [Frey and Backhaus, 2006]. + Dynamic framework allowing for with rich dynamics of credit- spreads

• Tractability, at least for non-homogeneous portfolios.

5

Our information-based approach

We model the default evolution of a portfolio of m firms; τi denotes the default time of firm i, Yt,i = 1{τi≤t} is the corresponding default indicator, and Yt = (Yt,1, . . . , Yt,m) gives current default state; default history is FY . We work directly under risk-neutral measure Q. Three layers of information: 1. Fundamental Model. Here default times τi are conditionally independent doubly stochastic random times driven by a finite-state Markov chain X with state space SX = {1, . . . , K}. Fundamental model is a theoretical device for model-construction.

6

2. Market information. Prices of traded credit derivatives are determined by informed market-participants. These investors

• bserve the default history and some process Z giving X in

additive Gaussian noise. Discounted prices of traded securities will be martingales wrt so-called market information FM := FY ∨ FZ; filtering results wrt FM will be used to obtain asset price dynamics.

• 3. Investor information The process Z represents an abstract form
• f ‘insider information’ and is not directly observable. We therefore

study pricing and hedging of (non-traded) credit derivatives from the viewpoint of secondary-market investors with information set FI ⊂ FM (investor information). We assume that FI contains the default history FY and (noisily

• bserved) prices of traded credit derivatives.

7

Advantages

• Prices

are weighted averages

• f

full-information value (the theoretical price wrt FX ∨ FY ) so that most computations are done in the full-information model. Since the latter has a simple structure, computations become straightforward.

• Rich credit-spread dynamics with spread risk (as credit spreads

fluctuate in response to fluctuations in Z) and default contagion (as defaults of firms in the portfolio lead to an update of the conditional distribution of X given FM

t ).

• Model has has a natural factor structure with factors given by the

conditional probabilities πk

t = Q(Xt = k | FM) , 1 ≤ k ≤ K.

• Great flexibility for calibration. In particular, we may view observed

prices as noisy observation of the state Xt and apply calibration via filtering.

8

Some related work

• [Frey and Runggaldier, 2006].

Relation between credit risk and nonlinear filtering and analysis of corresponding filtering problems; dynamics of credit risky securities is not studied

• [Gombani et al., 2005] Calibration via filtering for default-free

Gaussian term-structure models.

• [Frey and Schmidt, 2006] Firm-value models with unobservable

asset-value, extending [Duffie and Lando, 2001]

• and many more...

9

• 2. Model and Notation
• We work on probability space (Ω, F, Q) with filtration F.

All processes will be F adapted.

• Consider

portfolio

• f

m firms with default state Yt = (Yt,1, . . . , Yt,m) for Yt,i = 1{τi≤t}. Y i

t is obtained from Yt by

flipping ith coordinate. Ordered default times denoted by T0 < T1 < . . . < Tm; ξn ∈ {1, . . . , m} gives identity of the firm defaulting at Tn.

• Default-free interest rate r(t), t ≥ 0 deterministic.

Here we let r(t) ≡ 0.

10

The fundamental model

Consider a finite-state Markov chain X with generator QX and SX := {1, . . . , K} We assume that A1 The default times have (Q, F)-default intensity (λi(Xt)), i.e. there are functions λj : SX → (0, ∞), such that the processes Mt,j := Yt,j − t∧τj λj(Xs−)ds (1) are F-martingales, 1 ≤ j ≤ m. Moreover, τ1, . . . , τm are conditionally independent given FX

∞ = σ(Xs : s ≥ 0).

Define the full-information value of a FY

T -measurable claim H by

EQ e−

T

t rsdsH | Ft

• =: h(t, Xt, Yt) ;

(2) the last definition makes sense since (X, Y ) is Markov w.r.t. F.

11

Market information

Recall that the informational advantage of informed market participants is modelled via observations of a process Z. Formally, A2 FM = FY ∨ FZ, where the l-dim. process Z solves the SDE dZt = a(Xt)dt + dBt. Here, B is an l-dim standard F-Brownian motion independent of X and Y , and a(·) is a function from SX to Rl.

• Notation. Given a generic process U, we denote by

U the optional projection of U w.r.t. the market filtration FM; recall that U is a right continuous process and Ut = E(Ut|FM

t ) for all t ≥ 0.

12

• 3. Dynamics of Traded security

Traded securities. We consider N liquidly traded credit derivatives (eg. corporate bonds) with maturity T and FI

T-measurable payoff

PT,1, . . . , PT,N. A3 (Martingale modelling) The observed prices of traded securities are given by E

• PT,i|FM

t

• =:

pt,i (expectation wrt. Q). Market-pricing as a nonlinear filtering problem. Denote by pi(t, Xt, Yt) the full-information value of security i. We get from iterated conditional expectations we therefore obtain

• pt,i = E
• E(PT,i|Ft) | FM

t

• = E
• pi(t, Xt, Yt)|FM

t

• .

(3) Hence we need to obtain the conditional distribution of X given FM

t

(a nonlinear filtering problem).

13

Security-price dynamics

We introduce the innovations processes as follows:

• Mt,j := Yt,j −

t∧τj

• λj(Xs−)ds

for j = 1, · · · , m µt,i := Bt,i = Zt,i − t

• ai(Xs) ds

for i = 1, · · · , l. Note that Mj is an FM-martingale and µ is FM-Brownian motion. Lemma 1. Every square integrable FM-martingale ( Ut)t∈[0,T ] has the representation

• UT =

U0 + T γ⊤

s d

Ms + T α⊤

s d

µs, (4) for Rm respectively Rl-valued FM-predictable processes γ and α such that E T

0 |γs|2ds + E

T

0 |αs|2ds < ∞.

14

General filtering equations

Proposition 2 (General filtering equations). Consider a real- valued F-semimartingale ξt = ξ0 + t

0 Asds + ˜

Mt, where ˜ Mt is an F-martingale with [ ˜ M, B] = 0. Then the optional projection ξt has the following representation

• ξt =

ξ0 + t

• Asds +

t γ⊤

s d

Ms + t α⊤

s dµs.

(5) The square-integrable predictable processes γ and α are given by αt = ξta(Xt) − ξt a(Xt), (6) γt,j = (1 − Yt−,j)

• E(ξt|FM

t− ∨ {τj = t}) − E(ξt|FM t−)

• .

(7) The proof uses the standard arguments from the innovations approach to nonlinear filtering.

15

Security-price dynamics

Theorem 3. Under A1 - A3 the (discounted) price process of the traded securities has the martingale representation

• pt,i

=

• p0,i +

t γ

pi,⊤ s

d Ms + t α

pi,⊤ s

dµs, with α

pi t

=

• pt,i · at −

pt,i at γ

pi t,j

= (1 − Yt−,j)

• E
• pi(t, Xt, Y j

t−)|FM t−

• − E
• pi(t, Xt, Yt−)|FM

t−

• .

The predictable quadratic variations of the asset prices with respect to the market information FM satisfy d pi, pjM

t = vij t dt with

vij

t = m

• n=1

γ

pi t,n γ

• pj

t,n

λn +

l

• n=1

α

pi t,nα

• pj

t,n.

(8)

16

Filtering

Define the conditional probability vector πt = (π1

t , . . . , πK t )⊤ with

πk

t := Q(Xt = k|FM t ). πt is the natural state variable; under market

information FM all quantities of interest are functions of πt. Dynamics of πt.

• Updating at a default time τi. One has

Q(Xt = k|FM

t− ∨ {τi = t}) =

λi(k)πk

t−

K

n=1 λi(n)πn t−

.

• Kushner-Stratonovich equation (K-dim SDE-system for πt)

dπk

t = K

• ι=1

q(ι, k)πι

t−dt + γπ,⊤ t

d Mt + απ,⊤

t

dµt, with (9) γπ

t,i = πk t−

• λi(k)

K

n=1 λi(n)πn t− − 1

• , απ

t = πk t−

• a(k) − K

i=1 πi t−a(i)

• .

17

• 4. Secondary market investors

Recall that secondary market investors do not observe Z. Their information set is given by FI ⊂ FM; typically FI contains default history and noisy price information. Put νk

t := Q(Xt = k|FM t ),

1 ≤ k ≤ K.

• Pricing. Consider non-traded FY

T -measurable claim H. Define its

secondary-market value as E(H|FI

t ). Let ht(Xt) = E(H | Ft)

(full-information value of H). We get from iterated conditional expectations E(H|FI

t ) = E

• E(H|FM

t )

• FI

t

• = E

K

• k=1

πk

t ht(k)

• FI

t

• =

K

• k=1

νk

t ht(k),

i.e. pricing for secondary-market investors reduces to finding νt.

18

• Hedging. We look for risk-minimizing strategies under restricted

information in the sense of [Schweizer, 1994].

• Quadratic criterion combines well with incomplete information
• On credit markets it is natural to minimize risk wrt martingale

measure Q as historical default intensities are hard to determine. The risk-minimizing strategy θH can be computed by suitably projecting the Fm-risk-minimizing hedging strategy ξH

t

• n the set of

FI-predictable strategies. For instance we get with only one traded asset that θt is left-continuous version of E(vtξH

t | FI t )

• E(vt | FI

t ) .

Recall that vt and ξt are nonlinear functions of πt. ⇒ We need to determine conditional distribution of πt given FI

t .

19

Modelling FI and Calibration Strategies

Simple calibration. In this scenario FI = F

p ∨ FY (prices of traded

securities are observable). Recall that pt,i = K

k=1 πk t pi(t, k, Yt). If

N ≥ K (more securities than states) and if the matrix p(t, Yt) := (pi(t, k, Yt))1≤i≤N,1≤k≤K of fundamental values has full rank, the vector πt could be computed by simple calibration: πt = argmin{π≥0, K

k=1 πk=1}

N

• n=1

wn

• pt,n −

K

• k=1

pn(t, k, Yt)πk 2 , for suitable weights w1, . . . , wN. In that case pricing and hedging for secondary market investors and informed market participants coincides.

20

Calibration via filtering

Alternatively, assume that FI = FY ∨ FU where the N-dim process U solves the SDE dUt = ptdt + dWt = p(t, Yt)πtdt + dWt for a Brownian motion W independent of X, Y, Z. U can be viewed as cumulative noisy price information of the traded assets

• p1, . . . ,

pN; noise reflects observation errors and model errors. Recall that π solves the KS-equation (9). Hence computation of the conditional distribution of πt given FI

t is a standard nonlinear

filtering problem with signal process π and observation U. Note that π is typically high-dimensional; ⇒ particle filtering might be used (numerical analysis work in progress).

21

References

[Duffie and Garleanu, 2001] Duffie, D. and Garleanu, N. (2001). Risk and valuation of collateralized debt obligations. Financial Analyst’s Journal, 57(1):41–59. [Duffie and Lando, 2001] Duffie, D. and Lando, D. (2001). Term structure of credit risk with incomplete accounting observations. Econometrica, 69:633–664. [Frey and Backhaus, 2006] Frey, R. and Backhaus, J. (2006). Credit derivatives in models with interacting default intensities: a Markovian approach. Preprint, Universit¨ at Leipzig. available from www.math.uni-leipzig.de/~frey. [Frey and Runggaldier, 2006] Frey, R. and Runggaldier, W. (2006).

22

Credit risk and incomplete information: a nonlinear filtering

• approach. preprint, Universit¨

at Leipzig. [Frey and Schmidt, 2006] Frey, R. and Schmidt, T. (2006). Pricing corporate securities under noisy asset information. preprint, Universit¨ at Leipzig, submitted. [Gombani et al., 2005] Gombani, A., Jaschke, S., and Runggaldier,

• W. (2005). A filtered no arbitrage model for term structures with

noisy data. Stochastic Processes and Applications, 115:381–400. [Graziano and Rogers, 2006] Graziano, G. and Rogers, C. (2006). A dynamic approach to the modelling of correlation credit derivatives using Markov chains. working paper, Statistical Laboratory, University of Cambridge. [Hull and White, 2004] Hull, J. and White, A. (2004). Valuation of

23

a CDO and a nth to default CDS without monte carlo simulation. Journal of Derivatives, 12:8–23. [Jarrow and Yu, 2001] Jarrow, R. and Yu, F. (2001). Counterparty risk and the pricing of defaultable securities. J. Finance, 53:2225– 2243. [Laurent and Gregory, 2005] Laurent, J. and Gregory, J. (2005). Basket default swaps, CDOs and factor copulas. Journal of Risk, 7:103–122. [Sch¨

• nbucher, 2003] Sch¨
• nbucher, P. (2003).

Credit Derivatives Pricing Models. Wiley. [Schweizer, 1994] Schweizer, M. (1994). Risk minimizing hedging strategies under restricted information. Math. Finance, 4:327–342.

24