Pricing and Hedging of Credit Derivatives via Nonlinear Filtering R - - PowerPoint PPT Presentation

Pricing and Hedging of Credit Derivatives via Nonlinear Filtering

R¨ udiger Frey Universit¨ at Leipzig May 2008 ruediger.frey@math.uni-leipzig.de www.math.uni-leipzig.de/~frey based on work with

• T. Schmidt, W. Runggaldier, H. M¨

uhlichen and A. Gabih

Overview

• 1. Introduction: credit risk under incomplete information
• 2. Pricing and hedging credit derivatives via nonlinear filtering: the

[Frey et al., 2007] model. Main ideas:

• We model evolution of investors believes about credit quality, as

those are driving credit spreads.

• We use innovations approach to nonlinear filtering for deriving

dynamics of traded credit derivatives.

1

Attainable Correlations

sigma correlation 1 2 3 4 5

• 1.0
• 0.5

0.0 0.5 1.0

• min. correlation
• max. correlation

Attainable correlations for two lognormal variables, X1 ∼ Ln(0, 1), X2 ∼ Ln(0, σ2); (from McNeil, Frey, Embrechts, Quantitative Risk Management, Princeton University Press 2005)

2

• 1. Credit Risk and Incomplete Information

Basically we have two classes of dynamic credit risk models.

• Structural models: Default occurs if the asset value Vi of firm

i falls below some threshold Ki, interpreted as liability, so that default time is τi := inf{t ≥ 0: Vt,i ≤ Ki}. τi is (typically) predictable; dependence between defaults via dependence of the Vi.

• Reduced form models: Default occurs at the first jump of some

point process, typically with stochastic intensity λt,i. (τi is totally inaccessible.) Usually λt,i = λi(t, Xt), where X is a common state variable process introducing dependence between default times.

3

Incomplete information

In both model-classes it makes sense to assume that investors have

• nly limited information about state variables of the model
• Asset

value Vi is hard to

• bserve

precisely ⇒ consider firm-value models with noisy information about V (see for instance [Duffie and Lando, 2001], [Jarrow and Protter, 2004], [Coculescu et al., 2006] or [Frey and Schmidt, 2006]).

• In reduced-form models state variable process X is usually not

associated with observable economic quantities and needs to be backed out from observables such as prices.

4

Implications of incomplete information

• Under incomplete informations τi typically admits an intensity.
• Natural two-step-procedure for pricing: prices are first computed

under full information (using Markov property) and then projected

• n the investor filtration ⇒ Pricing and model calibration naturally

lead to nonlinear filtering problems.

• Information-driven default contagion. In real markets one frequently
• bserves contagion effects, i.e. spreads of non-defaulted firms

jump(upward) in reaction to default events. Models with incomplete information mimic this effect: given that firm i defaults, conditional distribution of the state-variable is updated, ⇒ default intensity of surviving firms increases ([Sch¨

• nbucher, 2004], [Collin-Dufresne et al., 2003], . . . .)

5

Some literature (mainly reduced-form models)

• Simple doubly-stochastic models with incomplete information such

as [Sch¨

• nbucher, 2004], [Duffie et al., 2006], extensions in recent

work by Giesecke.

• [Frey and Runggaldier, 2007].

Relation between credit risk and nonlinear filtering and analysis of filtering problems in very general reduced-form model; dynamics of credit risky securities not studied.

• Default-free term-structure models: [Landen, 2001]: construction
• f short-rate model via nonlinear filtering; [Gombani et al., 2005]:

calibration of bond prices via filtering.

• [Frey and Runggaldier, 2008] A general overview over nonlinear

filtering in term-structure and credit risk models.

6

• 2. Our information-based model
• Overview. Three layers of information:

1. Underlying default model (full information) Default times τi are conditionally independent doubly-stochastic random times; intensities are driven by a finite-state Markov chain X. 2. Market information. Prices of traded credit derivatives are determined by informed market-participants who observe default history and some (abstract) process Z giving X in additive Gaussian noise (market information FM := FY ∨ FZ); Filtering results wrt FM are used to obtain asset price dynamics. 3. Investor information. Z represents abstract form of ‘insider information’ and is not directly observable. ⇒ study pricing and hedging of credit derivatives for secondary-market investors with investor information FI ⊂ FM.

7

Advantages

• Prices are weighted averages of full-information values (the

theoretical price wrt FX ∨ FY ), so that most computations are done in the underlying Markov model. Since the latter has a simple structure, computations become relatively easy.

• Rich credit-spread dynamics with spread risk (spreads fluctuate in

response to fluctuations in Z) and default contagion (as defaults lead to an update of the conditional distribution of Xt 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

Notation

• We work on probability space (Ω, F, Q), Q the risk-neutral measure,

with filtration F. All processes will be F adapted.

• We consider portfolio of 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 r(t) ≡ 0.

9

The underlying full-information model

Consider a finite-state Markov chain X with SX := {1, . . . , K} and generator QX. A1 The default times are conditionally independent, doubly stochastic random times with (Q, F)-default intensity (λi(Xt)). Implications.

• The processes Yt,j −

t∧τj λj(Xs−)ds, 1 ≤ j ≤ m, are F- martingales.

• τ1, . . . , τm are conditionally independent given FX

∞; in particular no

joint defaults.

• The pair process (X, Y) is Markov wrt F .

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Examples

• 1. Homogeneous model (default intensities of all firms are identical).

Default intensities are modelled by some increasing function λ : {1, . . . , K} → (0, ∞) of the states of the economy. Elements of SX thus represent different states of the economy (1 is the best state and K the worst state). Various possibilities for generator QX; a very simple model takes X to be constant (Bayesian analysis instead of filtering).

• 2. Global- and industry factors. Assume that we have ¯

r different industry groups. Let SX = {1, . . . , κ} × {0, 1}r; write X0,. . . , X ¯

r

for the components of X, modelled as independent Markov chains. Xr is the state of industry r which is good (Xr = 0) or bad (Xr = 1); X0 represents the global factor. Default intensity of firm i from industry group r takes the form λi(x) = gi(x0) + fi(xr) for increasing functions fi and gi.

11

Full-information-values

Define the full-information value of a FY

T -measurable claim H (a

typical credit derivative) by EQ H | Ft

• =: h(t, Xt, Yt) ;

(1) the last definition makes sense since (X, Y ) is Markov w.r.t. F. Computation of full-information values. Many possibilities:

• Bond prices or legs of a CDS can be computed via Feynman-Kac
• For portfolio products such as CDOs we can use conditional

independence and compute Laplace transform of portfolio loss, (as in [Graziano and Rogers, 2006]) or use Poisson- and normal approximations, combined with Monte Carlo.

12

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 RCLL process U, we denote by

U the

• ptional projection of U w.r.t. the market filtration FM; recall that
• U is a right continuous process with

Ut = E(Ut|FM

t ) for all t ≥ 0.

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• 3. Dynamics of Security Prices

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

T -measurable payoff

PT,1, . . . , PT,N. We use martingale modelling: A3 Prices of traded securities are given by pt,i := EQ PT,i|FM

t

• .

Market-pricing. Denote by pi(t, Xt, Yt) the full-information value

• f security i. We get from iterated conditional expectations
• pt,i = E
• E(PT,i|Ft) | FM

t

• = E
• pi(t, Xt, Yt)|FM

t

• .

(2) Note that this is solved if we know the conditional distribution of Xt given FM

t

(a nonlinear filtering problem).

• Goal. Study the dynamics of traded security prices

pt,i; this is a prerequisite for hedging and risk management.

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Innovations processes

As a first towards determining the dynamics of the traded security prices step we introduce the innovations processes: Mt,j := Yt,j − t∧τj

• λj(Xs−)ds ,

j = 1, · · · , m µt,i := Zt,i − t

• ai(Xs) ds ,

i = 1, · · · , l. Properties.

• Mj is an FM-martingale and µ is FM-Brownian motion.
• Every FM-martingale can be represented as stochastic integral wrt

M and µ.

15

General filtering equations

Proposition 1 (General filtering equations). Consider a F- semimartingale of the form Jt = J0 + t

0 Asds + M J t , M J an F-

martingale with [M J, B] = 0. Suppose that [J, Yi]t = t

0 RJ,i s−dYs,i.

Then J has the representation

• Jt =

J0 + t

• Asds +

t γ⊤

s dMs +

t α⊤

s dµs;

(3) γ and α are given by αt = Jta(Xt) − Jt a(Xt), (4) γt,i = 1 ( λi)t−

• (

Jλi)t− + Jt−( λi)t− + ( RJ,iλi)t−

• .

(5) Proof based on innovations approach to nonlinear filtering.

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Security-price dynamics

Theorem 2. Under A1 - A3 the (discounted) price process of the traded securities has the martingale representation

• pt,i

=

• p0,i +

t γ

pi,⊤ s

dMs + t α

pi,⊤ s

dµs, with α

pi t

=

• pt,i · at −

pt,i at γ

pi t,j

= as in (5) with Rpi,j

t

= pi(t, Xt, Y i

t ) − p(t, Xt, Yt).

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

λt−,n +

l

• n=1

α

pi t−,nα

• pj

t−,n.

(6)

17

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. Kushner-Stratonovich equation. (K-dim SDE-system for π) Let q(ι, k), 1 ≤ ι, k ≤ K denote generator matrix of X. Then dπk

t = K

• ι=1

q(ι, k)πι

tdt + (γk(πt−))⊤ dMt + (αk(πt))⊤ dµt , with

(7) γk

j (π) = πk

• λj(k)

K

n=1 λj(n)πn

− 1

• ,

1 ≤ j ≤ m, (8) αk(π) = πk

• a(k) −

K

• n=1

πna(n)

• .

(9)

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Default contagion

• Updating at the default time τj.

∆πk

τj = πk τj−

• λj(k)

K

n=1 λj(n)πn τj−

− 1

• .
• Default contagion. At τj default intensity of firm i jumps:
• λτj,i −

λτj−,i =

K

• k=1

λi(k) · πk

τj−

• λj(k)

K

l=1 λj(l)πl τj−

− 1

• = covπτj−

λi, λj

• Eπτj−(λj)

.

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The filter in action

20

• 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.

• 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

• =

K

• k=1

E(πk

t | FI t ) ht(k),

i.e. pricing wrt FI reduces to finding E(πk

t | FI t ).

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• 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 νt(dπ), the conditional distribution of π given FI

t .

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Modelling FI and Calibration Strategies

Pragmatic calibration. Here prices of traded securities are

• bservable). 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)) of fundamental values has full rank, the vector πt could be implied by standard 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.

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Preliminary numerical results

Left: itraxx spreads from last winter for different maturities; Right: homogeneous model with 3 states and state probabilities calibrated to itraxx; note that probability of worst state increases

• ver time.

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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 (7). Hence computation of the conditional distribution of πt given FI

t is a nonlinear filtering

problem with signal process π and observation process U and Y .

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Filtering problem for secondary-market investors

Challenging problem:

• Observations
• f

mixed type; Joint jumps

• f

state process π and

• bservation

Y at defaults (see for instance [Frey and Runggaldier, 2007])

• Typically high-dimensional problem ⇒ use particle filtering as in

[Crisan and Lyons, 1999]

• Numerical analysis work in progress.

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References

[Coculescu et al., 2006] Coculescu, D., Geman, H., , and Jeanblanc, M. (2006). Valuation of default sensitive claims under imperfect information. working paper, Universit´ e d’ Evry. [Collin-Dufresne et al., 2003] Collin-Dufresne, P., Goldstein, R., and Helwege, J. (2003). Is credit event risk priced? modeling contagion via the updating of

• beliefs. Preprint, Carnegie Mellon University.

[Crisan and Lyons, 1999] Crisan, D. and Lyons, T. (1999). A particle approximation

• f the solution of the Kushner-Stratonovich equation. Probability Theory and

Related Fields, 115:549–578. [Duffie et al., 2006] Duffie, D., Eckner, A., Horel, G., and Saita, L. (2006). Frailty correlated defaullt. preprint, Stanford University. [Duffie and Lando, 2001] Duffie, D. and Lando, D. (2001). Term structure of credit risk with incomplete accounting observations. Econometrica, 69:633–664.

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[Frey and Runggaldier, 2007] Frey, R. and Runggaldier, W. (2007). Credit risk and incomplete information: a nonlinear filtering approach. preprint, Universit¨ at Leipzig,submitted. [Frey and Runggaldier, 2008] Frey, R. and Runggaldier, W. (2008). Nonlinear filtering in models for interest-rate and credit risk. preprint, submitted to Handbook of Nonlinear Filtering. [Frey and Schmidt, 2006] Frey, R. and Schmidt, T. (2006). Pricing corporate securities under noisy asset information. preprint, Universit¨ at Leipzig,forthcoming in Mathematical Finance. [Frey et al., 2007] Frey, R., Schmidt, T., and Gabih, A. (2007). Pricing and hedging

• f credit derivatives via nonlinear filtering. preprint, Universit¨

at Leipzig. available from www.math.uni-leipzig.de/%7Efrey/publications-frey.html. [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.

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[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. [Jarrow and Protter, 2004] Jarrow, R. and Protter, P. (2004). Structural versus reduced-form models: a new information based perspective. Journal of Investment management, 2:1–10. [Landen, 2001] Landen, C. (2001). Bond pricing in a hidden markov model of the short rate. Finance and Stochastics, 4:371–389. [Sch¨

• nbucher, 2004] Sch¨
• nbucher,

P. (2004). Information-driven default

• contagion. Preprint, Department of Mathematics, ETH Z¨

urich. [Schweizer, 1994] Schweizer, M. (1994). Risk minimizing hedging strategies under restricted information. Math. Finance, 4:327–342.

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