Portfolio Credit Derivatives with Markovian Default ... - CiteSeerX

Portfolio Credit Derivatives with Markovian Default Interaction R¨ udiger Frey Universit¨at Leipzig [email protected] www.math.uni-leipzig.de/ frey joint with Jochen Backhaus, Leipzig

c 2004 (Frey & McNeil)

1. Introduction • Portfolio credit derivatives are securities whose payoff is contingent

on credit events in a pool of firms (portfolio products). • Focus of this talk:

Pricing in portfolio credit derivatives in a Markovian model with interacting default intensities, which is an alternative to the market-standard Gauss copula model

• Overview • Introduction • Interacting intensities: a Markovian approach • Pricing (basket) default swaps in the Markov model • Pricing CDOs in the Markov model

Talk is based on Frey-Backhaus (2004); for background information see the forthcoming c 2004 (Frey & McNeil)

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Q U A N T I TAT I V E

RISK

MANAGEMENT Concepts Techniques Tools

Alexander J. McNeil Rüdiger Frey Paul Embrechts P R I N C E T O N


S E R I E S

I N

F I N A N C E

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Reduced Form Models for Credit Portfolios • Models with conditionally independent defaults and stochastic

intensity • Copula models such as Li (2001), Sch¨ onbucher-Schubert (2001).

Market standard since they allow for default contagion and are easily calibrated to defaultable term structure. Main drawback: unintuitive parametrization of dependence. Special case: factor copula models such as Laurent-Gregory (2003), Sch¨ onbucher (2003,2004), where contagion can be interpreted in terms of incomplete information. • Common

shock Kijima(2000)

models

such

as

Lindskog-McNeil(2001)

or

• Models with interacting intensities


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Basic Concepts and Notation Consider m firms with default times τi and default indicator process Yt = (Yt(1), . . . , Yt(m)) with Yt(i) = I{τi≤t}. • F i(t) = P (τi ≥ t) survival function of obligor i; joint survival

function: F (t1, . . . , tm) = P (τ1 ≥ t1, . . . , τm ≥ tm). • ordered default times denoted by T0 < T1 < . . . < Tm.

ξn ∈ {1, . . . , m} gives identity of the firm defaulting at time Tn • Filtrations. Hti = σ({Ys,i : s ≤ t}), Ht = Hti ∨ · · · ∨ Htm. {Ht} is

internal filtration of (Yt) (only information about default history). Default intensities. An {Ht}-adapted process R τi∧t(λt,i) is called the default intensity of τi (wrt {Ht}) if Yi(t) − 0 λs,ids is an {Ht}-martingale. Default intensities determine the law of the default indicator process (Yt). c 2004 (Frey & McNeil)

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Copula Models - the Market Standard. Background on copulas A copula is a df C on [0, 1]m with uniform margins. Copulas describe the dependence structure of a multivariate distribution with df F . If F has continuous margins F1, . . . , Fm and X ∼ F the copula C of X is the df of (F1(X1), . . . Fm(Xm)), and we have Sklars identity F (x1, . . . , xm) = C(F1(x1), . . . , Fm(xm)) . Similarly, the survival function of X can be written as F (x1, . . . , xm) = Cˆ F 1(x1), . . . , F m(xm) , where Cˆ is defined by ˆ 1, . . . , um) = C(1 − u1, . . . , 1 − um); Cˆ is called survival copula C(u Example. Gauss copula CPGa is the copula of X ∼ N (0, P ) where P is a correlation matrix.


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Copula models. Copula models are specified in terms of marginal distribution and survival copula (denoted C) of (τ1, . . . , τm) . Hence survival function of default times is given by F (t1, . . . , tm) = C F 1(t1), . . . , F m(tm) , (1) Specifying dependence structure C and marginal distribution F i separately is useful for calibration. Model is calibrated to given term structure of (single-name) CDS spreads by specifying F i; calibration of dependence structure (i.e. C) can then be done independently. √ Exchangeable Gauss-Copula Model. Here Xi = ρV + √ + 1 − ρi, for ’asset correlation’ ρ ∈ (0, 1) and V, (i)1≤i≤m iid standard normal rvs. Set Ui = Φ(Xi), so that U ∼ CPGa. Survival function of τ1, . . . , τm given by F (t1, . . . , tm) = P U1 ≤ F 1(t1), . . . , Um ≤ F m(tm) . c 2004 (Frey & McNeil)

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2. Models with Interacting Intensities (joint work with Jochen Backhaus). Basic idea. Default contagion is explicitly modelled. Default intensity is modelled as function λi(t, Yt) of time and and of default-state Yt of portfolio at time t. (Extension to stochastic state variables possible). Advantage. Intuitive and explicit parametrization of dependence between defaults; Markov process techniques available for analysis and simulation of the model. Disadvantage. Calibration to term structure of defaultable bonds or CDSs more difficult than with copula models, as marginal distribution of default times typically not available in closed form.


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Related work • Davis -Lo (2001): Default contagion, uses Markov chains. • Jarrow-Yu(2001): only very special types of interaction; model is

studied using Cox process techniques. Extensions by Yu(2004), who provides proper model construction and studies default correlations. • Kusuoka(1999) and Bielecki - Rutkowski (2002) study mathematical

aspects of the model • Collin-Dufresne-Goldstein-Hugonnier(2003): analytical evaluation of

certain derivatives using a particular change of measure. • Giesecke-Weber(2002/03):

application of interacting particle systems literature to default contagion.

• Herbertsson(2005);

Markovian

approach

using

phase-type

distributions. c 2004 (Frey & McNeil)

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Construction via Markov Chains. A model with interacting intensities is conveniently defined as time-inhomogeneous Markov chain with state space S = {0, 1}m and transition rate functions (from y to x) λ(t, y, x) =

( I{yi=0}λi(t, y), 0

if x = yi for some i ∈ {1 . . . , m}, else, (2)

where yi ∈ S is obtained from y ∈ S by flipping ith coordinate. Interpretation. The chain can jump only to neighbouring states, which differ from the current state Yt by exactly one default; if Yi(t) = 0, the probability of a jump in [t, t + h) to state Yti (default of firm i), is ≈ hλi(t, Yt).


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Model properties • The generator of (Yt) equals

G[t]f (y) =

m X

i

I{yi=0}λi(t, y) f (t, y ) − f (t, y) .

i=1

R t∧τi

• Yi(t) − 0

λi(s, Ys)ds is an {Ht}-martingale by the Dynkin formula, so that λi(s, Ys) is in fact the {Ht}-default intensity.

• Denote by p(t, s, x, y) = P(t,x)(Ys = y), s ≥ t, the transition

probabilities of the chain. They satisfy the Kolmogorov forward- and backward equation, here an ODE system.


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Kolmogorov equations Backward equation. m X ∂p(t, s, x, y) + (1 − xi) λk (t, x)(p(t, s, xi, y) − p(t, s, x, y)) = 0 , ∂t i=1 p(s, s, x, y) = δy (x) . (terminal condition) Forward equation. m X ∂ k p(t, s, x, y) = (y(k))λk s, y p(t, s, x, yk ) ∂s −

k=1 m X

(1 − y(k))λk (s, y) p(t, s, x, y), s > t,

k=1

p(t, t, x, y) = δx(y). (initial condition)


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Modelling Default Intensities The default intensities λi(t, y) are essential ingredient of the model. Some examples: • Jarrow-Yu (01): primary-secondary framework. • Frey-Backhaus (04): homogenous group model with

λi(t, Yt) = h (t, M (Yt)) , where M (y) =

m X

yi .

(3)

i=1

In exchangeable models λi(t, Yt) is necessarily of this form. Moreover, natural interpretation in terms of mean-field interaction. Extension to model with several groups possible. • Yu (04) claims that h(t, l) = 0.01 + 0.001 · I{l>0} is a reasonable

model for European telecom bonds. c 2004 (Frey & McNeil)

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Properties of Homogenous-Group Model • The process Mt := M (Yt) is itself a Markov chain with generator

GM [t] f

(l) = (1 − l)h (t, l) f (l + 1) − f (l) .

(4)

• State space of (Mt) is S M := {0, 1, . . . m} so that |S M | = m + 1

(instead of 2m). • Kolmogorov equations for (Mt) available in closed form. • P (Yi(T ) = 1) = 1/mE(MT ) and

P (Yi(T ) = 1, Yj (T ) = 1) = m−2E(MT (MT − 1)) etc. • Limit results for large portfolios available.


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Conditional Expectations Following results useful for (semi)analytic pricing of credit derivatives. Proposition. The density of τi0 equals X P (τi0 ∈ dt) = λi0 (t, y)P (Yt = y) . Moreover,

(5)

y:y(i0)=0

P Yt = y τi0 = t = y(i0)P (τi0 ∈ dt)−1λi0 (t, yi0 )P (Yt = yi0 ) . (6) Proof is based on Markov property. Corollary. In homogeneous-group model with default intensity h(t, l) m−1 X −1 P (τi0 ∈ dt) = m h(t, k)P (Mt = k)(m − k) and (7) k=0

(m − l + 1)h(t, l − 1)P (Mt = l − 1) P Mt = l τi0 = t = Pm−1 . k=0 (m − k)h(t, k)P (Mt = k) c 2004 (Frey & McNeil)

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3. (Basket) Default Swaps in the Markov model Credit Default Swap (CDS). Workhorse of market for credit derivatives. Three parties involved: reference entity C; protection buyer A; protection seller B.

C 6

?

premium payment at fixed dates until default or maturity

yes: default payment

B

A

default of C occurs ?no: no payment


-

-

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(Basket) Default Swaps Ordinary CDS. • Premium payments are due at times 0 < t1 < · · · < tN . If τC > tk

A pays in tk a premium of size x(tk − tk−1), where x denotes the swap spread; after τC premium payments stop. No initial payments. (accrued payments are neglected for simplicity) • Default payment. If τC < tN = T B pays A the LGD δ of C at τC . • Fair swap spread x∗. Since there are no initial payments x∗ is chosen

such that value at t = 0 of default payments equals value of premium payments. x∗ is the quantity which is quoted on the market. kth-to-default swap. Here default payment is triggered by credit events in a portfolio (the basket). Premium payments as before. If kth default time Tk < tN there is a default payment whose size depends on identity ξk of defaulting firm. c 2004 (Frey & McNeil)

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Pricing Results for Default Swaps Goal. Provide (semi)analytic pricing formulas which can be evaluated using Kolmogorov equations. Throughout we assume that model has been set up under equivalent martingale measure P and that R t risk-free interest rate r and LGD δi are deterministic; D(t) = exp(− 0 r(s)ds is default-free discount factor. CDSs. • Premium payments are terminal value claims of the form H = g(Ytk )

with g(y) = x∗(tk − tk−1)(1 − yi) ⇒ pricing via backward equation.

• Price of the default payments given by

E δi0 D(τi0 )I{τi

≤T } 0

Z

T

D(t)P (τi0 ∈ dt) dt ,

= δ i0

(8)

0

which can be evaluated numerically using (5) or (7). c 2004 (Frey & McNeil)

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Basket Default Swaps Default payments of kth-to-default swap. By definition P m V def := j=1 δj E D(τj )I{τj ≤T }InM =ko . Now τj

E

D(τj )I{τj ≤T }InM =ko τj

=

RT 0

D(t)P Mt = k | τj = t P (τj ∈ dt) dt.

Hence we get in homogeneous group model RT Pm def h(t, k − 1)P (Mt = k − 1)dt, V := j=1 δj 0 D(t) (m−k+1) m which can be evaluated by Kolmogorov. Similar formula also for non-homogenous case. Premium payments can be evaluated using {Tk ≤ t} = {Mt ≥ k}.


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4. CDOs: Pricing and Model Calibration Basic Structure of CDOs

Assets

SPV Interest and Principal

Liabilities Senior

Interest and Principal -

CDS Premiums

-

Protection Fee

Mezzanine Initial Investment

Default Payments on CDSs

Proceeds

Default Payments of CDO

Equity

Payments in a CDO structure; above arrow: asset-based structure; below arrow: synthetic CDO.


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Payoff of Tranches Consider portfolio of m loans with nominal Ni, relative LGD δi and default-indicator process Yt. Cumulative loss of the portfolio in t Pm given by Lt = i=1 δiNiYi(t). CDO-tranches. Maturity T . We have k tranches, characterized by Pm attachment points 0 = K0 < K1 < · · · < Kk ≤ i=1 Ni. The notional of tranche κ at time t is given by Nκ(t) = fκ(Lt) with fκ(l) = (Kκ − l)+ − (Kκ−1 − l)+ Note that fκ(l) corresponds to a put spread. Stylized CDO. Payoff of tranche κ given by Nκ(T ), the value of the notional at maturity. Real CDOs more complicated, as there is intermediate income.


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30

0.12

Default Correlation and CDO Tranches

10 0

0.0

5

0.02

Payoff of tranches

20 15

0.06

0.08

Equity Mezzanine Senior

0.04

Probability

25

0.10

Dependence Independence

0

20

40

60

80

100

Cumulative L loss

Payoff of stylized CDO with 3 tranches and attachment points 20 40 and 60 with two different loss distributions overlayed. Properties carry over to more realistic contracts. c 2004 (Frey & McNeil)

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Synthetic CDOs: Payment Description Consider CDO with attachment points K0 < · · · < Kk and notional of tranche κ given by Nκ(t) = (Kκ − Lt)+ − (Kκ−1 − Lt)+; define cumulative loss of tranche κ as Lκ(t) := Nκ(0) − Nκ(t). Default payments of CDO. Default payment of tranche κ at nth default time Tn < T given by ∆Lκ(Tn) = (Lκ(Tn) − Lκ(Tn−1)), i.e. by the part of cumulative loss at Tn which falls in the layer [Kκ−1, Kκ]. Protection fee or premium payments. Holder of tranche κ receives periodic premium payments at 0 < t1 < · · · < tN = T of size CDO xCDO (t − t )N (t ). No initial payments. x is called the (fair) n n−1 κ n κ κ CDO spread.


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Synthetic CDOs: Pricing Using partial integration we obtain for the value of the default payments of tranche κ Z T Z T def V =E D(t)dLκ(t) = D(T )E Lκ(T ) + rD(t)E Lκ(t) dt. 0

0

As Lκ(t) is a function of Lt this can be computed by one-dimensional integration if we know the distribution of Lt. Premium payments can also be expressed in terms of Lt. For deterministic LGD Lt is a function of Yt resp Mt. Hence in homogeneous group model computation via Kolmogorov equations possible; otherwise Monte Carlo must be used. Computation in Gauss-copula model can also be based on above representation.


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Explaining Market Quotes for CDOs Financial industry has developed CDS-indices for a variety of sectors; spreads for CDO-tranches on these indices are available. Observed CDS spreads. Consider the 5 year iTraxx EUR from August 4, 2004. The (average) index level was 42 bp. Assuming a constant recovery rate of 40% this leads to a marginal default probability P (τ ≤ t) = 1 − e−λt with λ = 0.007 and we obtain a 5-year default probability of 3.44%. Observed spreads for CDO tranches. On the market we observed the following tranche prices1. [0, 3]∗ [3,6] [6,9] [9,12] [12,22] 27.6 % 1.68% 0.70 % 0.43 % 0.20% ∗

1

The [0,3] tranche quoted on upfront + 5% per year.

Taken from Hull/White(2004)


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Implied Tranche- and Base Correlation Attempts to express CDO prices in terms of correlation parameter of Gauss copula model, similarly as option prices are expressed using implied volatilities • Implied tranche correlation is the value of ρ in a homogeneous

Gauss-copula model leading to the observed tranche price (generally not uniquely defined for mezzanine tranches). • Implied base correlation is the value of ρ explaining the price of an

equity tranche with the corresponding attachment point ([0,3], [0,6], [0,9], . . . ). Base correlation is unique. Moreover, (hypothetical) prices of equity tranches can be computed recursively from observed prices of CDO tranches.


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Implied Tranche Correlation and Base Correlation (2) In our example we have the following values for the tranche and base correlation. Tranche Base Correlation Correlation [0,3] 21.9% 21.9% [3,6] 4.2% 28.8% [6,9] 14.8% 33.7% [9,12] 22.3% 36.9% [12,22] 30.5% 44.8% This is a typical pattern of tranche and base correlation, called base correlation skew. In particular model based on Gauss copula cannot explain all prices simultaneously. c 2004 (Frey & McNeil)

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Explaining Base Correlation Skews Goal. Find a single model that explains (approximately) prices of all tranches or reproduces at least qualitative behavior of observed CDO prices. Needed for instance for pricing CDOs with non-standard attachment points. Idea. Markov model does yields base correlation skews if interaction between intensities is increasing and convex in number of defaults l, leading to infrequent but large clusters of defaults. We use n o h(t, l) = λ0 1 + λ1 · [eλ2l/m − eλ2µ¯t ]+ Here λ0, λ1, λ2 are parameters which should be calibrated to observed CDS and CDO spreads; µ¯t is a deterministic function giving the expected proportion of defaults.


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Numerical results We priced the observed CDO tranches by this model. increasing the parameter λ2 and calibrating the other parameters to the observed marginal 5-year default probability and the price of the equity tranche.

λ2 λ2 λ2 λ2 λ2 λ2

= 0.5 =1 =2 =3 =4 =6

Observed λ1 = 230 λ0 λ1 = 112 λ0 λ1 = 53 λ0 λ1 = 33.4 λ0 λ1 = 23.5 λ0 λ1 = 14 λ0

= 0.00427 = 0.00428 = 0.00429 = 0.00430 = 0.00432 = 0.00436

[0,3] 27.6% 27.6% 27.6% 27.6% 27.6% 27.6% 27.6%

[3,6] 1.68% 2.30% 2.28% 2.23% 2.17% 2.10% 1.99%

[6,9] 0.70% 1.12% 1.11% 1.08% 1.04% 1.00% 0.92%

[9,12] 0.43% 0.58% 0.58% 0.57% 0.56% 0.54% 0.51%

[12,22] 0.20% 0.16% 0.16% 0.18% 0.19% 0.20% 0.21%

• Model prices come much closer to observed prices. • Tentative interpretation of parameters: λ0 responsible for default

probability; product λ1λ2 responsible for default correlation/price of equity tranche; λ2 responsible for base correlation skew. c 2004 (Frey & McNeil)

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The Resulting Base Correlations


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References [Frey and Backhaus, 2004] Frey, R. and Backhaus, J. (2004). Portfolio credit risk models with interacting default intensities: a Markovian approach. working paper, Universit¨at Leipzig. [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, 2003] Laurent, J. and Gregory, J. (2003). Basket default swaps, cdos and factor copulas. working paper, University of Lyon and BNP Paribas. [Sch¨ onbucher, 2003] Sch¨ onbucher, P. (2003). Pricing Models. Wiley.

Credit Derivatives

[Yu, 2004] Yu, F. (2004). Correlated defaults and the valuation of defaultable securities. working paper, University of California, Irvine. c 2004 (Frey & McNeil)

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