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Credit Risk Modeling with Delayed Information Takanori Adachi ICS, Hitotsubashi University

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Contents

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Structural Credit Risk Models and their Problem

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Information Delay

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Market Time

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Market Filtration

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Conditional Expectation given Market Filtration

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Conclusion

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Further Research

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Structural versus Reduced-Form

A model of default is known as a structural model when it attempts to explain the mechanism by which default takes place. On the other hand, another important type of credit risk models known as a reduced-form leaves the precise mechanism leading to default unspecified. Our interest is on the mechanism and its influence to an increasing information system by which several pricing will be made. Therefore, we focus on structural models in this research. In the following, we see the framework of structural models initiated by Merton, and examine its problem.

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A problem of structural credit risk models Structural approach in credit risk modeling theory based on the default time is defined by τ := inf {t > 0 | Vt < L} where Vt is the firm value at time t ≥ 0, and L is a liability with L < V0 . If we have complete information on Vt in a real time base, the theoretical credit spread of a defaultable zero-coupon bond converges to 0 as the time goes to its maturity date. Why? Because the default time becomes a predictable stopping time. However, empirically observing credit spread does not become 0 even at just before its maturity time.

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Incomplete Models We need a kind of incompleteness on the information about Vt in the model, that makes its default time a totally inaccessible stopping time. Many authors have tried several formulation to make this happen. .. Duffie and Lando (2001) introduce noise into the market’s 1

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information set. They postulate that the market can only observe the firm’s asset value plus noise at equally spaced, discrete (non-continuous) time points.

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Kusuoka (1999) extends Duffie and Lando’s model to continuous time observations.

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C ¸ etin, Jarrow, Protter and Yildirim (2004) simply reduce the information the market can see instead of appending noise. They show the reduced information process is an Az´ema martingale.

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Giesecke’s model (2006) makes the default barrier be unobservable to the market.

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Market Delay Models

One of the ways to make the model incomplete is to introduce a delay of the information reaching to some parities as a source of the model incompleteness.

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Lindset, Lund and Persson (2008) introduce a model having constant lags for both managers and markets.

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Guo, Jarrow and Zeng (2009) propound a model having stochastic delay based on an increasing sequence of stopping times.

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Our Goal

We adopt the market delay approach and would like to investigate .1. What is a natural definition of randomly delayed time? .2. What kind of filtrations can we get through the randomly delayed time? .. How do we calculate conditional expectations given the filtration in 3

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order to make valuation?

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What is the market delay? First, we give a formulation of market delay. There are two types of time: t - Managers’ time (insiders’ time) mt - Market time The time market knows information is behind the time the managers know the information. That is, (in general) mt ≤ t. It is also natural to assume the following monotonicity to preserve the order of causality: s < t implies ms ≤ mt . Takanori Adachi (ICS, Hitotsubashi Univ.)

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Constantly Delayed case

Lindset, Lund and Persson investigate the situation in which t − mt is a constant (delay). . Definition .. [Constantly Delayed Case (LIndset, Lund, Persson, 2008) Let d be a positive constant. A market time m = {mt }t ≥0 is called a constantly delayed market time with delay d if for all t ≥ 0,

Takanori Adachi (ICS, Hitotsubashi Univ.)

mt := max{t − d, 0}.

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GJZ model

Guo, Jarrow, and Zeng treats mt as the following stochastic process. . Definition . .. [Time Change Process (Guo, Jarrow, Zeng, 2009)] For a given filtration G, a G-time change process is a stochastic process m = {mt }t ≥0 satisfying .. m0 = 0 a.s., 1 .2. mt ≤ t for all t ≥ 0 a.s., 3

ms ≤ mt

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Each mt is a G-stopping time.

for all s < t

Takanori Adachi (ICS, Hitotsubashi Univ.)

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Poisson Market Time 1

One of the natural examples of randomly delayed time is a Poisson market time. . Definition . .. [Poisson Market Time] .. Xn ∼ i.i.d.Exp (λ) for n = 1, 2, . . . , 1 2 3 4

Sn := ∑nk =1 Xk , Nt := sup{n | Sn ≤ t }, mt := SNt .

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Poisson Market Time 2

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Can Poisson Market time be a model of GJZ time change process?

No! since the fourth condition ” mt is a stopping time ” is too strong.

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Market Time

We introduce a wider family of stochastic processes than GJZ’s time change processes. . Definition .. [Market Time] A market time is a stochastic process m = {mt }t ≥0 satisfying .. m0 = 0 a.s., 1 2

mt ≤ t

3

ms ≤ mt

for all t ≥ 0 a.s., for all s < t

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Idempotent Market Time

. Definition .. [Idempotent Market Time] A market time m = {mt }t ≥0 is called idempotent if mmt = mt a.s. for every t ≥ 0. . .. Identity market time and Poisson market times are idempotent.

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GJZ model does not accept Idempotent Market Time

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. Proposition . .. Let G = {Gt }t ≥0 . be a filtration, and m = {mt }t ≥0 be an idempotent market time where mt is a G-stopping time. Then, for every pair t and s with t ≥ s, we have {mt = ms } ∈ Gs . . .. . This proposition says that at the current time s we are able to know if the information will have increased since now by any future time t, which is not realistic. This is why we exclude the condition of mt being a stopping time from the definition of market times.

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GJZ model does not accept Idempotent Market Time

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. Proposition . .. Let G = {Gt }t ≥0 . be a filtration, and m = {mt }t ≥0 be an idempotent market time where mt is a G-stopping time. Then, for every pair t and s with t ≥ s, we have {mt = ms } ∈ Gs . . .. . . Proof. . .. We have the following almost surely equivalent statements, mt ≤ s ⇔ mt ≤ s < t

Since mt is a G-stopping time, we have {mt = ms } ∈ Gs . . .. Takanori Adachi (ICS, Hitotsubashi Univ.)

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⇔ mmt ≤ ms ≤ mt ⇔ mt = mmt ≤ ms ≤ mt ⇔ mt = ms

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Characterization of Idempotent Market Times . Definition .. For a random set M ⊂ R + × Ω, define a positive process mM : R + × Ω → R + by

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.where we use the convention sup ∅ = 0. ..

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mtM (ω ) = sup{s ≤ t | (s, ω ) ∈ M },

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(Explain by using a white board) . Theorem . .. Let m : R + × Ω → R + be a positive process. Then, m is an idempotent market time iff there exists a random set M ⊂ R + × Ω such that m = mM . . .. .

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Honest Times and Idempotent Market Times

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. Definition . .. [Honest Times] A random time τ is called G-honest if there exists a G-adapted process {τt }t >0 such that for every t > 0, τ = τt on {τ ≤ t }, i.e. .τ1 {τ ≤t } = τt 1 {τ ≤t } . .. . . Theorem . .. Let G = {Gt }t ≥0 be a filtration. If m = {mt }t ≥0 is a G-adapted idempotent market time, then for every t ≥ 0, mt is a G-honest time. . .. .

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Another Example

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Definition (Starting Times for Excursions)

Then, the idempotent market time mZ picks the starting times for the excursions out of 0 of B. . ..

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.. Let B = {Bt }t ≥0 be a standard Brownian motion, and define a random set Z by Z := {(t, ω ) ∈ R + × Ω | Bt (ω ) = 0}.

(Explain by using a white board)

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Continuously Delayed Filtration We now turn to our focus on the filtration generated by delayed times. First, let us see the continuously delayed filtrations introduced by Guo, Jarrow and Zeng.

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. Definition . .. [Continuously Delayed Filtration(Guo, Jarrow, Zeng, 2009)] Let G = {Gt }t ≥0 . be a filtration, and m = {mt }t ≥0 be a G-time change process. Then, the continuously delayed filtration Gmt is defined by } { Gmt := A | (∀s ≥ 0)A ∩ {mt ≤ s } ∈ Gs . . .. . This definition works since mt is a G-stopping time.

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Market Filtration

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. Definition . .. Let G = {Gt }t ≥0 be a filtration and m = {mt }t ≥0 be a G-adapted market time. For t ≥ 0, the σ-field Gtm is defined by { } m G : = σ A ∩ { m ≥ u } | 0 ≤ u ≤ s ≤ t, A ∈ G . s u t . .. . . Theorem . .. [Market Filtration] Let G = {Gt }t ≥0 be a filtration, m = {mt }t ≥0 be a G-adapted market time, and define Gm := (Gtm )t ≥0 . Then, we have 1 2

Gm is a subfiltration of G. Assume that m is deterministic, i.e. there exists a deterministic function f : R + → R + such that for all t ≥ 0 and ω ∈ Ω, mt (ω ) = f (t ). Then, we have Gtm = Gf (t ) .

. .. We call Gm the market filtration modulated by the market time m. Takanori Adachi (ICS, Hitotsubashi Univ.)

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Relations between Market Filtrations and Continuously Delayed Filtration

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We are interested in the case when it happens to have the market time mt being a stopping time for every t. The following proposition says, in this case, our market filtration Gtm is a subfiltration of the continuously delayed filtration Gmt . . Proposition . .. Let G = {Gt }t ≥0 be a filtration, and m = {mt }t ≥0 be a G-adapted market time. If mt is a stopping time for all t ≥ 0, we have Gtm ⊂ Gmt . . .. . . Proposition . .. m In . general, Gmt ⊂ Gt does not hold. .. . See the handout for the counterexample. Takanori Adachi (ICS, Hitotsubashi Univ.)

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Optional Processes and Market Filtrations (1) . Definition . .. [Optional Processes] .. The optional σ-field with respect to G is the σ-field O G defined on 1 R + × Ω such that

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A process X = {Xt }t ≥0 is called G-optional if the map (t, ω ) 7→ Xt (ω ) is O G -measurable.

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For a random time τ, the σ-field G(τ ) is defined by

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G(τ ) := σ{Zτ | Z = {Zt }t ≥0 is a G-optional process. }.

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O G := σ{X | X = {Xt }t ≥0 is a G-adapted c´adl´ag process. }.

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Optional Processes and Market Filtrations (2)

. Theorem .. We have

. ∨ s ≤t

G ( ms ) .

The converse does not hold in general. . ..

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Conditional Expectation given Market Filtration (1)

Final Goal is to find a closed form of the conditional expectations given a general market filtration such as EQ [Z | Gtm ] For example, this is necessary to make valuations of credit spreads of defaultable bonds.

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Conditional Expectation given Market Filtration (2) . Definition .. .. For s ≥ 0, Ys := EQ [Z | Gs ], 1 .. A real-valued random variable Ytm is defined by for ω ∈ Ω, 2

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Ytm (ω ) = Ymt (ω ) (ω ).

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Theorem ( Cases of Stopping Times )

.. Let G = {Gt }t ≥0 be a right-continuous filtration, m be a G-adapted market time whose components are stopping times, and Z be a real-valued integrable random variable. Then, . .. Takanori Adachi (ICS, Hitotsubashi Univ.)

EQ [Z | Gtm ] = Ytm .

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Conditional Expectation given Market Filtration (3)

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Theorem ( Cases of Idempotent Market Times )

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Q (ms ≤ u | Gtm ) = Q (ms ≤ u | GT ).

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.. Let G = {Gt }t ≥0 be a filtration, and m = {mt }t ≥0 be a G-adapted idempotent market time with a horizon T . Then, for every 0 ≤ u < s ≤ t < T,

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Conclusions

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Provide a natural time delay examples that GJZ model is hard to accept.

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Introduce a more general delayed process, called market time that does not require its random times be stopping times.

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Introduce a class of market times, called a idempotent market times, which contains many practically interesting market times. Give a characterization of idempotent market times.

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Conclusions (cont.)

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Introduce a delayed filtration called market filtration that is the filtration modulated by a given market time.

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Determine the relationship between two filtrations, the market filtration and GJZ’s continuously delayed filtration coincide.

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Find a closed form of the conditional expectations given a market time whose components are stopping times.

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Further Research

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Find a closed form of the conditional expectations given a general market filtration.

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The End

Thank you for your attention.

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