Market Models for Forward Swap Rates and Credit Default Swap Spreads - - PowerPoint PPT Presentation

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models

Market Models for Forward Swap Rates and Credit Default Swap Spreads

Marek Rutkowski School of Mathematics and Statistics University of New South Wales Sydney, Australia Joint work with Libo Li Workshop on Stochastic Analysis and Finance City University of Hong Kong June 29-July 03, 2009

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models

Outline

1

Forward Swaps

2

Admissible Sets of Forward Swap Rates

3

Market Models of Swap Rates

4

Forward CDS Spreads

5

Market Models of CDS Spreads

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models

References on Modelling of Swap Rates

• M. Davis and V. Mataix-Pastor: Negative Libor rates in the swap market
• model. Finance and Stochastics 11 (2007), 181–193.
• S. Galluccio, J.-M. Ly, Z. Huang, and O. Scaillet: Theory and calibration
• f swap market models. Mathematical Finance 17 (2007), 111–141.
• F. Jamshidian: LIBOR and swap market models and measures. Finance

and Stochastics 1 (1997), 293–330.

• F. Jamshidian: Bivariate support of forward Libor and swap rates.

Mathematical Finance 18 (2008), 427–443.

• F. Jamshidian: Trivariate support of flat-volatility forward Libor rates.

Working paper, University of Twente, 2007.

• R. Pietersz and M. van Regenmortel: Generic market models. Finance

and Stochastics 10 (2006), 507–528.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models

References on Modelling of CDS Spreads

• N. Bennani and D. Dahan: An extended market model for credit
• derivatives. Presented at the international conference Stochastic

Finance, Lisbon, 2004.

• D. Brigo: Constant maturity credit default swap pricing with market
• models. Working paper, Banca IMI, 2004.
• D. Brigo and M. Morini: CDS market formulas and models. Working

paper, Banca IMI, 2005. S.L. Ho and L. Wu: Arbitrage pricing of credit derivatives. Working paper, HKUST, 2007.

• F. Jamshidian: Valuation of credit default swaps and swaptions.

Finance and Stochastics 8 (2004), 343–371.

• L. Schlögl: Note on CDS market models. Working paper, Lehman

Brothers, 2007.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models

Goals

We would like to address the following issues.

1

A unified approach to modelling of market rates.

2

A comparison of the top-down and bottom-up approaches.

3

Separate study of two related inverse problems.

4

Positivity of rates within market models.

5

Construction of default times consistent with CDS spreads.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models

Problems

The following problems arise in the context of market models.

1

The choice of admissible families of swaps and/or credit default swaps.

2

Computation of Radon-Nikodym densities for martingale measures.

3

Computation of dynamics of forward swap rates and CDS spreads under swap/CDS measures.

4

Existence and construction of default times consistent with CDS spreads.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Families of Forward Swaps

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Forward Swap

Terminology and notation:

1

Let 0 < T0 < T1 < · · · < Tn−1 < Tn be a fixed sequence of dates representing the tenor structure T.

2

We denote ai = Ti − Ti−1 for i = 1, . . . , n.

3

Let B(t, Ti) stand for the price of the zero-coupon bond maturing at Ti.

4

Let S = {S1, . . . , Sl} be any family of l distinct forward swaps associated with the tenor structure T.

5

Any reset or settlement date for any swap Sj in S belongs to T.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Forward Swap Rates

The forward swap rate κj for the forward swap Sj starting at Tsj and maturing at Tmj equals κj

t = κ sj ,mj t

= B(t, Tsj ) − B(t, Tmj ) mj

i=sj +1 aiB(t, Ti)

= P

sj ,mj t

A

sj ,mj t

, ∀ t ∈ [0, Tsj ]. where we set P

sj ,mj t

= B(t, Tsj ) − B(t, Tmj ), t ∈ [0, Tsj ], and A

sj ,mj t

=

mj

• i=sj +1

aiB(t, Ti), t ∈ [0, Tsj +1].

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Bond Numeraire

Let us choose the bond maturing at Tb as a bond numéraire. In terms of the deflated bond prices Bb(t, Ti) = B(t, Ti)/B(t, Tb), i = 1, . . . , n, we obtain κ

sj ,mj t

= Bb(t, Tsj ) − Bb(t, Tmj ) mj

i=sj +1 aiBb(t, Ti)

= P

b,sj ,mj t

A

b,sj ,mj t

, ∀ t ∈ [0, Tsj ∧ Tb]. We call the process Ab,sj ,mj , defined as A

b,sj ,mj t

=

mj

• i=sj +1

aiBb(t, Ti), t ∈ [0, Tsj +1 ∧ Tb], the deflated swap annuity or deflated swap numéraire.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Modelling Issues

We will deal with the following interrelated modelling issues.

1

Under which assumptions the joint dynamics of a given family of swap rates is supported by an arbitrage-free term structure model, where by a term structure model we mean the joint dynamics of deflated bond prices?

2

How to specify the joint dynamics for a given family of forward swaps in terms of “drifts” and “volatilities” under a single probability measure?

3

Under which assumptions the swap rates and/or CDS spreads follow positive processes and the default time can be constructed?

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Swap Equation and Inverse Problem

1

Each forward swap corresponds to the linear equation in which deflated bond prices are treated as “unknowns”.

2

Specifically, we deal with the following swap equation associated with the forward swap Sj and the numéraire bond B(t, Tb) Bb(t, Tsj ) −

mj −1

• i=sj +1

κ

sj ,mj t

aiBb(t, Ti) − (1 + κ

sj ,mj t

)amj Bb(t, Tmj ) = 0.

3

The following inverse problem is of interest: describe all families of forward swaps such that the knowledge of the corresponding family

• f swap rates is sufficient to uniquely specify the associated family
• f non-zero (or positive) deflated bond prices for any choice of the

numéraire bond.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Notation

1

Let xi stand for a generic value of the deflated bond price Bb(t, Ti) and let κj be a generic value of the forward swap rate κj

t = κ sj ,mj t

.

2

Since xi and κj are aimed to represent generic values of the corresponding processes in some stochastic model we have that (x0, . . . , xn) ∈ Rn+1 and (κ1, . . . , κl) ∈ Rl.

3

However, if the bond B(t, Tb) is chosen to be the numéraire bond then, by the definition of the deflated bond price, the variable xb satisfies xb = 1 and thus it is more adequate to consider a generic value (x0, . . . , xb−1, xb+1, . . . , xn) ∈ Rn.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Swap Linear System

1

We thus obtain, for every j = 1, . . . , l, κj = xsj − xmj mj

i=sj +1 aixi

.

2

For brevity, we write cj,i = κjai and cj,mj = (1 + κj)amj −xsj +

mj −1

• i=sj +1

cj,ixi + cj,mj xmj = 0.

3

For a given family S = {S1, . . . , Sl} of forward swaps with tenor T, any fixed b ∈ {0, . . . , n}, and an arbitrary (κ1, . . . , κl) ∈ Rl, we thus deal with the linear system Cb¯ xb = ¯ eb where ¯ xb = (x0, . . . , xb−1, xb+1, . . . , xn) is the vector of n unknowns and ¯ eb is some vector of zeros and ones.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Cycles

Definition Two swaps are called adjacent if the start date of one of them coincides with the maturity date of another. If distinct swaps Sj1, . . . , Sjk are such that Sjm is adjacent to Sjm+1 for m = 1, . . . , k − 1 then the pair (Tsj1 , Tmjk ) is termed the

• path. A path is called a cycle if the equality Tsj1 = Tmjk holds.

Lemma Let S = {S1, . . . , Sn} be a family of n swaps with the tenor structure T. (i) If there are no cycles in S then each date from T is either the start date or the maturity date of some forward swap Sj from S, that is, the equality T0(S) = T holds. (ii) Conversely, if T0(S) = T then there are no cycles in S.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Example

1

Let us consider an example of a family S of forward swaps without a cycle.

2

We assume here that b = 0 so that B(t, Tb) = B(t, T0). Let T = {T0, . . . , T7} and let S be given by the following linear system C0¯ x0 =           c1,1 c1,2 c1,3 c2,1 c2,2 c2,3 c2,4 c2,5 −1 c3,2 −1 c4,2 c4,3 c4,4 c4,5 c4,6 −1 c5,4 −1 c6,5 c6,6 c6,7 −1 c7,7                     x1 x2 x3 x4 x5 x6 x7           =           1 1           = ¯ e0.

3

The swaps S1 and S5 are adjacent and they yield the path (T0, T4).

4

The swaps S4 and S7 are adjacent and they yield the path (T1, T7).

5

No other swaps are adjacent and thus there are no cycles in S.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Problem (IP .1) and T-admissibility

Problem (IP.1) Provide necessary and sufficient conditions for a family S of forward swaps under which, for almost every (κ1, . . . , κl) ∈ Rl, there exists a unique non-zero solution (x0, . . . , xb−1, xb+1, . . . , xn) ∈ Rn to the linear system Cb¯ xb = ¯ eb. Definition A family S = {S1, . . . , Sl} of forward swaps associated with the tenor structure T is T-admissible if for any choice of b ∈ {0, . . . , n} the following property holds: for almost every (κ1, . . . , κl) ∈ Rl there exists a unique non-zero solution (x0, . . . , xb−1, xb+1, . . . , xn) ∈ Rn to the linear system Cb¯ xb = ¯ eb corresponding to S.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Admissibility

Galluccio et al. (2007) introduce the following definition of admissibility of S. Definition We say that a family S of forward swaps associated with T is admissible if the following conditions are satisfied: (i) the number of forward swaps in S equals n, i.e., l = n, (ii) any date Ti ∈ T coincides with the reset/settlement date of at least one forward swap from S, (iii) there are no cycles in S. Galluccio et al. (2007) claim that the admissibility of S is equivalent to the existence of a unique non-zero solution to the inverse problem.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Counter-example

1

The following counter-example shows that a family S with a cycle can be T-admissible.

2

Let n = 3 and let (sj, mj), j = 1, 2, 3 be given as (0, 2), (2, 3) and (0, 3), respectively.

3

The swaps S1 and S2 yield the path (T0, T3) and this path is also given by the swap S3, so that a cycle (T0, T0) exists.

4

For b = 0, we obtain the following linear system C0¯ x0 =   c1,1 c1,2 −1 c2,3 c3,1 c3,2 c3,3     x1 x2 x3   =   1 1   = ¯ e0.

5

A unique non-zero solution exists, for almost every (κ1, κ2, κ13) ∈ R3.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Notation

1

We say that a date Ti, i = 1, . . . , n is the relevant date for the swap Sj when the term cj,i is non-zero.

2

In addition, the date T0 is the relevant date for the swap Sj if it starts at T0.

3

Let T(Sj) stand for the set of all relevant dates for the swap Sj and let T0(Sj) = {Tsj , Tmj }.

4

We denote T0(S) – the set of all start/maturity dates for a family S, that is, T0(S) =

l

• j=1

T0(Sj) =

l

• j=1

{Tsj , Tmj }. T(S) – the set of all relevant dates (i.e., reset/settlement dates) for a family S, that is, T(S) =

l

• j=1

T(Sj).

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

(T, b)-inadmissibility

Definition For a fixed b ∈ {0, . . . , n}, we say that a cycle Sc ⊂ S is (T, b)-inadmissible if the number of dates in T(Sc) \ {Tb} is strictly less than the number of swaps in Sc. Lemma Let S be a family of forward swaps containing a (T, b)-inadmissible cycle Sc for some b ∈ {0, . . . , n}. Then the family S is not T-admissible. Lemma Assume that l = n and T(S) = T. If there is no (T, b)-inadmissible cycle in S for any b ∈ {0, . . . , n} then for any b ∈ {0, . . . , n} there exists a permutation

• f S = {S1, . . . , Sn} such that all entries on the diagonal of Cb are non-zero.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Forward Swaps Inverse Problem for Bonds

Sufficient Conditions

Lemma Assume that l = n, the equality T(S) = T holds and the graph associated with S is connected, that is, for every s, m ∈ {0, . . . , n} such that m < s there exist a path (Ts, Tm) generated by S. Then either: (i) T0(S) = T and there are no cycles in S, or (ii) T0(S) = T and there are no (T, b)-inadmissible cycle in S. Proposition Assume that l = n, the equality T(S) = T holds and the graph associated with S is connected. Then a unique solution to the linear system Cb¯ xb = ¯ eb exists and it is non-zero, for almost all (κ1, . . . , κn) ∈ Rn. Hence the family S

• f forward swaps is T-admissible.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Abstract Forward Swaps Inverse Problem for Numeraires

Density Processes Martingale Measures

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Abstract Forward Swaps Inverse Problem for Numeraires

Abstract Forward Swaps

1

By an abstract forward swap we mean the start date Tsj , the maturity date Tmj as well a pair P

sj ,mj t

, Asj ,mj of processes, where Asj ,mj is a positive process.

2

The forward swap rate κj in an extended forward swap starting at Tsj and maturing at Tmj is defined by the formula κj

t = κ sj ,mj t

= P

sj ,mj t

A

sj ,mj t

, ∀ t ∈ [0, Tsj ].

3

Let S = {S1, . . . , Sl} be a family of extended forward swaps associated with the tenor structure T, that is, such that T0(S) ⊂ T.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Abstract Forward Swaps Inverse Problem for Numeraires

Additivity and Swap Numeraires

1

The families Psj ,mj and Asj ,mj of processes corresponding to S are additive, in the sense that the following equations are satisfied for any cycle Sc ⊂ S

• j∈S1

c

A

s1

j ,m1 j

t

=

• k∈S2

c

A

s2

k ,m2 k

t

and

• j∈S1

c

P

s1

j ,m1 j

t

=

• k∈S2

c

P

s2

k ,m2 k

t

where S1

c and S2 c are any two paths which produce the cycle Sc.

2

Let us define, for a fixed d ∈ {1, . . . , l}, A

d,sj ,mj t

= A

sj ,mj t

Asd ,md

t

, ∀ t ∈ [0, Tsj ∧ Tsd ].

3

For an arbitrary choice of the swap numéraire Asd ,md the family Ad,sj ,mj , j = 1, . . . , l, of swap deflated annuities is additive as well.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Abstract Forward Swaps Inverse Problem for Numeraires

Inverse Problem for Numeraires

Lemma For any cycle Sc in S, the following equality holds

• j∈S1

c

A

d,s1

j ,m1 j

t

κ

s1

j ,m1 j

t

=

• k∈S2

c

A

d,s2

k ,m2 k

t

κ

s2

k ,m2 k

t

. Problem: Does a family of forward swaps S = {S1, . . . , Sl} uniquely specifies the family of swap deflated annuities Ad = {Ad,sj ,mj , j = 1, . . . , d − 1, d + 1, . . . , l} for almost all values of (κ1, . . . , κl) ∈ Rl.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Abstract Forward Swaps Inverse Problem for Numeraires

Inverse Problem for Numeraires

Problem (IP.2) Provide necessary and sufficient conditions for a family S of abstract forward swaps under which, for almost every (κ1, . . . , κl) ∈ Rl, there exists a unique non-zero solution (y1, . . . , yd−1, yd+1, . . . , yl) ∈ Rl to the following set of equations: for any cycle Sc in S

• j∈S1

c

yj =

• k∈S2

c

yk and

• j∈S1

c

κjyj =

• k∈S2

c

κkyk.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Abstract Forward Swaps Inverse Problem for Numeraires

Example: Swap Rates

1

Take n = 2 and S = {S1, S2, S3} with (sj, mj) equal to (0, 1), (1, 2) and (0, 2).

2

For b = 0 we obtain the following linear system parametrized by (κ1, κ2, κ3) ∈ R3 C0¯ x0 =   c1,1 −1 c2,2 c3,1 c3,2   x1 x2

• =

  1 1   = ¯ e0.

3

It is easily seen that no solution exists, for almost all (κ1, κ2, κ3) ∈ R3, and thus Problem (IP .1) has no solution.

4

The corresponding Problem (IP .2) has the form, for d = 1, 1 + y2 = y3, κ1 + κ2y2 = κ3y3.

5

For almost all (κ1, κ2, κ3) ∈ R3, the unique solution reads y2 = κ1 − κ3 κ3 − κ2 , y3 = κ1 − κ2 κ3 − κ2 .

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Abstract Forward Swaps Inverse Problem for Numeraires

Example: CDS Spreads

1

In the case of forward CDS spreads, Problem (IP .1) should be modified as follows: for almost all (κ1, κ2, κ3) ∈ R3, find a non-zero solution to the linear system C0¯ x0 =   c1,1 −1 c2,3 c3,1 c3,3     x1

• x1

x2   =   1 1   = ¯ e0.

2

The unique non-zero solution exists, for almost all (κ1, κ2, κ3) ∈ R3.

3

The dynamics of the CDS spreads (κ1, κ2, κ3) can be supported by some model in which we will deal with the following set of martingale measures associated swap numéraires Q

dP1 dQ

− − − − − → P1

y2∼ dP2

dP1

− − − − − → P2

y3∼ dP2

dP2

 

• P2

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Market Models for CDS Spreads

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Set-up and Notation

1

Let (Ω, G, F, Q) be a filtered probability space, where F = (Ft)t∈[0,T] is a filtration such that F0 is trivial.

2

We assume that the random time τ defined on this space is such that the F-survival process Gt = Q(τ > t | Ft) is positive.

3

The probability measure Q is interpreted as the risk-neutral measure.

4

Let 0 < T0 < T1 < · · · < Tn be a fixed tenor structure and let us write ai = Ti − Ti−1.

5

We denote ai = ai/(1 − δi) where δi is the recovery rate if default occurs between Ti−1 and Ti.

6

We denote by D(t, T) the default-free discount factor over the time period [t, T].

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Top-down Approach under Deterministic Interest Rate

1

Assume first that the interest rate is deterministic.

2

The pre-default forward CDS spread κi corresponding to the single-period forward CDS starting at time Ti−1 and maturing at Ti equals 1 + aiκi

t =

EQ

• D(t, Ti)1{τ>Ti−1}
• Ft
• EQ
• D(t, Ti)1{τ>Ti }
• Ft

, ∀ t ∈ [0, Ti−1].

3

Since the interest rate is deterministic, we obtain, for i = 1, . . . , n, 1 + aiκi

t = Q(τ > Ti−1 | Ft)

Q(τ > Ti | Ft) , ∀ t ∈ [0, Ti−1], and thus Q(τ > Ti | Ft) Q(τ > T0 | Ft) =

i

• j=1

1 1 + ajκj

t

, ∀ t ∈ [0, T0].

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Auxiliary Probability Measure P

We define the probability measure P equivalent to Q on (Ω, FT) by setting, for every t ∈ [0, T], ηt = dP dQ

• Ft

= Q(τ > Tn | Ft) Q(τ > Tn | F0). Lemma For every i = 1, . . . , n, the process Z κ,i given by Z κ,i

t

=

n

• j=i+1
• 1 +

ajκj

t

• ,

∀ t ∈ [0, Ti]. is a (P, F)-martingale.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

CDS Martingale Measures

1

For any i = 1, . . . , n we define the probability measure Pi equivalent to P (and thus also to Q) on (Ω, FT) by setting dPi dP

• Ft

= Z i

t = ciZ κ,i t

= Q(τ > Ti) Q(τ > Tn)

n

• j=i+1
• 1 +

ajκj

t

• 2

Note that Z κ,n

t

= 1 and thus Pn = P.

3

Assume that the PRP holds under P = Pn with the Rk-valued spanning (P, F)-martingale M.

4

Then PRP is also valid with respect to F under any probability measure Pi for i = 1, . . . , n.

5

Hence the positive process κi satisfies, for i = 1, . . . , n, κi

t = κi 0 +

• (0,t]

κi

sσi s · dΨi(M)s

where σi is an Rk-valued, F-predictable process and Ψi(M) is the Pi-Girsanov transform of M.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Dynamics of Forward CDS Spreads

Proposition Assume that the PRP holds with respect to F under P with the spanning (P, F)-martingale M = (M1, . . . , Mk). Assume that the positive processes κi, i = 1, . . . , n are such that the process Z κ,i

t

=

n

• j=i+1
• 1 +

ajκj

t

• is a (P, F)-martingale for i = 1, . . . , n. Then there exist Rk-valued,

F-predictable processes σi−1 such that the joint dynamics of processes κi, i = 1, . . . , n under P are given by dκi

t = k

• l=1

κi

tσi,l t dMl t − n

• j=i+1
• ajκi

tκj t

1 + ajκj

t k

• l,m=1

σi,l

t σj,m t

d[Ml,c, Mm,c]t − 1 Z i

t−

∆Z i

t k

• l=1

κi

tσi,l t ∆Ml t .

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Bottom-up Approach: First Step

Proposition Let M = (M1, . . . , Mk) be an arbitrary (P, F)-martingale and let σi, i = 1, . . . , n be Rk-valued, locally bounded, F-predictable processes. Assume that the processes Z i, i = 1, . . . , n are (P, F)-martingales, where Z i

t =

n

j=i+1

• 1 +

ajκj

t

• EP

n−1

j=i

• 1 +

ajκj

t

. Then the joint dynamics of processes κi, i = 1, . . . , n under P are given by the previous proposition. For every i = 1, . . . , n, the process κi is a (Pi, F)-martingale, where the probability measure Pi is given by dPi dP

• Ft

= ci

n

• j=i+1
• 1 +

ajκj

t

• .

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Bottom-up Approach: Second Step

1

We will now construct a default time τ consistent with the dynamics of forward CDS spreads. Let us set Mi−1

Ti−1 = i−1

• j=1

1 1 + ajκj

Ti−1

Mi

Ti = i

• j=1

1 1 + ajκj

Ti

.

2

Since the process aiκi is positive, we obtain, for every i = 0, . . . , n, GTi := Mi

Ti =

Mi−1

Ti−1

1 + aiκi

Ti

≤ Mi−1

Ti−1 =: Gi−1 Ti−1.

3

The process GTi = Mi

Ti is thus decreasing for i = 0, . . . , n.

4

We make use of the canonical construction of default time τ taking values in {T0, . . . , Tn}.

5

We obtain, for every i = 0, . . . , n, P(τ > Ti | FTi ) = GTi =

i

• j=1

1 1 + ajκj

Ti

.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Top-down Approach under Independence

Assume that we are given a model for Libors (L1, . . . , Ln) where Li = L(t, Ti−1) and CDS spreads (κ1, . . . , κn) in which:

1

The default intensity γ generates the filtration Fγ.

2

The interest rate process r generates the filtration Fr.

3

The probability measure Q is the spot martingale measure.

4

The H-hypothesis holds, that is, F

Q

֒ → G, where F = Fr ∨ Fγ.

5

The PRP holds with the (Q, F)-spanning martingale M. Then it is possible to determine the joint dynamics of Libors and CDS spreads (L1, . . . , Ln, κ1, . . . , κn) under any martingale measure Qi.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Bottom-up Approach under Independence

To construct a model we assume that:

1

A martingale M = (M1, . . . , Mk) has the PRP with respect to (P, F).

2

The family of process Z i given by Z L,κ,i

t

:=

n

• j=i+1

(1 + ajLj

t)(1 +

ajκj

t)

are martingales on the filtered probability space (Ω, F, P) so that there exists a family of probability measure Pi, i = 1, . . . , n on (Ω, FT) with the density function given by dPi dP = ciZ L,κ,i.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Dynamics of LIBORs and CDS Spreads

The dynamics of Li and κi under Pn with respect to the spanning (P, F)-martingale M are given by dLi

t = k

• l=1

ξi,l

t dMl t − n

• j=i+1

aj 1 + ajLj

t k

• l,m=1

ξi,l

t ξj,m t

d[Ml,c, Mm,c]t −

n

• j=i+1
• aj

1 + ajκj

t k

• l,m=1

ξi,l

t σj,m t

d[Ml,c, Mm,c]t − 1 Z i

t

∆Z i

t k

• l=1

ξi,l

t ∆Ml t

and dκi

t = k

• l=1

σi,l

t dMl t − n

• j=i+1

aj 1 + ajLj

t k

• l,m=1

σi,l

t ξj,m t

d[Ml,c, Mm,c]t −

n

• j=i+1
• aj

1 + ajκj

t k

• l,m=1

σi,l

t σj,m t

d[Ml,c, Mm,c]t − 1 Z i

t

∆Z i

t k

• l=1

σi,l

t ∆Ml t .

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Top-down Approach: One- and Two-Period Spreads

1

Let (Ω, G, F, Q) be a filtered probability space, where F = (Ft)t∈[0,T] is a filtration such that F0 is trivial.

2

We assume that the random time τ defined on this space is such that the F-survival process Gt = Q(τ > t | Ft) is positive.

3

The probability measure Q is interpreted as the risk-neutral measure.

4

Let 0 < T0 < T1 < · · · < Tn be a fixed tenor structure and let us write ai = Ti − Ti−1.

5

We no longer assume that the interest rate is deterministic.

6

We denote by D(t, T) the default-free discount factor over the time period [t, T].

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

One-Period CDS Spreads

The one-period forward CDS spread κi = κi−1,i satisfies, for t ∈ [0, Ti−1], 1 + aiκi

t =

EQ

• D(t, Ti)1{τ>Ti−1}
• Ft
• EQ
• D(t, Ti)1{τ>Ti }
• Ft

. Let Ai−1,i be the one-period CDS annuity Ai−1,i

t

= ai EQ

• D(t, Ti)1{τ>Ti }
• Ft
• and let

Pi−1,i

t

= EQ

• D(t, Ti)1{τ>Ti−1}
• Ft
• − EQ
• D(t, Ti)1{τ>Ti }
• Ft
• .

Then κi

t = Pi−1,i t

Ai−1,i

t

, ∀ t ∈ [0, Ti−1].

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

One-Period CDS Spreads

Let Ai−2,i stand for the two-period CDS annuity Ai−2,i

t

= ai−1 EQ

• D(t, Ti−1)1{τ>Ti−1}
• Ft
• +

ai EQ

• D(t, Ti)1{τ>Ti }
• Ft
• .

and let Pi−2,i

t

=

i

• j=i−1
• EQ
• D(t, Tj)1{τ>Tj−1}
• Ft
• − EQ
• D(t, Tj)1{τ>Tj }
• Ft

. The two-period CDS spread κi = κi−2,i is given by the following expression

• κi

t = κi−2,i t

= Pi−2,i

t

Ai−2,i

t

= Pi−2,i−1

t

+ Pi−1,i

t

Ai−2,i−1

t

+ Ai−1,i

t

, ∀ t ∈ [0, Ti−1].

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

One-Period CDS Measures

1

Our aim is to derive the semimartingale decomposition of κi, i = 1, . . . , n and κi, i = 2, . . . , n under a common probability measure.

2

We start by noting that the process An−1,n is a positive (Q, F)-martingale and thus it defines the probability measure Pn on (Ω, FT).

3

The following processes are easily seen to be (Pn, F)-martingales Ai−1,i

t

An−1,n

t

=

n

• j=i+1
• aj(

κj

t − κj t)

• aj−1(κj−1

t

− κj

t)

= an

• ai

n

• j=i+1
• κj

t − κj t

κj−1

t

− κj

t

.

4

Given this family of positive (Pn, F)-martingales, we define a family of probability measures Pi for i = 1, . . . , n such that κi is a martingale under Pi.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Two-Period CDS Measures

1

For every i = 2, . . . , n, the following process is a (Pi, F)-martingale Ai−2,i

t

Ai−1,i

t

=

• ai−1EQ
• D(t, Ti−1)1{τ>Ti−1}
• Ft
• +

aiEQ

• D(t, Ti)1{τ>Ti }
• Ft
• EQ
• D(t, Ti)1{τ>Ti }
• Ft
• =

ai−1

• Ai−2,i−1

t

Ai−1,i

t

+ 1

• =

ai

• κi

t − κi t

κi−1

t

− κi

t

+ 1

• .

2

Therefore, we can define a family of the associated probability measures

• Pi on (Ω, FT), for every i = 2, . . . , n,

3

It is obvious that κi is a martingale under Pi for every i = 2, . . . , n.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

One and Two-Period CDS Measures

We will summarise the above in the following diagram Q

dPn dQ

− − − − − → Pn

dPn−1 dPn

− − − − − → Pn−1

dPn−2 dPn−1

− − − − − → . . . − − − − − → P2 − − − − − → P1

d Pn dPn

 

• d

Pn−1 dPn−1

 

• 



• d

P2 dP2

 

• Pn
• Pn−1

. . .

• P2

where dPn dQ = An−1,n

t

dPi dPi+1 = Ai−1,i

t

Ai,i+1

t

= ai+1

• ai

κi+1

t

− κi+1

t

κi

t −

κi+1

t

• d

Pi dPi = Ai−2,i

t

Ai−1,i

t

= ai

• κi

t − κi t

κi−1

t

− κi

t

+ 1

• .

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Top-down Approach: Joint Dynamics

1

We are in a position to calculate the semimartingale decomposition of (κ1, . . . , κn, κ2, . . . , κn) under Pn.

2

It suffices to use the following Radon-Nikodým densities dPi dPn = Ai−1,i

t

An−1,n

t

= an

• ai

n

• j=i+1
• κj

t − κj t

κj−1

t

− κj

t

d Pi dPn = Ai−2,i

t

An−1,n

t

= an

• κi

t − κi t

κi−1

t

− κi

t

+ 1

• n
• j=i+1
• κj

t − κj t

κj−1

t

− κj

t

= an  

n

• j=i
• κj

t − κj t

κj−1

t

− κj

t

+

n

• j=i+1
• κj

t − κj t

κj−1

t

− κj

t

  = ai−1 dPi−1 dPn + ai dPi dPn

3

Explicit formulae for the joint dynamics of one and two-period spreads are available.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Bottom-up Approach: Postulates

1

The processes κ1, . . . , κn and κ1, . . . , κn are F-adapted.

2

For every i = 0, . . . , n the process Z κ,i Z κ,i

t

= cn ci

n

• j=i+1
• κj

t − κj t

κj−1

t

− κj

t

is a positive (P, F)-martingale where c1, . . . , cn are normalizing constants.

3

For every i = 0, . . . , n the process Z

κ,i given by the formula

Z

κ,i = Z κ,i + Z κ,i−1 = κi−1 − κi

κi−1 − κi Z κ,i is a positive (P, F)-martingale.

4

The process M = (M1, . . . , Mk) is the (P, F)-spanning martingale.

5

Probability measures Pi and Pi, i = 1, . . . , n have the density processes Z κ,i and Z

κ,i, i = 1, . . . , n. In particular, Pn = P since Z κ,n = 1.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Bottom-up Approach: Lemma

Lemma For any i = 1, . . . , n, the process X i admits the integral representation have that κi

t =

• (0,t]

σi

s · dΨi(M)s

and

• κi

t =

• (0,t]

ζi

s · d

Ψi(M)s where ζi = (ζi,1, . . . , ζi,k) is an Rk-valued, predictable process and the (Pi, F)-martingale Ψi(Ml) is given by Ψi(Ml)t = Ml

t −

• (ln Z i)c, Ml,c

t −

• 0<s≤t

1 Z i

s

∆Z i

s∆Ml s.

An analogous formula yields Ψi(Ml).

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Bottom-up Approach: Joint Dynamics

Proposition The semimartingale decomposition of the (Pi, F)-spanning martingale Ψi(M) under the probability measure Pn = P is given by, for i = 1, . . . , n, Ψi(M)t = Mt −

n

• j=i+1
• (0,t]

(κj−1

s

− κj

s) ζj s · d[Mc]s

( κj

s − κj s)(κj−1 s

− κj

s)

−

n

• j=i+1
• (0,t]

σj

s · d[Mc]s

• κj

s − κj s

−

n

• j=i+1
• (0,t]

σj−1

s

· d[Mc]s κj−1

s

− κj

s

−

• 0<s≤t

1 Z i

s

∆Z i

s∆Ms.

An analogous formula holds for Ψi(M). Hence the joint dynamics of the process (κ1, . . . , κn, κ2, . . . , κn) under P = Pn are explicitly known.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models One-Period Case One- and Two-Period Case

Towards Generic Swap Models

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Postulates Comments Volatilities Conclusions

Towards Generic Swap Models

Let (Ω, F, P) be a filtered probability space. We are given a family of swap rates S = {κ1, . . . , κl} and a family of (P, F)-martingales {Z 1, . . . , Z l} such that:

1

For each j = 1, . . . , l, the process κj is a positive special semimartingale.

2

For each j = 1, . . . , l, the process κjZ j is a (P, F)-martingale.

3

For each j = 1, . . . , l, Z j is uniquely expressed as a C2 function of some subset of swaps in S. That is the process Z j = fj(κn1, . . . , κnk ) where fj is a C2 function in variables belonging to Sj := {κn1, . . . , κnk } ⊂ S.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Postulates Comments Volatilities Conclusions

Comments

1

Assumption 1 forces the semi-martingale decomposition of κj to be uniquely determined.

2

Assumption 2 gives the existence of a family of probability measures {P1, . . . , Pl}, for which κj is a martingale under Pj.

3

Assumption 3 together with the fact that the process Z j is a (P, F)-martingale implies that Z j has the following integral representation Z j

t = nk

• i=n1
• [0,t)

∂fj ∂xi (κn1

s , . . . , κnk s )d(κi)m s ,

where (κi)m stands for the (unique) martingale part of κi.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Postulates Comments Volatilities Conclusions

Volatilities

1

We claim that the semi-martingale decomposition of a swap rate process κn ∈ S can be chosen under P by choosing a family of “volatility” processes.

2

Hence κj = Nj ∈ M(Pj, F) and by inverse Girsanov’s transform the martingale part of κn must have the following representation under P (κj)m = Nj

t −

• (0,t]

Z j

s d

1 Z j , Nj

s = Nt

• r, equivalently, the semi-martingale decomposition of κn under P is

given by κj = Nj

t = Nt +

• (0,t]

Z j

s d

1 Z j , Nj

s

where N is unique since the Girsanov transform is a bijection.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads

Market Models for Forward Swaps Density Processes of Martingale Measures Market Models for CDS Spreads Towards Generic Swap Models Postulates Comments Volatilities Conclusions

Conclusions

1

For the purpose of modelling, Nj is defined under Pj as follows Nj

t =

t κj

sσj s · dΨj(M)s.

2

Therefore, specifying Nj is equivalent to specifying the “volatility” σj.

3

The martingale part of κj can be expressed as (κj)m = t κn

sσj s·dΨj(M)s−

• (0,t]

Z j

sκj sσj s· d

1 Z j , dΨj(M)s

• s =

t κj

sσj s·dMj s

where Mj is a (P, F)-martingale.

4

The process Z j has the following decomposition Z j

t = nk

• i=n1
• [0,t)

∂fj ∂xi (κn1

s , . . . , κnk s )κni s σni s · dMni s .

5

Choice of processes Z j is equivalent to the choice of the family of “volatilities”.

6

We conclude that the choice of “volatilities” specifies the model.

Marek Rutkowski and Libo Li Models for Swap Rates and CDS Spreads