Defaultable Bonds. Thought of the Day 133.0

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Defaultable bonds are bonds that have a positive possibility of default.  Most corporate bonds and some government bonds are defaultable.  When a bond defaults, its coupon and principal payments will be altered.  Most of the time, only a portion of the principal, and sometimes, also a portion of the coupon, will be paid. A defaultable (T, x) – bond with maturity T > 0 and credit rating x ∈ I ⊆ [0, 1], is a financial contract which pays to its holder 1 unit of currency at time T provided that the writer of the bond hasn’t bankrupted till time T. The set I stands for all possible credit ratings. The bankruptcy is modeled with the use of a so called loss process {L(t), t ≥ 0} which starts from zero, increases and takes values in the interval [0, 1]. The bond is worthless if the loss process exceeds its credit rating. Thus the payoff profile of the (T, x) – bond is of the form

1{LT ≤ x}

The price P(t, T, x) of the (T, x) – bond is a stochastic process defined by

P(t, T, x) = 1{LT ≤ x}e−∫tT f(t, u, x)du, t ∈ [0, T] —– (1)

where f (·, ·, x) stands for an x-forward rate. The value x = 1 corresponds to the risk-free bond and f(t, T, 1) determines the short rate process via f(t, t, 1), t ≥ 0.

The (T, x) – bond market is thus fully determined by the family of x-forward rates and the loss process L. This is an extension of the classical non-defaultable bond market which can be identified with the case when I is a singleton, that is, when I = {1}.

The model of (T, x) – bonds does not correspond to defaultable bonds which are directly traded on a real market. For instance, in this setting the bankruptcy of the (T, x2) – bond automatically implies the bankruptcy of the (T, x1) – bond if x1 < x2. In reality, a bond with a higher credit rating may, however, default earlier than that with a lower one. The (T, x) – bonds are basic instruments related to the pool of defaultable assets called Collateralized Debt Obligations (CDOs), which are actually widely traded on the market. In the CDO market model, the loss process L(t) describes the part of the pool which has defaulted up to time t > 0 and F(LT), where F as some function, specifies the CDO payoff at time T > 0. In particular, (T, x) – bonds may be identified with the digital-type CDO payoffs with the function F of the form

F(z) = Fx(z) := 1[0,x](z), x ∈ I, z ∈ [0,1]

Then the price of that payoff pt(Fx(LT)) at time t ≤ T equals P(t, T, x). Moreover, each regular CDO claim can be replicated, and thus also priced, with a portfolio consisting of a certain combination of (T, x) – bonds. Thus it follows that the model of (T, x) – bonds determines the structure of the CDO payoffs. The induced family of prices

P(t, T, x), T ≥ 0, x ∈ I

will be a CDO term structure. On real markets the price of a claim which pays more is always higher. This implies

P(t, T, x1) = pt(Fx1(LT)) ≤ pt(Fx2(LT)) = P(t, T, x2), t ∈ [0, T], x1 < x2, x1, x2 ∈ I —– (2)

which means that the prices of (T, x) – bonds are increasing in x. Similarly, if the claim is paid earlier, then it has a higher value and hence

P(t, T1, x) = pt(Fx(LT1)) ≥ pt(Fx(LT2)) = P(t, T2, x), t ∈ [0, T1], T1 < T2, x ∈ I —– (3)

which means that the (T, x) – bond prices are decreasing in T. The CDO term structure is monotone if both (2) and (3) are satisfied. Surprisingly, monotonicity of the (T, x) – bond prices is not always preserved in mathematical models even if they satisfy severe no-arbitrage conditions.

Ricci-flow as an “intrinsic-Ricci-flat” Space-time.

A Ricci flow solution {(Mm, g(t)), t ∈ I ⊂ R} is a smooth family of metrics satisfying the evolution equation

∂/∂t g = −2Rc —– (1)

where Mm is a complete manifold of dimension m. We assume that supM |Rm|g(t) < ∞ for each time t ∈ I. This condition holds automatically if M is a closed manifold. It is very often to put an extra term on the right hand side of (1) to obtain the following rescaled Ricci flow

∂/∂t g = −2 {Rc + λ(t)g} —– (2)

where λ(t) is a function depending only on time. Typically, λ(t) is chosen as the average of the scalar curvature, i.e. , 1/m ∱Rdv or some fixed constant independent of time. In the case that M is closed and λ(t) = 1/m ∱Rdv, the flow is called the normalized Ricci flow. Starting from a positive Ricci curvature metric on a 3-manifold, Richard Hamilton showed that the normalized Ricci flow exists forever and converges to a space form metric. Hamilton developed the maximum principle for tensors to study the Ricci flow initiated from some metric with positive curvature conditions. For metrics without positive curvature condition, the study of Ricci flow was profoundly affected by the celebrated work of Grisha Perelman. He introduced new tools, i.e., the entropy functionals μ, ν, the reduced distance and the reduced volume, to investigate the behavior of the Ricci flow. Perelman’s new input enabled him to revive Hamilton’s program of Ricci flow with surgery, leading to solutions of the Poincaré conjecture and Thurston’s geometrization conjecture.

In the general theory of the Ricci flow developed by Perelman in, the entropy functionals μ and ν are of essential importance. Perelman discovered the monotonicity of such functionals and applied them to prove the no-local-collapsing theorem, which removes the stumbling block for Hamilton’s program of Ricci flow with surgery. By delicately using such monotonicity, he further proved the pseudo-locality theorem, which claims that the Ricci flow can not quickly turn an almost Euclidean region into a very curved one, no matter what happens far away. Besides the functionals, Perelman also introduced the reduced distance and reduced volume. In terms of them, the Ricci flow space-time admits a remarkable comparison geometry picture, which is the foundation of his “local”-version of the no-local-collapsing theorem. Each of the tools has its own advantages and shortcomings. The functionals μ and ν have the advantage that their definitions only require the information for each time slice (M, g(t)) of the flow. However, they are global invariants of the underlying manifold (M, g(t)). It is not convenient to apply them to study the local behavior around a given point x. Correspondingly, the reduced volume and the reduced distance reflect the natural comparison geometry picture of the space-time. Around a base point (x, t), the reduced volume and the reduced distance are closely related to the “local” geometry of (x, t). Unfortunately, it is the space-time “local”, rather than the Riemannian geometry “local” that is concerned by the reduced volume and reduced geodesic. In order to apply them, some extra conditions of the space-time neighborhood of (x, t) are usually required. However, such strong requirement of space-time is hard to fulfill. Therefore, it is desirable to have some new tools to balance the advantages of the reduced volume, the reduced distance and the entropy functionals.

Let (Mm, g) be a complete Ricci-flat manifold, x0 is a point on M such that d(x0, x) < A. Suppose the ball B(x0, r0) is A−1−non-collapsed, i.e., r−m0|B(x0, r0)| ≥ A−1, can we obtain uniform non-collapsing for the ball B(x, r), whenever 0 < r < r0 and d(x, x0) < Ar0? This question can be answered easily by applying triangle inequalities and Bishop-Gromov volume comparison theorems. In particular, there exists a κ = κ(m, A) ≥ 3−mA−m−1 such that B(x, r) is κ-non-collapsed, i.e., r−m|B(x, r)| ≥ κ. Consequently, there is an estimate of propagation speed of non-collapsing constant on the manifold M. This is illustrated by Figure

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We now regard (M, g) as a trivial space-time {(M, g(t)), −∞ < t < ∞} such that g(t) ≡ g. Clearly, g(t) is a static Ricci flow solution by the Ricci-flatness of g. Then the above estimate can be explained as the propagation of volume non-collapsing constant on the space-time.

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However, in a more intrinsic way, it can also be interpreted as the propagation of non-collapsing constant of Perelman’s reduced volume. On the Ricci flat space-time, Perelman’s reduced volume has a special formula

V((x, t)r2) = (4π)-m/2 r-m ∫M e-d2(y, x)/4r2 dvy —– (3)

which is almost the volume ratio of Bg(t)(x, r). On a general Ricci flow solution, the reduced volume is also well-defined and has monotonicity with respect to the parameter r2, if one replace d2(y, x)/4r2 in the above formula by the reduced distance l((x, t), (y, t − r2)). Therefore, via the comparison geometry of Bishop-Gromov type, one can regard a Ricci-flow as an “intrinsic-Ricci-flat” space-time. However, the disadvantage of the reduced volume explanation is also clear: it requires the curvature estimate in a whole space-time neighborhood around the point (x, t), rather than the scalar curvature estimate of a single time slice t.

Sheafification Functor and Arbitrary Monetary Value Measure. Part 3

Here are Part 1 and Part 2

Let A be a fixed set of axioms. Then for a given arbitrary monetary value measure Ψ can we make a good alternative for it? In other words, can we find a monetary value measure that satisfies A and is the best approximation of the original Ψ? For a Grothendieck topology J on χ, define Sh(χ, J) ⊂ Setχop to be a full sub-category whose objects are all sheaves for J. Then, it is well known that  a left adjoint πJ in the following diagram.

Sh(χ, J) → Setχop

Sh(χ, J) ←πJ Setχop

πJ (Ψ) ← Ψ

The functor πJ is known as Sheafification functor, which has the following limit cone:

img_20170117_125305

for sieves I, K and U. This also satisfies the following theorem.

1.0 If πJ (Ψ) is a sheaf for J

1.1 If Ψ is a sheaf for J, then for any U ∈ χ,  πJ (Ψ)(U) ≅ L(U)

The theorem suggests that for an arbitrary monetary value measure, the sheafification functor provides one of its closest monetary value measures that may satisfy the given set of axioms. To make this certain, we need a following definition.

2.0 Let A be a set of axioms of monetary value measures

2.1 M(A) := the collection of all monetary value measures satisfying A

2.2 MO := collection of all monetary value measures

2.3 A is called complete if

πJM(A) (MO) ⊂ M(A)

3.0 Let A be a complete set of axioms. Then, for a monetary value measure Ψ ∈ MO, πJM(A(Ψ) is the monetary value measure that is the best approximation satisfying A.

Let us investigate if the set of axioms of concave monetary value measures is complete in the case of Ω = {1, 2, 3} with a σ-field F := 2Ω

We enumerate all possible sub-σ-fields of Ω, that is, the shape of the category χ = χ(Ω),

img_20170117_132606

where,

U := F := 2Ω

U1 := {Φ, {1}, {2, 3}, Ω}

U2 := {Φ, {2}, {1, 3}, Ω}

U3 := {Φ, {3}, {1, 2}, Ω}

U4 := {Φ, Ω}

The Banach spaces defined by the elements of χ are

L := L := L(U) := {a, b, c | a, b, c ∈ ℜ}

L1 := L(U1) := {a, b, b | a, b ∈ ℜ}

L2 := L(U2) := {a, b, a | a, b ∈ ℜ}

L3 := L(U3) := {a, a, c | a, c ∈ ℜ}

L0 := L(U0) := {a, a, a, | a ∈ ℜ}

Then a monetary value measure Ψ : χop → Set on χ is determined by the following six functions

img_20170117_142341

We will investigate its concrete shape one by one by considering axioms it satisfies.

For Ψ1 : L → L1, we have by the cash invariance axiom,

Ψ1 (a, b, c) = Ψ1 ((0, b – c, 0) + (a, c, c))

= Ψ1 ((0, b – c, 0)) + (a, c, c)

= (f12 (b – c), f11 (b – c), f11 (b – c)) + (a, c, c)

= (f12 (b – c) + a, f11 (b – c) + c, f11 (b – c)+ c)

where f11, f12 : ℜ → ℜ are defined by (f12(x), f11(x), f11(x)) = Ψ1 (0, x, 0).

Similarly, if we define nine functions

f11, f12, f21, f22, f31, f32, g1, g2, g3ℜ → ℜ by

(f12(x), f11(x), f11(x)) = Ψ1(0, x, 0)

(f21(x), f22(x), f21(x)) = Ψ2(0, 0, x)

(f31(x), f31(x), f32(x)) = Ψ3(x, 0, 0)

(g1(x), g1(x), g1(x)) = Ψ01(x, 0, 0)

(g2(x), g2(x), g2(x)) = Ψ02(0, x, 0)

(g3(x), g3(x), g3(x)) = Ψ03(0, 0, x)

We can represent the original six functions by nine functions

Ψ1(a, b, c) = (f12(b – c) + a, f11(b – c) + c, f11(b – c) + c),

Ψ2(a, b, c) = (f21(c – a) + a, f22(c – a) + a, f21(c – a) + a),

Ψ3(a, b, c) = (f31(a – b) + b, f31(a – b) + b, f32(a – b) + c),

Ψ01(a, b, b) = (g1(a – b) + b, g1(a – b) + b, g1(a – b) + b),

Ψ02(a, b, a) = (g2(b – a) + a, g2(b – a) + a, g2(b – a) + a),

Ψ03(a, a, c) = (g3(c – a) + a, g3(c – a) + a, g3(c – a) + a)

Next by the normahzation axiom, we have

f11(0) = f12(0) = f21(0) = f22(0) = f31(0) = f32(0) = g1(0) = g2(0) = g3(0) = 0

partially differentiating the function in Ψ1(a, b, c), we have

Ψ1(a, b, c)/∂a = (1, 0, 0)

∂Ψ1(a, b, c)/∂b = (f’12(b – c), f’11(b – c), f’11(b – c))

Ψ1(a, b, c)/∂c = (- f’12(b – c), 1 –  f’11(b – c), 1 –  f’11(b – c))

Therefore, by the monotonicity, we have f’12(x) = 0 and 0 ≤ f’11 ≤ 1. Then by the result of the normalization axiom, we have

x ∈ ℜ, f12(x) = 0. Hence, ∀ x ∈ ℜ,

f12(x) = f22(x) = f32(x) = 0

With this knowledge, let us redefine the three functions f1, f2, f3 : ℜ → ℜ by

(0, f1(x), f1(x)) = Ψ1(0, x, 0)

(f2(x), 0, f2(x)) = Ψ2(0, 0, x)

(f3(x), f3(x), 0) = Ψ3(x, 0, 0)

Then, we have a new representation of the original six functions

Ψ1(a, b, c) = (a, f1(b – c) + c, f1(b – c) + c)

Ψ2(a, b, c) = (f2(c – a) + a, b, f2(c – a) + a)

Ψ3(a, b, c) = (f3(a – b) + b, f3(a – b) + b, c)

Ψ01(a, b, b) = (g1(a – b) + b, g1(a – b) + b, g1(a – b) + b)

Ψ02(a, b, a) = (g2(b – a) + a, g2(b – a) + a, g2(b – a) + a)

Ψ03(a, a, c) = (g3(c – a) + a, g3(c – a) + a, g3(c – a) + a)

Thinking about the composition rule, we have

Ψ0 = Ψ01 o Ψ1 = Ψ02 o Ψ2 = Ψ03 o Ψ3

g1(a – f1(b – c) – c) + f1(b – c) + c

= g2(b – f2(c – a) – a) + f2(c – a) + a

=g3(c – f3(a – b) – b) + f3(a – b) + b

………..

 

Monetary Value Measure as a Contravariant Functor (Category Theory) Part 1

category_concepts

Let us get a very brief review of dynamic risk measure theory and category theory.

1.0 A one period monetary risk measure is a function ρ = LP (Ω, F, P) → ℜ satisfying the following axioms

1.1 Cash Invariance (∀X) (∀ a ∈ ℜ) ρ (X + a) = ρ (X) – a

1.2 Monotonicity (∀ X) (∀ Y) X ≤ Y ⇒ ρ (X) ≥ ρ (Y)

1.3 Normalization ρ (0) = 0

where ρ = LP (Ω, F, P) is the space of equivalence classes of ℜ-valued random variables which are bounded by the || . ||P form.

2.0 For a σ-field U ⊂ F, L(U) = L (Ω, U, P | U) is the space of all equivalence classes of bounded ℜ-valued random variables, equipped with the usual sup form.

3.0 Let F = {Ft}t∈[0,T] be a filtration. A dynamic monetary value measure is a collection of functions ψ = {Ψt : L(FT) → L(Ft)}t∈[0,T] satisfying

3.1 Cash Invariance (∀ X ∈ L(FT))(∀ Z ∈ L(FT)) Ψt (X + Z) = Ψt (X) + Z

3.2 Monotonicity (∀ X ∈ L(FT))(∀ Y ∈ L(FT)) X ≤ Y ⇒ Ψt (X) ≤ Ψt (Y)

3.3 Normalization Ψt (0) = 0

Note that the directions of some inequalities in Definition 1.0-1.3 are different from those of Definition 3.0-3.3, because we now are dealing with monetary value measures instead of monetary risk measures.

Since dynamic monetary value measures treat multi-period situations, we may require some extra axioms to regulate them toward the time dimension.

Axiom 4.0 Dynamic Programming Principle: For 0 ≤ s ≤ t ≤ T, (∀ X ∈ L(FT)) Ψs (X) =  Ψst (X))

Axiom 5.0 Time Consistency: For 0 ≤ s ≤ t ≤ T, (∀ X, ∀ Y ∈  L(FT)) Ψt (X) ≤ Ψt (Y) ⇒ Ψs (X) ≤ Ψs (Y)

Category theory is an area of study in mathematics that examines in an abstract way the properties of maps (called morphisms or arrows) satisfying some basic conditions.

A Category C consists of a collection of OC of objects and a collection of MC of arrows or morphisms such that

6.0 There are two functions MCdom OC & MC →cod OC

When dom(f) = A and cod (f) = B, we write f : A → B

We define a so-called hom-set of given objects A and B by

HomC(A, B) := {f ∈ MC | f : A → B}

6.1 For f : A → B & g : B → C, there is an arrow g o f : A → C called the composition of g and f. 

6.2 Every object A is associated with an identity arrow 1A : A → A, such that f o 1A = f and 1A o g = g where dom(f) = A & cod(g) = g

7.0 Functors: Let C and D be two categories. A functor F: C → D consists of two functions

FO : OC → OD and FM : MC → MD

satisfying

7.1 f : A → B ⇒ F(f) : F(A) → F(B)

7.2 F(g o f) = F(g) o F(f)

7.3 F(1A) = 1F(A)

8.0 Contravariant Functors: A functor F : Cop → D is called a contravariant functor if 7.1 and 7.2 are replaced by

8.1 f : A → B ⇒ F(f) : F(B) → F(A)

8.2 F(g o f) = F(f) o F(g)

9.0 Natural Transformations: Let C →F D and C →G D be two functors. A natural transformation α : F →. G consists of a family of arrows 〈αC | C ∈ OCmaking the following diagram commute

C1          F(C1) —>αC1 G(C1)

f↓       F(f) ↓             G(f)↓

C2         F(c2) —>αC2 G(C2)

10.0 Functor Categories: Let C and D be categories. A functor category DC is the category such that

10.1 ODC := collection of all functors from C to D

10.2 HomDC (F, G) := collection of all natural transformations from F to G.

Now, for defining monetary value measures with the language of category theory, we introduce a simple category that is actually a partially-ordered set derived by the σ-field F.

11.0 Let χ := χ(F) be the et of all sub-fields of F. It becomes a poset with the set inclusion relation χ becomes a category whose hom set Homχ(V, U) for U, V ∈ χ is defined by

Homχ(V, U) := iVU if V ⊂ U

:= Φ otherwise.

The arrow iVU is called the inclusion map. 

12.0 ⊥ := {Ω, Φ}, which is the least element of χ. 

13.0 Monetary Value Measure is a contravariant functor

Ψ : χop → Set

satisfying the following two conditions

13.1 for U ∈ χ, Ψ(U) := L(U)

13.2 for U, V ∈ χ, such that V ⊂ U, the map ΨVU := Ψ(iVU) : L(U) → L(V) satisfies

13.3 cash invariance: (∀ X ∈ L(U))(∀ Z ∈ L(V)) ΨVU (X + Z) =  ΨVU (X) + Z

13.4 monotonicity: (∀ X ∈ L(U)) (∀ Y ∈ L(U)) X ≤ Y ⇒ ΨVU(X) ≤ ΨVU(Y)

13.5 normalization: ΨVU(0) = 0

At this point, we do not require the monetary value measures to satisfy some familiar con- ditions such as concavity or law invariance. Instead of doing so, we want to see what kind of properties are deduced from this minimal setting. One of the key parameters from 13.0 is that Ψ is a contravariant functor, and thus for any triple σ-fields W ⊂ V ⊂ U in χ, we have

13.6 ΨUU = 1L(U) and ΨWV o ΨVU = ΨWU

14.0 Concave monetary value measure: A monetary value measure Ψ is said to be concave if for any V ⊂ U in χ, X, Y ∈ L(U) and λ ∈ [0,1],

ΨVU(λX + (1- λ)Y) ≥ λΨVU(X) + (1-λ)ΨVU(Y)

An entropic value measure is concave.

Here are some properties of monetary value measures.

15.0 Proposition: Let Ψ : χop → Set be a monetary value measure, and W ⊂ V ⊂ U be σ-fields in χ.

15.1 (∀ X ∈ L(V)) ΨVU(X) = X

By cash invariance and normalization, ΨVU(X) = ΨVU(0 + X) = ΨVU(0) + X = X

15.2 Idempotentness: (∀ X ∈ L(U)) ΨVUΨVU(X) = ΨVU(X)

Since, ΨVU(X) ∈  L(V), it is obvious by 15.1

15.3 Local property: (∀ X ∈ L(U))(∀ Y ∈ L(U))(∀ A ∈  V) ΨVU (1AX + 1ACY) = 1AΨVU(X) + 1AC ΨVU(Y) 

First we show for any A ∈ V,

1AΨVU(X) = 1AΨVU(1AX)

Since, X ∈ L(Ω, U, P), we have |X| ≤ ||X||

therefore,

1AX – 1AC||X|| ≤ 1AX + 1ACX ≤ 1AX + 1AC||x||

hence, by cash invariance and monotonicity,

ΨVU(1AX) – 1AC||x|| = ΨVU(1AX – 1AC||x||) ≤ ΨVU(X) ≤ ΨVU(1AX) + 1AC||x||)

then,

1AΨVU(1AX) = 1AVU(1AX) – 1AC||x||) ≤ 1AΨVU(X) ≤ 1AVU(1AX) + 1AC||x||) = 1AVU(1AX)

getting 15.3

Using 15.3 twice, we have

ΨVU (1AX + 1ACY) = 1AΨVU(1AX + 1ACY) + 1ACΨVU(1AX + 1ACY)

1AΨVU(1A(1AX + 1ACY)) + 1ACΨVU(1AX + 1ACY))

1AΨVU(1AX) + 1ACΨVU(1ACY)

1AΨVU(X) + 1ACΨVU(Y)

15.4 Dynamic programming principle: (∀ X ∈ L(U)) ΨWU(X) = ΨWUVU(X))

by way of dynamic risk measure and monetary value measure,

ΨWU(X) = ΨWVVU(X)) =  ΨWVVUVU(X))) = (ΨWV o ΨVU)(ΨVU(X)) = ΨWUVU(X))

15.5 Time consistency: (∀ X ∈ L(U))(∀ Y ∈ L(U)) (ΨVU(X)) ≤ (ΨVU(Y)) ⇒ ΨWU(X) ≤ ΨWU(Y)

Assuming ΨVU(X) ≤ ΨVU(Y), then, by monotonicity and monetary value measure,

ΨWU(X) = ΨWVVU(X)) ≤ ΨWVVU(Y)) = ΨWU(Y)

……………